Analytical and Numerical Methods for Investigation of Flow Fields with Chemical Reactions, Especially Related to Combustion

Transcription

1 P N9001 AGARD-CP-164 S o DC < O AGARD CONFERENCE PROCEEDINGS No. I64 on Analytical and Numerical Methods for Investigation of Flow Fields with Chemical Reactions, Especially Related to Combustion NORTH ATLANTIC TREATY ORGANIZATION DISTRIBUTION AND AVAILABILITY ON BACK COVER

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3 AGARD-CP-164 f : NORTH ATLANTIC TREATY ORGANIZATION ADVISORY GROUP FOR AEROSPACE RESEARCH AND DEVELOPMENT (ORGANISATION DU TRAITE DE L'ATLANTIQUE NORD) AGARD Conference Proceedings No. 164 ANALYTICAL AND NUMERICAL METHODS FOR INVESTIGATION OF FLOW FIELDS WITH CHEMICAL REACTIONS, ESPECIALLY RELATED TO COMBUSTION I ' '/ Papers and discussions presented at the Specialists Meeting on "Analytical and Numerical Methods for Investigation of Flow Fields, with Chemical Reactions, Especially Related to Combustion" held on 1 and 2 April 1974 during the PEP 43rd Meeting at Liege, Belgium.

4 THE MISSION OF AGARD The mission of AGARD is to bring together the leading personalities of the NATO nations in the fields of science and technology relating to aerospace for the following purposes: - Exchanging of scientific and technical information; - Continuously stimulating advances in the aerospace sciences relevant to strengthening the common defence posture; Improving the co-operation among member nations in aerospace research and development; Providing scientific and technical advice and assistance to the North Atlantic Military Committee in the field of aerospace research and development; Rendering scientific and technical assistance, as requested, to other NATO bodies and to member nations in connection with research and development problems in the aerospace field; Providing assistance to member nations for the purpose of increasing their scientific and technical potential; Recommending effective ways for the member nations to use their research and development capabilities for the common benefit of the NATO community. The highest authority within AGARD is the National Delegates Board consisting of officially appointed senior representatives from each member nation. The mission of AGARD is carried out through the Panels which are composed of experts appointed by the National Delegates, the Consultant and Exchange Program and the Aerospace Applications Studies Program. The results of AGARD work are reported to the member nations and the NATO Authorities through the AGARD series of publications of which this is one. Participation in AGARD activities is by invitation only and is normally limited to citizens of the NATO nations. Published May 1975 Copyright AGARD :536.46: Set and printed by Technical Editing and Reproduction Ltd Harford House, 7-9 Charlotte St. London, W1P 1HD

5 AGARD PROPULSION AND ENERGETICS PANEL OFFICERS CHAIRMAN: Mr F.Jaarsma, National Aerospace Laboratory, Amsterdam, The Netherlands DEPUTY CHAIRMAN: ICA M.Pianko, Service Technique Aeronautique, Paris, France PROGRAM COMMITTEE Mr M.L.Barrere, ONERA, Chatillon-sous-Bagneux, France Professor J.Chauvin, VKI, Rhode-St-Genese, Belgium Professor A.Ferri, New York University, Westbury, Long Island, NY, USA Professor A.H.Lefebvre, Cranfield Institute of Technology, Bedford, UK Professor R.Monti, Istituto di Aerodinamica, Universita di Napoli, Italy Mr J.Surugue, ONERA, Chatillon-sous-Bagneux, France Dr Ing. G.Winterfeld, DFVLR, Institut f. Luftstrahlantriebe, Porz-Wahn, Germany HOST COORDINATOR Professor J.Ducarme, Universite de Liege, Institute de Mecanique, Liege, Belgium PANEL EXECUTIVE Major John B.Catiller, USAF, AGARD, Neuilly-sur-Seine, France The Propulsion and Energetics Panel wishes to express its thanks to the hosts, the Belgian National Delegates to AGARD, for the invitation to hold the meeting at Liege, and for the provision of the necessary facilities and personnel to make the meeting possible.

6 FOREWORD by Marcel L.Barrere Program Committee Chairman I would like first to thank Prof. Ducarme, an esteemed member of our Panel, as well as the Belgian personalities who kindly invited us to the magnificent city of Liege. As for myself, I have kept an excellent memory of the second AGARD Colloquium which took place in the same city some twenty years ago. Today, this meeting of specialists is devoted to analytical and numerical methods used in the study of flows with chemical reactions, with particular application to flows with combustion. This is a broad field that widely extends beyond combustion problems. It is not possible to discuss all aspects within two days, so we had to limit ourselves to those which seem most important. Before giving some indications on the program, we would like to pay homage to a great scientist, who had a determining impact on the evolution of the problems we are discussing today: I mean Prof. Theodore Von Karman, the founder of AGARD who, in the 1950's and in spite of his heavy load of work, devoted part of his time to the study of combustion phenomena, in particular to the calculation of the propagation rate of laminar flames. Thanks to his works, his lectures, his discussions during our Panel Meetings, he provided a new impulse to the studies of flows with chemical reactions, showing the unity of this science which he defined as "Aerothermochemistry". Several of us, present at this meeting, have been marked by his ideas; as a token of gratefulness for all that he gave us, I think we should dedicate these two days of common work to him. This meeting of specialists is spread over four sessions, and the first criticism that could be made is that we did not divide them in a classical manner, into large resolution methods, such as: finite difference methods, - methods making use of series, in particular techniques of asymptotic developments, finite elements methods, but, in view of the papers offered, it appeared simpler to make use of another system which will enable us to insist on some points: - A first session is thus devoted to classical integration methods, using more or less finite difference techniques based on Prof. D.B.Spalding's works; in many laboratories these techniques are used to solve problems of laminar in turbulent (mean flow) combustion. In view of the success of this work, it appeared necessary to devote to it the first session and part of session IV concerning the application of combustor calculation methods. - The second session is devoted to calculation methods used in the study of turbulent flames. It is a very important problem on both fundamental and practical viewpoints, which recently showed a marked progress; that is why we have insisted on this point. - The third session approaches more general and more varied methods permitting the analysis of chemical reactions: for instance the techniques deriving from particular thermodynamic conditions of equilibrium and near equilibrium. - The last session deals with methods more directly applicable to combustors and their operation, and also to the analysis of pollutant formation. All the models proposed rest on a better knowledge of flows with chemical reactions, and that is why it seemed interesting to introduce a paper on the measurements within multi-reaction turbulent flows. This program is far from being complete and leaves aside the finite element method, still little used in the combustion field. The analysis of techniques implemented for the study of unsteady flows with chemical reactions was also left aside. Limited to two days, this meeting of specialists could not be comprehensive; the choices made should allow us to study in depth some aspects of aerothermochemistry, and to orient conveniently our future research.

7 PREFACE par Marcel L.Barrere Program Committee Chairman Je voudrais tout d'abord remercier le Professeur Ducarme, membre estime de notre Panel ainsi que les Personalites Beiges qui ont bien voulu nous accueillir dans cette magnifique ville de Liege. J'ai pour ma part garde un excellent souvenir du deuxieme Colloque AGARD sur la Combustion qui s'est tenu dans cette meme cite, il y a de cela 20 ans. Aujourd'hui cette reunion de specialistes est consacree aux methodes analytiques et numeriques utilisfes dans 1'etude des ecoulements avec reactions chimiques avec des applications plus particulieres aux ecoulements en combustion. C'est la un vaste domaine qui deborde largement le cadre de la combustion. II n'est pas possible de debattre en deux jours de tous les problemes, c'est pourquoi nous nous sommes limites a quelques uns qui nous paraissent les plus importants. Avant de donner quelques indications sur le programme, nous voudrions rendre hommage a un grand savant, qui a eu une action determinante dans 1'evolution des problemes qui nous preoccupent aujourd'hui, je veux parler du Professeur Theodore Von Karman, fondateur de 1'AGARD qui, entre les annees , malgre ses lourdes charges, a consacre une partie de son temps a 1'etude des phenomenes de combustion, en particulier au calcul de la vitesse de propagation des flammes laminaires. Grace a ses travaux, a ses conferences, a ses discussions au cours de reunions de notre Panel, il a donne une nouvelle impulsion aux etudes des ecoulements avec reactions chimiques en montrant une certaine unite dans cette science qu'il a definie sous le vocable "Aerothermochimie". Plusieurs d'entre nous, presents a cette reunion ont etc marques par ses idees, en reconnaissance pour tout ce qu'il nous apporte, je pense que nous devons lui dedier nos deux journees de travail. Cette reunion de specialistes a ete repartie sur quatre sessions et une premiere critique que Ton peut nous faire, est de n'avoir pas utilise un decoupage classique en grandes methodes de resolution comme: les methodes des differences finies, - les methodes utilisant des series et en particulier les techniques des developpements asymptotiques, les methodes des elements finis, mais, compte tenu des contributions, il nous a paru plus simple d'utiliser une autre decoupage qui nous permettait d'insister sur quelques points. Une premiere session est done consacree a des methodes classiques d'integration utilisant plus ou moins les techniques des differences finies et qui ont pour base les travaux du professeur D.B.Spalding; dans de nombreux laboratoires ces techniques sont employees pour resoudre des problemes de combustion laminaire ou turbulente (ecoulement moyen). Devant le succes de ces travaux il nous a paru necessaire d'y consacrer une premiere session et une partie de la session IV relative aux applications des methodes de calcul aux foyers. La deuxieme session est consacree aux methodes de calcul utilisees dans 1'etude des flammes turbulentes. C'est un probleme tres important aussi bien sur le plan fondamental que pratique et qui a fait ces derniers temps de serieux progres, c'est pourquoi nous avons mis 1'accent sur ce sujet. - La troisieme session aborde des methodes plus generates et plus variees permettant 1'analyse des ecoulements avec reactions chimiques, c'est par exemple les techniques qui decoulent des conditions thermodynamiques particulieres de 1'equilibre et du proche equilibre. La derniere session traite des methodes plus directement appliquees aux foyers, a leur fonctionnement et, egalement, a 1'analyse de la formation des polluants. Tous les modeles proposes reposent sur une meilleure connaissance des ecoulements avec reactions chimiques, c'est pourquoi il nous a paru interessant d'introduire une contribution sur les mesures dans les ecoulements turbulents multireactifs. Ce programme est loin d'etre complet et laisse de cote la methode des elements finis encore peu adoptee dans le domaine de la combustion. L'analyse des techniques mises en oeuvre pour 1'etude des ecoulements instationnaires avec reactions chimiques est aussi laissee de cote. Limitee a deux jours cette reunion de specialistes ne pouvait etre complete, les choix qui ont ete fails vont nous permettre d'approfondir certains domaines de I'aerothermochimie et de mieux orienter nos futures recherches.

8 CONTENTS Page AGARD PROPULSION AND ENERGETICS PANEL OFFICERS FOREWORD by M.L.Barr re PREFACE by M.L.Barrdre Hi iv v SESSION I - CLASSICAL METHODS FOR NUMERICAL COMPUTATION OF LAMINAR FLOW AND MEAN FLOW WITH CHEMICAL REACTION SESSION CHAIRMAN: A.Ferri, USA Reference 1. NUMERICAL COMPUTATION OF PRACTICAL COMBUSTION CHAMBER-FLOWS by D.B.Spalding M 2. THEORETICAL ANALYSIS OF NONEQUILIBRIUM HYDROGEN AIR REACTIONS BETWEEN TURBULENT SUPERSONIC COAXIAL STREAMS by H.R6rtgen LA SIMULATION DE LA TURBULENCE DANS LES MODELES PETULA par J.P.Huffenus ANALYSE NUMERIQUE DE LA PHASE D'INFLAMMATION DANS UNE COUCHE DE MELANGE TURBULENTE par O.Leuchter 1-4 SESSION II - MODERN METHODS OF ANALYSIS OF TURBULENT FLAMES SESSION CHAIRMAN: R.Monti, Italy 1. A REVIEW OF SOME THEORETICAL CONSIDERATIONS OF TURBULENT FLAME STRUCTURE by F.A.Williams 2. KINETIC ENERGY OF TURBULENCE IN FLAMES by K.N.C.Bray II-l II-2 3. A NUMERICAL SPECTROSCOPIC TECHNIQUE FOR ANALYZING COMBUSTOR FLOWFIELDS by M.E.Neer H-3 4. METHODE ANALYTIQUE DE PREVISION DES TAUX DE REACTION CHIMIQUE EN PRESENCE D'UNE TURBULENCE NON HOMOGENE. (APPLICATION A LA COMBUSTION TURBULENTE) par R.Borghi H-4 5. STUDIES RELATED TO TURBULENT FLOWS INVOLVING FAST CHEMICAL REACTIONS by P.A.Libby II-5 SESSION HI - GENERAL METHODS OF ANALYSIS OF FLOWS WITH NUMEROUS CHEMICAL REACTIONS SESSION CHAIRMAN: M.L.Barrere, France 1. METHODE DE QUASI-EQUILIBRE POUR L'ETUDE DES ECOULEMENTS RELAXES par R.Prud'homme HI-1

9 Reference 2. CALCULATION OF THE EFFECT OF AFTERBURNING IN EXTERNAL SUPERSONIC FLOW BY MEANS OF A METHOD OF CHARACTERISTICS WITH HEAT ADDITION AND MIXING LAYER ANALYSIS by P.Mittelbach 3. SUPERSONIC MIXING AND COMBUSTION IN PARALLEL INJECTION FLOW FIELDS by J.S.Evans and G.Y.Anderson IH-2 IH-3 SESSION IV - NUMERICAL AND ANALYTICAL METHODS APPLIED TO COMBUSTORS AND TO THE STUDY OF POLLUTION SESSION CHAIRMAN: A.H.Lefebvre, UK 1. TURBULENT BOUNDARY LAYER IN HYBRID PROPELLANTS COMBUSTION by R.Monti 2. SOME PROBLEMS AND ASPECTS IN COMBUSTOR MODELLING by F.Suttrop 3. MEASUREMENTS IN TURBULENT FLOWS WITH CHEMICAL REACTION by N.AChigier 4. SOME MEASUREMENTS AND NUMERICAL CALCULATIONS ON TURBULENT DIFFUSION FLAMES by Th.T.A.Paauw APPENDIX A - LIST OF PARTICIPANTS IV-1 IV ' 2 IV-3 IV-4 App.A-1

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11 II NUMERICAL COMPUTATION OF PRACTICAL COMBUSTION-CHAMBER FLOWS by D.B.Spalding Department of Mechanical Engineering Imperial College London, SW7 2BX

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13 11-1 NUMERICAL COMPUTATION OF PRACTICAL COMBUSTION-CHAMBER FLOWS D.B.Spalding Department of Mechanical Engineering Imperial College London SW7 2BX RESUME Des m6thodes nume'riques pour la provision des Ecoulements dans les chambres de combustion reposent sur deux bases: des modules mathe'matiques reprdsentant des processus physiques, et des programmes d'ordinateur pour re"soudre les Equations diffeventielles qui en r^sultent. Ce rapport donne un re"sum6 des modeles de la turbulence, du transfer de chaleur par rayonnement, de la cine"tique chimique et des effets de deux phases. Egalement le rapport ddcrit les champs d'application des programmes d'ordinateur PASS, EASI, STABL et TRIC, dont tous utilisent 1'algorithme SIMPLE. La maniere avec laquelle les programmes et les modeles mathe'matiques peuvent etre utilises pour des applications pratiques est illustre'e par les exemples suivants:- propagation stationnaire de flammes dans une conduite, foyer axi-syme'trique, propagation nonstationnaire de flammes dans une conduite; jet de sortie incline 1 d'une fus^e,section d'un foyer en forme d'anneau, et zone re'actionnelle d'un laser chimique. II est montrd que les nece'ssite's actuelles portent sur des experiences et peus nombreuses sur 1'exploitation des me'thodes numdriques de previsions fournies par les recherches r6centes. ABSTRACT Numerical procedures for predicting combustion-chamber flows rest on two foundations: mathematical models of physical processes; and computer programs for solving the resulting differential equations. The paper briefly reviews the models for turbulence, radiation, chemical kinetics and two-phase effects; and it also describes the fields of applicability of the computer programs PASS, EASI, STABL and TRIC, all of which employ the SIMPLE algorithm. The ways in which the computer programs and the mathematical models can be used for practical purposes are illustrated by reference to:- steady flame spread in a duct; the axisymrhetrical combustor; unsteady flame spread in a duct; the inclined rocket exhaust; the annular-combustor sector; and the reaction region of a chemical laser. It is argued that the main current needs are for testing and exploitation of the numerical prediction procedures which recent research has made available.

14 11-2 Fig. 1 Numerical computation of practical combustion processes Figures Problem statement Mathematical models of physical processes Procedures for solving the equations Examples of practical calculations Final remarks The purpose of this lecture is to review the current status of the numerical approach to predicting combustion phenomena. This approach has been gaining support in recent years because of advances in computer technology, in numerical ' analysis, and in physics and chemistry; its motive has been the increasing need of engine designers to limit the soaring costs of new-product development. For the gas-turbine designer, the imposition of stringent limits on pollutant formation has necessitated an even more detailed understanding of the processes occurring in the combustor. Now, and in addition, emphasis is inevitably being placed on the saving of fuel, during development as well as in service. The numerical approach to combustor design will never make experimental testing unnecessary; but it can already increase the information obtained from each experiment; and, as experience of mathematical modelling increases, the reduction in the costs of experimental programs may be significant. The attraction of the numerical approach is primarily an economic one. Like all good applied science, it maximises the ratio of benefit to cost. Fig.2 The mechanism of prediction Geometry, operating conditions \ Prediction of u, v, w, p, k~, m., etc fields; fluxes, balances 4 Computer Program Particular material properties ***+ **** 44*444** Mathematical models Problem <;ategories 4- ^ Physics, chemistry Applied mathematics ^ Whether one performs scale-model experiments, applies "rules-of-thumb", utilises a mathematical model, or simply makes a guess, the result is a prediction, ie a prognostication, preferably quantitative, of the outcome of a hypothetical future experiment. In order to make the prediction, one must know what will be the conditions of the future experiment, and.what are the propetries of the fuel, airstream, combustor-wall and other materials to be used. These items are indicated on the slide in the two boxes flanking the desired outcome - the prediction. If the prediction is being made in a numerical manner, the input conditions lead to the prediction by way of a suitable computer program. This is illustrated by the long horizontal box on the slide. The computer program is no mere logical mechanism; it also incorporates two kinds of scientific input: the teachings of physics and chemistry, codified as "mathematical models" of the various relevant phenomena; and the techniques of numerical analysis, which provide solution algorithms for the equations through which the mathematical modelling is expressed.

15 11-3 Fig.3 Mathematical models - 1: Turbulent transport and reaction Nature: Differential & algebraic equations for: /3 U \ k = iu'i u'j, e = v( i'f, u'j u'j.u'i m'i, m', 2, etc \ OUi/ Auxiliary relations such as: u'j u'j =k 2 e '3uj/9xj, reaction rate of substance = const (m 1 / 2 ) 1/2 pe/k. Purposes: To provide quantifiable expressions for diffusion, source and sink terms in equations for time-average Uj, m/, T, etc. Basis: Rigour + intuition + empiricism Status: Hydrodynamics and concentrations of inerts well predicted; concentration of reactants less so. This is the first of four figures about mathematical models; it concerns turbulence. A turbulence model is a set of differential and auxiliary algebraic equations connecting statistical properties of turbulence, the so-called correlations, with each other and with terms appearing in the time-averaged equations of motion, continuity, energy, and conservation of chemical species. It is needed to allow the latter equations to be solved. The alternative path via the time-dependent equations is impracticable, because the computer storage and time would exceed practical limits. The figure indicates some quantities which appear as dependent variables of common turbulence models, eg: k, the turbulence energy; e, the rate of dissipation of this energy, and m'/ 2 the time-average value of the square of the deviations of mass fraction m/ from the time-average value, mf. These quantities are needed for calculating the hydrodynamical and chemical behaviour of turbulent flows. Although turbulence modelling has been actively practised for only a few years, it can already respond to the demands of the engineer, especially in respect of hydrodynamics. When the turbulence interacts with the chemical reaction, however, still considerable uncertainty remains. Fig.4 Mathematical models - 2: Radiative transport Nature: EITHER (Hottel) full-matrix equations for interzone transfer, OR (flux method) sparse-matrix equations for 1,2 or 3 flux sums, PLUS data on ' emissivities etc of gases, clouds, walls. Purposes: To provide radiative sources and sinks in energy equation. Basis: Finite-interval simplification of rigorous equations; material properties from experiment. Status: Mathematics: Full-matrix method is too expensive for practical use; sparse-matrix method is not always sufficiently accurate; improvement needed. Property data are sufficient only for non-reacting gases. Heat transport by turbulent motion can be described by differential equations fairly well, even though typical eddy sizes are not small compared with apparatus dimensions. The "mean free path" of radiative transfer, by contrast, is normally so much greater than these dimensions that an Integra-differential formulation must be used. Every part of the flow can communicate with every other part; and the mathematical model must allow for this fact.

16 The numerical form of the integro-differential equations is provided by Hottel's "zone method" of radiativetransfer analysis. If the number of zones is large enough, this method is rigorously correct. Unfortunately, the algebraic representation of the method involves a large and full matrix of coefficients, each of which must be calculated repeatedly as the composition distribution of the gases.changes; consequently, perhaps no-one has ever been able to combine the zone method with a hydrodynamic calculation. There is another approach, the "flux method". It is less rigorous, but simple enough to be employed. What we do not yet know is whether its errors will be serious in practice; but certainly a method combining the advantages of the "zone" and "flux" methods is to be desired. Fig.5 Mathematical models - 3: Chemical kinetics Nature: Algebraic relations between reaction rate R/, and and::m/, m m, etc, p, T and other local properties. Purpose: To provide sources and sinks in differential equations for m/, etc. Basis: Physico-chemical concepts (collision frequency, activation energy) plus empiricism. Status: For fast reactions, controlled by mixing, detailed kinetic knowledge not needed. Many species participate in combustion; rate constants are often uncertain For NOX constants are known. For soot they are not. Chemical processes also require to be modelled. When we focus attention on the mass fractions of various chemical species in a mixture, and suppose that these and the mixture temperature completely specify the state, we are of course simplifying the situation. Thus, some members of a given species have a high energy at a given location, and some a low energy; and the partition of the energy is important. Thus, the Arrhenius reaction-rate expression is only a roughly-correct representation of what occurs. Yet we accept its form, and turn to experiment to provide us with best-fit values of the "constants" appearing in it. The science of chemical kinetics is, looked at from one point of view, just a collection of such best-fit constants, and of their associated uncertainties. These expressions and constants provide us with the "source" and "sink" terms of the differential equations for individual-species calculations. These equations are numerous; but often it is practicable to neglect a large number of them. This is true for the main fuel-air reaction for example; and only a few reactions need be considered in the prediction of NOX. Knowledge of the soot-forming reactions is, unfortunately, less satisfactory. Fig.6 Mathematical models - 4: Two-phase phenomena Nature: Differential and algebraic equations for population densities of discrete groups of particles, differing in size, composition, velocity, temperature. Purpose: To predict fuel burning, soot formation, influences on hydrodynamics, radiation, kinetics. Basis: Physics of nucleation, size change, flocculation, drag, fragmentation. Status: Computer size permits only 1 or 2 particlegroup distinctions. Little use so far. Knowledge of particle-particle interaction is still small.

17 11-5 The fourth group of phenomena requiring "modelling" concerns two-phase effects. In combustion technology, these appear because fuel is often injected in the form of a spray of small droplets, and because products of combustion such as soot and aluminium oxide are present as solid particles. The processes requiring representation in the model are: the momentum, heat and mass transfers between the particulate and 'continuous phases; particle creation by nucleation, collision and disruption; and inter-particle exchanges of mass and momentum. Some treatment of these processes, even if it is only neglect, must underly every spray-combustion calculation. Because the presence of turbulent mixing causes particles entering the combustor together to undergo differing histories, analysis of particle behaviour must be statistical: one focusses attention on the density of the particle population in given intervals of particle size, velocity, temperature etc. Usually however, computer limitations restrict what one can consider at any one time: one common practice is to presume that all particles possess the local gas-phase velocities, and then to solve the differential equations for the distribution of material in particle-size space. This is an important area of modelling, easily amenable to analysis, yet it has still been very little explored. Fig.7 Computational procedures - 1: Two-dimensional parabolic Typical equation:^ 30 3x, + + b 3 (c / 3*\ 1 = 3x 2 3x 2 \ 3x 2 / Examples: Axi-symmetrical steady jets, wakes, diffusion flames, boundary layers on walls, flows in pipes, diffusers, nozzles, solid-propellant rockers, after burner systems. Author's preferred method: Patankar-Spalding (GENMIX, PASS), with marching integration, expanding grid, nondimensional stream-function coordinates. Status: Computer program is generally available. Method is economical, and has been widely used. Recent versions handle lateral momentum equation. We turn now to procedures for solving the differential equations which the mathematical models supply. The most important statement to be made is this: procedures do now exist for solving all the equations, even in the most complex circumstances; some of the methods developed and used by the author and his colleagues will be mentioned if on the next few figures. All combustor problems entail numerous equations, linked together, and highly non-linear. Each of the following variables is important, and requires an equation of its own: each of three velocity components; pressure; stagnation enthalpy; concentration. Fortunately, the relevant equations are all similar in form so the same algorithm can be used for them all. It is convenient to class methods of solution according to their dimensionality: this involves enumerating the independent variables, and stating whether the flow is parabolic (ie possessing one direction without second-order differentials) or elliptic. This figure refers to two-dimensional parabolic systems: the typical equation is given; examples are mentioned; and a method of solution and computer code are briefly described. Problems of this kind are now very easy to solve; for the program is available; and there are very many users who can be consulted throughout the world.

18 Fig.8 Computational procedures - 2: Two-dimensional elliptic / 30 \ 3 / 30 \ Typical equation: a,- + a b,- Cl -) +b 2 (c 2 =d 3x, 3x 2 3x, \ 3x,/ 3x 2 \ 3x 2 / Examples: Axi-symmetrical steady flows with recirculation; flame stabilisation; rocket-base flow; swirling flow in cyclindrical chamber. Author's preferred method: SIMPLE ( = semi-implicit method for pressure-linked equations) incorporated into computer program EASI (= elliptic axi-symmetrical integrator). Status: "Teaching" version of the program available. Many practical flows now successfully computed. The equation on this figure differs from that on the previous one in that is possesses second-order differential coefficients with respect to both Xi and x 2. This is one feature which makes it elliptic. Such equations must be solved whenever the flow exhibits recirculation in both the space directions. The author and his colleagues formerly used a method employing the stream function and the vorticity as the main hydrodynamic variables. The resulting computer program has been widely used, and is sometimes called the Spalding method, particularly when it has been giving indifferent results (Otherwise the user normally gives credit to the particular detailed innovation which he has supplied). Although the i// ~ u method has given good service, the preferred one is now SIMPLE, which focusses attention directly on the pressure and the three velocity components. Major reasons are:- SIMPLE can be used for 3D as well as 2Dproblems;SIMPLE handles compressible-flow problems as well as incompressible ones; boundary conditions and variable properties are more accurately and realistically handled in SIMPLE. Although this method is only a few years old, it has already been used by many people with success. It is economical of computer time, a fact that is to be attributed to its "implicit" nature. Most other methods approach the steady state by way of "time-marching"; SIMPLE proceeds more directly. Fig.9 Computational procedures - 3: Three-dimensional parabolic / 30 \ / / 90\\ Typical equation: [a- I + [b.- [c.- ) = d \ l3 Vi=l,3 \ l9x i V l3x i//i=l,2 Examples: (i) Transient versions of examples on slide 8. (ii) Steady flow in ducts of non-circular section and/or nonsymmetric boundary conditions. Author's preferred method: SIMPLE (Incorporated into EASI (transient, or STABL (= steady analysis of boundary layers). Status: Program not yet public. Successful computations made for ducts, missiles, effects of chemical reaction, buoyancy, compressibility. 9 Of course, it may be the unsteady behaviour that one is interested in; the equations then become parabolic, a third first-order differential coefficient being introduced. The EASI program can also operate in a transient mode in order to solve problems of this kind. However,it is not only transient problems with two space dimensions that fall into this category: certain steady three-dimensional flows, namely those with one pronounced direction of flow, possess a similar character. This pronounced direction takes the place of the time dimension; only two-dimensional computer storage is required; and integration takes place by "marching" from upstream to downstream. The flow in a rectangular-sectioned diffuser is of this category; and so is that in a pipe bend. Recirculation in the main direction of flow must not occur.

19 11-7 There is a special computer program for this second class of problem. It is called STABL; and indeed it is a very well-behaved program. In addition to problems arising in mechanical engineering, it is also useful for those in the environment, for example the thermal pollution of rivers. Fig. 10 Computational procedures - 4: Three-dimensional elliptic and parabolic / 30\ / 3 / 30^ Typical equation: I*.- }. + (b. c. - 1). \i3xj/i=l,4 V i 3x; \i 3xj//i=l,3 = d Examples: Steady (elliptic) and transient (parabolic) 3D flows in: gas-turbine combustion chambers; mixing chambers; stalled diffusers; elbows; rocket motors; compressor cascades. Author's preferred method: SIMPLE, incorporated into TRIC or TRIP (three-dimensional recirculating flow in cartesian or polar coordinates). Status: Program not yet public. Practical applications made to: nuclear-reactor rod bundles; steam boilers; simple combustors. In the most general case, the differential equations contain four first-order terms (3 for space and 1 for time) and three second-order terms. They are parabolic in the time-dependent case, and otherwise elliptic. These are the equations which must be solved whenever there are both positive and negative velocities in all three directions. As the figure shows, some of the most interesting aero-engine phenomena fall into this category. The SIMPLE algorithm has been incorporated into two computer codes for fully-3d flows: TRIC and TRIP, the one being for a Cartesian and the other for a polar co-ordinate system. These codes have already been employed successfully for predicting the practical flow and heat-transfer phenomena associated with nuclear reactors and associated plant; environmental flows, such as those in bays and estuaries, are also being predicted. The time has now arrived for using them for the design of combustors. Of course, the computer storage must be three-dimensional; and, since, currently-available computers still have rather small core stores, it is often necessary to use a finer grid than one would like for high accuracy. Developments in the computer industry should soon remove this limitation. Fig.l 1 Examples - 1: Flame spread in a duct Description: Downstream of flame-holder, flame spreads through air fuel-droplet stream. Models employed: Turbulence, including reaction-rate influence; radiation; chemical kinetics (perhaps); two-phase effects. Problem category: 2D, parabolic Computer code: GENMIX or PASS Status: Successful comparisons with experiment have been made for single-phase uniform-upstream-composition flows.

20 11-8 The final section of the paper will be devoted to describing, by way of example, how the mathematical models and the computational procedures can be put to use. The examples will be of progressively advancing dimensionality. The slide shows a two-dimensional parabolic problem, namely, steady-state flame-spread downstream of a flameholder in an axi-symmetrical duct. The immediate vicinity of the flame-holder is omitted from the analysis, because there is recirculation there; as a consequence, only one-dimensional storage is required, and a single marching integration suffices. How many differential equations will have to be solved? One each for the following variables:- longitudinal velocity; stagnation enthalpy; gaseous fuel concentration; oxygen concentration; fuel particle concentration in from 5 to 10 particle-size intervals; at least one radiant-flux sum; and at least two statistical properties of the turbulence. Probably the minimum number is 12 and the maximum could lie above 30; Despite this, the computer code is economical enough for computer costs to be no serious deterrent to the industrial user. Fig. 12 Examples - 2: Axi-symmetrical combustor Description: Pre-vaporised fuel is injected with swirling air into a film-cooled combustor. fuel Models employed: Turbulence, including concentration fluctuations; radiation; combustion kinetics (simple scheme); NOX kinetics. Problem category: 2D, elliptic Computer code: EASI (steady-state mode). Status: Comparison with experiment is in progress. Qualitative agreement is obtained, but experimental air injection not quite axi-symmetrical. If we are concerned with the flame-stabilisation region of the previous example, or with the main combustor of the engine, a computer code is needed that will handle recirculation. The problem is elliptic, in the steady state; and a grid must be used which allows simultaneous storage of fluid variables over the whole field of flow. The figure shows a two-dimensional parabolic proble, namely, steady-state flame-spread downstream of a flameregarded as axi-symmetrical. Pre-vaporised fuel and air enter separately, in a swirling manner, a film-cooled cylindrical combustor with conical ends. The equations governing the flow, mixing, reaction and heat transfer are solved by way of the program EASI. This uses two-dimensional storage, and operates iteratively. The steady-state version of the code is used. Real combustors of this type almost invariably possess some features which are not axi-symmetrical. Thus, "secondary air" is commonly injected through a few large holes in the downstream part of the outer wall; the resulting jets tend to penetrate to the combustor axis, where they collide and set up axially-directed streams. The figure shows how this feature can be partially represented by inclusion of a central upstream-directed additional air supply. Of course, this is only an incomplete representation of reality.

21 11-9 Fig. 13 Examples - 3: Unsteady axi-symmetrical flame spread Description: As for example 1 (figure 11), but with oscillatory input conditions. The task is to determine what makes small oscillations grow. Models employed: Turbulence, including reaction-rate influence; simple chemical kinetics; two-phase effects. Problem category: 3D, parabolic Computer code: EASI (transient version) Status: Preliminary demonstration of oscillatory effects has been made. Much testing and refinement may be needed before "buzz" phenomenon is elucidated. The next step along the road to increased dimensionality is a three-dimensional parabolic problem, of which the first kind has two space dimensions and a variation in time. As our example, we here consider the flame-spread situation of figure 11, but allow for time-dependence. The transient flame-spread phenomenon is of great importance because, under some circumstances, oscillatory burning takes place; then noise, and excessive heat transfer to solid surfaces, may render the engine unusable. The transient version of EASI will solve this problem; or rather it will solve the relevant equations. Whether the so-called "buzz" phenomenon can be adequately represented, so that the practical problem of designing buzz-free systems can also be solved, remains still to be seen. What could prevent adequate representation? The main obstacles are likely to be firstly that a fine grid will be needed to take adequate account of geometrical features, and secondly that there is still considerable uncertainty about the proper way of modelling the interactions of turbulence and chemical reaction. However, the only way to advance is to go forward hopefully, and overcome the obstacles as they arise. This is what we are doing. Fig. 14 Examples - 4: Inclined rocket exhaust Description: Relative to the rocket, both the exhaust and the air flow at supersonic speed, but in slightly different directions. Models employed: Turbulence; complex kinetics; radiation; two-phase effects. Problem category: 3D, parabolic Computer code: STABL (supersonic version). Status: Simple demonstrations have been made. Development of the complete program awaits funding. This figure also concerns a three-dimensional parabolic flow; but it is a steady one, the "parabolic direction" being that of the general flow. The problem is that of predicting the conditions within the exhaust plume of a rocket, the axis of which is inclined to the direction of flight. This inclination causes the plume to "bend"; and of course it eliminates the possibility of axial symmetry.

22 11-10 The computer code for this problem is STABL; this employs two-dimensional storage and a marching integration starting at the nozzle exhaust and proceeding downstream. Because the flow is supersonic, the accuracy of prediction is likely to be greater than for the corresponding subsonic flow; for the coupling of all three momentum equations via the pressure can be rigorously taken into account, in supersonic flows, without numerical instability. The status of this example is as reported on the figure. There are certainly no obstacles to successful prediction of flows of this kind; all we have to do is to employ established techniques. Fig. 15 Examples - 5: Annular-combustor sector Description: Air and atomised fuel flow steadily into annulus, mixing and burning simultaneously. Models employed: Turbulence, simple kinetics; radiation; two-phase effects (size distribution). Problem category: 3D, elliptic Computer code: TRIP (steady-state version, with cyclic boundary conditions). Status: Single-phase demonstrations have been performed. Development of complete program awaits new resources. The flow in the main combustor of a gas-turbine engine is three-dimensional, steady and elliptic. In order to predict this flow, and its practical effects such as the total NOX production and the temperature distributions on the walls, we need a computer code such as TRIP; this name, you may remember, stands for tfiree-dimensional.recirculating flow in polar coordinates. Polar coordinates are needed because the geometry is axi-symmetrical in many of its features. Often attention can be confined, as is indicated on the figure, to just one sector of an annulus. Then the boundary conditions possess a cyclic character: what goes out from one boundary comes in at the other; but neither is known a priori. To be able to design gas-turbine combustors through the use of TRIP as an optimising device is one of those objectives of such attractiveness that enthusiasts are inclined to confuse the first timid explorations with actual achievement, while pessimists feel compelled to ridicule the prospect. Realists can take note of the fact that feasibility has been demonstrated, that computer storage and time requirements remain heavy, that comparisons with experiment remain to be made; and they can reasonably conclude that the potential benefit justifies the cost of continued development.

23 Fig. 16 Examples - 6: Chemical laser Description: Supersonic streams of hydrogen and helium + fluorine mix in a fixed-area He + F cavity and react forming mixture allowing laser action. Models employed: Turbulence (with low-reynoldsnumber effects); complex chemical kinetics; radiation. Problem category: 2D, parabolic Computer code: PASS (parabolic axi-symmetrical systems) with lateral momentum equations. Status: Demonstration runs complete. Pressure waves well predicted. This final example brings us back to two-dimensional parabolic flows, and possesses two distinct points of interest. The first is that the chemical reactions are not those of fuel combustion but of the chemical laser. You will know that, when hydrogen and fluorine are appropriately mixed and reacted, it is possible for their intermediate reaction products to exhibit those states of disequilibrium which give rise to laser effects. There is much practical interest in such phenomena, both for military and industrial purposes. The second point of interest is that, because the flow is supersonic and the two streams enter with a finite mutual angle of incidence, solution of the lateral momentum equation is essential; for it is this which allows the pressure variations across the flow to be calculated, without which the reaction rates could not be predicted. The computer code PASS has been quite recently extended to permit this pressure-variation calculation, still within the framework of a single marching integration. Because only one-dimensional storage is needed, and no iteration, a fine grid can be employed at little cost. Fig. 17 Final remarks Progress: All computational difficulties have been solved (though 3D elliptic problems remain expensive). Mathematical models exist for all major component processes. Research needs: (i) A major experiment-plus-computation effort is needed, to verify predictions for full-scale & laboratory equipment, (ii) Guided by the results of (i), models can then be refined, especially: turbulence-reaction effects; influence of temperature fluctuations on NOX; soot formation. What then are the conclusions that can legitimately be drawn from thisrsurvey? Certainly the situation is radically different from that of ten years ago. Then one had to perform experiments, in almost every case, in order to establish how a new design was likely to perform. The best theoretical aid was a set of difficult-to-apply similarity rules. Now we can fairly say that the computational problems are solved; but it is not yet certain that the mathematical

24 11-12 models of physical processes are adequate. Research is needed at two levels. First and foremost, we need many more comparisons of the computer predictions with practical experiments; these are required both for the simplified, but therefore illuminating, experiments of the research laboratory, and for the complex but ultimately decisive conditions of the full-scale equipment. Secondly the models of turbulence, radiation, chemical kinetics and two-phase phenomena all require improvement, I hope however that enough of the potential utility of the computer programs has appeared from this lecture to have made it seem likely that the costs of this research will be repaid many times. APPENDIX The following pages supplement the text of the lecture by providing lists of references, with some explanatory notes. The task of providing a comprehensive lists of relevant references is a large one; consequently, it is ignorance rather than policy which has caused so many of the references to be to the publications of the author and his colleagues. The author would be glad to be told of notable omissions. Notes for Figure 1 Some references concerned with the mathematical modelling of combustion processes and other fluid-flow phenomena of similar complexity. 1. Mellor, A.M. 2. Patankar, S.V. Spalding, D.B. 3. Spalding, D.B. 4. Spalding, D.B. 5. Spalding, D.B. 6. Spalding, D.B'. 7. Spalding, D.B. 8. Spalding, D.B. 9. Spalding, D.B. 10. Swithenbank, J. Poll, I. Vincent, M.W. Wright, D.D. Current Kinetic Modelling Techniques for Continuous How Combustors. In - Emissions from Continuous Combustion Systems. Edited by W.Cornelius and W.G.Agnew. Plenum Press, New York, Mathematical Models of Fluid Flow and Heat Transfer in Furnaces: A Review. Paper 2. Presented at the Fourth Symposium on Flames and Industry: Predictive Methods for Industrial Flames. September, Published by the Institute of Fuel, Predicting the Performance of Diesel-Engine Combustion Chambers. Closing Lecture at I Mech E Symposium on Diesel-Engine Combustion, London, Mathematical Models of Continuous Combustion. In - Emissions from Continuous Combustion Systems. Edited by W.Cornelius and W.G.Agnew. Plenum Press, New York, pp 3-21, Combustion as Applied to Engineering. J. Institute of Fuel, pp , April, The Mathematical Modelling of Rivers. Imperial College London. Mechanical Engineering Department Report. Number HTS/74/4, January, Heat and Mass Transfer in Aircraft Propulsion. Imperial College London. Mechanical Engineering Department Report. Number HTS/73/55, December, Heat and Mass Transfer in the Environment. Imperial College London. Mechanical Engineering Department Report. Number HTS/73/56, December, New Ways of Predicting Heat-Transfer Performance. Imperial College London. Mechanical Engineering Department Report. Number HTS/73/49, October, Combustion Design Fundamentals. 14th Symposium (International) on Combustion. The Combustion Institute. Pittsburgh, pp , Notes for Figure 2 The following references provide a partial introduction to the literature on modelling. The first section concerns the theory and practice of performance prediction by way of scale-model experiments. Subsequent sections concern mathematical models of increasing dimensionality.

25 ,11-13 a) Physical Models 1. Chesters, J.H. J. Iron and Steel Institute. Vol.162, p.385, Howes, R.S. Halliday, I.M.D. Philip, A.R. 2. Clarke, A.E. Some Experiences in Gas Turbine Combustion Practice Using Water Flow Visualization Gerrard, A.J. Techniques. Ninth Symposium (International) on Combustion. Academic Press. Holliday, L.A. New York, p.878, Damkohler, G. Der Chemie-Ingenieur (A.Eucken and M.Jakob eds). Band III. Teil 1. Akademische Verlag, Reprinted by VDI-Fachgruppe Verfahrenstechnik Leverkusen, Groume-Grjimailo, W.E. The Flow of Gases in Furnaces. Wiley, New York, Hottel, H.C. Modelling Studies of Baffle-Type Combustors. Ninth Symposium (International) on Williams, G.C. Combustion. Academic Press, p.923, Jensen, W.P. Tobey, A.C. Burrage, P.M.R. 6. LeFebvre, A.H. Simulation of Low Combustion Pressures by Water. Injection. Seventh Symposium Halls, G.A. (International) on Combustion. Butterworths, London, p.654, Putnam, A.A. Application of Dimensionless Numbers to Flash-Back and Other Combustion Phenomena. Jensen, R.A. Third Symposium (International) on Combustion. Williams and Wilkins, Baltimore. pp.89-98, Spalding, D.B. Analogue for High-Intensity Steady-Flow Combustion Phenomena. Proceedings of Institution of Mechanical Engineers. Vol.171, No.10, pp , Spalding, D.B. The Art of Partial Modelling. Ninth (International) Symposium on Combustion. Academic Press, New York, pp , Spalding, D.B. Some Thoughts on Flame Theory. Combustion and Propulsion. Third AGARD Colloquium.Pergamon Press, London, pp , Stewart, D.G. Scaling of Gas Turbine Combustion Systems. In Selected Combustion Problems. Vol.2, Butterworth's, London, pp , Traustel, S. Modellgesetze der Vergasung und Verhuttung. Akademie-Verlag, b) Zero-Dimensional Mathematical Models 13. Bragg, S.L. The Influence of Altitude Operating Conditions on Combustion Chamber Design. In Holliday, J.B. Selected Combustion Problems. Vol.2, Butterworth's, London, pp , Hottel, H.C. Space Requirement for the Combustion of Pulverized Coal. Ind.Eng.Chem. Vol.32, Stewart, I.McC. pp.719-7/o, Longwell, J.P. Flame Stability in Bluff-Body Recirculation Zones. Ind.Eng.Chem. Vol.45, p. 1629, Frost, E.E Weiss, M.A. 16. Spalding, D.B. The Stability of Steady Exothermic Chemical Reactions in Simple Non-Adiabatic Systems. Chemical Engineering Science. Vol.11, pp.53-60, Van Heerden, C. Autothermic Processes. Ind.Eng.Chem. Vol.45, p.1242, c) One-Dimensional Mathematical Models 18. Chesters, J.H- Comparison of Calculated and Actual Heat Transfer. Iron and Steel Institute. Special Hoyle, K.H. Report. Number 59. p.65, Pearson, S.W. Thring, M.W.

26 Csaba, J. Leggett, A.D. 20. Field, M.A. Gill, D.W. Prediction of the Temperature Distribution Along A Pulverised-Coal Flame. Journal of the Institute of Fuel. Vol.37, pp , A Mathematical Model of the Combustion of Pulverized Coal in a Cylindrical Combustion Chamber. BCURA Members Information Circular. Number 318, Spalding, D.B. The One-Dimensional Theory of Furnace Heat Transfer. Society Journal. Vol.10, pp.8-17, University of Sheffield Fuel 22. Thring, M.W. 23. Thring, M.W. Smith, D. The Effect of Emissivity and Flame Length on Heat Transfer in the Open-Hearth Furnace. J.Iron and Steel Institute. Vol.171, pp , An Improved Model for the Calculation of Heat Transfer in the Open-Hearth Furnace. J.Iron and Steel Institute. Vol.179, p.227, 1955 d) Two-Dimensional Mathematical Models 24. Gibson, M.M. Morgan, B.B. 25. Kent, J.H. Bilger, R.W. 26. Lowes, T.M. Heap, M.P. Michelfelder, S. PAI, B.R. 27. Pun, W.M. Spalding, D.B. 28. Richter, W. Quack, R. 29. Sala, R Spalding, D.B. 30. Schorr, C.J. Berman, K. Worner, G. 31. Schorr, C.J. Worner, G.A. Schimke, J. 32. Spalding, D.B. Mathematical Model of Combustion of Solid Particles in a Turbulent Stream with Recirculation. J.Institute of Fuel. Vol.43, pp , Turbulent Diffusion Flames. Fourteenth Symposium (International) on Combustion The Combustion Institute. Pittsburgh, pp , Mathematical Modelling of Combustion Chamber Performance. Proceedings Fourth Flames and Industry Symposium, London, A Procedure for Predicting the Velocity and Temperature Distributions in a Confined, Steady, Turbulent, Gaseous, Diffusion Flame. XVIII International Astronautical Congress, Belgrade, Pergamon Press, pp.3-21, A Mathematical Model of a Low-Volatile Pulverized-Fuel Flame. In - Seminar of International Centre of Heat and Mass Transfer To be Published by Scripta Technica. A Mathematical Model for an Axi-Symmetrical Diffusion Flame in a Furnace. La Rivista dei Combustibili. Vol.XXVH, pp , April/May, Modelling of High-Energy Gaseous Combustors for Performance Prediction. Bell Aerospace Company, Analytical Modelling of a Spherical Combustor Including Recirculation. Fourteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh. pp , Mixing and Chemical Reaction in Steady Confined Turbulent Flames. Thirteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh. p.649, f) Three-Dimensional Mathematical Models 33. Patankar, S.V. Spalding, D.B. 34. Patankar, S.W. Spalding, D.B. 35. Zuber, I. A Computer Model for Three Dimensional Flow in Furnaces. Fourteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, pp , Simultaneous Predictions of Flow Pattern and Radiation for Three-Dimensional Flames. In - Seminar of International Centre of Heat and Mass Transfer To be published by Scripta Technica. Ein Mathematisches Modell des Brennraums. Monographs and Memoranda No. 12. Staatliche Forschungs Institut fiir Maschinenbau Bechovice Czechoslovakia, 1972.

27 11-15 Notes for Figure 3 The literature on mathematical models is increasing rapidly. The following classified list of references may assist the newcomer to approach it effectively. a) General Works on Turbulence and Turbulence Models 1. Davies, J.T. 2. Harlow, F.H. (Editor) 3. Hinze, J.O. 4. Launder, B.E. Spalding, D.B. 5. Launder, B.E. Spalding, D.B. 6. Launder, B.E. Spalding, D.B. 7. Launder, B.E. Spalding, D.B. Whitelaw, J.H. 8. Leslie, D.C. 9. Mellor, G. Herring, H. 10. Monin, A.S. Yaglom, A.M. 11. Spalding, D.B. 12. Tennekes, H. Lumley, J. Turbulence Phenomena. Academic Press, New York, Turbulence Transport Modelling. AIAA Selected Reprint Series. Vol.XIV, Turbulence. McGraw Hill, New York, Mathematical Models of Turbulence. Academic Press, London and New York, The Numerical Computation of Turbulent Flows. Computer Methods in Applied Mechanics and Engineering. To be published. Turbulence Models and Their Application to the Prediction of Internal Flows. Institute of Mechanical Engineering Symposium on Internal Flows. In - Heat and Fluid Flow. Vol.2, No.l, pp April, Turbulence Models and Their Experimental Verification. A Course of Lectures at Imperial College Mechanical Engineering Department, Recorded in Heat Transfer Section Reports. Numbers HTS/73/ 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28. Developments in the Theory of Turbulence. Oxford University Press, A Survey of the Mean Turbulent Field Closure Models. AIAA Journal. Vol.11, p.590, Statisticheskaya Gidrodinamika. Nauka Moscow, Mathematical Models of Free Turbulent Flows. Institute Nazionale di Alta Matematica Symposia Mathematica. Vol.IX. pp , A First Course in Turbulence. MIT Press, b) Turbulence Models without Special Differential Equations 13. Clauser, F.H. 14. Prandtl, L. 15. Von Karman, Th The Turbulent Boundary Layer. In - Advances in Applied Mechanics. Vol.4, parti. Academic Press, New York, Bericht uber Untersuchungen zur ausgebildeten Turbulenz. Z.Angew Math.Mech. (ZAMM). Vol.5. No.2, pp , Mechanische Ahnlichkeit und Turbulenz. Proceedings Th.ird International Congress Applied Mechanics. Stockholm. Part 1. p.85, c) Turbulence Models Employing One Special Differential Equation 16. Beckwith, I.E. Bushnell, D.M. 17. Bradshaw, P Ferriss, D.H. Atwell, N.P. 18. Emmons, H.W. Detailed Description and Results of a Method for Computating Mean and Fluctuating Quantities in Turbulent Boundary Layers. NASA Report. Number TN D Washington, Calculation of Boundary-Layer Development Using the Turbulent Energy Equation. J.Fluid Mechanics. Vol.28, part 3, pp , Shear Flow Turbulence. Proceedings Second U.S. National Congress of Applied Mechanics. ASME. p.l, 1954.

28 Glushko, G.S. 20. Nee, V.W. Kovasnay, L.S.G. 21. Nevzglyadov, V. 22. Prandtl, L. 23. Spalding, D.B. Turbulent Boundary Layer on a Plane Plate in an Incompressible Fluid. Izv Akad Nauk SSSR Ser Mech. No.4, pp , (A full account of this is given by Beckwith and Bushnell, 1968). Simple Phenomenological Theory of Turbulent Shear Flows. Physics of Fluids. Vol.12, No.3, pp , A Phenonemological Theory of Turbulence. Journal of Physics (USSR) Vol.IX, No.3, pp , Uber ein neues Formelsystem fur die ausgebildete Turbulenz. Nachr Akad der Wissenschaft in Gottingen: van den Loech und Ruprecht. pp.6-19, Heat Transfer from Turbulent Separated Flows. J Fluid Mechanics. Vol.27, part 1, pp , d) Turbulence Models Employing Two Special Differential Equations 24. Gibson, M.M. Spalding, D.B. 25. Harlow, F.H. 26. Harlow, F.H. Hirt, C.W. 27. Harlow, F.H. Nakayama, P.I. 28. Jones, W.P. Launder, B.E. 29. Jones, W.P. Launder, B.E. 30. Kolmogorov, A.N. 31. NG K.H. Spalding, D.B. 32. Rodi, W. Spalding, D.B. 33. Saffman, P.G. 34. Spalding, D.B. 35. Spalding, D.B. 36. Spalding, D.B. A Two-Equation Model of Turbulence Applied to the Prediction of Heat and Mass Transfer in Wall Boundary Layers. ASME. 72-HT-15, Transport of Anisotropic or Low-Intensity Turbulence. Los Alamos Scientific Laboratory Report. Number. LA-3947, Generalised Transport Theory of Anisotropic Turbulence. Los Alamos Scientific Laboratory Report. Number LA-4086, Turbulence Transport Equations. Physics of Fluids. Vol.10, No.l 1, pp , November, The Prediction of Laminarisation with a Two-Equation Model of Turbulence. International Journal of Heat and Mass Transfer. Vol.15, p.301, Prediction of Low Reynolds Number Phenomena with a Two-Equation Model of Turbulence. International Journal of Heat and Mass Transfer. Vol.16, pp , Equations of Turbulent Motion of an Incompressible Fluid. Izv Akad Nauk. SSSR. Ser Phys. Vol.6, No.1/2, pp.56-58, (Translated into English as Imperial College Mechanical Engineering Department Report. Number ON/6, 1968). Turbulence Model for Boundary Layers Near Walls. Physics of Fluids. Vol.15, No.l, pp.20-30, A Two-Parameter Model of Turbulence and its Application to Free Jets. Warme und Stoffubertragung. Vol.3, No.2, pp.85-95, A Model for Inhomogeneous Turbulent Flow. Proceedings of Royal Society, London, Vol.A317, pp , The Calculation of the Length Scale of Turbulence in Some Shear Flows Remote from Walls. In - Progress in Heat and Mass Transfer. Vol.2. Edited by T.F.Irvine et al Pergamon Press, pp , The Prediction of Two-dimensional Steady, Turbulent, Elliptic Flows. International Seminar on Heat and Mass Transfer in Flows with Separated Regions. Yugoslavia. Sep, Imperial College, London. Mechanical Engineering Department Report. Number EF/TN/A/16, A Two-dimensional Model of Turbulence. VDI Forschungsheft 549 pp.5-16, e) Turbulence Models with Three or More Special Differential Equations 37. Chou, P.Y. 38. Daly, B.J. Harlow, F.H. On Velocity Correlations and the Solution of the Equations of Turbulent Fluctuation. Quarterly J of Applied Mathematics. Vol.3, No.l, pp.38-54, Transport Theory of Turbulence. Los Alamos Scientific Laboratory Report. Number LA-DC-11304, 1970.

29 39. Davidov, B.I. On the Statistical Dynamics of an Incompressible Turbulent Fluid. Dokl AN. SSSR. Vol.136, No.l, pp.47-50, Hanjalic, K. Two-Dimensional Asymmetric Turbulent Flow in Ducts. PhD Thesis. London University, Hanjalic, K A Reynolds Stress Model of Turbulence and Its Application to Asymmetric Shear Flows. Launder, B.E. J Fluid Mechanics. Vol.52, p.609, Kolovandin, B.A. Statistical Transfer Theory in Turbulent Shear Flows. International Seminar on Heat and Mass Transfer in Flows with Separated Regions, Yugoslavia, Sep, Kolovandin, B.A. On the Statistical Theory of Non-Uniform Turbulence. International Seminar on Heat Vatutin, LA. and Mass Transfer in Flows with Separated Regions, Yugoslavia, Sep, Rotta, J. Statistische Theorie Nichthomogener Turbulenz. Z.Physik, Vol.129, p , (1951), and Vol.131, pp.51-57, (1953). Translated into English by W.Rodi as - Imperial College Mechanical Engineering Department Technical Notes. Numbers TWF/TN/38 and TWF/TN/ Spalding, D.B. Concentration Fluctuations in a Round Turbulent Free Jet. Chemical Engineering Science. Vol.26, pp , f) Turbulence Models Concerned with Interactions of Turbulence and Chemical Reaction 46. Mason, H.B. Prediction of Reaction Rates in Turbulent Pre-Mixed Boundary Layer Flows. Spalding, D.B. Combustion Institute European Symposium, Sala, R. A Mathematical Model for an Axi-Symmetrical Diffusion Flame in a Furnace. La Spalding, D.B. Rivista dei Combustibili. Vol.XXVII, pp Spalding, D.B. Mathematische Modelle Turbulenter Flammen. In - Vortrage der VDI-Tagung. Karlsruhe, Verbrennung und Feuerungen. VDI-Berichte. No. 146, Dusseldorf: VDI-Verlag. pp.25-30, Spalding, D.B. Mixing and Chemical Reactions, in Steady Confined Turbulent Flames. Thirteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, p.649, g) Comparison of Turbulence-Model Prediction with Experiment 50. Launder, B.E. The Prediction of Free Shear Flows - A Comparison of Six Turbulence Models. Morse, A.P. NASA Free Shear Flows Conference, Virginia, July NASA Report. Number Rodi, W. SP-311, Spalding, D.B. 51. NG, K.H. A Comparison of Three Methods of Predicting the Hydrodynamic Behaviour of Two- Spalding, D.B. Dimensional Turbulent Boundary Layers. Imperial College London, Mechanical Engineering Department. Report. Number BL/TN/A/27, Rodi, W. The Prediction of Free Turbulent Boundary Layers by Use of a Two-Equation Model of Turbulence. PhD Thesis, London University, Notes for Figure 4 a) General Works on Radiative Heat Transfer 1. Beer, J.M. Heat Transfer from Flames. Seminar of International Centre for Heat and Mass (Editor) Transfer. To be published by Scripta Technica, Hottel, H.C. Radiative Transfer, McGraw Hill, New York, Sarofim, A.F. 3. Ozisik, M.N. Radiative Transfer. Wiley-Interscience, New York, Sparrow, E.M. Radiation and Heat Transfer. Brooks/Cole, Belmont California, Cess, R.D.

30 Viskanta, R. Radiation Transfer and Interaction of Convection With Radiation Transfer. In - Advances in Heat Transfer. Vol.3, Edited by T.F.Irvine and J.P.Hartnett, Academic Press, New York, b) References on the Hottel Zone Method 6. Beer, J.M. 7. Cannon, P. 8. Hirose, T. Mitunaga, A. 9. Hottel, H.C. Cohen, E.S. 10. Hottel, H.C. Sarofim, A.F. 11. Johnson, T.R. Beer, J.M. 12. Johnson, T.R. Beer, J.M. 13. Osuwan, S. 14. Steward, F.R. Cannon, P. 15. Steward, F.R. Guruz, H.K. 16. Steward, F.R. Osuwan, S. 17. Steward, F.R. Osuwan, S Picot, J.J.C. Methods for Calculating Radiative Heat Transfer from Flames in Combustors and Furnaces. In - Heat Transfer from Flames. Seminar of International Centre of Heat and Mass Transfer. To be published by Scripta Technica, The Calculation of Radiative Heat Flux in Furnace Enclosures Using the Monte Carlo Method. MSc Thesis in Chemical Engineering,University of New Brunswick, Investigation of Radiant Heat Exchange in a Boiler. Bulletin. S.M.E. Vol.14, p.829, Radiant Heat Transfer in a Gas-Filled Enclosure: Allowance for Non-Uniformity of Gas Temperature. J.Am.Inst.Chem.Engrs, Vol.4, p.3, The Effect of Gas Flow Patterns on Radiative Transfer in Cylindrical Furnaces. International Journal of Heat and Mass Transfer. Vol.8, pp Pergamon Press, The Zone Method of Analysis of Radiant Heat Transfer: A Model for Luminous Radiation. Fourth Symposium on Flames in Industry, Imperial College, London, Further Development of the Zone Method. Fourteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, pp , Radiative Heat Transfer in a Cylindrical Test Furnace. PhD Thesis. University of New Brunswick, The Calculation of Radiative Heat Flux in a Cylindrical Enclosure Using the Monte Carlo Method. International Journal of Heat and Mass Transfer. Vol.14, p.245, Mathematical Simulation of An Industrial Boiler by the Zone Method of Analysis. In Heat Transfer from Flames. Seminar of International Centre for Heat and Mass Transfer. To be published by Scripta Technica, A Mathematical Simulation of Radiative Heat Transfer in a Cylindrical Test Furnace. Canadian J of Chemical Engineering. Vol.50, p.450, Heat Transfer Measurements in a Cylindrical Test Furnace. Fourteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, pp , c) References on the Flux Method 18. Eddington, A.S. 19. Chu, C.M. Churchill, S.W. 20. Chen, J.C. Churchill, S.W. 21. CHEN, J.C. Churchill, S.W. 22. Evans, L.B. Chu, C.M. Churchill, S.W. 23. Gibson, M.M. Monahan, J.A. The Internal Constitution of Stars. Dover, New York, Institute Radio Engineers. Vol.AP-4, No.2, p.142, AIChE Journal. Vol.9, No.l, p.35, AIChE Journal. Vol.10, No.2, p.253, J Heat Transfer. Vol.87, p.381, A Simple Model of Raidation Heat Transfer from a Cloud of Burning Particles in a Confined Stream. International Journal of Heat and Mass Transfer. Vol.14, pp , 1971.

31 Gibson, M.M. Morgan, B.B. 25. Gosman, A.D. Lock wood, F.C. 26. Hamaker, H.C. 27. Larkin, B.K. Churchill, S.W. 28. Lockwood, F.C. Spalding, D.B. Mathematical Model of Combustion of Solid Particles in a Turbulent Stream with Recirculation. J Institute of Fuel, p.517, December, Incorporation of a Flux Model for Radiation into a Finite-Difference Procedure for Furnace Calculations. Fourteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, pp , Radiation and Heat Conduction in Light-Scattering Material. Philips Research Report. Vol.2, pp.55-67, p.103, p.l 12, p.420, AIChE Journal. Vol.5, No.4, p.467, Prediction of a Turbulent Reacting Flow with Significant Radiation. Imperial College. Mechanical Engineering Department. Report Number HTS/71/21, To be published in Proc Colloques d'evian J de Physique. 29. Lowes, T.M. Bartelds, H Heap, M.P. Michelfelder, S. Pai, B.R. The Prediction of Radiant Heat Transfer in Axi-Symmetrical Systems. Flame Research Foundation Document. Number G 02/a/25, International 30. Milne, E.A. 31. Patankar, S.V. Spalding, D.B. 32. Patankar, S.V. Spalding, D.B. 33. Roesler, F.C. 34. Schuster, A. In - Handbuch der Astrophysik. Edited hy G.Eberland et al. Vol.3, part 1, pp Springer, Berlin, A Computer Model for Three-Dimensional Flow in Furnaces. Fourteenth Symposium (International) on Combustion. Combustion Institute, Pittsburgh, pp , Simultaneous Predictions of Flow Pattern and Radiation for Three-Dimensional Flames. In Heat Transfer from Flames. Seminar of International Centre for Heat and Mass Transfer. To be Published by Scripta Technica, Imperial College London. Mechanical Engineering Department. Report Number HTS/73/39, Chemical Engineering Science. Vol.22, p.1325, Astrophysics Journal. Vol.21, pp.1-22, Siddall, R.G. Flux Methods for the Analysis of Radiant Heat Transfer. Flames in Industry. Institute of Fuel, London, In Fourth Symposiun'of 36. Siddall, R.G. Selchuk, N. The Application of Flux Methods to Prediction of the Behaviour of a Process Gas Heater. In - Heat Transfer from Flames. Seminar of International Centre for Heat and Mass Transfer, To be Published by Scripta Technica. Notes for Figure 5 a) General References on Chemical Kinetics 1. Benson, S.W. 2. Baulch, D.B. Drysdale, D.D. Home, D.G. 3. Baulch, D.L. Drysdale, D.D. Home, D.G. Lloyd, A.C. 4. Baulch, D.L. Drysdale, D.D. Home, D.D. Thermochemical Kinetics. Wiley, New York, Evaulated Kinetic Data for High Temperature Reactions. Vol.11. Homogeneous Gas- Phase Reaction of the N 2 /O 2 System. In preparation, Evaluated Kinetic Data for High-Temperature Reactions. Vol.1. Homogeneous Gas- Phase Reactions of H 2 /O 2 System, Butterworth's, London, An Assessment of Rate Data for High-Temperature Systems. Fourteenth Syposium (International) on Combustion. The Combustion Institute, Pittsburgh, pp , 1973.

32 Baulch, D.L. High Temperature Reaction Rate Data. The University, Leeds 2. Drysdale, D.D. Number 1, Number 2, Number 3, Number 4, Number 5, Lloyd, A.C. 6. Trol,T. Wagner, H.G. Physical Chemistry of Fast Reactions. Plenum Press, London. To be Published. 7. Wagner, H.G. Elementary Reactions in the Combustion of Small Inorganic Molecules. Fourteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh. pp.27-36, b) References on Hydrocarbon-Air Reactions 8. Bowman, B.R. Effects of Turbulent Mixing and Chemical Kinetics on Nitric Oxide Production in a Jet- Pratt.D.T. Stirred Reactor. Fourteenth Symposium (International) on Combustion. The Corn- Crowe, C.T. bustion Institute, Pittsburgh, pp , Edelman, R. A Quasi-Global Chemical-Kinetic Model for the Finite-Rate Combustion of Hydrocarbon Fortune, O. Fuels. AIAA Paper , Peelers, J. Reaction Mechanisms and Rate Constants of Elementary Steps in Methane-Oxygen Flames. Mahnen, G. Fourteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, pp , Wilson, W.E. Study of the Reaction of Hydroxy I Radical with Me thane by Quantitative ESR. Westenberg, A.A. Eleventh Symposium (International) on Combustion. The Combustion Institute, Pittsburgh. p.1143, c) References on NOX Reactions 12. Bowman, C.T. Kinetics of Nitric Oxide Formation in Combustion Processes. Fourteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, pp , Caretto, L.S. Modeling Pollutant Formation in Combustion Process. Fourteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, pp , Fletcher, R.S. A Model for Nitric Oxide Emissions from Aircraft Gas Turbine Engines. AIAA Paper. Heywood, J.B. Number , Westenberg, A.A. Combustion Science Technology. Vol.4, p.59, Zeldovich, Y.B. Acta Physicochim USSR. Vol.21, p.577, d) Reference on Soot Reactions 17. Lee, K.B. On the Rate of Combustion of Soot in a Laminar Soot Flame. Combustion and Flame. Thring,M.W. Vol.6, p.137, Beer, J.M. 18. Linden, L.H. Smoke Emission from Jet Engines. Combustion Science and Technology. Vol.2, p.401, Heywood, J.B Khan, I.M. Formation and Combustion of Carbon in a Diesel Engine. Proceedings Institute of Mechanical Engineers. Vol.184, part 3J, Sjogren, A. Soot Formation by Combustion of An Atomised Liquid Fuel. Fourteenth Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, pp , Wessborg, B.L. Physical Mechanisms in Carbon Formation in Flames. Fourteenth Symposium (Inter- Howard, J.B. national) on Combustion. The Combustion Institute, Pittsburgh, pp , Williams, G.C. Notes for Figure 6 a) References Concerned with Single Particles 1. Spalding, D.B. The Combustion of Liquid Fuels. Fourth Symposium (International) on Combustion. Williams and Wilkins, Baltimore, pp , 1953.

33 Spalding, D.B. Convective Mass Transfer. Arnold, London, Spalding, D.B. The Combustion of Liquid Fuels. PhD Thesis. Cambridge University, Spalding, D.B. Combustion of a Single Droplet and of a Fuel Spray. In - Selected Combustion Problems. Butterworth's, London, pp , b) References Concerned with the Behaviour of Clouds of Particles 5. Gibson, M.M. Mathematical Model of Combustion of Solid Particles in a Turbulent Stream with Re- Morgan, B.B. circulation. J Institute of Fuel. Vol.43, Spalding, D.B. The Calcualtion of Combustion Processes. Lecture 7: Some Special Applications of Combustion Theory. Imperial College London, Mechanical Engineering Department Report. Number HTS/71/74, Notes for Figure 7 Numerous practical applications have now been made of the method referred to on the figure. The advantageous features have proved to be: ( i) The "stretching" of the grid to enclose only the region of significant transfers of heat, mass and momentum. (ii) The use of the non-dimensional stream function as the lateral independent variable allows the lateral convective fluxes to be determined without iteration; the use of the cross-stream distance, whether nondimensionalised or not, lacks this advantage, often fatally. (iii) The computer programs have been organized so as to allow flexible adaptation to a large variety of processes. Many of the references on earlier pages report work in which the method has been used. Among these are:- Figure 2 References 1. Kent and Bilger Spalding Figure 3 References 1. Gibson and Spalding Jones and Launder 1972, Ng and Spalding Rodi and Spalding Spalding Hanjalic Hanjalic and Launder Spalding Spalding 1970, Mason and Spalding Sala and Spalding Launder, Morse, Rodi and Spalding 1972.

34 Ngand Spalding Rodi Figure 4 References 1. Lockwood and Spalding The following references cite one report of the use of the method for a complex combustion process (Jensen and Wilson) and then three sources of information about the method in successive stages of development. 1. Jensen, D.E. Rapid Computation of Physical and Chemical Structures of Rocket Exhaust Flames.. Wilson, A.S. In - Proceedings of Combustion Institute European Symposium. Edited by F.Weinberg. Academic Press, London, Patankar, S.V. Heat and Mass Transfer in Turbulent Boundary Layers.^ Morgan-Grampian, London, Spalding, D.B Patankar, S.V. Heat and Mass Transfer in Boundary Layers. Intertext, London, 2nd Edition, Spalding, D.B. 4. Spalding, D.B. A General Computer Program for Two-Dimensional Boundary-Layer Problems. Imperial College London. Mechanical Engineering Department Report. Number HTS/73/48, Notes for Figure 8 Although the SIMPLE method has now been in use for a number of years, it appears that only one publication has been made of its applications to two-dimensional flows (Spalding and Tetchell). The major reference to the SIMPLE algorithm is that of Patankar and Spalding, International Journal of Heat and Mass Transfer (see notes to figure 9) The method for two-dimensional steady flows used earlier by the author and his colleagues is that of Gosman et al (see below). It has been used by many authors for studies reported earlier in this paper namely: Figure 2 Reference 1. Gibson and Morgan Lowes et al Pun and Spalding Richter and Quack Schorr et al 1971, Figure 3 References 1. Spalding Figure 4 References 1. Gosman and Lockwood. 2. Lowes et al 1973.

35 11-23 References for Figure 8 1. Gosman, A.D. Pun, W.M. Runchal, A.K. Spalding, D.B. Wolfshtein, M. 2. Spalding, D.B. Tatchell, D.G. Heat and Mass Transfer in Recirculating Flows.. Academic Press, London, A Prediction Procedure for Flow, Combustion and Heat Transfer Close to the Base of a Rocket. Imperial College London. Mechanical Engineering Department Report. Number HTS/7 3/42. July, References for Figure 9 1. Caretto, L.S. Curr, R.M. Spalding, D.B. 2. McGuirk, J.J. Spalding, D.B. 3. Patankar, S.V. Spalding, D.B. 4. Sharma, D. Spalding, D.B. 5. Patankar, S.V. Spalding, D.B. 6. Rastogi, A.K. Whitelaw, J.H. Two Numerical Methods for Three-Dimensionaf Boundary-Layers. Computer Methods in Applied Mechanics and Engineering. Vol.1, pp.39-57, The Treatment in the 3D Boundary-Layer Program of Walls which do not Pass Through Grid Nodes. Imperial College London. Mechanical Engineering Department Report. Number HTS/73/13, A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Parabolic Flows. International Journal of Heat and Mass Transfer. Vol.15, pp Pergamon Press, Laminar Flow Heat Transfer in Rectangular Sectioned Ducts with One Moving Wall. First National Heat and Mass Transfer Conference. IIT; Madras, India. Paper Number HTM-26-71, Heat Transfer in the Entry Region of a Square-Sectioned Duct in the Presence of a Lateral Gravitational Field. First National Heat and Mass Transfer Conference. IIT, Madras, India. Paper Number HMT-40-71, The Calculation of Combustor-Wall Temperature Downstream of Total-Heat Cooling Rings. Imperial College London. Mechanical Engineering Department.Report. Number HTS/73/38, References for Figure Amsden, A.A. Harlow, F.H. 2. Caretto, L.S. Gosman, A.D. Patankar, S.V. Spalding, D.B. 3. Chorin, A.J. 4. Gosman, A.D. Herbert, R. Patankar, S.V. Potter, R. Spalding, D.B. 5. Patankar, S.V. 6. Patankar, S.V. Spalding, D.B. The SMAC Method. Los Alamos Scientific Laboratory Report. Number LA-4370, Two Calculation Procedures for Steady, Three-Dimensional Flows with Recirculation. Proceedings of Third International Conference on Numerical Methods in Fluid Mechanics. Edited by J.Ehlers, K.Hepp and H.A.Weidenmuller. Published by Springer, Germany. Vol.11, pp.60-68, Numerical Solution of the Navier-Stoke Equations. Mathematics of Computation. Vol.22, pp , No.104, Prediction of Coolant Flows and Temperatures in Pin Bundles Containing Blockages. Imperial College London. Mechanical Engineering Department Report. Number HTS/73/47. October, Calculation of Unsteady Compressible Flows Involving Shocks. Imperial College London, Mechanical Engineering Department Report. Number HTS/71/28, Numerical Predictions'of Three-Dimensional Flows. Imperial College London. Mechanical Engineering Department Report. Number HTS/72/15, 1972.

36 11-24 DISCUSSION Prof. Ferri: I would like to add a few remarks to Prof.Spalding's presentation. I think this type of analysis is very useful for extrapolating or interpolating experimental data. The implication here is that we do not really understand too well the physics of the problem and I suspect that the equations that are used are not really satisfactory insofar, for instance, they do not take into account the fluctuation of pressure due to chemical reactions. My point is that, since we do not have a very detailed understanding neither of the turbulence characteristics nor of its stability, we should try to understand the physics more than writing more complex equations. For instance working in combustion one of the things which worried me for many years has been the interaction of turbulence with droplets and the transport properties: there seem to be a lack of good papers dealing with this subject. In conclusion I think that one should devote more time trying to understand, in more details, the physics; only then the analyses may become much more quantitative than they are now.

37 12 THEORETICAL ANALYSIS OF NONEQUILIBRIUM HYDROGEN AIR REACTIONS BETWEEN TURBULENT SUPERSONIC COAXIAL STREAMS by H.Rortgen Lehrstuhl fiir Technische Thermodynamik Rhein.-Westf. Technische Hochschule Aachen 5100 Aachen, W.-Germany

38 12 RESUME Une 6tude analytique est faite de melange de jets turbulents, la combustion prenant place dans la zone de melange entre un jet central supersonique d'hydrogene froid et un ecoulement d'air supersonique p6riph6rique prechauffe. La meihode developpee est celle des differences finies qui decouple les equations principals par des techniques de linearisation locale jointe des m6thodes iteratives. Les modeles utilises pour d6crire les processus de transport en 6coulement turbulent reposent sur 1'introduction d'un coefficient de viscosite turbulent. Deux modeles sont employes pour prgdire le processus de combustion: (1) 6quilibre chimique local; (2) cine'tiques chimiques de l'6tat stationnaire suivant les travaux de Classman. La partie experimental des flammes hydrogene-air a et conduite par Cohen et Guile. Les predictions num6riques sont compares aux rdsultats experimentaux.

39 12-1 THEORETICAL ANALYSIS OF NONEQUILIBRIUM HYDROGEN AIR REACTIONS BETWEEN TURBULENT SUPERSONIC COAXIAL STREAMS H.Rortgen Lehrstuhl fur Technische Thermodynamik Rhein.-Westf.Technische Hochschule Aachen 5100 Aachen, W.-Germany ' SUMMARY An analytical study is made of the free turbulent mixing and combustion taking place in the mixing layer between a cold supersonic central hydrogen jet and a pre-heated supersonic coaxial air strem. Finite difference approximations have been developed that uncouple the governing equations by local linerization techniques together with iterative methods. Eddy viscosity models were used to describe the turbulent transport processes. Two models were investigated for the prediction of the combustion process: (1) local chemical equilibrium and (2) steady-state kinetics according to the work of Classman 1. Experimental investigations of the studied hydrogen-air flame have been carried out by Cohen and Guile 2. The numerical predictions are compared with the experimental results. 1. NOMENCLATURE A k = molecular formular of species k hfc = static enthalpy of species k H = total enthalpy L = number of third bodies in a trimolecular reaction M = Mach number M)j = molecular weight of species k NE = number of dependent flow variables NK = number of species considered in the chemical model NR = number of elementary reactions considered in the chemical model P = static pressure Pr t = turbulent Prandtl number r = radial coordinate Sc t = turbulent Schmidt number u = vector of instantaneous velocity Wfc = net mass rate of production of the species k x = axial coordinate Yfc = mass fraction of species k Greek a kj> a kj = stoichiometric coefficients of the species k in the j-th reaction /^t = turbulent coefficient of viscosity p = mixture mass density i// = stream function w = dimensionless stream function

40 12-2 Subscripts E I r t x 1 2 oo reference to the outer boundary of the mixing region reference to the inner boundary of the mixing region vector component in r-direction turbulent quantity vector component in x-direction reference to the exit of the inner nozzle reference to the exit of the outer nozzle reference to free stream condition Superscripts (n) = reference to n-th iteration = time-averaged turbulent quantity 2. THE FLAME MODEL The present investigation is concerned with the prediction of the axisymmetric free turbulent mixing and combustion between a central hydrogen jet and a coaxial air stream. Figure 1 is a schematic representation of the flow situation considered. The inner stream is a cold supersonic hydrogen jet and the outer stream a hot annular super-sonic air jet. Combustion will take place in the inner mixing region between the two streams where oxygen and hyrodgen are brought together at a temperature which is higher than the self ignition temperature. / / REGION OF INTEREST TEMPERATURE AIR AND REACTION PRODUCTS JO mm 50 mm POTENTIAL CORE INNER MIXING ZONE VELOCITY PROFILE Fig. 1 Schematic diagram of the axisymmetric turbulent free mixing flow field with combustion This particular flow configuration was chosen because experimental data for the turbulent mixing of hydrogen and air with combustion were found in the literature 2, and further data will be available in the near future from the DFVLR (Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt) in Cologne where similar experiments are carried out. To predict this free turbulent mixing process with finite rate chemical reactions a computer program was developed. The phenomenological approach for free turbulent mixing was used whereby the governing equations were reduced to a set of coupled highly nonlinear parabolic partial differential equations of the boundary layer type. These equations were then solved by numerical techniques. For the prediction of the combustion process two models were investigated. First, it was assumed that, according to Classman 1, steady-state kinetics applies and the finite rate chemical reactions can be treated as homogeneous gas reactions. Secondly, the chemical process was predicted with the assumption of local chemical equilibrium.

41 Governing Equations The governing equations for the coaxial mixing of two dissimilar gases including heat release by chemical reactions in the mixing region were derived assuming that the potential core region and the outer jet region consisted of chemically frozen, inviscid, and uniform flow (Figure 1). The Navier-Stokes equations of motion, the conservation of mass, the conservation of species, and the energy equation for a compressible flow were considered in the instantaneous form. Steady axisymmetric flow with body forces was assumed. The governing equations were then reduced to the boundary layer type form by making use of the following assumptions. (a) (b) (c) Viscous shear stresses depend primarily on the radial gradient of axial velocity. Diffusion in the axial direction is negligible compared to that in the radial direction. Energy transferred in the axial direction by conduction and diffusion is negligible compared to that transferred in the radial direction. The turbulent forms of the governing equations were obtained by using the concept of time-averaged values for the flow variables. Also, assuming that turbulent mixing predominates to the extent that molecular transport becomes negligible, the governing equations take the final form. Continuity Axial momentum (pu x )+- (rpu r ) = 0 (1) ox r or Conservation of k" 1 species _ x 1 3 / 3uL\ pu r ^=- + [n t T *> ) (2) 3x 3r 3d r 3r \ 3r / pu 9Y k 9Y k 1 3 /Mt x + pu r S. = - ^tr -r- 3x 3r r 3r \Sc t 3r 3Y k -r w k (3) k = 1,...,NK Conservation of Energy ^L + ML = _L JL [fl M x r 3x 3r r 3r pr t 3r - 9Y k. >x -^1 (4) In equation (3) the species production term vv k is calculated using the time-averaged flow quantities. According to Classman 1 at conditions where steady-state kinetics applies, the chemical rate may be calculated with sufficient accuracy using mean quantities. This assumption can be used if steady-state concentration of the free radicals important to the chain reactions is achieved in the dominant laminar eddies. Under this condition the chemical production terms \v k may be expressed through the same law of mass action that is used for laminar flows. In this case the finite-rate chemical reactions in a gas mixture can be described as a branched chain mechanism with several simultaneous elementary steps. Any of the reaction steps in this reaction mechanism can be expressed as A k ^M» ^ «kj A k j = 1,...,NR

42 The law of mass action then gives for the net mass rate of production of the species k by all elementary reactions w = - 7hk ' j (6) The specific reaction rate coefficients Kf. and Kj,. are given by the well-known Arrhenius equation. To predict the combustion of hydrogen with air the reaction mechanism given in Table 1 was used. This mechanism consists of 12 elementary reaction steps recommended in the literature 12, and the nine chemical species H, O, H 2, 0 2, H 2 O, HO 2, OH, H 2 O 2, N 2 are assumed with N 2 treated as a chemically inert component. For the chemical model with the assumption of local chemical equilibrium, the reaction rates ^k in the species conservation equations are replaced by the well known algebraic relations for chemical equilibrium. TABLE 1 Reaction Mechanism for the Hydrogen-Air System j H O + H 2 H 2 + OH 2OH H 2 +M H 2 O + M OH + M O 2 +M Reaction = = = = = = = = H = H M = 2OH + M HO 2 + H 2 UNITS: D f. in Ef. = = OH-f O OH+H H + H 2 O O + H 2 O 2H + M H + OH + M O + H + M 2O + M 2OH HO 2 +M H 2 O 2 +M H 2 O 2 +H NK+L 1 X^ ' «. /kmol\ i=i sec ' 1 1 \ m 3 / dimensionless F f. in KELVIN The third body M is the sum over all species. Coefficient for Arrhenius Law Df. Ef. Ff. J J j 2.24E E E E E E E E+16 l.ooe E E E E E E E E E E E E E E E+4 The phenomenological turbulent transport properties are the turbulent viscosity coefficient Mt. the turbulent Prandtl number Pr t, and the turbulent Schmidt-number Sc t, which are not only depending on the fluid properties but are also related to the structure of the turbulence in the mean flow. For this study the turbulent transport properties were primarily based upon experimental data for hydrogen and air mixing flows, reported by Peters et al 3 and Chriss 4. From this experiments Paulk 5 has determined the following simple viscosity model, _ u t = k p, (x x 0 ) (7) k = \ 0 = m From the same experiments the turbulent Prandtl number and the turbulent Schmidt number were found equal to the value These turbulent transport properties have been used in the present study.

43 According to Patankar and Spalding 13 now the von Mises Transformation defining a stream function i// and the linear transformation = *-* t (x) 12-5 have been used to transform the governing equations. The first transformation eliminates the flow variable u r and gives the advantage of not having to solve the continuity equation. The second transforms the triangle mixing region into a rectangular region. This is quite efficient for finite difference methods because it allows to use a rectangular grid which always fits the mixing region. The resulting equations are of the following form ^ + [a(x) + b(x) co] ^L = ± (c, *} + d, (9) 3x 3co 3co \ 3co/ i = 1,...,NE Here the quantities 0, represent the dependent flow variables u x, H, Y k, the properties Cj describe the transport processes and the coefficients d, are the source terms which describe the "local production" of the quantities 0j. The equations (9) are parabolic, nonlinear partial differential equations. Thus, one initial and two boundary conditions must be specified for each dependent variable (u x, H, and Y k ). These conditions have been taken from experiments NUMERICAL METHODS Regardless of the numerical scheme employed, the solution of the governing equations represents a difficult numerical integration problem due to the "stiffness" caused by the source terms in the species conservation equations. Today implicit finite-difference procedures are used for the numerical solution of such problems 6 because of their better stability behaviour compared with explicit finite-difference procedures. To get a good approximation of the source terms in the governing equations this term should be implicitly approximated. The result is a nonlinear system of difference equations. Numerical solution procedures for such systems are very time consuming compared with linear systems. An alternative to the complete implicit approximation of the non-linear problem is a procedure which can be called successive approximation by linearization 9. Here the nonlinear problem is replaced by a linearized problem which satisfies the same initial and boundary conditions. By an iteration procedure then a sequence of solutions of the linearized problem is developed which converges to the solution of the nonlinear problem. The implicit finite-difference approximation of the linearized problem results in a system of linear difference equations which can be solved by very efficient numerical methods. The simplest successive approximation procedure is Picard's method 9. For the problem studied here Picard's method gives the iteration procedure 1 ),,,, ApjV"* 1 ) 3 / 3«pA( n+1 ),, + [a(n) + b(n)cy] (_ ) = _ ( Cj ) + di(n) (10) i= 1,...,NE with i^( n ) being the solution of the linearized system at the n-th iteration step. Another approximation procedure can be constructed by replacing dj by the first two terms of its Taylor series expanded over a finite integration interval Ax. These two approximation methods define sequences [<p } of solutions of the linearized problem: Unfortunately, the convergence of these sequences is only linear. Thus, it is useful to search for successive approximations with higher order convergence at least with quadratic convergence. Now we will study a procedure called successive approximation via quasilinearization which has been used successfully in solving problems of dynamic programming 10 ' u. Starting point of the procedure is the maximum operation of a technique first used by Bellman 15. Kalaba 16 has developed this method, in some detail, for both nonlinear and partial differential equations.

44 12-6 The maximum operation uses the fact that a strictly convex and twice differentiable function dj (0,, 0 2,...,0 NE ) may be represented in the form d; (0i, 0 2,.-.,0 NE ) = djgv«!> 2,...,t? NE ) NE,*. _^ 3dil (H) where the maximization is over all d. The maximum is attained when # = 0. For strictly concave functions a corresponding representation exists. Using, (11) the nonlinear problem (9) can be transformed to a linearized problem with the solution $. This equation has the form * + (a + bco) - = (ci - 3x 3co 3co V 3" k=l i= 1,...,NE with the same initial and boundary conditions as the nonlinear problem (9). Because of the linearity in ^ of the approximated source term d; this procedure is called quasilinearization. Form the maximum operation it follows 0 = m^n yj(t?) (13) Thus, the solution 0 of the nonlinear problem can be found by developing a sequence {#(")} of functions r? that make the solution <p of the linearized problem to converge to its minimum. This procedure is called successive approximation. Now it is a reasonable method to use the solution (f of the linearized system itself as a good approximation for t?. Starting with a reasonable approximation t> 0 we then get the following successive approximation». JL 3cj i= with the initial and boundary conditions taken from the nonlinear problem (9). This is precisely the recurrence relation we would obtain if we applied a Newton-Raphson-approximation scheme to the source term dj. Thus, we can expect that the sequence of solutions {0j( n )} of (14), where convergent is usually quadratically convergent. To show the advantage of this approximation procedure some numerical experiments have been made. First, the partial derivatives in the governing equations (9) were discretized using the implicit finite-difference approximations proposed by Patankar and Spalding 13. For each of the successive approximation procedures discussed, this finitedifference method produces a system of linear difference equations with a tridiagonal coefficient matrix. These equations then were solved for the discussed axisymmetric hydrogen-air flame. The initial and boundary conditions were taken from the experiments of Cohen and Guile RESULTS AND DISCUSSION The Figures 2, 3, 4 and 5 shows some numerical results computed with successive approximation via quasilinearization. Figure 2 shows some computed lines of constant flow quantities in the mixing region. In this case the model of steady-state kinetics was used to describe the combustion process.

45 12-7 Above the centerline the stream lines are shown and below it the lines of constant temperature COMPUTED BOUNDARIES STREAMLINES \' T=300K 500 K IOOOK 1250K \ 2000 K 1500K LINES OF CONSTANT TEMPERATURE ISO x /mm 200 Fig.2 Computed stream lines and lines of constant temperature The diagram shows that the problem under study is really a parabolic one because the angle between the centerline and the lines of constant flow quantities varies very slowly. This numerical result is in accordance with the experimental investigations. Figure 3 shows the mole fractions of the stable species H 2 and H 2 O as a function of the radial distance from the centerline, again computed with the steady-state kinetics reaction model. These profiles belong to a location x = 10.2 cm downstream of the nozzle exit MOLE FRACTION W, Fig.3 Concentration profiles of H 2 and H 2 O at x = 10.2 cm In this diagram the computed profiles indicated by full lines, are compared with the experimental data from Cohen and Guile 2. The results show a good agreement. Figure 4 shows the corresponding profiles for the stable components O 2 and N 2. Again quite a good agreement is shown of the computed profiles with the experimental results.

46 12-8 *- 10Jem O 02\ EXPERIMENTS BY A Afef COHEN AND GUILE MOLE FRACTION V/ Fig.4 Concentration profiles of O 2 and N 2 at x = 10.2 cm Figure 5 shows the profiles of the two unstable species OH and HO 2 at the location x = 10.2 cm. In this diagram the full lines again represent the numerical solution for the steady-state kinetics reaction model while the dashed lines have been computed with the assumption of local chemical equilibrium. The results clearly show that the gas in the combustion region is in the state of chemical nonequilibrium. Fig.5 Concentration profiles of OH and HO 2 at x = 10.2 cm

47 12-9 The numerical results shown in the previous figures have been obtained by successive approximation of the source terms via quasilinearization with the Newton-Raphson procedure. Similar calculations have been made with Picard's approximation procedure and with the approximation procedure using a Taylor expansion of the source term. The calculations with all the three approximation procedures were made with varying step-sizes to investigate the accuracy and computing time of the various schemes. Starting with a large step-size the step-size was decreased until there was no "visible" change in the solution of the problem. All the solution procedures converged to the same solution. All the calculations were made on a CDC 6400 digital computer. Table 2 shows some results of this "numerical experiment". Picard Approximation Procedure Taylor expansion Quasilinearization with Newton-Raphson method Mean specific computing time sec/mm Mean step-size mm 5 x 10~ 6 6 x ID- 5 3 x 10~ 5 Table 2 Comparison of successive approximation procedure. The mean specific computing time is the mean time to compute 1 mm flame length. The second quantity in table 2 is the mean step-size- of the finite difference procedure. Both quantities show the advantage of the successive approximation via quasilinearization with the Newton-Raphson procedure. Especially Picard's method showed very bad convergence, but also the Taylor expansion approximation yet gives large computing times. 5. REFERENCES 1. Classman, I Eberstein, I.J. 2. Cohen, L.S. Guile, R.N. 3. Peter, C,E, Chriss, D.E. Paulk, R.A. 4. Chriss, D.E. 5. Paulk, R.A. 6. Blottner, F.G. 7. Bellman, R Juncosa, M. Kalaba, R. 8. Bellman, R.E. Kalaba, R.E. 9. AmesW.F. Reaction Kinetics in Turbulent Flows, American Rocket Society, 17th Annual Meeting and Space Flight Exposition, Los Angeles, California, Report , Nov Investigation of the Mixing and Combustion of Turbulent, Compressible Free Jets. NASA Contractor Report, NASA CR-1473, Dec Turbulent Transport Properties in Subsonic Coaxial Free Mixing Systems. AIAA paper No , Presented at the 2nd Fluid and Plasma Dynamics Conference, June 1969, San Francisco, California. Experimental Sfudy of the Turbulent Mixing of Subsonic Axisymmetric Gas Streams AEDC TR Experimental Investigation of Free Turbulent Mixing of Nearly Constant Density Coaxial Streams, Masters Thesis, The University of Tennessee, Knoxville, Finite-Difference Methods for Solving the Boundary Layer Equations with Second- Order Accuracy, Proceedings of the 2nd Int. Conf. on Numerical Methods in Fluid Dynamics, University of California, Berkeley, Sept Some numerical experiments using Newton's Method for Nonlinear Parabolic and Elliptic Boundary-value Problems, Comm. ACM, Vol. 4, 1961, pp Quasilinearization and Nonlinear Boundary-value Problems, American Elsevier Publish. Comp., New York Nonlinear Partial Differential Equations in Engineering, Academic Press, New York, London 1965.

48 Bellman, R.E. 11. Angel, E. Bellman, R.E. 12. Baulch, D.L. Drysdale, D.D., Home, D.G. Lloyd, A.C. 13. Patankar, S.V. Spalding, D.B. 14. Brevig, O. Shahrokni, F. 15. Bellman, R 16. Kalaba, R. Methods of Nonlinear Analysis, Vol. II, Academic Press, New York, London Dynamic Programming and Partial Differential Equations, Academic Press, New York, London Evaluated Kinetic Data for High Temperature Reactions, Vol. 1, London 1972 Heat and Mass Transfer in Boundary Layers. Intertext, London On the Free Turbulent Mixing and Combustion Between Coaxial Hydrogen and Air Streams. AIAA paper No. 71-5, 9th Aerospace Sciences Meeting, New York, Proc. Natl. Acad. Sci. U.S. 41, pp and (1955) J. Math. Mech. 8, p. 519 (1959).

49 13 LA SIMULATION DE LA TURBULENCE DANS LES MODELES PETULA par LP.Huffenus ALSTHOM-Grenoble France

50 13 SUMMARY The extension of numerical methods of flow calculation makes it necessary to improve turbulence simulation systems. The aims of these systems must be examined. In a two-parameters simulation, there arises problem of the determination of a turbulence characteristic scale; a.critical examination of the possibilities offered by a partial differential equation, shows that this approach is not sufficient and brings one to take into account, in addition to the local characteristics, a characteristic of general environment. A proposal for a model based on these data has been drawn for constant density flows and for flows with density differences, in which the force of gravity can modify the turbulence to a considerable extent. An example of the possibilities offered by this calculation system is given, in spite of the very summary character of the turbulence simulation system which has been used. RESUME L'extension des methodes numeriques de calcul des 6coulements rend necessaire le developpement de proc6d6s de simulation de turbulence. Les objectifs auxquels doivent tendre ces precedes sont examines. Dans le cadre d'une simulation a deux parametres se pose le probleme de la determination d'une 6chelle caracteiistique de la turbulence; un examen critique des possibility offertes par une Equation aux de"rivees partielles montre 1'insuffisance d'une telle demarche et amene a prendre en compte, en plus des caracteristiques locales, une caracteristique d'environnement g6neial. Un modele fonde sur ces bases est esquissd pour les ecoulements de fluide i density homogene et pour les ecoulements de convection mixte dans lesquels les forces de gravit peuvent modifier la turbulence de facon importante. On donne un exemple des possibilites offertes par ce type de calcul malgre le caractere tres sommaire du proc6d6 de simulation de turbulence utilise.

51 13 LISTE DE SYMBOLES C terme de convection D terme de dissipation diff terme de diffusion E energie turbulente F force exterieure g acceleration de la pesanteur ou "gradient" de vitesse k constante k s rugositd / echelle de turbulence P terme de production Pr nombre de Prandtl p pression r rayon R t s t nombre de Reynolds turbulent abscisse curviligne temps V vecteur vitesse de composantes u; x, coordonne'es y distance a diffusivit6 thermique ou de matiere v diffusivite des quantites de mouvement p masse specifique indice a = apparent indice t = turbulent = partie fluctuante

52 1

53 13-1 LA SIMULATION DE LA TURBULENCE DANS LES MODELES PETULA J.P.Huffenus ALSTHOM-Grenoble France Des 1965, la Societe SOGREAH (Grenoble) entreprenait le developpement de programmes de calcul permettant d'obtenir une description fine des champs d'ecoulements incompressibles bidimensionnels, grace a une resolution numerique des equations de NAVIER STOKES dans des systemes de coordonnees curvilignes orthogonales adaptes aux frontieres du domaine. Visant bien entendu des problemes industriels, il etait indispensable de repr6senter d'une fa$on ou d'une autre les effets de la turbulence de maniere a restituer correctement les champs de vitesses et de pressions moyennes. L'outil mis au point debut 1968, principalement axe sur des applications ou pr6dominaient les phenomenes de turbulence de paroi, permettait deja de couvrir un certain champ d'utilisation des equations (1,2,3), ce qui entraina un ralentissement dans les travaux de perfectionnement qui restaient necessaires. Repris en 1973 par le Groupe Techniques des Fluides de la Soci6te ALSTHOM, ces programmes connus sous le nom de PETULA* servent actuellement de cadre a une recherche visant a generaliser le precede de simulation de turbulence pour tous les 6coulements de fluide homogene et a 1'etendre aux ecoulements dits "de convection mixte" (dans lesquels les forces de gravite engendrees par les differences de densite jouent un role aussi important que les forces d'inertie). Cette recherche, soutenus par la Delegation Generate a la Recherche Scientifique et Technique, est actuellement en cours d'achevement et nous voudrions ici, sinon en tirer les conclusions, ce qui serait premature, tout au moins en rapporter les idees maitresses et les aspects essentiels. 1. LES OBJECTIFS DE LA SIMULATION DE LA TURBULENCE II convient tout d'abord de souligner qu'un precede de simulation de la turbulence, meme lorsqu'il est base sur la resolution d'equations complexes issues des equations de REYNOLDS, ne constitue pas a priori un moyen d'investigation fondamental en matiere de turbulence. II ne peut done evidemment pas pretendre inventer ce qui est encore relativement mal connu en matiere de physique des phenomenes turbulents. Tout au plus, une bonne simulation pourrait-elle etre une synthese des connaissances actuelles, mais 1'absence d'une theorie g6n6rale coherente rend tres problematique une telle realisation. Aussi le seul objectif raisonnable de la simulation sera de permettre au calcul de prendre en compte de facon relativement exacte les effets du champ turbulent sur le champ de vitesses et de pressions moyennes. Tant mieux si de surcroit, on peut en meme temps atteindre des valeurs assez representatives de certaines grandeurs qui caracterisent le mouvement fluctuant. La realisation d'un tel objectif introduit un facteur supplemental de qualite du fait du recours obligatoire a un "precede numerique de resolution des equations du mouvement", veritable support et raison d'etre de toute 1'affaire, sans lequel la simulation ne presenterait rigoureusement aucun interet. Or ce proc6d6 est lui-meme susceptible d'amener des erreurs non negligeables qui vont s'ajouter aux imperfections de la simulation. II devient ainsi inutile de pousser les "performances" techniques de cette simulation au-dela de ce que peut mettre en evidence le precede numerique general de calcul. Bien entendu la finesse du maillage a sur ce point une importance capitale et compte tenu des limitations impos6es dans ce domaine par la taille meinoire et le temps de calcul, le traitement numerique des termes de convection devient 1'un des elements essentiels. Ce traitement regit en effet 1'inevitable diffusion numeiique qui peut eventuellement, si Ton n'y prend garde, dans bien des cas, arriver a masquer completement 1'effort que Ton aura fait pour determiner au mieux la diffusion physique par une bonne simulation de la turbulence. A titre d'exemple, le classique traitement des termes de convection par le schema amont introduit une diffusivite parasite d'origine purement nume'rique dont 1'ordre de grandeur est iu5x, soit pour un ecoulement d'eau a une vitesse U = 2 m/s, avec une maille 6x = 1 cm, une diffusivite egale a fois la diffusivite mol6culaire. Une telle diffusivit^ numerique sera ainsi la plupart du temps superieure a la diffusivite turbulente et ce sera elle qui regira 1'ecoulement calcule. C'est dire que dans les problemes industriels ou Ton n'est pas toujours libre de limiteur autant qu'on veut le domaine de calcul et de mettre dans ce dernier autant de points qu'on le voudrait, ces aspects nume'riques presentent une incidence extreme et que par suite il serait vain de vouloir considerer in abstracto la qualite d'une simulation de turbulence. Programmes d'ecoulements TUrbulents et LAminaires

54 13-2 Ceci dit, le choix et 1'eiaboration d'un precede de simulation peuvent etre bases sur les criteres suivants: la simplicite, a la fois au niveau des concepts que Ton fait intervenir et au niveau de nombre d'operations de calcul necessaires la fiabilite, c'est-a-dire la faculte de la simulation de bien reagir a toutes les situations d'ecoulement, meme les plus imprevues 1'exactitude de reproduction des effets de diffusion dans toutes les zones ou le gradient de vitesse est important. Cette "exactitude" peut etre basee sur le respect de lois et processus fondamentaux bien connus dans la mesure oil ils sont susceptibles d'etre integres dans des concepts coherents et pas trop compliques. 2. LE CHOIX D'UN TYPE DE SIMULATION La simulation de la turbulence utilisee dans PETULA repose sur 1'approximation classique d'une viscosite apparente scalaire definie a partir de deux parametres, 1'energie des fluctuations E et une longueur / caracteristiques du spectre s'apparentant a 1'echelle macroscopique aux grands nombres de Reynolds R t = \felfv et a 1'echelle microscopique aux petits nombres de Reynolds R t. Les equations du mouvement moyen s'ecrivent ainsi: div V = 0 3V 1 = - V grad V - -grad p* + 2 div (v DV) + F (1) 3t p avec La variation de 1'energie E est definie a partir des equations de Reynolds par une equation aux derives partielles exprimant sous forme semi-empirique le bilan Convection, Production, Dissipation et Diffusion en chaque point de Fecoulement. 3E E = - V grad E + 2k, v R t DV..DV - Kk 3 + k 4 R t ) - + v div ( ^ + k 2 R t J grad E (2) ou plus simplement: = C + P - D + diff 3t.. designe le produit doublement contracte. La determination de / dont la valeur intervient dans la viscosite apparente et dans liquation de 1'energie est en realite toute la difficulte du probleme. II est d'ailleurs interessant de noter que ceci n'est pas une particularite des modeles de simulation a 2 parametres (E, /). En effet, que ce soit dans les theories phenomenologiques assez anciennes (Prandtl, Reichart..) ou dans des modeles tres eiabores depassant largement le concept de viscosite scalaire (Launder, Harlow..), on a toujours besoin de caracteriser le spectre des fluctuations par (au moins) une longueur dont la determination est delicate et revet un aspect essentiel. De nombreux auteurs definissent 1'echelle de turbulence / par une nouvelle equation aux derivees partielles E E 3 ' 2 deduite elle aussi des equations de Reynolds, decrivant la variation d'un parametre tel que El, ou... Cette equation presente bien entendu une forme analogue a celle de 1'equation de 1'energie avec des termes de convection, production, dissipation et diffusion. En dehors des problemes poses par le traitement numerique de 1'ensemble des deux equations decrivant E et /, et par la determination de nouvelles constantes universelles, ce procede de simulation de turbulence presente divers inconvenients:

55 Tout d'abord 1'equation aux derivees partielles que Ton peut tirer pour / generalement sous la forme: a partir des deux equations, se presente = - V grad 7 (ap - bd) + diffusion 3t E (3) ou a et b sont des constantes positives, b etant nettement superieure a a. Cette equation fait ressortir le caractere essentiellement convectif de / qui demeure cependant une grandeur intensive. On con?oit bien la convection de 1'energie E mais beaucoup plus mal celle d'une longueur... Quoiqu'il en soit, negligeons un moment le terme de diffusion ainsi que celui de 1'equation de 1'energie qui devient: 3t = - V grad E + P - D L'ensemble de ces deux equations s'integre analytiquement le long d'une ligne de courant en regime permanent et Ton obtient: -b exp (4) ou / 0, E 0 sont les conditions au point M 0 amont a 1'abscisse curviligne s 0 /, E sont les conditions au point M d'abscisse curviligne s > s 0 On voit ainsi ressortir en dehors des zones de production (P = 0) une loi simple: et Ton peut montrer qu'une telle loi permet effectivement de rendre compte de fa9on satisfaisante d'experience telles que celle de la decroissance de la turbulence derriere une grille dans un champ uniforme. /3E\ Par contre 1'existence d'un taux de production I I = P genere un terme multiplicatif toujours superieur \ 3t /p a 1 qui peut s'ecrire: exp (b-a) M 0 M (3E/3t)P ou t est le temps de passage d'une particule en un point donne dt / ds\ * («- v) Ainsi sous la seule hypothese de negliger les termes de diffusion, nous arrivons a la conclusion suivante: l'6chelle en un point M est proportionnelle a 1'echelle / 0 en un point amont M 0 quelconque, la coefficient de proportionnalite dependant du rapport des energies E/E 0 corrige par un terme, toujours superieur a 1, dans lequel intervient la quantite relative d'energie turbulente produite sur le parcours M 0 M. Cette dependance indelebile de 1'etat turbulent en un point a 1'etat turbulent d'un point quelconque en amont, parait contredire 1'existence experimentale d'ecoulements dits d'equilibre dans lesquels les conditions amont ou perturbations diverses s'effacent graduellement pour laisser place a un equilibre qui peut etre decrit independamment d'elles. Bien entendu il sera possible a 1'aide d'equation (4) de decrire convenablement 1'evolution d'un regime d'equilibre en supposant que Ton s'est au d6part place dans des conditions d'equilibre 4 mais il parait douteux que, partant d'un regime perturbe, on puisse tendre a 1'aval vers les conditions d'equilibre telles que nous les decrit 1'experience. Un autre aspect troublant concerne les phenomenes de transition et la coexistence dans un meme champ d'ecoulement de zones laminaires et de zones turbulentes. Les zones laminaires vont evidemment etre d crites par des energies turbulentes tres petites, mais 1'echelle 1 perd alors toute signification. Or selon 1'expression de 1'equation (4), le calcul devra continuer a accorder une attention soutenue a la valeur quelconque quasiment nulle de E et a la valeur fantaisiste de /, car au moment de la transition, ces valeurs (amont) peuvent constituer la base de depart principale servant a 1'elaboration des caracteristiques de la zone turbulente.

56 134 - Enfin on peut noter que 1'utilisation d'une equation aux derivees partielles pour decrire 1'echelle de turbulence donne a la determination de ce parametre / un caractere local puisque, mis a part les termes de diffusion, on tient essentiellement compte des grandeurs locales P et D et de 1'influence priviiegiee de 1'amont due a la convection. Or il parait intuitivement desirable d'introduire par le biais de / quelque chose qui caracterise I'environnement du point auquel on s'interesse. Dans la reference 5, les auteurs developpent un concept pour / appartenant a ce dernier type: ils definissent / en un point donne M proportionnel a une longueur X tel que: W g 4 g dv (5) ou g designe le "scalaire gradient de vitesse" g = >/DV..DV au point courant N centre de Felement de volume dv W 'designe une fonction de ponderation, par exemple exp( r 2 /X 2 ) ou r designe la distance MN Cette demarche presente des inconvenients lies a la complexity de la determination de X en chaque point par 1'expression implicite de 1'equation (5). Par ailleurs on s'attend a trouver du fait de la forme integrate de 1'equation (5) une application universelle aussi bien pour des problemes. plans que pour des problemes de revolution et a ne pas etre oblige de faire appel a une "non universalite" de constantes comme c'est le cas 4 avec une equation aux derivees partielles pour /. Des calculs sommaires ont et assez decevants sur ce chapitre. Neanmoins 1'idee d'une caracdu/dy terisation de I'environnement par une generalisation spatiale de 1'expression / = K reste interessante. d z u/dy z Notons de plus que dans le cas d'une turbulence de paroi, il est bien sur tentant de simplifier en faisant appel directement a la distance a la paroi. Aussi nous avons axe nos efforts sur une determination de 1'echelle de turbulence basee principalement sur la geometric du domaine et la topologie du champ des vitesses. La longueur ainsi elaboree pourra etre corrigee, si besoin est, par une fonction appropriee du nombre de Reynolds turbulent local R t (et le cas echeant du nombre de Richardson local pour les problemes de convection mixte). Ces corrections permettront en particulier de faire apparaltre des phenomenes de transition laminaire-turbulent. Pour tenter ensuite de depasser le cadre des phenomenes quasi d'equilibre et pour rester capable de rendre compte d'aspects evolutifs tels que ceux mis en evidence par la decroissance d'une turbulence derriere une grille, nous introduirons en plus un aspect convectif de / tel qu'il ressort d'equations du type des equations (3) ou (4). Cet aspect sera principalement pris en consideration dans les zones dites de convection ou la production P n'est pas le facteur preponderant. Avant de parcourir rapidement les grandes lignes de I'algorithme general projete sur ces bases, algorithme qui n'est d'ailleurs pas totalement fige et qui comporte encore diverses variantes en- cours d'etude, nous ferons une petite pause en regardant les resultats d'une confrontation experience-calcul dans un cas "industriel" pour lequel a ete utilisee en attendant mieux la simulation PETULA Dans cette simulation / est uniquement basee sur la geometric (distance aux parois) et Ton s'accommode done fort mal de tout phenomene de turbulence libre: L'ecoulement est celui qui existe entre plafond et ceinture dans une turbine Francis dont on a conserve le distributeur mais enleve la roue. La figure 1 montre le maillage retenu dans le plan meridien et la position des deux lignes de sondage amont et aval utilisees experimentalement. On remarquera que 1'axe de revolution a ete entoure pour simplifier d'un petit tube de rayon 5 mm; le diametre au niveau du distributeur est de 420 mm. Compte tenu des vitesses de 1'eau assez elevees (quelques 5 m/s), conduisant a un nombre de Reynolds important, les lignes sont tres serrees pres des parois et ne sont pas toutes visibles. A Tentree (distributeur), on impose le vecteur vitesse et 1'energie turbulente. Par centre a la sortie du diffuseur, on impose au vecteur vitesse une direction quasi axiale (en fait legerement divergente comme les lignes du maillage) et une condition de regularite sur les vitesses de rotation et 1'energie turbulente..du plus, on suppose que la pression dans la section de sortie au rayon r est uniforme aux effets centrifuges pres, soit: P(r) = / / V6 2 dr. o r. Sur les figures 2, 3 et 4 sont portes les resultats du calcul pour ce qui est des vitesses meridiennes, des vitesses de rotation (representees par des vecteurs inclines comme pour une perspective) et des lignes de courant. Dans une experience fait ulterieurement, on a pu comparer les previsions du calcul avec la realite. La turbine est alimentee par le distributeur avec Tangle et le debit utilises pour le calcul. A Taval le cone du diffuseur plonge

57 dans un plan d'eau pratiquement infini. On voit sur la figure (5) que les conditions ideates considerees a Famont pour le calcul ne permettent pas de respecter une concordance tout a fait exacte experience-calcul dans le plan de sondage amont. La figure 6 qui montre la comparaison experience-calcul dans le plan de sondage aval permet d'apprecier les les ecarts mis en evidence. On notera qu'il a ete pratiquement impossible de realiser des mesures correctes de vitesses dans la recirculation centrale compte tenu des fluctuations tres importantes dans cette zone. La perte de charge entre le plan de sondage amont et le plan de sondage aval ressort a 50 cm de hauteur d'eau, le resultat etant identique par le calcul ou d'apres les mesures a 2% pres. Get exemple rapidement decrit montre que meme pour un cas assez complexe, une simulation de turbulence basee sur des concepts relativement sommaires donne deja un resultat presque acceptable. Tout 1'objet des complications que nous aliens maintenant faire intervenir, sera bien sur d'ameiiorer un peu un tel resultat, mais surtout d'etendre la validite du calcul a des ecoulements ou la turbulence libre peut jouer un role beaucoup plus determinant et oii les forces de gravite viennent apporter des elements nouveaux d'une importance capitate LA SIMULATION DE LA TURBULENCE POUR LES ECOULEMENTS DE FLUIDE HOMOGENE 3.1 Echelle caracteristique des zones de production Cette echelle est eiaboree a partir de concepts physiques que nous indiquerons d'abord a propos de phenomenes eiementaires avant d'examiner leurs interactions et de traiter le cas general Phenomenes eiementaires (monodimensionnels) Turbulence de paroi I est proportionnel a la distance a la paroi. L'adjonction d'une fonction de R t permet une reproduction satisfaisante de la sous-couche visqueuse et du phenomene de transition. i- ^^ (» Une eventuelle rugosite, definie par Tequivalent sable Nikuradze k so peut etre representee en rempla?ant y par une fonction plus complexe de y et de: k s = Turbulence libre Une zone dite de turbulence libre est reperee par Texistence d'un gradient de vitesse maximum g max au point M. De part et d'autre de ce gradient, on recherche des points M, et M 2 ou le gradient tombe a a - (par exemple A = 5). / est alors pris uniforme dans toute la zone MiM 2 et proportionnel a la A distance M t M 2. La correction par une eventuelle fonction de R( peut etre ici omise. On remarquera que cette determination de / se rapproche de la formulation evoquee plus haut a propos de 1'expression (5); M,M par exemple fait intervenir le rapport entre g^ ~~ gmi Q U1 est proportionnel du dg d 2 UU UK U uu (c'est-a-dire ), et la valeur moyenne de sur MM] (c'est-a-dire : 3 5-)- Mais la encore, la dy dy dy recherche directe de M : M 2 integre d'une certaine maniere les choses dans I'environnement du point M et permet de concretiser ce que Ton nomme souvent d'une maniere intentionnellement tres floue "1'epaisseur de la zone de gradient" Aspects complementaires (monodimensionnels) Plusieurs amenagements doivent etre introduits dans la prise en compte de phenomenes eiementaires ci-dessus. Ainsi la loi donnee pour la turbulence de paroi n'est appliquee que dans une zone limitee ou le gradient decroit de gjnax a la paroi a gmax/b (B =a 500). / est ensuite maintenu uniforme (partie exterieure des couches limites) jusqu'a Tannulation du gradient.

58 13-6 Par ailleurs, des interactions entre zones de production doivent etre prises en compte. Ainsi, il est facile de voir que dans un jet libre, par exemple, la consideration separee de chacune des deux couches de melange dont il est forme n'amene pas une echelle de turbulence correcte. On est done amene a penser que les zones de gradient maximum assez proches les unes des autres reagissent entre elles en augmentant mutuellement leur echelle propre. Un processus de coefficients d'influence a done ete mis au point qui fait intervenir a la fois la distance entre gradients maxima et 1'importance relative de ces gradients, importance chiffree sur la base de 1'ecart de vitesse 6U entre M, et M 2. Ceci permet de rendre compte de certains phenomenes physiques: nous avons parte plus haut du jet libre; il en est de meme pour le jet de paroi ou Tinteraction mutuelle des zones de turbulence libre et de turbulence de paroi parait un fait acquis, mis en evidence par exemple par une variation du coefficient K de 1'equation (6) qui intervient directement dans la loi logarithmique de paroi. Mais ceci permet aussi de s'affranchir de longueurs / peu significatives generees par des gradients peu importants (faibles 6U ), voire des gradients parasites lies au traitement numerique plus ou moins imparfait Aspects hi et tridimensionnels Nous avons developpe ci-dessus les principes de determination de / pour des ecoulements relativement simples, quasi monodimensionnels, c'est-a-dire dans lesquels les gradients de vitesse appreciates se trouvent perpendiculairement a la vitesse elle-meme (g = ). Pour etre appliques a des cas bidimensionnels, ces principes demandent une deterdy mination des lignes de courant, penible et couteuse si elle doit etre effectuee frequemment. Pour un point donne M, on s'est done oriente vers- une application des principes ci-dessus selon chacune des lignes de coordonnees Mj, M;. en elaborant une formule de combinaison des deux longueurs /j et /j d6terminees suivant chacune de ces lignes Mj, M:. De nombreux problemes ont ete rencontres tels que: Choix d'un scalaire representant grad V : le scalaire g = ^DV..DV semble le plus simple. C'est lui d'ailleurs qui intervient dans le terme de production P. On a pu cependant accorder une certaine attention a, derivee du module de la vitesse par rapport a la normale a la ligne de courant, qui presente dn 1'avantage de changer eventuellement de signe Necessite de tenir compte de 1'importance relative des gradients ayant genere /; ou /j, par le biais des 6U Necessite de tenir compte de la direction de la ligne de courant par rapport a i et j Prise en compte "logique" de Teffet tridimensionnel pour les problemes de revolution Etc... L'etat inacheve de la recherche entreprise sur ces bases et le nombre de details a prendre en consideration nous obligent a reporter a une publication ulterieure la description complete de la mecanique du calcul de / pour les zones de production. 3.2 Introduction des aspects convectifs La longueur / p eiaboree ci-dessus caracterise le processus de production de la turbulence. Elle n'a pas de signification precise dans les zones de faible production. On calcule alors une autre longueur / c a partir des aspects purement convectifs deduits d'equation (3) ou d'equation (4) pour une production P negligeable. /p et / c Un parametre de ponderation p permet alors de fixer la longueur qui sera definitivement retenue a partir de / = P / + (/ - p)/ c La valeur p a ete dans un premier stade deduite de Tintensite relative du terme de production "potentielle" par rapport au terme de convection de Tenergie. II semblerait en fait aujourd'hui que Ton dispose de suffisamment d'etements lors de la determination de /_ pour fixer a priori en chaque point p = 1 ou p = 0. La encore une solution definitive n'est pas arretee. 4. LA SIMULATION DE LA TURBULENCE POUR LES ECOULEMENTS DE CONVECTION MIXTE Nous supposerons par exemple que les differences de densite sont d'origine thermique.

59 13-7 La prise en compte des forces de gravite introduit deux elements nouveaux essentiels: 1'equation (1) du mouvement moyen avec F=g et p non uniforme ne pose pas de problemes particuliers -T-T par contre 1'equation (2) de Tenergie turbulente comporte un terme supplemental g ou pv P est la correlation vitesse-densite. De plus rien n'indique qu'il ne faille pas modifier /. Tequation d'etat donne p en fonction de la temperature T, mais cette derniere doit etre decrite par une equation de transport faisant apparaitre un terme de diffusion par turbulence V'T' qu'il convient d'expliciter. 4.1 Influence des forces de gravite sur la turbulence Une analyse de la litterature (refs 6 8 et bien d'autres) sur les recherches qui ont ete et sont actuellement effectuees dans ce domaine, soit du point de vue meteo, soit a Taide d'experiences "in vitro", montre qu'il regne encore un certain desordre dans les connaissances si bien qu'il serait tres risque de vouloir en deduire une philosophic qui echappe encore aux specialistes. Trois points neanmoins paraissent se degager et peuvent conduire a une simulation "la moins fausse possible". p'v' Le terme g peut etre decrit par une forme empirique du type: P ga t gradp P a t etant la diffusivite turbulente de chaleur consideree comme scalaire en toute premiere- approche. Ce terme ne peut cependant representer dans des conditions de stabilite tres prononcee qu'une part relativement faible (10 a 15%) du terme de production P, ce qui signifie qu'il ne peut a lui seul expliquer par exemple la diminution voire la disparition des effets de la turbulence dans un ecoulement stratifie stable. Oj Le rapport des diffusivites de chaleur et de quantite de mouvement, varie bien avec le nombre Richard- "t son dans le sens indique par Ellison, mais dans des proportions relativement faibtes (de 1,4 pour Ri = 0 a 0,6 pour Ri tres grande dans le cas de la stabilite). L'affaiblissement ou la disparition des effets turbulents dans un ecoulement stratifie stable semble etre provoque par une "decorrelation" du meme type que celle qui est observee dans les processus dits "de relaminarisation", c'est-a-dire que 1'energie turbulente u'j 2 persiste un moment alors que les correlations Ujuj sont diminuees et tendent vers zero. Un tel processus ne peut etre decrit dans un modele a deux parametres (E, /) si bien que Ton est reduit a modifier 1'echelle / en fonction du nombre de Richardson pour avoir une repercussion vraisemblable sur «> t sans se preoccuper des valeurs que prendra Tenergie E. Pour ce faire, on doit s'inspirer des donnees experimentales qui semblent montrer que la decorrelation apparait totale pour des nombres de Richardson atteignant 0,25 (resultat concordant par ailleurs avec certaines analyses de stabilite). 4.2 Diffusivites de chaleur (ou de matiere) Bien que Ton puisse ajouter a Tequation de transport de la temperature moyenne en un point, une equation relative aux fluctuations, nous avons prefere avoir recours directement au vieux concept du nombre de Prandtl turbulent pour ne pas compliquer une simulation qui se veut relativement simple. Outre les variations relevees ci-dessus a propos de Tinfluence des forces de densite sur la turbulence, ce nombre Pr t depend du type de turbulence (ce que nous avons neglige) et du nombre de Prandtl moteculaire lorsque ce dernier differe notablement de / et que de plus R t n'est pas tres grand. Nous avons sur ce point retenue 1'expression donnee par A.T.Wassel et I.Catton 9 Pr = 2i?l 1 - exp( - Six) r * Pr 1 - exp(- 5,25/x Pr) avec x = = k,r t que nous avons trouve assez coherentes avec des essais a bas nombre de Prandtl 10 correcte de la simulation pour les metaux liquides.. Ceci permet une extension

60 13-8 En ce qui concerne les valeurs eievees du nombre de Prandtl, la simulation pourrait a la limite s'appliquer egalement; mais la difficulte est alors de decrire convenablement les instabilites de la sous couche visqueuse qui sont de peu d'importance sur le mouvement et sur le transfert a nombre de Prandtl modere, mias qui deviennent le facteur essentiel du transfert a haut nombre de Prandtl. 5. CONCLUSION La simulation de turbulence dont les caracteres essentiels viennent d'etre decrits, ne pretend pas constituer un outil de recherche pour mieux comprendre la nature des phenomenes turbulents. II nous est en effet apparu que le travail entrepris, pour ne pas devier de son objectif industriel, devait respecter les trois regies suivantes: simplicite fondamentale et relation tres directe avec des elements empiriques bien etablis rester adapte aux possibilites relativement limitees du calcul numerique de Tecoulement dans un cas industriel (ne pas rechercher la performance qui ne pourra jamais etre percues avec les lunettes plus ou moins deformantes du calcul en differences finies) prendre en compte de fa9on tres schematique et simplifee le peu que Ton sait de pratique, sur Tinfluence des differences de densite de fa9on a pouvoir aborder un domaine ou le calcul pourrait presenter des debouches indiscutables. Cette simulation de la turbulence en cours de mise au point ne peut evidemment etre jugee aujourd'hui assez operationnelle pour etre soumise aux tests tres severes d'une utilisation dans des problemes industriels complexes. Or seule cette utilisation repetee, confrontee a des informations d'origine experimentale, permet a la longue de bien cerner les domaines d'application interessants et de definir la confiance qu'il convient d'accorder aux resultats que fournit le modele. BIBLIOGRAPHIE 1. Huffenus J.P. Calculs d'ecoulements en fluide reel. La Houille Blanche 6 (1969). 2. Gauthier M.F. PETULA, un exemple de modele mathematique complexe. AGARD Lecture series 48 (1971). 3. Biesel F. Calculs numeriques d'ecoulements a nombres de Reynolds eleves. XIV Congres AIRH (1972). 4. Rodi W. A two parameter Model of Turbulence. Warme und Stoffubertragung Bd 3 (1970). Spalding D.B. 5. Gawain T.H. A unified heuristic model of Fluid Turbulence. Journal of Computational Physics Pritchett J.W. 5 (1970) 6. Stewart R.W. The problem of diffusion in a stratified fluid. Advances in Geophysics 6 (1959). 7. Arya S.P.S. The.critical condition for the maintenance of turbulence in stratified flows. Quart J.R. Met.Soc. 98 (1972). 8. Turner J.S. Buoyancy effects in fluids Cambridge Univ. Press (1973) 9. Wassel A.T. Calculation of turbulent boundary layers over flat plates. International Journal of Catton I. Heat and Mass Transfer 16, 8 (1973). 10. Sleicher C.A. Temperature and eddy diffusivity in Nak. International Journal of Heat and Mass Awad AS. Transfer 16, 8 (1973). Notter R.H.

61 13-9 Axe de revolution Sondage amont 00 o Sondage aval Figure 1.

62 13-10 Axe de revolution Sondage amont < H H W W W 5S M Z CO tt co Sondage aval Figure 2.

63 13-11 Axe de revolution Sondage amont H > CO id W Sondage aval Figure 3.

64 13-12 Axe de revolution Sondage amont o o G 2! a M > w '8 Sondage aval Figure 4.

65 13-13 en. myi 3 2 o o *'..-b 0 PLAFoW i 1 I 3 * S f eft C"l CAUCUL ViTtSSE HERIDIENNE (RAJMtEJ... o o o Fig. 5 Sondage amont CALCUL EXPERIENCE VITESSE MEKIDIE.WE (AXMLE) o o o VITESSE T/>NSEKTIELLE (ROTATIOV AXE. BE. REVOLUTION ^ ~ ~ VITESSE TAH/SEKTIEUE. (KOTATIOW) >-»--) H. \ + \ \ '..o \ + "x\+ o-*-. I Illllllfll/llll/ rayon en cm. "' Pa.ro L ± curfuseur 1 ;///// 3 V Vi-ttsse, C.1. / '^-/>l Fig.6 Sondage aval

66 13-14 DISCUSSION Dr Barrere : Comment se trouve pris en compte la correlation p'v? Author's reply : en partie dans le terme additif de 1'equation de 1'energie. Mais cela ne suffit pas et il faut une correction compldmentaire sur 1. Prof. Spalding : Do you use a second partial derivative equation for "/"? Author's reply : Non, compte tenu des defauts qu'elle pr6sente pour la description des regimes etablis.

67 14 ANALYSE NUMERIQUE DE LA PHASE D'INFLAMMATION DANS UNE COUCHE DE MELANGE TURBULENTE par Otto Leuchter Office National D'Etudes et de Recherches Aerospatiales (ONERA) 92 Chatillon France

68 14 RESUME La presente etude a pour objet le traitement numerique du processus d'inflammation dans la zone de melange turbulente entre un jet de combustible (melange H 2 Ar) et I'ecoulement exterieur (air) suppose uniforme, les deux fluides etant a la meme temperature. Les conditions initiates a la confluence sont caracterisees par le rapport des vitesses et I'epaisseur des couches limites initiates. La description du champ des grandeurs moyennes et de celui des correlations d'ordre 2 est effectuee au moyen d'equations de bilan construites a partir des theoremes generaux de conservation. Des hypotheses simplificatrices sont introduites au niveau de la cinetique chimique, compte tenu du comportement specifique de la reaction en chaine H 2 O 2 dans le domaine des delais courts. De ce fait le nombre des equations decrivant la production chimique et 1'action de la turbulence sur celle-ci peut etre considerablement reduit. Les solutions numeriques ont mis en evidence que les effets de ralentissement dus a la turbulence sont peu affectes par les conditions initiates, mais dus essentiellement au comportement de la cinetique dans la couche de melange, alors que la reduction des longueurs d'inflammation en presence de couches limites est imputable en premier lieu a la distorsion du champ de vitesse moyenne. Un critere simple d'inflammabilite des jets est etabli compte tenu de ces resultats. SUMMARY The present paper concerns the numerical description of the inflammation process in the turbulent mixing region between a fuel jet (mixture of H 2 and AR) and an external air stream, the two fluids being at the same temperature and of the same density. The initial conditions at the confluence are characterized by the velocity ratio and the initial boundary layer thicknesses. The description of the fields of mean quantities and second order correlations is performed by means of balance equations constructed from the general conservation theorems. Simplifying assumptions are introduced for the chemical kinetics, accounting for the particular behaviour of the hydrogen-oxygen chain reaction in the region of short ignition delay. The number of equations describing the chemical production and the effects of turbulence on it may thus be considerably reduced. The numerical solutions have revealed that the slowing effects of the turbulence are little affected by the initial conditions but depend essentially on the behaviour of the kinetics in the mixing layer, whenever the reduction of the inflammation length in the presence of boundary layers is due mainly to the distorsion of the mean velocity field. A simple criterion for the inflammability of jets is established taking into account these results.

69 INTRODUCTIOH.-. La zone de melange engendree par la confluence de deux ecoulements reactifs est generalement le siege de phenomenes complexes d 1 interaction entre la turbulence et la cinetique chimique affectant 1'eVolution des grandeurs moyennes liees a la transformation chimique [V] aussi bien que le developpement de la structure turbulente de la couche de melange. Un des problemes particuliers se rattachant a cette configuration et interessant notamment les applications pratiques est celui de la prevision des conditions d 1 autoinflammation dans la zone de melange d'un jet (Fig. 1), en fonction des conditions initiales : vitesses, temperatures, couches limites et turbulence dans les deux ecoulements. Le cas des jets (p. ex. jets de combustible dans un milieu oxydant). merite en effet une attention particuliere du fait qu'en general le processus chimique conduisant a 1' inflammation doit tre amorce" avant que le processus du melange ait trop dilue le jet dans I 1 ecoulement ambiant. Dans un grand nombre d 1 applications pratiques les schemas reactionnels regissant 1'evolution chimique au cours du processus d 1 inflammation spontanee s'apparentent aux schemas multireactifs des reactions en chalne, dont la reaction H2-0 constitue un exemple particulierement representatif. Dans ce type de mecanisme la reaction globale directe A-2 + B2» Produits est remplacee par un ensemble de reactions ramifiees caracterise par I 1 apparition et la croissance rapide de radicaux intermediaires A, B, AB,...etc. Un cas frequemment rencontre en pratique et particulierement interessant pour les applications numeriques est celui ou 1'une des reactions de branchement, sensiblement plus lente que les autres reactions de la chalne, regie pratiquement a elle seule le taux d'accroissement des radicaux determinant directement le delai d 1 inflammation. Tel est le cas p. ex. pour la reaction Hg G dans le domaine des delais courts, ou le r8le de la reaction de branchement Og + H»OH + 0 predomine. L' introduction de la notion de pseudo-equilibre permet alors de simplifier sensiblement la description du mecanisme reactionnel et de faciliter considerablement la discussion des effets de la turbulence sur les taux moyens de production chimique. La inethode numerique developpee dans la presente etude est basee sur ces simplifications. Son application portera sur I 1 auto-inflammation d'un jet chaud contenant un melange Ar + Efe en presence d'un ecoulement d'air de m$me temperature, les dimensions du jet etant ajustees, compte tenu des conditions initiales, pour que 1'auto-inflammation soit situee dans la zone de melange du noyau potentiel. L'hypothese restrictive portant cur 1'isotherrnie des deux ecoulements a ete adoptee provisoirement afin de mettre plus clairement en evidence, dans ce type de reaction, le rfile de la turbulence et des couches limites initiales sur les distances d'ini'la-:i;::ation. La prise en compte de ces effets s'ei'fectue, dans le cadre de la presente etude, au noyen d 1 equations de comportement relatives aux correlations d'ordre 2, dont le nombre a pu Stre reduit au minimum compte tsnu des simplifications iiitrod^oites au niveau de la cinetiqus chimique. Dans une etude suiterieure relative au>: conditions d'inflaiit.ation dans une couche.de melaivte }ij r drbgene-air [2] il etait appara que 1'activite chinique se concentre dens une zone relativement etroite placse autour de la surface noyerine stoechionotrinue, elle-mfice.situee, compte tenu du faible rapport laassinus stoochionetrique, dans la region exterieure de la scr.e de melange, ou los effets de I'intermittence sont tres sensibles. Le clioix d'voi mulang-3

70 14-2 I Ar - Hg retenu dans la presente etude etait motive essentiellement par le souci d'eliminer les forts gradients de densite dans le melange et de ramener la region stoechiometrique dans la zone centrale de la couche, ou les effets d'intermittence sont ne'gligeables, afin d'assurer une description des effets de la turbulence la plus conforme possible avec les modeles de turbulence utilises. 2 - MISE m EQUATION DU PROBLEMS Equations de base. L 1 evolution des grandeurs aerothermochimiques est regie par les lois de conservation de la masse et des N + 4- quantites div (pu) m 0 (0 p pu.grad ipj = S-, + div F (2) 3 t representant respectivement les 3 composantes du vecteur vitesse, 1'enthalpie totals et les fractions massiques >$ des N especes en presence. Les S; designent les termes de source propres aux~ f; et les Fj les vecteurs flux dus au transport moleculaire et supposes de la forme F - ( grad «p. (3) ou les<*idesignent les coefficients d'echange usuels supposes independants de la composition chimique. Aux equations (O et (2) on associe les equations (4) et (f) decrivant le champ moyen des (p. suppose stationnaire : pu =. 0 pu grad *ft + div p U' <p/ = S- -+ div Fj La -correlation pu'if.' qui represente le vecteur flux de *f. dft aux echanges turbulents est exprime au moyen de la formule du gradient en fonction de la correlation q'' = U'* * V 2 * w /2 (representant le double de 1'energie cinetique de la turbulence) et d'une echelle caracteristique de la turbulence L; associee a la grandeur *f>.» Grace a cette formulation qui apparalt comme une generalisation de la relation de Prandtl - Kolmogorov, le nombre de correlations a prendre en compte pour la resolution de (5} subit une reduction tres sensible. L' action du vecteur flux moyen F; dans (5) est negligee par suite de la valeur supposee tres elevee du nombre de Reynolds turbulent. _

71 14-3 Quant aux termes de source S\ seuls ceux relatifs aux concentrations seront consideres, compte tenu de I'hypothese d'un gradient de pression moyen nul dans la zone de melange. Dans un milieu multireactif de R les termes Sj s'expriment par 5i = W,, A, E ( TVi - VH ) reactions symbolisees par ^ AJ (r < --- R) ou les vitesses des reactions directes sont supposees de la forme P = A r (T) e * et les vitesses des reactions inverses reliees a celles-ci par fc' P = t r /K r CP,T; ^rcp/ 1 ") etant la constarxte d'equilibre (relative aux concentrations molaires) de la r ieme reaction. En effectuant les moyennes dans (7) on fait apparaltre, compte tenu de 1'absence de gradients de densite" et de temperature,des expressions de la forme dans lesquelles apparaissent des termes de correction Vp» fonctions des correlations doubles // /«' relatives aux especes participant a la r ieme reaction. L' evolution spatiale des correlations doubles de la forme if/ *f-' figurant dans (6) et (i) est regie par- des equations de bilan que 1'on peut construire a partir des equations de conservation (2 ).On obtient, par une procedure classique : tp;. 9 rad (f> + pu' y'.^ro* (g. ^div/ pu' f. 7 if/ - I UL 5ZL = o (9) Seloii la terminologie habituellement employee les diffisrents termes I < a 2T de cette equation s'interpre'tent coimae : I : terme de convection U : " production

72 144 H : terme de diffusion turbulente H : " d'echange (ou d' interaction) -gr j " de diffusion laminaire (generalement negligeable) 31 : " de dissipation. Seuls les termes JK et TZE necessitent une modelisation particuliere en raison de leur forme specifique. Le terme de diffusion turbulente sera modeuse en analogie avec 1'equ. (6) par j' = - p Ljj grad ^ ou' L;J est une longueur d 1 echelle associe"e a la correlation Quant au terme de dissipation, on negligera la contribution des fluctuations des coefficients <Xj et «Xj de sorte que et on posera Fj' grad <(?' + FJ' grad <p/ = («j + 5?j ) grad if.' grod ip.' ou A,-; est une echelle de longueur caracteristique, du processus dissipatif de la correlation tfl'fj L' evolution des grandeurs moyennes dans le champ turbulent est ainsi coiiipletemeivt determinee par les relations (4) a (11) Reduction des equations de diffusion. Avant d'aborder la raise en forme finale des equations de conservation on remarque qu'une diminution tres sensible du nombre des equations portant sur les y et les correlations associees > '// peut tre effectue en tenant compte des bilans massiques des elements chimiques en presence. En effet, si ceux-ci sont au nombre de K, on dispose de K relations lineaires de la forme oil les y. designent les fractions massiques des elements en presence et lea ^^i les coefficients stoechiometriques correspondents. En negligeant dans (S) les contributions moleculaires et en adiaettant pour les N especes une m$me valeur A. de I 1 echelle de diffusion turbulente (equ. (6) ) on obtient par combinaison linccire K equations du type (5) :

73 14-5 pu.grad7 w. div (p \q A grad / k ) («) desquelles les termes de source ont disparu en vertu de la conservation des elements dans les reactions chimiques : OC k. W; = 0 Ces K equations peuvent Stre ramenees a une seule equation pu.grad$ = div (p fq^ A grad portant sur la fonction 4> definie par L 1 introduction de cette fonction qui prend la valeur 0 dans 1' ecoulement exte"rieur et la valeur 1 dans I 1 ecoulement interne permet ainsi d'eliminer K-4 des equations (S) portant sur les concentrations en les remplacant par des relations lineaires du type (w La reduction du nombre d'equations (8) portant sur les /[ )$' est encore plus appreciable, puisque leur nombre se reduit de </2N (N+1) yz(n-k+2)(n-k+'t). Celui-ci comprend : a equations portant sur les y/% (i <N-k, j < N-K) N - K equations portant sur les y/ ' i equation portant sur 4> Mise en forme finale des equations. En raison de la configuration particuliere examinee (couche de melange sur le noyau potentiel d'un jet) le formalisme de simplification habituellement adiais pour les ecoulements du type couche limite bidiniensionnelle peut $tre adopte. Les variables spatiales se reduisent alors a x et r et le nombre des equations portant sur les quantites moyennes se reduit a N-K*3 - vitesse longitudinale :

74 temperature : - concentrations ( N-K equations) : 4* («> - fonction de concentration <J> La viscosit^ turbulente ^ est exprimee, conformemait a I 1 equation (6) par ( 22 ) ou -T&r est Equivalent aux L ; de (6) pour f.» U Pour la temperature et les concentrations, les longueurs de diffusion sont supposees proportionnelles a L et dans le rapport des inverses des nombres de Prandtl et de Schmidt turbulent s/ ceux-ci ehant consideres comme constants dans tout I 1 Ecoulement ( f^. = 5 T =0,7). Par suite de I'hypothese d'isothermie 1'equation (49) n'est prise en compte qu'au voisinage de la zone d'inflammation. Aux equations precedentes on adjoint les equations (9) portant sur les correlations doubles de vitesse et de concentration, en tenant compte des reductions operees au paragraphe precedent : - 2 p *, (.ZS) -*^ («)

75 14-7 Les taux de dissipation, TJy / "tf; et ~6 sont exprimes conf ormement a (41) au moyen des echelles de dissipation A (pour la vitesse) et A m (pour les concentrations) : ~~ "' ^ (27) ou D m est le coefficient de diffusion moyen suppose 6tre le m6me pour toutes les correlations massiques. Dans I'hypothese de nombres de Reynolds turbulents R T = S-j- /» eleves, I 1 echelle dissipative X peut tre reliee a I 1 echelle de diffusion L (equ. (22)) par : X = 4 " 5L. (2») Pour les fluctuations de concentration, nous adoptons pour le rapport X m /X une valeur moyenne de 1,5, compte tenu des resultats experimentaux obtenus par Fulachier et Dumas [3] pour la turbulence thermique d'une couche limits de paroi chauffee. Les taux de dissipation des fluctuations de concentration s'expriment alors en fonction de celui de 1'energie cinetique turbulente par les relations simples : S etant le nombre de Schmidt S Dans le cas de fortes variations transversales de 1'echelle de la turbulence L, notamment dans la phase initiale du melange en presence de couches limites epaisses a la confluence, il convient d'adjoindre aux equations precedentes une equation supplementaire portant sur I 1 echelle L ou sur une grandeur turbulente qui la contient. L 1 equation portant sur le produit q/ 2 L etablie par Rotta [4], [5] a partir des correlations spatiales des fluctuations de vitesse par une procedure analogue a celle conduisant a 1'equ. CB), semble convenir particulierement pour le traitement des couches de cisaillement libres \6] et specialenient pour 1'etude des effets de coucheslimites initiales sur le developpement du melange [?] Cette equation est semblable a (23) : * r (30) Les constantes de diffusion a\_ et a' L (de m nie que la const ante a^ dans 1'equ. (23) ) et la constante de production Cp ont ete evaluees dans [?] a partir dv. comportement as;/raptotique de la couche de melange libre (<3^s\,4,QL = H,2, Q'L = O, Cpsr4,96 ), alors que la cbnstante de dissipation resultant du comportement de la turbulence de grille est fixee a 1,16.

76 L' application numerique de ce modele a la region asymptotique d'une couche de melange fait apparaltre, en ce qui concerne 1' evolution du parametre d 1 expansion 0* en fonction du rapport des vitesses (Fig. 2), une concordance globale satisfaisante avec 1' ensemble des resultats experimentaux disponibles [8] a [15J En particulier, la relation simple proposee par Sabin [16] se trouve bien confirmee. 3 - MODELISATION DE LA REACTION EN CHAINE $2-2* - Schema reactionnel complet.- Parmi les nombreuses reactions eiementaires participant au schema reactionnel de la combustion de 1'hydrogene (voir p. ex. [17] ) les reactions suivantes apparaissent comme les plus significatives : H =^ OH + H2 * H * *~ 0 +H 2 ^^ H M =^ H02 + H 2 «^ HgOg + M *- OH + OH "«* H2 2 +H = H OH * *- H + H + M 5= M 5 " H M 3=S H + OH 4- M S " OH + OH H H OH + 0 OH + H H0 2 + M H2(>2 + H OH + OH + M H H20 + OH H H0 2 H 2 + M M OH + M H20 + M (R1) (R2) (R3) (R4) (R5) (R6) (R7) (R8) (R9) (R10) (R11) (R12) (R13) (R14) L 1 importance des reactions d'initiation du type R1 est relativement reduite dans les problemes d 1 inflammation par confluence et melange, en raison de la presence de produits de dissociation (radicaux H et 0) dans les deux ecoulements chauds, susceptibles d'accelerer le demarrage de la chalne principale (reactions R2 a R7 ). Ces reactions, du 1er ordre par rapport aux radicaux crees, determinent pratiquement k elles seules 1'evolution chimique pendant la phase d 1 induction.

77 14-9 En raison du caractere termoleculaire de R5, le rdle des reactions R5" a R7 s'amenuise au fur et a mesure que la pression diminue ou la temperature crolt, ce qui correspond a la region des delais courts en dessous de la 2eme limite d 1 explosion, ou les 3 reactions R2 a R4- predominent. Lorsque a la fin de la phase d'induction la concentration en radicaux atteint un. certain seuil critique, les reactions inverses ainsi que les reactions R8 a R44-, qui sont toxvtes du 2eme ordre par rapport aux radicaux, rentrent en jeu et interrompent la chalne. Par suite de la forte exothermicite de ces reactions la rupture de la chalne correspond pratiquement a 1'inflammation du melange. Les constantes des reactions directes retenues pour les applications numeriques sont celles preconisees dans la compilation de Baulch, Drysdale et Lloyd DeO pour les reactions R2 a R^O et pour R44-. Pour les reactions restantes on peut se referer a [J9J1 La'figure 3 illustre, dans le cas d'un melange stoecmometrique hydrogene air, I 1 evolution de la production chimique et de la temperature pendant les phases principales d'induction et de combustion. Le schema reactionnel complet ( R 4 a R 4 4- ) est pris en compte dans ce calcul et le melange suppose porte" instantanement a une temperature de K (conditions du type tube a choc). Les constantes adoptees pour R1 sont celles preconisees par Schott et Kinsey [20] qui semblent conduire, d'apres une etude recente de Schmalz [21J / a un meilleur accord avec les experiences au tube a choc que celles proposees par Ripley et Gardiner [22]. La figure 3 fait clairement apparaltre le trait caracteristique de ce type de reaction j apres une periode d 1 initiation assez breve, la chalne R2 a R7s'installe dans un etat quasi-stationnaire caracterise par une croissance exponentielle de tous les produits crees et par le fait que leurs concentrations restent pratiquement proportionnelles entre elles. L'exemple presente fait etat d'une concentration en H0 2 relativement elevee pendant la phase d'induction soulignant 1'importance des reactions R5 a R7 dans la chalne pour les conditions initiales considerees ( -jo = 1 b, T = K). Comme critere pour la caracterisation de la phase d'induction nous avons retenu une augmentation de la temperature egale a lo^o de 1'elevation totale ATnax jusqu'a I'equilibre alors que la phase de combustion vive est schematisee par une augmentation de T entre 10/6 et 90^> de AT W ox Simplification du schema reactionnel.- Compte tenu des remarques precedentes et de 1'allure generale des evolutions de la fig. 3 on admettra desonnais pour la phase d'induction les hypotheses simplificatrices suivantes : Les concentrations des especes de depart H 2 et 0 2 restent constantes. - Par suite du faible niveau des concentrations en radicaux, les reactions du 2eme ordre sont inactives. - La variation de la temperature est negligeable.

78 14-10 L'accroissement des concentrations molaires des radicaux actifs H, 0, OH, H02, H est alors gouverne" par le systems lineaire suivant j (32) (33) [H 2 ][OH] t k 3 [0 2 1[H] + k 4 [H 2 ][0] 4 2 k 7 MfoOfl (34) - kg [M][0 2 ][H] -k c CHg][H02] (35) - U 6 [H 2 ][H02] - k 7 [M][ri20 2 ] (30 admettant pour solution generale des expressions de la forme ou les A.^ sont les racines de I 1 equation aux valeurs propres associe"e au systeme (32) a (36). La comparaison numerique des differents termes suggere par ailleurs de ne"gliger dans les equations (34) k (36) les termes soulignes, de sorte que le systeme precedent se reduit au systeme des 3 equations (32) a (34) L' equation cubique en A associe' a ce systeme a ete derivee par Kondratiev [23] et discute" egalement par Brokaw 124] dans le cas des delais brefs ( U 5 [w]/2 k 3 <4 ). Elle fait apparaltre dans ce domaine une seule racine positive responsable de la croissance exponentielle des radicaux en dehors de la phase d' initiation. L' introduction de I'hypothese de pseudo-e"quilibre (steady-state) pour les radicaux OH et 0, suivant laquelle les de"rivees v^"' et ^1 sont tres faibles devant les termes dont elles representent la difference, permet de remplacer (33) et (34) par des relations lineaires entre les radicaux et de decrire leur evolution par la seule equation (32) faisant alors apparaltre la racine avec «x o k 5 [M]/2k 3 Cette procedure simplificatrice semble trouver une certaine justification par le fait que divers auteurs (p. ex. Schott et Kinsey, [20J ) ont pu correler leurs mesures du delai d'inflammation a I 1 aide du seul parametre [02]. TJ, notaminent pour les melanges relativement riches. Elle s'avere cependant inadaptee sous la forme ci-dessus pour le traitement de zones de melange du fait de la variation importante de la composition locale.

79 14-11 En variante de la procedure ci-dessus, nous posons done plus generalement : ««, MM (39) *o k 3 MM (40) Of ou les coefficients ^o et OM, non necessairement tres petits, sont des fonctions de la composition locale du melange. En portant ces relations dans (33) et (34) et en eliminant [o] et [OH] dans (32) on obtient ^M = 2 ^ [H 2 ][ k 3 [02] M dt avec la racine A = 2 k 3 f [02], ou (4tf 5 o 1-«*- 0 - OH /2 (42) L' equation (41) est alors suffisante pour decrire Involution chimique dans la phase d'induction, a condition d'y introduire les fonctions correctrices <?«(4>) et f OH C<t)). Leur determination peut s'effectuer de facon simple par un calcul iteratif. En effet, on peut remarquer que si on ne prend en compte que la relation (39), on reduit le systeme (32) a (34) a 2 equations et = A/2 W 3 [Oz] se deduit alors immediatement de 1'equation caracteristique associee : OU <X = <X 4 Or, en vertu de (39), (40) et (42) avec on obtient pour OH ( ) : [OH] ^ a[qh]/d[h] THTar/~ * ~ 2? avec

80 14-12 Le calcul iteratif de OH et i? au moyen de (43) et (44) est amorce en posant cx'-w. Sa convergence est rapide du fait que O H est generalement assez petit dans la region des delais courts. L'effet des hypotheses precedemment envisagees sur le comportement de A en fonction de la richesse est illustre Fig. 4 pour les conditions examinees dans la presente etude : T = K? = 1 b melange Air - H 2 + Ar (~ 30$ vol H 2 ) On remarque en particulier que 1'hypothese initiale (pseudo-equilibre en OH et en 0) conduit a une evaluation inexacte de X (<p) f sauf pour des richesses extr$mement elevees, alors que 1'hypothese de pseudo-equilibre en OH aeul fournit deja des resultats assez proches de ceux obtenus par la me'thode iterative, d'ailleurs en parfait accord avec les valeurs de X tirees de la resolution numerique du systeme complet. Cette derniere comparaison permet de justifier a posteriori les hypotheses simplificatrices introduites au debut du paragraphe 3.2. On peut observer d'autre part que la vitesse de production chimique est maximale dans la region sous-stoechiometrique du melange ( If ^ 0,35) alors que 1'Elevation maximale de la temperature se produit pour ^ = 1. Ce decalage des deux regions principales d'activite chimique a deja ete mis en evidence dans le cas du melange H 2 - air [2]. L 1 evolution des coefficients de pseudo-equilibre 0 et OH en fonction de V precisee sur la figure 5 met en evidence 1'importance relative de ces coefficients, notamment en ce qui concerne 0, et justifie la demarche adoptee pour la formulation du pseudo-equilibre, equ. (39) et (40). La variation appreciable de la fonction correctrice qui en resulte se repercutera. en particulier sur le calcul des corrections de production chimique dues a la turbulence, aborde au paragraphe 3.4. La determination complete des concentrations Y\ (k 1'exception de )J,, determine a partir de (41) ) s'effectue au moyen de relations lineaires entre les Xj et X H resultant des equations (33) a (36), auxquelles il faut adjoindre et des equations (39) et (4-0). On obtient ainsi : 0 k 2 [H 2 ][OH] (45) '. "^^t u 4- g/2 - foh/z,, ^ XH

81 14-13 (49) a v«c (50) La figure 6 precise 1'evolution de ces rapports en fonction du rapport de melange et confirme de nouveau la bonne concordance entre la methode simplifiee et le calcul numerique direct a partir du schema complet R-1 a Les concentrations de H 2 et 0 2 resultent des relations precedentes compte tenu des bilans massiques (12J pour I'hydrogene et 1'oxygene. avec y 02 - >o - F 2/ H (sz) y =y.(j>-t > y (4-4>) (f$) ^o -^o,** L^-V Fi et FJ2 sont deux fonctions auxiliaires donnees par : ' (5A) L'application du schema simplifie au calcul des delais d'inflammation conduit egalement a un accord satisfaisant avec le calcul complet, comme en temoigne la figure 7. Alors que dans la region pauvre les delais sont legerement sousestimes par la methode simplifiee, 1'accord est parfait pour les melanges riches. L'elevation de temperature indiquant la fin de la phase d 1 induction selon le critere introduit plus haut resuite de I'application de 1'equation (19) ou les enthalpies Hj contieraient les chaleurs massiques de formation des especes. On peut remarquer egalement que la position du minimum du delai d'inflammation ( *P ft 0,4) correspond approximativement a celle du ma::imum de la racine X de la figure 4-, alors que celle du minimum du delai total (induction + combustion) se trouve sensiblement decale en direction des melanges riches ( «0,8).

82 Application du calcul du melange turbulent.- La prise en compte des simplifications pre"cedentes dans le schema de calcul developpe au paragraphe 2 conduit a une reduction tres sensible du nombre d 1 equations a integrer. En particulier, les ( N-K ) equations (20) portant sur les ) (ici N-K =10-4 = 6) se ramenent dans le cas present a une seule equation pour > : (S7) ou W H s'ecrit, conformement k (41) J 1^0 et i\ representent les corrections dues a la turbulence (equ» (8)X Elles peuvent s'exprimer en fonction des correlations "y^, y' (j)' et 4> qui sont ici les seules a prendre en consideration." En tenant compte de (51) et (52) et en linearisant les fluctuations des fonctions f (4>) selon il vient alors t avec 7j = JJ J^ * ft \/ *_* t ( et pour Tl., en negligeant les correlations d'ordre superie'or- a 2 : avec -JX. L (60)

83 14-15 y 7 'o ' / Enfin, les equations portant sur X M ' et <1> X H ' se deduisent immediatement de (24) et (25), 1'equation (26) rest ant inchange'e : Les termes d'echange / ' ^It et $ w^ peuvent s'exprimer egalement en fonction des correlations (j? 1 f *"' y -* et "^ y 2 'H 7 H (63) avec - s c - y Hi - d RI 1 X.- F, F 2 c 2 = -^i- - -=± y y ' '02

84 14-16 Le systeme des equations aux derive'es partielles gouvernant I 1 evolution des principales grandeurs moyennes et fluctuantes dans la couche de melange est ainsi constitue". II comprend les equations (18) a (21), (23), (26), (30), (61) et (62). 4 - METHODS NUMERIQUE DE RESOLUTION.- Le systeme des equations a integrer subit d'abord un. changement de variables par introduction d'une fonction de courant "V satisfaisant 1'^quation (4) : La resolution nume'rique des Equations transformees s'effectue au moyen d'un schema de differences finies explicite de"ja utilise" dans [2]. L 1 integration est effeature le long des lignes de courant "Y = Cte, le pas d'integration e'tant regie" soit par un parametre lie a la production chimique soit, si celle-ci est peu active, par un critere de stabilite lie" au schema de calcul. L'extension du maillage aux frontieres du domaine d'integration est effectuee de maniere k respecter les conditions aux limites imposees. Lorsque le nombre de mailles atteint une limite superieure fixee arbitrairement, celui-ci est reduit de moitie et le pas A Y double. 5 - RESULTATS DB CALCUL.- Une premiere application numerique de la methode porte sur une configuration de melange de jet compose, comme indique" plus haut, d'un melange H2 + Ar dans des proportions telles que la presence de gradients de masse specifique en amont de la zone d'inflammation est pratiquement evitee (~ 30? vol. de H 2 ). Les conditions initiales suivantes ont ete adoptees : T; = T e = K p = 1 b Uj = 200 m/s U e = 100 m/s Le diametre du jet est fixe a D = 1 cm, assurant une inflammation dans la region'du noyau du jet. Les distances sont en principe normalisees par le noyau du jet Rs-S. et les vitesses par Uj. On notera en particulier x = x/r, r = r/r, 5 = S/R, J N = -? N /R, U c = U c /Ui Une premiere comparaison entre la methode simplifiee du paragraphe 3 et le calcul complet est effectuee sur la figure 8 pour ce qui concerne les effets des evolutions chimiques dans la zone de melange. Le calcul complet prend en compte 1'ensemble des equations de diffusion (20) pour Y H, y o, ^OH / XM o / ^Hoj * yma0 4 ai- ns i <l ue 1 schema reactionnel complet R1 a FM4* Afin de faciliter cette comparaison, les termes correctifs de la production

85 14-17 chimique dus k la turbulence sont provisoirement negliges. Quant aux lignes iso- X H, la methode simplifiee repre"sente tres correctement 1'evolution chimique dans la position inferieure de la couche de melange alors qu'elle surestime I'accroissement de Y H dans la region exterieure. Cette difference se traduit par un leger decalage vers 1'amont du front de flamme dans la partie exterieure du melange et confirme les observations formulees plus haut au sujet de la figure 7. Au voisinage de la ligne de jet "Mf a "Yj par centre les deux methodes de calcul fournissent des resultats tres voisins quant au point d'inflammation. L 1 allure des lignes isothermes dans la zone de combustion fournies par le calcul complet permet d'etendre k la couche de melange turbulente la constatation faite plus haut quant au decalage des maxima de temperature vers 1'interieur de la zone de melange, ou les richesses sont plus e"leve es (voix aussi [2] ). L'effet direct de la turbulence sur la production chimique apparalt sur la figure 9, ou sont portes en fonction de <f> les termes correctifs definis par (59) et (60). Alors que le terme "*\ 0 relatif a la reaction d'initiation est tcujours negatif dans le melange (conformement aux resultats obtenus dans [1] ), il n'en est pas de m me du terme 1\ affectant la reaction predominante R3. En effet, "r\ presente dans les regions exterieures du melange des ^ contributions fortement positives provenant des correlations y^ % et?'y^, comme I'indique la decomposition effectuee en bas de la figure 9 Les valeurs positives de celles-ci sont dues a la forme particuliere des distributions transversales de y H (figure 8). L'action de la turbulence sur la cinetique caracterisee essentiellement par 1\ parait done relativement limitee dans la region centrale du melange, ou la production chimique est la plus intense. Elle y a pour effet une legere diminution, de 1'ordre de 10^ au maximum. Ce niveau, lie essentiellement aux lois gouvernant la cinetique chimique dans la couche de melange, est relativement peu affecte par les conditions initiales (rapport de vitesse, couches limites initiales). L'action indirecte de la turbulence s'exercant sur la croissance des radicaux au travers des echanges turbulents (terme de diffusion) se superpose a ces actions directes. L'ensemble de ces effets est illustre sur la figure 10 pour I'accroissement relatif de X H dans_ le cas d'une couche limite epaisse de part et d'autre de la confluence ( S =0,5, figure du bas) et dans le cas d'epaisseurs nulles (figure du haut). L'action de la diffusion, turbulente, tres semblable dans les deux figures, a pour effet, comme dans le cas precedent, de reduiie le taux d'accroissement des radicaux dans la region de.croissance maxiraale et de 1'augmenter dans les regions marginales. On'remarque cependant qu'au voisinage de la ligne de jet ( Y = *% ) ces influences sont iiotablement attenuees. On montre d'ailleurs facilement qu'en absence d 1 effets de couches limites 1'action retardatrice de la diffusion turbulente ne depend pratiquement, pour le type de reaction examine, que de la loi X C$) et qu'elle peut tre consideree comme independante du rapport des vitesses. La presence des couches limites a la confluence ne modifie pas fohdamentalement cette propriete, comme on a pu constater a partir des resultats numeriques, de msme qu'elle n'altere pas profondement le niveau des grandeurs turbulentes sur la ligne de jet, k 1'exception du voisinage iiamediat de la confluence.

86 14-18 L'existence de couches limites Epaisses a la confluence affecte par contre de facon beaucoup plus directe la position du front de flamme, comme indique sur la figure 11, ou deux criteres differents ont ete utilises. Le raccourcissement tres sensible de la longueur d'inflammation indique par ces deux criteres est du 1 essentiellement k la modification apportee par les couches limites initiales au champ moyen de vitesse, comme on peut juger d'apres le fait que le temps de parcours T* entre le decollement et le point d'inflammation sur la ligne de jet est sensiblement le meme dans les deux cas : T* ~ 1.2 T;. t 0 est le delai d'inflammation en milieu premelange pour la valeur de reference <J> =0,5 C^= 0,23 ms dans le present exemple d'apres la figure 7). Le rapport T'/T? represente non seulement les effets de retardement dus k la turbulence mis en evidence plus haut (figure 10), mais aussi 1'Evolution sur la ligne de jet de la fonction ($> dont depend la production chimique (figures 4 et 7). Get effet s'avere $tre quasiment indifferent k la presence de couches limites initiales. Compte tenu de ces remarques il est maintenant possible d'etablir un critere d'inflammabilite des jets en adoptant les conventions suivantes : on considere que 1'inflammation du jet est effectivement realisee si le point d'inflammation se situe en amont de la fin du noyau de concentration ( x = < ). Le choix de ce critere est motive par le fait qu'en aval du noyau (J> diminue rapidement dans tout le champ et en particulier au voisinage de la ligne de jet, ou la reaction est la plus avancee. Il deviendrait alors de plus en plus difficile d'amorcer la combustion (voir figure 7).11 convient done de poser comme critere : ou fw est le temps de parcours sur la ligne de jet entre le decollement et la fin du noyau et T* le temps chimique precedemment defini. Alors que f* ne depend que du comportement de la cinetique, %«est directement proportionnel, pour des conditions de confluence fixees, aux dimensions du jet. On est ainsi en mesure de deduire du critere ci-dessus la valeur minimale du diametre du jet, en dessous duquel 1'inflammation ne peut plus tre assuree suivant les conditions fixees. En rapportant D m -, 0 a une longueur de reference formee avec la vitesse de 1'ecoulement interne Uj et le temps chimique de reference t, on obtient une expression dans laquelle n'interviennent plus que les grandeurs sans dimension U e et o s Cette quantite qui s'apparente a un nombre de Damkohler de premiere espece est presentee en haut de la figure 12. La diminution tres sensible de f en fonction de U e dans le cas de la confluence sans couches limites est imputable k 1'augmentation iciportante de la longueur du noyau avec la vitesse de 1'Ecoulement exterieur, corome

87 14-19 I 1 indique la figure inferieure. Ces variations sont fortement attenuees par la presence de couches limites initiales dont les effets se font ressentir notamment dans le domaine des rapports de vitesse ElevEs. Comme il results de ces traces, une valeur ded m ; n /Ui t* de I'ordre de 0,1 assurera 1'inflammation des jets, independamment du rapport de vitesse et de 1'Epaisseur de couche limite t Dmin «0,1 Ui t. Pour les conditions initiales considerees dans la presente Etude ce oritere conduirait k une valeur de Dmin =» 5 mm. 6 - CONCLUSION.- Le probleme de 1'inflammation spontanee des jets a ete aborde dans le cas particulier ou les deux Ecoulements sont initialement k la msme temperature et ou la transformation chimique est dominee par un mecanisme de reaction en chalne quasi-isotherme k 2 branchements. L'application particuliere a la reaction H 2-02 a mis en Evidence le benefice que I 1 on pouvait tirer du comportement specifique de ce type de reaction pour la description des effets de la turbulence sur les reactions chimiques. En particulier, 1'introduction de la notion.de pseudo-equilibre et la prise en compte des proprietes de linearite de la chalne ont permis d'apporter une simplification notable k la prooedure numerique en reduisant le nombre des equations k integrer k 2 pour les concentrations moyennes et a 3 pour les correlations doubles des concentrations, Dans cette premiere approche, les correlations de concentration d 1 ordre superieur k 2 ont EtE supposees suffisamment petites devant celles-ci pour 8tre negligees. Les conclusions d'ordre pratique qui se degagent des premieres applications numeriques peuvent se resumer ainsi : - Les effets de la turbulence sur la production chimique paraissent pour ce type de reaction relativement moderes. Us sont essentiellement determines par les lois de comportement de la cinetique dans la couche de melange et semblent e*tre peu influences par le champ de vitesses ou les couches limites initiales. II en est de m me des effets de la diffusion turbulente, I'ensemble de ces deux effets etant maximal au voisinage du maximum de la production chimique. - Lteffet des couches limites sur la longueur d'inflammation se repercute de facon plus marquante au travers de la deformation du champ de vitesse. Les applications numeriques ont permis de degager un critere' d'inflammabilite pour le cas ou la zone de production maximale est localisee dans la partie centrale de la couche de melange. D'apres 'ce critere le diametre minimal Dmin du jet est lie a la vitesse du jet et a un temps chimique caracteristique ou interviennent la composition moyenne du melange et le niveau therinique des deux ecoulements. Le cas ou les temperatures des deux ecoulements sont differentes n'a pas ete envisage dens le cadre de cette premiere approche. Ce cas necessiterait, en particulier, la prise en compte d'autres correlations contenant les fluctuations de temperature qui pourraient avoir des effets beaucoup plus marques sur la production chimique en raison de la variation importante des constantes de reaction avec la temperature.

88 14-20 REFERENCES.- M, BARRERE CinEtique chimique en milieu turbulent. R. BORGHE C.R. Acad. Sc. Paris, t. 276 (8 Janvier 1973). 0. LEUCHTER Etude des Evolutions chimiques dans une couche de melange hydrogene-air. Colloques de la SociEte Francaise de Physique - Evian Mai 1971, O.N.E.R.A. T.P. N 981 (Traduction anglaise : NASA TT F ). [3] L. FULACHIER Repartitions spectrales des fluctuations thermiques R. DUMAS dans une couche limite turbulente. Proceedings of Agard Conference on Turbulent Shear Flows, London (1971). J.C. ROTTA J.C. ROTTA Statistische Theorie nichthomogener Turbulenz. 1 und 2. Mitteilung. Zeitschr. f. Physik 129, p. 547 r J31, p. 51 (1951). Recent attemps to develop a generally applicable calculation method for turbulent shear flow layers. Proceedings of Agard Conference on Turbulent Shear Flows, London (1971). [6] W. RODI A two parameter modsl of turbulence and its application D.B. SPALDING to free jets. Warme-und Stoff-^bertragung_3,85 (1970). 0. LEUCHTER Ueber das Verhalten der turbulenifen freien Strahl" grenze mit Initialgrenzschicht. REunion annuelle de la DGLR, Innsbruck Septembre B.W. SPENCER Statistical investigation of pressure and velocity E.G. JONES fields in the turbulent two-stream mixing layer. AIAA Paper N (1971). [10] A.J. YULE J.B. MILES J»S. SHIH Two-dimensional self-preserving turbulent mixing layers at different free stream velocity ratios. University of Manchester - Reports and Memoranda N Mars Similarity parameter for two stream turbulent jet mixing region. AIAA Journal Vol.' 6, pp H30 (1968). _11J R.P. PATEL An experimental study of a plane mixing layer. AIAA Journal Vol. 11, H 1, p (1973). [12] H.W. LIEPMANN Investigations of free turbulent mixing. J. LAUFER NACA TN 1257 (1947).

89 14-21 M [15] [16] [17] M D9] [20] [21] [22] G. BROWN A. ROSHKO I. UTGNANSKI H.E. FIEDLER K. DANG TRAN 0. LEUCHTER C.M. SABIN J. ALGERMISSEN D. HOETZOLD D.L. BAULCH D.D. DRYSDALE A.C. LLOYD R. PRUD'HOMME C. LEQUOY C.L. SCHOTT J.L. KENSEY F. SCHMALZ D.L. RIPLEY W.C. GARDINER The effect of density difference on the turbulent mixing layer. Proceedings of Agard Conference on Turulent Shear Flows, London (1971). The two-dimensional mixing region. J.F.M. Vol. 41, pp (1970). Resultats k publier dans la Recherche Aerospatiale. An analytical and experimental study of the plane, incompressible, turbulent free shear layer with arbitrary velocity ratio and pressure gradient. AFOSR W 5443 (1963). Der zeitliche Ablauf der Verbrennung von Wasserstoff im Ueberschall - Luftstrom Forsch. Ing. - Wes. Bd. 36 (1970), N 6, p Critical evaluation of rate data for homogeneous, gas - phase reactions of interest in high - temperature systems (High temperature reaction rate data N 2 et 3) Department of Phisical Chemistry - The University LEEDS 2 ENGLAND (1969). Tableau des vitesses specifiques de reactions chimiques utilisables en aerothermochimie. O.N.E.R.A. Note technique N 147 (1969). Kinetic studies of hydroxyl radicals in shock waves. II - Induction times in the hydrogen-oxygen-reaction. J. Chem. Phys. 29 (1958), p Me s sung und theoretische., Bereclmng von Zundverzugszeiten in Wasserstoff -Luft-Gemischen bei Temperaturen urn K und Drucken unter 1 at DLR F (1971). Shock-tube study of the hydrogen-oxygen-reaction. II. Role of exchange initiation - J. Chem. Phys ), p [23] M V.N. KONDRATIEV R.S. BROKAW Chemical Kinetics of gas reactions.- Pergamon Press, (1964). Analytic solutions to the ignition kinetics of the hydrogen-oxygen reaction. 10 th. Symposium (int.) on combustion. Pittsburgh : The Combustion Institute, (1965) p

90 14-22 u; Representation schematique de la configuration examinee Figure 1 1 o 0,1 O SPENCER-JONES [«] + YULE [9] MILES-SHIM [10] PATEL [11] LIEPMANN-LAUFER [12] BROWN - ROSHKO [ij] WYONANSKY-FIEDLER [14] O.N.E.R.A. [15] 0,05 calcul numtriqu* Ui Taux d'expansion du melange en fonction du rapport des vitesses Figure 2

91 14-23 Figure ,1 0,2 0,3 [fnfl Phase d'inflammation d'un melange H2~Air ( Stoechiomerrique, p=1b ) pseudo-equilibre en OH at 0 m.thod. it.rahvt o resolution numtrjqu. du lyilim. compl.t (t.ti/2) Figure 4 0, V. P max. du systeme lineaire associe (phase d'induction) A T ,01 0,1 Temperature de fin de reaction (equilibre)

92 14-24 methode it.rahve o resolution numerique du system, complet Figure 5 0, ,01 0,1. Fonctions correctness de pseudo-equilibre HjO 1 0,1 0,01 o methode iterative resolution numerique du lysteme complet (t.tj/2) 0,5 0,9 -r- 0,01 0, *P Rapport des fractions massiques dans la zone de melange Figure 6

93 14-25 Melange Hj + Ar - Air p. 1 b r T. 10OO'K resolution numtrtqut du lysrtmc compltt - mtthod* du pseudo-equilibrt f IMS 0,6 Figure 7 0, ,1 0,5 0,9 Temps chimiques en fonction du rapport de melange lyst.me complet ot_0,1at max ps.udo-equilibre 1,5 0,5 Lignes iso-y^ et iso-t dans le plan physique (sons correction de turbulence) Figure 8

94 14-26 Figure 9 Termes correctifs dus a la turbulence (sans couches limites initiales) TOTAL '{Oirrusion»Production) sans couche limite initial* Figure 10 sans correction f TOTAL l/tdifrusiontproduclion) 0,5 1 "$ avec couche limite initiale (6.0.SR) Effet des couches limites initiales sur I'accroissement de y H

95 ,SR 1,5 8 = 0 Crit.r. I : AT _ 0,1 ATffl Critere I : AT. 0,05 T o 0,5 Ul 10 Influence des couches limites initiales sur la position du front de flamme Figure 11 Cmm] 0,5 1 Diametre minimum * Figure ,5 1 Ue "Of Longueur d inflammation reduife Critere d'inflammabilite de jets avec couches limites

96 14-28 DISCUSSIONS Prof. Ferri: I want to make a general comment about the equations and the transport properties for these kinds of problems. In this type of problems where one has very large temperature rises due to the combustion processes there are also very large variations of density and pressure so that one has a convection mechanism that is connected with pressure variation that cannot be neglected. For example if you make an experiment with this type of jet and for this type of problem you find that the turbulence with combustion at the axis of the jet, where the combustion does not take place, varies tremendously due to the combustion itself. So that the convection mechanism is through pressure waves and not through diffusion and I believe that this should be taken into account. We have done a simple experiment on an axisymmetric jet producing random fluctuations at the axis and measuring the fluctuations at the periphery of the jet. The turbulence at the periphery was found to be much different from what one would expect. In conclusion we are convinced that this diffusion mechanism is connected with waves and pressure fluctuations in combustion, but this is not included in your equations (as in many other's equations as well). Prof. Classman: In your reaction scheme you took the first 8 reactions. It seems to me later, when I observed the temperatures which you got within the flame in the order of to K, I became very concerned of the presence of the Hydroperoxyd radical and of the possibility of H2 -f O2 -* 2 OH at those temperatures. On the contrary, I think that the more important initiating reaction would be the inversion of reaction R 11 which is the dissociation of the Hydrogen molecule. Author's reply: La restriction aux 7 premieres reactions du schema considere n'a ete retenue que pour la description de la phase isotherme en amont du front de flamme. Par contre, les lignes isothermes entre K et K de la figure 8 resultent d'un calcul pour lequel le schema reactionnel complet a ete pris en compte, en considerant toutes les reactions comme reversibles. En ce qui concerne la presence du radical d'hydroperoxyl a ces temperatures, on peut observer que sa concentration diminue rapidement au debut de la zone d'inflammation (voir p. ex. figure 3) de sorte que la presence de ce radical, qui contribue fortement a 1'evolution chimique dans la phase d'induction, peut etre considere comme negligeable dans la phase des reactions exothermiques. Quant a la reaction RI, son action consiste a contribuer, au cours de la phase d'initiation, a la production de radicaux necessaires au demarrage de la chaine, pourvu que les deux ecoulements confluants n'en contiennent pas deja en nombre suffisant. Par contre, pendant les phases de croissance exponentielle et de combustion vive la contribution de cette reaction devient negligeable par suite de la faible valeur de sa vitesse specifique. D'autre part, la reaction inverse de R 11 ne contribue pas au processus d'initiation dans la presente configuration, etant donne que chacun des deux ecoulements est suppose en equilibre chimique en amont de la confluence. II en serait autrement pour les conditions du type tube a choc (cas de la figure 3) oil les reactions inverses de R 11 et de R 12 s'associent a la reaction RI pour initier la chaine. Prof. Libby: Are you perfectly comfortable with your handling of the effects of turbulence on chemistry? I ask that question because I and others have been struggling with very simple chemical systems (like A + B = C) under temperature and pressure conditions where molecular kinetics are infinitely fast and I must say that this treatment is not in very good shape yet. Therefore I am unable to understand how one can be confident in the case of Hydrogen and Oxygen systems which involve many reactions in a turbulent flow. Author's reply: La description des evolutions chimiques dans la region d'induction et des effets de la turbulence sur celle-ci se ramene, compte tenu des simplifications introduites au niveau de la cinetique, a 1'etude d'un systeme chimique egalement tres simple ne comportant en particulier qu'une seule espece active, representative pour 1'evolution de 1'ensemble des radicaux de la chaine. Ces simplifications impliquent evidemment que la vitesse de mise en equilibre des radicaux est tres grande devant la vitesse devolution de 1'ensemble des radicaux qui determine le delai d'inflammation dans le melange. La mise en evidence des effets de la turbulence sur le comportement de ce systeme chimique tres simplifie fait appel a la description statistique de la turbulence avec 1'hypothese classique de nombres de Reynolds de la turbulence

97 tres eleves. Dans le present cas, la simplicite relative de cette description est due essentiellement aux hypotheses restrictives introduites au niveau de la procedure de fermeture, en particulier pour les moments statistiques des concentrations. En depit de ces simplifications qu'il etait necessaire d'adopter dans un premier temps pour des raisons de facilite de traitement numerique, il est possible de caracteriser globalement le comportement de la cinetique d'inflammation dans une couche de melange turbulente et de mettre en lumiere les effets de la turbulence sur celle-ci. Bien entendu, la presente analyse n'est a considerer que comme une premiere tentative de caracterisation de ces effets et devra etre raffinee par la suite, notamment en adoptant des hypotheses de fermeture moins restrictives Prof. Libby: My second question is related to pressure waves and intermittency which are not contained in the equations. There are experimental results which indicate that probably the density fluctuation, which is the essential thing that you try to describe when you have all these correlations, is enormously affected by intermittency; now, in the case of the velocity field and the velocity correlation, that neglect may not be quite so severe because after all the pressure field pushes the velocity around in the external stream. But this is not true of the scalars, such as the concentrations and the temperature. It seems to me that this difference in the behaviour between the scalars and the velocity fluctuations is very significant and may raise some doubt about the modelling which is very frequently used with respect to the scalars. Author's reply: Les fluctuations de pression n'apparaissent en effet pas de fa9on explicite dans les equations de bilan, mais elles sont contenues implicitement dans certains termes comme p. ex. dans le terme de diffusion dans 1'equation de 1'energie cinetique turbulente. Ailleurs elles sont negligees, compte tenu de 1'absence d'effets marques de la compressibilite pour les conditions initiales envisagees dans la presente analyse. Quant aux effets de 1'intermittence, ceux-ci n'ont pas ete pris en compte explicitement par suite des difficultes de simulation de ces effets dans le cadre de la description statistique. II a ete admis que ces effets sont relativement faibles pour les conditions de la presente etude du fait que la region ou 1'activite chimique est la plus intense coincide avec la region centrale du melange, ou le facteur d'intermittence n'est que peu different de 1. Dans le cadre de raffinements ulterieurs de ce type de description il conviendrait bien entendu de porter un effort particulier sur la modelisation des phenomenes d'intermittence, compte tenue de leur importance pour la caracterisation des taux moyens de production chimique dans les ecoulements turbulents a frontiere libre.

98

99 Ill A REVIEW OF SOME THEORETICAL CONSIDERATIONS OF TURBULENT FLAME STRUCTURE by F.A. Williams Department of Applied Mechanics and Engineering Sciences University of California, San Diego La Jolla, California 92037

100 Ill SOMMAIRE Trois aspects de la theorie des flammes turbulentes sont examines. (a) La structure d'une flamme laminaire dans un ecoulement cisaille est consideree. Les resultats relatifs a la fois aux flammes premelangees et aux flammes de diffusion sont resumes. II est 6tabli que le taux de deformation est un element important de Pecoulement turbulent qui agit sur la structure de flamme. On identifie et on oppose les differences des effets de la distorsion et de la compression de la flamme. Le rapport de 1'epaisseur de la flamme laminaire a 1'echelle de Kolmogorov est un element essentiel pour les flammes premelangees, des vitesses de propagation de la flamme negatives ou des disparitions de flamme se produisant si 1'ordre de grandeur de ce rapport est au dela de un. Pour les flammes de diffusion, 1'inverse du premier nombre de similitude de Damkohler apparait, 1'extinction de la flamme se produisant pour les valeurs de ce parametre suffisamment elevees. On arrive a la conclusion que les flammes turbulentes premelangses sont rarement compose'es d'un ensemble de flammes laminaires premelangees et cisaillees, alors que les flammes turbulentes de diffusion sont souvent composees d'une collection de flammes laminaires de diffusion. (b) (c) Le principe d'une description statistique de la structure des flammes turbulentes premelangees est present^, dans le cas d'une turbulence de faible intensite. On montre qu'il existe des zones de convection-diffusion et de reaction-diffusion. Les fluctuations de vitesse agissent sur les gradients moyens dans la zone de convection-diffusion pour produire des fluctuations de concentrations et de temperature qui augmentent la moyenne quadratique du niveau de fluctuation dans la zone de reaction-diffusion. Cette augmentation modifie la vitesse de propagation de la flamme. Une formule pour cette vitesse est presentee. On montre que la vitesse de propagation de la flamme augmente avec 1'echelle de la turbulence, a condition que l'e"chelle int6gr6e soit eleve'e par rapport a 1'epaisseur de la flamme laminaire. Cette prediction est en accord avec des resultats expeiimentaux recents pour une turbulence de faible intensite. On examine la possibilite d'utiliser des fonctions de couplage pour decrire la structure des flammes turbulentes de diffusion qui sont faites d'une ensemble statistique de flammes laminaires de diffusion, pour lesquelles 1'approximation de la flamme mince est applicable. On demontre que, pour des conditions tres g6n6rales les champs de temperature et de concentrations du combustible, de 1'oxydant et du produit de la reaction peuvent etre relies a un champ de concentration normalisee d'inerte, dans un plan espace-temps. En consequence, les proprietes statistiques des champs consideres peuvent etre deduites des proprietes statistiques de 1'inerte. On montre comment ce resultat peut etre utilis6 pour obtenir de maniere simple le taux volum6trique local moyen de production d'oxyde d'azote dans la flamme turbulente de diffusion, avec la seule connaissance de la fonction de densite de probability locale pour 1'inerte, evaluee pour une concentration d'inerte qui correspond la position du front de flamme. L'utilite des mesures de fonctions de densitd de probabilite pour les inertes est soulignee. On montre que le probleme du calcul du taux volurn6trique de production de chaleur dans les flammes turbulentes, par 1'utilisation de fonction de couplage, est difficile. Enfin, on souligne le besoin d'etudes theoriques sur Pinfluence des flammes sur les champs de vitesses turbulentes.

101 A REVIEW OF SOME THEORETICAL CONSIDERATIONS OF TURBULENT FLAME STRUCTURE F.A. Williams Department of Applied Mechanics and Engineering Sciences University of California, San Diego La Jolla, California ABSTRACT Three different aspects of turbulent flame theory are discussed. (a) The structure of a laminar flame in a shear flow is considered. Results are summarized for both premixed and diffusion flames. The strain rate is established as a key turbulent-flow factor influencing flame structure. Differences in effects of flame stretch and flame compression are identified and contrasted. The ratio of laminar flame thickness to Kolmogorov scale is demonstrated to be critical for premixed flames, with negative flame speeds and flame annihilation occurring if the order of this parameter exceeds unity. For diffusion flames, in place of this parameter, the reciprocal of Damkohler's first similarity group appears, and flame extinction occurs at sufficiently large values of the new parameter. It is concluded that premixed turbulent flames seldom will be composed of an ensemble of sheared, premixed, laminar, flames, while turbulent diffusion flames often will be composed of a collection of laminar diffusion flames. (b) A statistical description of premixed turbulent flame structure is outlined for turbulence of low intensity. It is shown that convective-diffusive and reactivediffusive zones exist. Velocity fluctuations act on mean gradients in the convective-diffusive zone to produce concentration and temperature fluctuations which increase the mean-square fluctuation level in the reactive-diffusive zone. This increase modifies the flame speed. A formula for the turbulent flame speed is given. It is shown that the flame speed increases with increasing turbulence scale, provided that the integral scale is large compared with the laminar flame thickness. This last prediction is found to agree with recent experimental results for turbulence of low intensity. (c) The use of coupling functions is discussed for describing the structure of turbulent diffusion flames that consist of a statistical collection of laminar diffusion flames for which the flame-sheet approximation is applicable. It is demonstrated that under remarkably general conditions fuel, oxidizer, product and temperature fields all can be related to the field of a normalized inert, on a space-time resolved basis. Therefore, statistical properties of the fields of interest can be inferred from statistical properties of the inert. It is shown how this result can be used to obtain in a simple manner the average local volumetric production rate of nitric oxide in the turbulent diffusion flame, requiring as input only the local probability density function for the inert, evaluated at an inert concentration which corresponds to the flame-sheet position. The usefulness of measurements of probability distribution functions for inerts is emphasized. The problem of coupling functions is shown to be difficult. Finally, the need for theoretical studies of the influence of flames on turbulent velocity fields is expressed.

102 II1-2 LIST OF SYMBOLS A variance factor; constant B skewness factor; constant C coupling function c inert mass fraction Cp specific heat at constant pressure D diffusion coefficient D! Damkohler number E activation energy F reaction-rate function f distribution function I flame-speed integral J flame-speed integral K equilibrium constant k rate constant / integral scale n reaction order P probability Q heat released per unit mass of fuel consumed R turbulent Reynolds number R universal gas constant T temperature t time u nondimensional velocity v velocity V L laminar flame speed v~ T turbulent flame speed w reaction rate x nondimensional distance Y nondimensional temperature YJ mass fraction a nondimensional temperature rise across the flame 0 nondimensional activation energy 7 flame stretch 5 flame thickness e nondimensional turbulence intensity (f> nondimensional flame stretch spatial coordinate in the flow direction 77 Kolmogorov scale A burning-rate eigenvalue X Taylor scale H stoichiometric oxidizer mass fraction v diffusivity v\ stoichiometric coefficient

103 II INTRODUCTION From a basic point of view, turbulent flame theory is very difficult. In approximately isobaric turbulent flames, ten-fold density decreases occur locally, at rates exponentially dependent on the local density. This statement puts the complexity into a nutshell. It implies that investigators must consider not only the highly nonlinear, local, instantaneous effects of turbulent velocity and temperature fluctuations on rates of heat release, but also the influence of the local, instantaneous spots of heat release on the turbulent fields of velocity and temperature. Although estimates suggest a strong, nonlinear coupling, no one has given thorough consideration to the second part of the problem, the influence of the flame on the turbulence. Therefore of necessity in the present paper discussion is focused on the influence of the turbulence on the flame. It will be seen that our understanding of even this aspect of the structure of turbulent flames is rudimentary. However, certain bits of fundamental knowledge recently have been uncovered. Some of these are reviewed herein. The classical division into premixed and nonpremixed combustion remains useful for turbulent flames. We begin by considering premixed flames, first discussing the structure and motion of a laminar flame in a potentially turbulent shear flow. Necessary conditions for a turbulent flame to be composed of an ensemble of these stretched laminar flames will be defined. Next, the problem of developing a statistical description of the structure of a turbulent flame, which may or may not be composed of a collection of laminar flames, will be addressed. A formula for the turbulent flame speed in low-intensity, grid turbulence will be derived, and comparison with a recent flame-speed experiment will be made. Turning to nonpremixed systems, we first give a discussion of the structure of a laminar diffusion flame in a shear flow, paralleling the earlier discussion of the premixed laminar flame. Differences in behaviors for the premixed and nonpremixed cases will be emphasized. The idea of using coupling functions to bypass analysis of the reaction rate in obtaining descriptions of turbulent diffusion flames next will be addressed, with emphasis on its utility and limitations. The way in which the coupling functions can be employed to calculate NO (nitric oxide) production in turbulent diffusion flames will be outlined, and the problem of calculating mean rates of heat release will be approached. It will be seen that potentially promising first steps can be taken toward theoretical analysis of the structure of turbulent diffusion flames, provided that data is available on probability distribution functions in corresponding nonreacting turbulent flows. 2. PREMIXED LAMINAR FLAME IN A SHEAR FLOW 2.1 The One-Dimensional Laminar Flame The structure of one-dimensional, steady, laminar deflagrations has been studied extensively in the past 1. It has been established that for many purposes the complex chemistry which occurs can be approximated well as a one-step, overall, Arrhenius, rate process, and molecular diffusivities for the principal chemical species and for heat may be set equal. Under these conditions, by use of coupling functions it can be shown that the flame structure is described by one ordinary differential equation of the general form vdy/dg - ^d 2 Y/d 2 = -(vj»f(y). (1) Here is the spatial distance in the flow direction, v is the flow velocity, v the diffusivity and F(Y) a nondimensional reaction-rate function. The dependent variable Y can be viewed either as reactant mass fraction or as nondimensional temperature, a suitable definition for it being Y = (T f T)/(Tf T 0 ), where T is temperature, the subscript o denotes upstream conditions and the subscript f final downstream conditions. Boundary conditions are Y -» 1 as -* and Y -> 0 as ->. On physical grounds it is believed that a solution to Equation (1) satisfying the boundary conditions, will exist only for a particular approach-flow velocity V L, the laminar flame speed. To simplify discussion, Equation (1) has been written in a form applicable only in the case of constant properties. Since this approximation is not good in flames, it is important to state that through a suitable coordinate transformation, realistic property variations can be included without altering any of the results discussed in the present section. In particular, the original analysis 2 of the flame in a shear flow includes variable properties. On the other hand, when the turbulent statistics are considered in Section 3, the constant-property approximation will be needed, since analyses have not been performed to account for density variations. The terms appearing in Equation (1) represent convection, diffusion and reaction, respectively. It is known 3 that laminar flames are composed of two zones, an upstream convective-diffusive zone in which the reaction term is negligible and a thinner downstream reactive-diffusive zone in which convection is negligible in a first approximation. It is evident from Equation (1) that the characteristic thickness of the convective-diffusive zone, i.e., the flame thickness, is 6 = f/v L. Physically, for the heat to diffuse upstream, the molecular diffusivity must equal the product of the flame thickness and the progagation speed. It is true but not evident from Equation (1) that the thickness of the reactive-diffusive zone is 6/0, where 0 = E(T f - T 0 )/R T ~ 10, E being the overall activation energy for the reaction and R the universal gas constant.

104 111-4 Strictly speaking, it is only in an asymptotic sense, 0 -*, that V L and the structure just described exist. The function F(Y) needs a suitable form for the results to follow. Qualitatively, F(Y) must be a nonnegative single-peaked function that goes to zero as Y approaches zero and that becomes negligibly small as Y approaches unity. Furthermore, F(Y) must depend parametrically on the nondimensional activation energy 0, in such a way that the peak becomes increasingly narrow and moves progressively closer to Y = 0, while the magnitude of the function continually decreases for values of Y appreciably greater than zero, as 0 approaches infinity. The Arrhenius rate function, F(Y) = A0 n+1 Y n exp[-0y/(l - ay)], (2) possesses these properties. Here a = (T f T 0 )/T f (0 < a < 1) is the increase in temperature across the flame divided by the adiabatic flame temperature, n > 0 is the order of the reaction, and A is the burning-rate eigenvalue, which is inversely proportional to the square of the flame speed. The factor 0 n+1 has been inserted in Equation (2) for the purpose of producing as the proper asymptotic expansion of A for 0 ->. A = AO + A,0-' + A 2 r 2 + ' ' (3) An analysis of the one-dimensional laminar flame is outlined in Appendix A. It is of the same type as an earlier analysis 4, but the presentation differs in many respects. The development in the Appendix is valid for any positive reaction order, whereas the earlier results were restricted to n = 1. The entire range 0 < a < 1 is covered, whereas the earlier theory was developed for a = 1. Results of the present development must be interpreted either in terms of a thermal theory or in terms of the approximation that the Lewis number is unity; the earlier work explicitly considered the influence of the value of Lewis number. Retention of x as the independent variable, instead of employing 4 the phase plane, helps to clarify the means of proceeding to introduce effects of turbulence. 2.2 Length Scales of Turbulence For discussing the structure of a laminar flame in a turbulent shear flow, it is useful to have at hand a summary of various length scales of turbulence, as provided in Table 1. Although these scales have been defined and discussed thoroughly only for nonreacting turbulent flows 6, there is no reason to question their approximate relevance to reacting flows. The integral scale / is the largest. The Taylor scale X is defined in such a way that the rate of dissipation of turbulent kinetic energy, (v' 2 ) 3/2 //, is of order v v' 2 /X 2. Here v' is the turbulent fluctuation velocity and v' 2 its mean square. The symbol v now denotes the kinematic viscosity; with the possible exception of hydrogen flames, to the accuracy to which turbulent flames can be analyzed it is pointless to_distinguish between this and the diffusivity introduced in Equation (1). To be precise, we define X by (W) 2 = v' 2 /X 2 and deduce approximately X = //,/R, where R = ^/v^l/v is the turbulent Reynolds number. For large R turbulence intensity decreases with decreasing scale and cuts off at the Kolmogorov scale T? = f> 3 /[(v' 2 ) 3/2 //]}" 4 = //R 3/4 = X/R 1/4. The preceding relationships hold only if R ^ 1, which usually is satisfied in practice. For R ^ 1 much of the complexity of turbulence goes away, and all scales become of the same order. For grid turbulence, the condition R 1 defines the "final stage of decay". Although magnitudes of /, X and 17 can differ greatly in different turbulent flows, representative values are listed in Table 1 for orientation; the estimates / = 1 cm and R = 100 were used. Depending on pressure, temperature, size and turbulence intensity (v' 2 ), the flame thickness 6 may bear differing relationships to the turbulent scales. This immediately suggests the existence of a number of different regimes of turbulent flame propagation. A typical situation in practice is r? S> 6 ^ X < I. 2.3 Approach to Analysis of the Influence of Shear on the Flame If K 8, then a turbulent flame cannot be composed of a collection of sheared laminar flames, since there would be many turbulent eddies inside the laminar flame. In this case all fluctuations occur at scales so small that statistics must be introduced before attempting to describe the flame structure. On the other hand, for 6 < I, on the scale of the major turbulent fluctuations it may be possible to treat the flame as a discontinuity. To define the limitation of this view, it is necessary to analyze the structure of a laminar flame in a shear flow. There are two distinct ways in which fluid motions on scales large compared with the flame thickness can modify laminar flame structure. One is by producing flame curvature and the other by producing flame stretch, a time rate of increase of area of an element of flame bounded by fluid particles. The curvature effect, which has been studied in some detail 7, leads to corrections of order 6//. For R > 1 it turns out that the stretch effect is larger than the curvature effect. We shall consider the flame to be locally planar and ignore the curvature effect. The component of the vorticity vector normal to the flame produces no change in flame area. Since modifications in planar flame structure will be seen to arise only through area change, only the component of vorticity parallel to the flame needs to be considered. Adopt a coordinate system in which the flame is normal to the

105 x, y plane, and consider the projections of fluid and flame motions onto the x, y plane. Allow the origin of the x, y system, initially taken to be a point on the flame surface, to move with the projected fluid motion. Rotate the coordinates so that in the absence of the flame the derivative of the y component of velocity with respect to the x coordinate vanishes. Let the derivative of the x component of velocity with respect to the y coordinate, in the absence of the flame, be a. Then the situation illustrated in Figure 1 is achieved. In Figure 1 the flame locally lies along the f axis initially. In a time dt, the shear moves a fluid element on the flame surface through a distance aydt, as illustrated. This increases the local area A of the flame surface by an amount which can be calculated from simple geometrical considerations. We find that II1-5 7 = d/na/dt = axy/(x 2 + y 2 ), (4) where 7 denotes the flame stretch. It has been shown 2 that the one-dimensional, time-dependent flame structure under these conditions of shear is described by the equation 9Y 3Y 3Y 3 2 Y /v?\ 7? +v v = - ) F(Y), (5) at as w ag 2 \i>/ where V L is the normal (unsheared) flame speed, while 7 and v may depend on t. The function F(Y) still is given by Equation (2), and the boundary conditions are the same as those for Equation (1). Equation (5) contains two terms not present in Equation (1). One is the transient term accounting for energy accumulation; this is needed because in some cases a steady-state solution does not exist in the presence of shear, even in the asymptotic sense 0 -*. The other is the shear or stretch term. The dependence of the solution to Equation (5) on 7 has been worked out in detail Results for Influence of Shear on the Flame The character of the solution to Equation (5) depends on the value of the nondimensional flame stretch (f = 76/v L. The flame quantity used for nondimensionalization is simply the transit time 6/v L of a fluid element passing through the flame. Stretch effects are small for lip I < 1, significant for \<p\ ~ 1 and large for \(p\ > 1. Flame stretch is useful for investigating stabilization of laminar flames on burner ports; in this case 7 is the velocity gradient at the wall. We shall not discuss stabilization of laminar flames. In a turbulent flow, 7 is simply the strain rate. Its order of magnitude is \y\~ [(Vv') 2 ] 1/2 ~ y^/x ~ (Ri///)/(//v/R) ~ R 3 ' 2^/ 2. Since 6/v L ~ 5 2 /v, we find that lip I ~ R 3/2 6 2 // 2 ~ (8/r?) 2. Thus, the non-dimensional flame stretch is of the order of magnitude of the square of the ratio of the flame thickness to the Kolmogorov scale. Since the Kolmogorov scale typically is small (see Table 1), it is clear that lip I <J 1 is likely to be encountered in turbulent flames, considerably more likely than the condition 5/1 ^ 1, needed for large effects of flame curvature. The nature of the solution to Equation (5) depends on the sign of ip. Stretch corresponds to \p > 0 and flame compression to >p <. 0. Table 2 summarizes the qualitative behavior that has been derived 2. Some of the results are nearly self-evident. For example, since, according to continuity, positive stretch implies a tendency for material to flow toward the flame from distances far upstream or downstream from it, the shear-produced convection tends to steepen and thin the temperature and composition profiles in the flame for 7 > 0. This should be evident from the sign of the 7 term in Equation (5). On the other hand, profiles are smoothed and thickened by convection in the case of flame compression (7 < 0). Since the convective-diffusive zone of a steady laminar flame consists of a region in which there is a balance between convection and diffusion, it is clear that in this upstream region convective and diffusive fluxes of heat must be in opposite directions for a steady state to exist. Since heat always is conducted in the upstream direction, it must be convected in the downstream direction for there to be a steady state. For 7 > 0 it is, but for 7 < 0 heat is convected upstream where ( y)( ) > v in Equation (5), and therefore no steady-state solution exists for 7 < 0. Thus, the flame must evolve and propagate in an essentially transient manner until it encounters a region in which 7 > 0. Although a flame speed cannot be defined for 7 < 0, nevertheless the flame tends to move more rapidly in this case than it does for 7 > 0. For small negative values, of <p, a steady-state solution and a flame speed can be defined approximately 2. For large positive values of ip, the flame speed is negative, in the sense that the convective velocity in the reactive-diffusive zone is directed from the burnt gas toward the unburnt mixture. This can be seen from Equation (5) by observing that in this case 7^ + v becomes negative at relatively small positive values of ; since convection is a second-order effect in the reactive-diffusive zone, the negative flame speed is entirely consistent with a coherent flame structure. The negative flame speed for large positive values of ip provides a mechanism 2 for annihilation of hot spots of burnt gas in a turbulent flow. Suppose that on opposite sides of a hot spot there are plane flames, each in a region of large positive <p. As time goes on, because of the overwhelming effect of shear as compared with consumption of fresh reactant, these flames will move toward each other and eventually meet and extinguish each other, leaving, for example, a torus of flame.

106 II1-6 In turbulent flows, stretch statistically predominates over compression. Therefore, for lu?l» 1, hot-spot annihilation is expected to occur relatively often, although many rapidly moving, highly transient flames also are to be expected. It is seen that the picture of a turbulent flame, which emerges from analysis of a laminar flame in a shear flow, is rich in content. The results imply that a classical wrinkled laminar flame model could be good if 6 <?C T?, provided that the statistics of describing the continuous wrinkled surface are worked out properly. On the other hand, for the more common case of 5 ^ T?, a statistical approach which does not begin with the idea of there being a collection of laminar flames in shear flows, may prove more fruitful for obtaining a simple and accurate description of various mean properties of turbulent flame structure, including turbulent flame speeds. 3. TURBULENT FLAME SPEEDS AND MACROSTRUCTURE 3.1 General Status of Theory For 5 «: T?, wrinkled laminar flame models appear to be applicable. For 6» /, Damkohler's original idea of viewing turbulence as effectively enhancing transport properties within the flame may be good, at least for the convective-diffusive zone, provided that reaction-zone influences do not dominate the effect. In the range of greatest practical interest, r? < 6 < /, justifiable mechanistic views for calculating turbulent flame speeds do not appear to be available. Some promise is offered by the formal approach of averaging the conservation equations and working with moments 8. Theoretically, such formalism could be applicable irrespective of the validity of particular mechanistic results concerning microstructure. However, if the formalism is to be viewed as deductive rather than guesswork, then each step in the analysis must be justifiable. This requirement can severely limit the range of usefulness of formal statistical methods, although the restrictions which arise in general do not correspond to those obtained from studies of microstructure. For example, it is possible to develop a statistical analysis for low turbulence intensity, with no restriction on the ratio of flame thickness to turbulence scale. Recently work has been completed on the analysis of a flame in grid turbulence. The system is steady and one-dimensional in the mean, while the turbulence is stationary and axisymmetric. The analysis, performed for the limit of low turbulence intensity, i.e., small values of e = v' 2 /v[, and for 0 ->, provides explicitly the mean structure and a formula for the turbulent flame speed V T. Since the work will not be published elsewhere, it is of interest to outline the theory here. 3.2 Conservation of Mean and Fluctuating Temperature For turbulence of low intensity, it is reasonable to expand F(Y) about the local mean value Y, to obtain F(Y) = F(Y) + F (1) (Y)Y' + $F (2) (Y)Y' 2 +,. (6) where the superscripts on F denote derivatives and the prime on Y signifies the deviation from the mean. If the mean of the three-dimensional, time-dependent conservation equation for Y, viz., - = - (7Y 7 ) - F(Y) - S [F(k+0(Y)/(k + l)!]y'k+l, (7) dx dx 2 dx k=l is subtracted from that equation, and use is made of Equation (6), then the equation for conservation of temperature fluctuations is found to be ^ + IX! + u-jh _ -1 (U^Y') + v. (U'Y') - V 2 Y' br 3x dx dx k=l 1)!] (Y*+l - Y' k+1 ). (8) Here x is the nondimensional distance variable defined in Appendix A, T = tv /i> is the corresponding nondimensional time, u' = V'/V L is the nondimensional vector velocity fluctuation, u' is the x component of u', and the y and z coordinates in the gradient operator have been nondimensionalized in the same way as x. Comparison of Equations (1) and (7) reveals the presence of two additional effects in the equation for the mean. One is the streamwise turbulent transport, the first term on the right-hand side, which can be generalized by incorporation of additional terms to include influences of variable density 8. The other is the turbulent modification to the mean reaction rate, the final term in Equation (7), which will be found to be extremely important, thereby raising severe doubts concerning the use of "laminar kinetics", i.e., setting F(Y) = F(Y). The objective is to extract from Equation (7) the mean structure, and especially the turbulent flame speed, which should emerge as an eigenvalue of the equation. In view of the closure problem, it is necessary to consider Equation (8) as well in order to proceed. When the intensity of turbulence is sufficiently low, Equation (8) reduces to the linear equation

107 II 1-7 where the laminar value Y L replaces Y on the right-hand side. The formal restrictions u' <S 1 and Y'/Y «C 1, employed in obtaining Equation (9), typically are reasonably accurate for grid turbulence. Comparison of Equations (8) and (9) reveals that the principal physical phenomena neglected in Equation (9) are the spectral transfers associated with convection and reaction. By estimating the magnitudes of the second and fifth terms in Equation (8), it can be reasoned that unless the flame is thicker by a factor 1/y/e" than the scale / of the energy-containing eddies - an unusual experimental situation - the first of these spectral transfers will be negligible within the flame. The second always is negligible within the diffusion zone but is found to be negligible within the reaction zone only if the velocity fluctuations are in their final stage of decay. Since the process is stationary in r and homogeneous in the spatial coordinates y and z normal to x, may introduce a Fourier-Stieltjes decomposition of Y'/Ve" and u'l^/e, viz., and Y 'fv v 7 t} = /#T Ip^WT+k-jy+k-iz)^!/*(v v \f / \\ (\(\\ (\, y, L, i) ye jev u i /av/^x, K 2, K 3, co; (.1^^ u'(x, y, z, t) = v /^/ e ' ( - wrfk2y+k3z - ) dip(x, k 2, k 3, w). (11) This is not essential, but it does greatly facilitate solving Equation (9). The transform of Equation (9) becomes one where k 2 = k?, + k 2,. Since the thickness 5 of the laminar flame has been selected as the natural unit of length, for large-scale turbulence di// and dip have appreciable contributions only from ranges where k and oj are both small compared with unity. Conversely, for small-scale turbulence di// and dtp will be negligible unless k and co are large. 3.3 Character of the Temperature Fluctuations The central mathematical task is to solve Equation (12) for the x dependence of di//, in the form of an asymptotic expansion for large values of 0. This analysis parallels that of Appendix A and is outlined in Appendix B. The key aspect of the analysis is recognition of the importance of the functional form of the asymptotic solution of Equation (B8) for y 0 -» oo. The functional dependence is distinctive because, through linearization, the first derivative F (1) of the rate function appears in Equation (12), instead of F itself. This turns out to dictate the ordering shown in Equation (B5), to achieve matching. The order of di// (in 0~') is forced to be the same in the reaction zone as it is in the diffusion zone, in contrast to the laminar case, wherein Y L becomes of order 0" 1 in the reaction zone. The lowest-order solution for di// in the reaction zone is given by Equation (BIO), with da 0 expressed in terms of approach-flow quantities by Equation (B 12), and with T defined in Equation (A6). Corresponding diffusionzone solutions are found in Equations (B2) and (Bl 1). The statistics of the temperature fluctuations are recovered from these results by use of Equation (10), since the statistics of da and di// are presumed known from specified upstream conditions. From Equation (B2) it is seen that the probability distribution of Y' changes appreciably across the diffusion zone. However, Equation (BIO) shows that to lowest order the probability distribution of Y' is unmodified within the reaction zone, where each Fourier amplitude merely decreases proportionally to zero. From these results one can obtain any jlesired statistical property of Y'. Of particular interest are profiles of the variance (the mean-square fluctuation, Y' 2 ) the skewness Y' 3 and the streamwise turbulent transport v'y' Composite expansions of these are plotted in Figure 2 for a particular case. Computation of these curves entails definition of autocovariances and their spectra for u' and Y' 0 as well as the cross-covariance spectrum for u' and YO. The principal observation to be made from the curves is that for large-scale turbulence (/ S> 8) there is a great amplification of fluctuations in the convective-diffusive zone. This is traceable ultimately to the GO in the denominator of Equation (B14). Within the flame, the quantities Y 72, Y 73 and u'y 7 turn out to be of order e[l + (//6) 2 ], e 3/2 [l + (//5) 3 ] and e[l + (//5)], respectively. In general in the diffusion zone velocity fluctuations act on the mean temperature gradient to generate temperature fluctuations. In particular, even in the absence of temperature fluctuations in the approach flow, velocity fluctuations produce mean-square temperature fluctuations in the flame. For I ^> 8, approaching temperature fluctuations are amplified so as to increase Y' 2 within the flame, even in the absence of velocity fluctuations. When both velocity and temperature fluctuations are present in the approach flow, then streamwise turbulent transport tends to increase Y' 2 if it is in the downstream direction and to decrease Y 72 if in the upstream direction. Production of Y' 2 in the flame vanishes only if nondimensional velocity and temperature

108 II 1-8 fluctuations in the approach flow are equal in magnitude and have perfect negative correlation. The stationarity requirement that u' f u'dr = 0 may be used to show that unlike Y' 2, the streamwise turbulent transport is J 00 only amplified within the flame; it is not produced when none of it is present in the approach flow. Because of the strong generation of fluctuations within the flame, the perturbation analysis which produces the preceding results turns out to be valid only if 0 2 e[l + (// ) 2 ] <S 1, which also can be written as 0 2 R 2 [1 + (5//) 2 ] "C 1. If this restriction is not satisfied, then Equation (9) is not a suitable approximation to Equation (8), and higher moments in the reaction-rate term in Equation (8) do not become progressively smaller, i.e., the expansion in Equation (6) is not justified. This restriction limits the analysis to the final stage of decay. Turbulence specialists would be inclined to believe that the breakdown stems from neglect of u'y' in moving from Equation (8) to Equation (9), but this is false. Instead, the inconsistency first appears only in the reactivediffusive zone, and physically it means that spectral transfers of temperature fluctuations through chemical reactions become a dominant aspect of the reaction-zone structure. These transfers are larger than the transport effect by the factor 0 2. Although precise estimates are difficult to make, it appears that typically the results cited in this and the following section can be employed if the rms velocity fluctuation is less than roughly two percent. 3.4 Mean Temperature Profile and Turbulent Flame Speed The analysis of the turbulent perturbation to the mean temperature in the reaction zone is summarized in Appendix C. According to Equation (C4), to lowest order this perturbation is proportional to both the chemical reaction rate and the mean-square fluctuation level entering the reaction zone. Since g 0 is non-negative, turbulence always tends to increase the average reactant concentration and decrease the average temperature in the reaction zone. Representative profiles of the mean Y are shown in Figure 2. In using Figure 2. to obtain profiles in the physical coordinate, one must realize that v/v has been used as the characteristic length, so that a flame-speed increase resulting from turbulence produces a contraction of the physical scale. Thus, turbulence steepens the gradient of Y in the upstream part of the flame and makes it shallower in the downstream part. This is in qualitative agreement with an earlier result 8. The expansion of the burning-rate eigenvalue, given in Appendix C, shows that the way in which turbulence first influences the flame speed is by perturbing the profile of mean temperature in the reaction zone. The largest nonvanishing turbulent influence on the flame speed arises from either e07, or e 3/ , whichever is larger. An explicit flame-speed formula can be written, including both of these terms. Since A is inversely proportional to the square of the flame speed, we find from the expansion of A and from Equations (C6) and (C7) that the difference between the turbulent flame speed V T and the laminar flame speed V L is - V = iv L [e0(//5) 2 AI(n, a) + e 3/2 0 3 (//6) 3 BJ(n)]. (13) The quantities A and B are given in Equations (C5) and (C8). The functions J(n) and I(n, a), defined in Equations (C6) and (C7), were calculated at the same time as yi, by augmenting the array in the integration routine to include their integrands as dependent variables. The results are plotted in Figure 3. As can be verified analytically, divergence occurs at n = 1/5, due to turbulence-induced development of a distinguished zone downstream from the reaction zone for n < \/5. Figure 3 should not be used for n near 1/5. The flame-speed formula contains contributions from the variance A and the skewness B of the probability distribution function for the time integral of the sum of the nondimensional velocity and temperature fluctuations in the approach flow, if 8/1 <S 1. In this large-eddy limit, the quantities (//5) 2 A and (//5) 3 B equal the variance and skewness of the probability distribution function for fluctuations in the normalized nondimensional temperature at the downstream edge of the diffusion zone, based on length and time scales derived from the thickness and speed of the laminar flame. With this last interpretation, Equation (13) remains valid whenever 8/1 is small compared with 0. Since A is non-negative and the integral I(n, a) is positive (see Figure 3), the A term in Equation (13) always tends to make the turbulent flame speed exceed the laminar flame speed. The integral J(n) also is positive but since the skewness B may be positive or negative, the B term in Equation (13) tends to produce either an increase or a decrease in flame speed, depending on the sign of B. If Y' 3 leaving the diffusion zone is positive, then the B term increases the flame speed. Therefore concentration-fluctuation histories containing spikes in the direction of high reactant concentration (low temperature) tend to enhance the flame speed. This phenomenon will in fact occur if small pockets of unburnt reactants remain at the downstream side of the diffusion zone. The relative importance of the variance and skewness terms in Equation (13) can be estimated by using Figure 3 if A and B are presumed to be of order unity. Figure 3 shows that for most values of the reaction order n and of the heat-release parameter a, J(n) «I(n, a)/5. Numerical estimates in Equation (13) then show, quite roughly, that at the largest value of e for which the formula is applicable, the two terms will be of the same magnitude for 0 = 20 and the A term will be larger, perhaps by one order of magnitude, for 0 «2. Probably the variance usually will be more important, although the relative importance of the skewness increases with increasing activation energy, and the value of the skewness factor might be larger than A.

109 In Equation (13) it is possible for the B term to be negative and larger in magnitude than the A term. In such cases, the theory predicts that turbulence decreases the flame speed. However, it seems from estimates of the terms in Equation (13) that this situation is unlikely to occur in real flames. Nevertheless, it cannot be ruled out. To contrast the new results with those of earlier work, note that the way in which factors 1/8 and 0 appear in Equation (13) shows that an increase in turbulence intensity, an increase in eddy size, or an increase in activation energy will increase the departure of the flame speed from its laminar value. If the skewness term is small compared with the variance term, then Equation (13) agrees with an earlier result 8 concerning the dependence of flame speed on intensity (V T V L proportional to e ), although other models' usually have given a somewhat weaker dependence on e. Earlier work has not predicted the dependence on /, since the two-zone structure needed to achieve amplification of fluctuations to a level proportional to / was not included. Also, there are no earlier theoretical results concerning the dependence of V T on the activation energy. 3.5 Comparison of Theoretical and Experimental Turbulent Flame Speeds There are many experimental results on the dependence of turbulent flame speeds on turbulence intensity 1, but few for intensities low enough for the preceding theoretical considerations to be applicable. The relationship V T ~ >/e" often is observed for large e. There is little data on the dependence of flame speed on turbulence scale. In early work, some authors have reported that V T is independent of /, while others have reported that V T increases with /'. This may be consistent with the fact that the preceding model predicts an increase with / only if 1/8 ~^> 1 ; otherwise (//6) 2 A and (//6) 3 B are independent of 1/8, and there is predicted to be no dependence on /. Recently, a relatively detailed experimental study of the dependence of V T on e and / has been completed for propane-air flames 9. Results are shown in Figure 4. It is seen that the scale dependence is distinctly different at high and low intensities The strong increase in flame speed with increasing scale at the lower intensities is qualitatively consistent with the prediction of the theory. II LAMINAR DIFFUSION FLAME IN A SHEAR FLOW The type of analysis reviewed in Sections 2.3 and 2.4 also can be completed for diffusion flames. To proceed, it is necessary to understand in advance the structure of a one-dimensional, laminar, diffusion flame in the absence of shear. The processes of diffusion and reaction are basically the same as those discussed for the premixed flame. The differences in structure stem from the boundary conditions, fuel and oxidizer initially being separated for the diffusion flame. Usually both fuel and oxidizer are cold, and the temperature peaks at a thin reaction zone and decreases in the broad diffusion zones which exist on each side of the reaction zone. Unlike the premixed flame, the diffusion flame is not influenced by the rate of the chemical reaction, except in the narrow reaction zone, which often can be approximated well as sheet of negligible thickness. Thus, an expression like Equation (1) describes diffusion-flame structure, but in a first approximation the rate term may be removed and replaced by suitable continuity conditions for coupling functions. More details may be found in References 1 and 3. An important result of the differing boundary conditions for premixed and diffusion flames is that in the strictly one-dimensional case in infinite space the latter possesses no steady-state solution; diffusion flames evolve transiently. The reason is that in the absence of shear or boundaries, convective-diffusive balances cannot be established on both sides of the reaction zone. Thus, for 7 = 0 the diffusion flame is unsteady. For 7 = 0 Equation (5) again describes the flame structure. The reasoning of Section 2.4 now shows that in the presence of positive flame stretch, convective-diffusive balances can be established on each side of the reaction zone in the diffusion flame. In view of the existence of the counterflow diffusion flame, physically this must be true. The situation, summarized in Table 3, bears a close correspondence to that for the premixed flame, given in Table 2. Stretched diffusion flames are steep, thin and steady, compressed ones thick and unsteady. The inflow into a stretched diffusion flame increases the reactant flux to the reaction zone. This in turn decreases the residence time in the reaction zone. It is known 3 that if the ratio of the residence time to the chemical reaction time T C^, i.e., Damkohler's first similarity parameter Dj, becomes too small, then extinction occurs due to insufficient time for heat release. Thus, too much stretch will extinguish a diffusion flame; compression will not. There is a critical Damkohler number, D, ext, defining extinction conditions. In "flame-strength" experiments, 7 is gradually increased until Dj reaches Dj ext. Applying these results to turbulent flows, one sees immediately that, for turbulence scales large compared with diffusive-zone thicknesses for the flame, if the strain rate 7 becomes too large, then local extinctions of sheared, laminar, diffusion flames will occur in the turbulent diffusion-flame brush. For sufficiently small values of 7, a turbulent diffusion flame of large scale can be viewed as a collection of sheared laminar diffusion flames, but for sufficiently large values of 7, it cannot. It will be seen in the following section that if the turbulent flame is an ensemble of laminar diffusion flames, then some progress can be made in theoretical analysis of it. For this reason it is important to know when the

110 condition 7r ch < D^xt will be satisfied. From recently completed work on laminar diffusion-flame extinction, it appears that for typical hydrocarbon-air mixtures, if T ch is based on bulk concentrations and flame temperature, then D^xt ~ 10. The result is that the condition 7T ch < D7{. xt is appreciably less restrictive than the condition \*p\ -C 1, needed for treating the premixed turbulent flame as a collection of laminar flames. This is not contradictory, since the extinction mechanism for the diffusion flame differs from the annihilation mechanism for the premixed flame. Thus, it appears on preliminary theoretical grounds that real turbulent diffusion flames often will be composed of sheared laminar diffusion flames. 5. MACROSTRUCTURE OF TURBULENT DIFFUSION FLAMES 5.1 Use of Coupling Functions Our discussion of the structure of turbulent diffusion flames will be restricted entirely to the case yr^ < D^xt Thus, the turbulent flame is viewed as being composed of laminar flames for which the flame-sheet approximation is valid. The chemical mixture then is described in terms of fuel F, oxidizer O and products P, interpreted to include inerts. As reasonably good approximations, it is assumed that diffusivities D for all species and heat are equal, and (not quite so good, and avoidable, at the expense of more complicated notation, by working with thermal enthalpy) that the specific heat at constant pressure c p, is constant. Attention is restricted to low-speed flows for which kinetic energy is negligible, and radiant energy transfer is assumed to be negligibly small (thereby possibly ruling out sooty fires). Our objective will be a limited one, i.e., to relate turbulent flame structure to the structure of a simpler, nonreacting flow. A frontal attack, ab initio, on the problem of describing the turbulent flame structure would be much too ambitious. The equations for conservation of chemical species and energy become IXl +vvyi = -^ + -V-tpDVYj), i = F,0,P,T,- (14) at p P where Y ; denotes the mass fraction of species i for i = F, O, P, and Y T denotes c p T/Q, in which T is temperature and Q is the heat released in the reaction per unit mass of fuel consumed. Other symbols appearing in Equation (14) are the density p, velocity v, mass rate of consumption of fuel per unit volume w, and stoichiometric coefficients I>J(J> T = 1). We are interested in turbulent flows, described by Equation (14), having two different types of nonfluctuating boundary streams or initial values, one (subscript 1) containing fuel but no oxidizer, and the other (subscript 2) containing oxidizer but no fuel. A simple example would be the twodimensional turbulent mixing layer between F and O. The mass fraction c of an inert in a reacting or nonreacting flow obeys the equation -^ + v. V C = V (pdvc). (15) at p Suppose that we set up a nonreacting flow in the same geometry as the reacting flow, and suppose that in the boundary streams c, = 1 and c 2 = 0. Define a general coupling function as v } (16) Then it is clear that every C- t -. satisfies the same equation and boundary conditions as c. We thus deduce that everywhere in the turbulent flow Cjj = c. This is a very strong statement, It applies at every point and at every instant of time throughout the turbulent flow. It is easy to make the statement invalid, e.g., by introducing turbulent velocity fluctuations into one of the boundary streams. Yet, the statement indeed is applicable for many of the turbulent flames of practical interest. Elements of the idea of using coupling functions in this manner for turbulent flows are traceable to the work of Hawthorne, Weddell and Hottel 10. The concept was developed extensively by Toor 11. In the flame-sheet approximation, the coupling functions determine all Y, separately. Consider C FO. Since Y F = Y 0 = 0 at the flame, the location of the flame sheet is defined by C FO = M, where u = Y 02 /[0> 0 /i> F )Y F1 + Y 02 ]. The function C FO> measures Y F if C FO >M (i.e., where Y o =0) and Y o for C FO < M (where Y F = 0 ). Other coupling functions may be used for obtaining the other Yj. In general, we have Yj =, (17) A i2 +B i2 c, c<n where the A's and B's are constants easily evaluated from stoichiometry and boundary conditions.

111 Since Equation (17) holds for every realization of the flow, it is clear that all statistical properties of the Yj can be obtained from the corresponding statistical properties of c. For example, if at a given position and time the probability that c < x is called f(x), i.e., P(c < x) = f(x), then and n*f ^y; = PCV <i \i\ r(. Y o ^ y> = r i IYFI + *2i Y 2 + /' V "o/ f i r/ Y 02 - y ( Y *O2 ^+-^-Y Y F1 S, *F P(Y T < y) = f ( ^ ) f ^Y T1 - YT, - 3- Y v f Two terms arise in this last expression because of the nonmonoticity of the temperature profile, as illustrated in Figure 5. From these probability distributions, instantaneous local mean values can be calculated; for example, Y F = f y=0 F1 ydp(y F <y). Calculation of means in this manner for various position enables one to plot mean concentration and temperature profiles. Of course the whole procedure requires knowledge of the statistics of the inert field c. While such knowledge would be very difficult to develop purely theoretically, experimental measurement of the statistics of c may be much easier than measurement of the statistics of Yj. Thus, the advantage of the development is that it enables properties of all Yj to be obtained from measurements of statistics of only c. For example, the fuel may be seeded with an inert whose concentration is detected by a suitable instrument in the turbulent portion of the flow. It should be clear that one cannot go directly from c" to Y F, for example. For the procedure to be useful, the minimum.amount of statistical information needed for c is the probability distribution function. Moreover, strictly speaking this information must be obtained in a flow with the same density and velocity fluctuations as the flame, e.g., in the flame itself. This is evident from the fact that p and v appear in Equations (14) and (15). No rigorous justification can be provided for applying the approach to calculate hot-flow quantities from cold-flow experiments, since the density changes associated with the heat release at the contorted flame surfaces can modify the fluctuating velocity field locally, thereby removing the correspondence between c and Cj:. This is a rather stringent limitation. Since the magnitude of the local influence of the flame on the turbulence is not known well at present, possibly the correspondence could provide good results for hot-flow Yj's by use of cold-flow data for c. Since cold-flow experiments are so much easier to perform, it seems reasonable to try. 5.2 Production of NO in Turbulent Diffusion Flames There are problems for which the procedure just described can lead directly to practical results, and others for which ambitious work remains to be done even after use of coupling functions. As an example of a problem of the first type, consider NO production. It is known that there exist cases in which NO production is kinetically controlled by the Zeldovich mechanism. In its simplest form, this mechanism involves equilibrium dissociation of oxygen, O 2 ^ 2O, with known equilibrium constant K(T), and participation of the 0 atoms so produced in the rate-controlling step O + N 2 -» NO + N, with known rate constant k(t), followed by the fast step N + O 2 -* NO + O. According to this mechanism, the net production rate of NO (moles per unit volume per second) is w = 2[N 2 ] [O]k = 2[N 2 ] % /TO 2 lkk. In this last expression; [N 2 ] will be approximately constant in a turbulent fuel-air diffusion flame, but [O 2 ] will vary, and so will the product Kk. The concentration [O 2 ] will be approximately proportional to Y o of the previous section, while Kk is proportional to e~ E / R T, the effective activation energy obtained from Kk being E = kcal/mole. With the symbols introduced in the previous section, these results may be expressed by the proportionality exp[-(ec p /R Q)/Y T ].

112 In this simplest version of the kinetics, there are no O atoms on the fuel side of the flame, because of the absence of O 2 atom's there according to the flame-sheet approximation. Therefore co is nonzero only on the oxidizer side. On this side, local instantaneous application of equality of coupling functions implies that Y T is related linearly to Y o (see Equation (17) for c < p. with i = O and i = T ). If Y Tf is the nondimensional flame temperature, then we have Y o = const. (Y Tf - Y T ). Define y = Y Tf - Y T, and the expression for co can be written as where B is a constant of proportionality. LVR 0 Q/\Y Tf Now, the coefficient inside the exponential in Equation (18) is large. Therefore it is a very good approximation to write Y-' f - (Y Tf - y)" 1 = -y/y 2 f, and with 0 = (E/R T f )(Q/c p T f ), we obtain Equation (19) holds for c < p. (y > 0), while co = 0 for c > p.. Since Equation (17) implies that y = A(ju - c) where A is a simply determined constant, we see that Equation (19) allows the local, instantaneous, production rate to be related directly to the local, instantaneous, mass fraction of the inert. The ensemble average of the production rate is the quantity of practical interest. It is clear that this can be computed if the probability distribution function, P(c < x) = f(x), is known. The fact that 0A is large greatly simplifies the calculation, since this means that the production rate is very strongly peaked near the flame sheet, much more strongly peaked than f(x) could be. Therefore the average is (19) u = fcodp = BVA r v /M=Te-< 3A (' J - x ) + '"f'(x)dx» n -*dz = (V7T/2)(B/A)0-3/2 f'(m), (20) o where the prime on f denotes its derivative. Equation (20) shows that aside from previously known quantities, the NO production rate depends only on the value of the density function f (x), for the inert, evaluated at x = p., the value of c corresponding to the flame sheet. This calculation has yielded the local mean production rate. To obtain the total production rate, co can be integrated over the total volume of the turbulent flame. We see that the information needed for use of the method is the density function for the inert at a number of positions within the turbulent flame. In fact, not even the complete density function is required, since only its value at c = M appears. The approach given here was suggested by Lilian 5. It bears very little resemblance to currently fashionable techniques of numerically integrating complicated sets of differential conservation equations for means, containing laminar chemical kinetics. The present approach is much simpler and yet will be much more accurate if 5.3 Volumetric Heat Release Rates in Turbulent Diffusion Flames An important quantity of practical interest is the average rate of heat release per unit volume, as a function of position within a turbulent diffusion flame. When exothermic reactions occur only at flame sheets, this quantity is related to average rates of diffusion of reactants into the sheets. Since a flame sheet is a surface of constant c, its normal will lie in the direction of Vc. Therefore the flux into the flame, hence the rate of heat release per unit flame area, will be proportional to pdlvcl, evaluated at c = p.. The distribution function for this quantity can be obtained from the joint distribution function for c and IVcl, viz., P(c < x, IVcl < y) = f(x, y). The conditional distribution function for IVcl is simply f x (M, y)/f'(r«) Therefore, if pd is evaluated at the flame f temperature Tf, the average rate of heat release per unit flame area is proportional to pd J yf xy (M, y)dy/f'(m) Subscripts on f denote partial derivatives. From this it is seen that knowledge of the joint density function for c and IVcl enables one to calculate the local average value of the rate of heat release per unit area of the flame sheet. Measurement of this joint density is significantly more difficult than measurement of f(x). Of course, the joint density is needed only at c = M, and therefore a conditioned density would serve as well. Although it could be simpler computationally to obtain in the conditioned mean f yf v (ylc = ju)dy directly, instrumentation for sensing simultaneously c and IVcl is J o y essential.

113 Even after finding the conditioned mean, one has only the heat release rate per unit flame area. The average heat release rate per unit volume can be obtained from this only if there is independence between the rate per unit area and the area per unit volume. Under this last restriction one may multiply the average rate per unit area by the average area per unit volume to obtain the average rate per unit volume. Calculation of the flame area per unit volume is a zero-crossing problem of a kind that has been solved only for Gaussian processes. Unfortunately, the processes which we are considering here certainly are not Gaussian; for example, not even f'(x) is Gaussian since necessarily f'(x) = 0 for x < 0 or for x > 1. Recent measurements in nonreacting mixing layers 12 demonstrate that f'(x) is not Gaussian. Therefore it is seen that the problem of obtaining volumetric heat release rates in turbulent diffusion flames is difficult. Coupling functions directly provide answers to certain practical problems but not to others CONCLUDING REMARK The discussion in the present paper has been devoted entirely to questions concerning the effects of turbulence on flames. The methods considered do not provide information on the effects of the flame on the turbulence. It is observed 13 that premixed flames tend to intensify turbulence and diffusion flames to lessen the intensity. Theoretical studies directed toward explaining these effects would be of interest. ACKNOWLEDGEMENT Portions of this work have been supported by the Office of Naval Research Contract N (Subcontract No ) as part of Project SQUID and by the Air Force Office of Scientific Research, Office of Aerospace Research, US Air Force under Grant AFOSR REFERENCES 1. Williams, F.A. 2. Klimov, A.M. 3. Williams, F.A. 4. Bush, W.B. Fendell, F.E. 5. Linan, A. 6. Batchelor, G.K. 7. Markstein, G.H. 8. Williams, F.A. 9. Ballal, D.R. Lefebvre, A.H. 10. Hawthorne, W.R. Weddell, D.S. Hottel, H.C. 11. Toor, H.L. 12. Stanford, R.A. Libby, P.A. 13. Gunther, R. Combustion Theory, Addison-Wesley Publ. Co., Reading, Mass., Laminar Flame in a Turbulent Flow, Zhur. Prikl. Mekh. i Tekhn. Fiz., Vol.3, pp.49-58, Theory of Combustion in Laminar Flows, Annual Reviews of Fluid Mechanics, Vol.3, pp , Asymptotic Analysis of Laminar Flame Propagation for General Lewis Numbers, Combustion Science and Technology, Vol.1, pp , 1970) Personal communication. Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge, Non-Steady Flame Propagation, Macmillan Publ. Co., New York, An Approach to Turbulent Flame Theory, Journal of Fluid Mechanics, Vol.40, pp , Turbulence Effects on Enclosed Flames, Fourth International Colloquium on Gas Dynamics of Explosions and Reactive Systems, San Diego, Calif., July 10-13, 1973; to appear in Acta Astronautica. Mixing and Combustion in Turbulent Gas Jets, Third Symposium on Combustion, Flame and Explosion Phenomena, Williams and Wilkins Co., Baltimore, Md., pp , Mass Transfer in Dilute Turbulent and Nonturbulent Systems with Rapid Irreversible Reactions and Equal Diffusivities, A.I.Ch.E. Journal, Vol.8, pp.70-78, Further Applications of Hot-Wire Anemometry to Turbulence Measurements in Helium-Air Mixtures, to appear in Physics of Fluids. Measurements of Flame Turbulence : Aims, Methods and Results, Journal of the Institute of Fuel, Vol.43, pp , 1970.

114 APPENDIXA ANALYSIS OF THE ONE-DIMENSIONAL LAMINAR FLAME From Equations ( 1 ) and (2) it may be seen that the reaction rate becomes exponentially small in 0 as 0 -> when Y is larger in order of magnitude than 0" 1. In this upstream convective-diffusive zone, the solution to Equation (1) to all algebraic orders is simply Y L = l-e x, - <x<0, (Al) if translational invariance is employed to place the reaction zone at x = 0. Here the subscript L emphasizes that we are considering the laminar case, and the nondimensional distance, defined as x = v/v, reduces to x = /6. In the reaction zone, the stretched variables i? = 0x and y L = 0Y L are used. The expansion y L = y 0 + y!0-1 + y 2 0~ then reduces Equation (1) to dr? 2 d7? 2 _ dr? J ) (A2) for the rate function given in Equation (2), with A given by Equation (3). Here f(y 0 ) = yge~y, and primes on f denote derivatives with respect to y 0. A logical boundary condition to apply to Equation (A2) is y L ~ 0 as rj -» 0, and the matching condition can be shown from Equation (Al) to be y L ~0S (-l) k+1 (-T)/0) k /k! as T? ->-«.. k=l (A3) To lowest order, a first integral of Equation (A2), satisfying the condition dy 0 /di? ~ 0 as TJ -», is (A4) The condition dy 0 /di7 ~ 1 as 17 -» «>, implied by Equation (A3), then yields A 0 = (2n!)! for the lowest approximation to the burning-rate eigenvalue. When Equation (A4) is integrated again, the integration satisfies y 0 ~ 0 as 17 -> o automatically for n > 1, but for 0 < n < 1, y 0 reaches zero with zero slope at a finite value of 77, forcing one to presume that y 0 is identically zero for all larger values of r?. To satisfy Equation (A3) with the inverse solution,y 0 r.y "1-1/2... r, = - J 2A 0 / f(x)dx dy, (AS) the constant a must be selected to obey a = J (P" 1 - l)dy, where 3 r pvo "1 1/2 T = -dy 0 /dt? = J f(y)dy/n! (A6) is the square root of a normalized incomplete gamma function of arguments n and y 0. For n = 1, T = [1 (1 + y 0 )e~ y ] 1/2 and 5 a = An alternative to calculating a in this way is to select a arbitrarily, to insert a constant before e x in Equation (Al), and to determine this constant through matching to the modified version of Equation (A3), thereby effecting a small translation in the position of the reaction zone. Although the same kind of thing can be done in higher order, instead we treat y 0 as the independent variable, express Equation (A3) in terms of y 0, and thereby remove the necessity of calculating a or its higher-order counterparts. The first correction term y, turns out to be needed for the turbulent-flame-speed calculation reviewed in Section 3. From Equations (A2), (AS) and (A6) it can be shown that the equation for y, may be written as Proceeding roughly as outlined above, we obtain -T-fc-^} = Ao-r-friO + A.f-AoaySf-r. (A7) dy 0 V dy 0 / dy 0 «v r" "I y, = -r r" (" + ncn + 2) _ + lpa + r-3 d (A8) JK L L9 J

115 where b is a constant of integration, F, denotes the integral of F with respect to y 0 from zero to y 0, F 2 denotes the value obtained for F if n is replaced by n + 2, the next correction to the burning-rate eigenvalue.00 is A, = 2A 0 [a(n + l)(n + 2)/2 I], and I = I (1 F)dy 0. Since the integral I, which depends only on 'o n, is used in calculation of the turbulent flame, it has been evaluated numerically and is given in Table 4. The table can be used to obtain the laminar flame speed to second order in 0" 1 for a wide range of reaction orders. The entry for n = 1 has been obtained previously 4 ' 5. A necessary boundary condition for y,, stemming from Equation (A3), is that the constant vanish in the asymptotic expression yl/2 + const, for y[ as y 0 -». There are combinations of n and a such that this condition can be used to determine a positive value for b in Equation (A8). For other values of n and a it is found that b is zero, and in these cases the boundary condition effectively selects one from an infinite number of solutions with b = 0. So that all cases may be treated in a uniform manner, and to avoid evaluating the integral for F,, we have employed a subroutine for integration of a system of ordinary differential equations, to integrate Equation (A7) numerically, starting from a large value of y 0 (10 or 30) with the correct asymptotic solution. Two typical solutions for y, are plotted, along with rj and dy 0 /d77, as functions of y 0 in Figure 6. These curves can be used to construct the second-order approximation to the structure of the laminar flame. Expansion of Equation (A8) about y 0 = 0 reveals that y t approaches zero as y only if n < 3. For n > 3, the reaction rate is too low at small values of y 0 to consume all of the reactant in the flame zone. An amount of reactant Y L of order 0~ 2 leaks through the flame and is consumed in a thick downstream zone, where x/0 n ~ 3 is of order unity and where only convection and chemical reaction occur to lowest order. The solution for y, in this downstream zone is y, = [(n - l)a 0 x/0 n const.]-ian-i), (A9) in which the constant must be determined through suitable matching to Equation (A8) for small values of y 0. The presence of the downstream zone does not alter the burning-rate eigenvalue.

116 APPENDIX B ANALYSIS OF THE LOW-INTENSITY TURBULENT FLAME In the diffusion zone the reaction rate is negligible, and Equation (12) becomes d d 2 (k 2 + ico)(d<//) + (di//) (d*) = e x (dip), (Bl) dx dx 2 where use has been made of Equation (Al) for Y L. For simplicity we write the solution to Equation (Bl) only for the case in which dip is constant across the flame; at small e this is always true for the initial stage of decay, but it is true in the final stage only if 8/1 <C 1. The general solution to Equation (Bl) becomes where di// = dip(k 2 + icor! e x + dc e x ' x + da e x ' x, (B2) X,, 2 = iu ± H + 4(k 2 -f ico)] 1/2 }. (B3) The transforms dc and da are independent of x. Since Re{X,}>0 and Re{X 2 } < 0, in Equation (B2) dc remains arbitrary and must be determined from matching conditions, while the statistics of da are determined by the upstream statistics of the fluctuation field. The expansion dc = dc 0 + dc,0"' + dc 2 0~ may be introduced for the purpose of matching with the reaction zone. If the stretched variables defined in Appendix A are employed in Equation (12), then the equation d d 2 dy, (k 2 + ico) (d!//) + (d*) (d>//) = - - (d*) - drj drj drj -0 2 A(d4/)[n-y L (l ay L r 2 ] y n- 1 exp[-y L /(l -0-1 ay L )] (B4) is obtained for the reaction zone, where Equation (2) has been used for the evaluation of F (1). Presuming that di/> possesses the expansion di// = d^o + d^- 1 + di we obtain from Equation (B4) the sequence of equations H d (di//. d yk-2,,,, r> i T CRA-> k _j) + (k 2 + ico)(di// k _ 2 ) + (dip), k = 0, 1, 2 (.Bb) d?7 dr? in which di// k and y k should be taken to vanish identically for k = 1 or 2. Here dr k is a linear function of (di// 0, di//,,..., dv> k _!, y k, A,/A 0, A 2 /A 0,..., A k /A 0 ) which depends algebraically on (y 0, y,,..., y k _!, n, a). Clearly dr 0 = 0, and fa, /n-1 n+l-y 0 \ n y 0 ~ dr, = + )y, ay 2 di// 0 - (B7) L A o \ yo n-y 0 / n-y 0 J Suitable boundary conditions for Equation (B6) are obtained from the requirements that di/> ~ 0 as TJ -> and that di// match with Equation (B2) as T? -». A first integral of Equation (B6), analogous to Equation (A4), can always be found. Thus, p y r d (d) = A(dtf/)f+ dy 0 J 0 L A 0 (dr k )f'-r dy o ico)(di// k _ 2 )-(d p)f^:1 1 dy 0, k = 0, 1, 2,... (B8) dyo J

117 The condition for matching with Equation (B2) can be expressed as S (d^k)0~ k ~ da (-l)kxkyk0-k/k! + d^(k 2 + ico)" 1 Z (-I)kyk0-k/ k i + k=o k=o k=o _ as y 0^. (B9) k=0 " ' JLk=0 The asymptotic solution to Equation (B8), for y 0 ->, is d< / ~ da 0, di/>, ~ da, + (db,)y 0, di// 2 ~ da 2 + (db 2 )y 0 + (dc 2 )y 2,/2, etc., where da k, db k, dc k,... are "constants", independent of y 0. At each stage all of these constants are related to earlier ones, through Equation (B8) itself, except for da k, which are determined along with dc k, through Equation (B9). It turns out that in general to obtain a solution completely through order k, it is necessary to consider matching through order k -I- 1 and to use certain properties of the solution to the inner equation of order k + 1. Since the details become algebraically complicated, we merely state results. Through Equation (B8), without considering matching, one can show that in the reaction zone d!// 0 = FdA 0, (BIO) and that db, = da 0 and dc 2 = (1 + k 2 + ico)dao dip. Consideration of Equation (B9) as well then produces dc 0 = -(X,/X 2 )da, da 0 = dip(k 2 + ico)" 1 - da(x, - X 2 )/X 2, (Bll) (B12) and in higher order da k = dc k. From Equation (D8) one can show that for the reaction zone di//, = da 0 S(r?) + da,f, (B13) where 8(77) is the particular solution to the non-homogeneous equation with the property that the constant vanishes in the asymptotic expression S ~ [dip(k 2 + ico)" 1 + X 2 da (X 2 /X 2 )da]t7 + const, as T? -*. We note that for small-scale turbulence in the final stage of decay, these results remain valid provided that in Equation (B12) and in the asymptotic expression for S, dip is interpreted as the value of the transform at the downstream edge of the diffusion zone. In the limit of large-scale turbulence, the composite expansion for d\// 0, constructed from Equations (B2), (BIO), (Bll) and (B12), is especially simple. The approximations (k 2 + ico)" 1» (ico)" 1, X, «1, X 2 «0 in Equation (B2) and X 2 «ico in Equations (Bll) and (B12) are employed. To lowest order, the composite expansion becomes A\l> = da + (e x + T - l)(d.p + da)/(ico), (B14) where it is understood that e x is to be replaced by unity for x > 0. Note that the second term in Equation (B14), which describes the profile of dip within the flame for a particular realization, is large compared with the first by order 1/8. In addition, from Equations (10) and (11) it follows that the second term is the product of a deterministic function of x with the transform of / (u' + Yo)dr, where Yj, denotes the value of Y' in the 00 approach flow. Therefore for large-scale turbulence, the statistics of temperature fluctuations in the reaction zone are expressible directly in terms of statistics of the time integral of velocity and temperature fluctuations in the approach flow.

118 APPENDIX C CALCULATION OF THE TURBULENT FLAME SPEED To the lowest two orders, Y = Y L in the diffusion zone. To obtain suitable equations for the turbulent perturbation to the mean temperature in the reaction zone, we transform from x to the variable T? and introduce the expansions Y' 2 = e(g 'G, +...), Y' 3 = e 3/2 (H ), Y = 0- ] y 0 + 0" 2 y, e0g 0 + eg, e a/2 0 2 h and A = A 0 + 0" 1 A, +... e e07, e 3/ By substituting these expansions into Equation (7) and collecting terms of like powers of e and 0, one obtains d 2 g 0 A 0 f'g 0 + 7of + 4A 0 f"g 0, (Cl) Arj drj 2 = A 0 fg, + T, f + - dr? + A,fg 0 + A ogo [f"y,- a (y 2 0'] + ^f'g, + i A, f "Go + i A 0 G 0 [f'y, - «(y 0 f)"], (C2) and -^ = A 0 fh f+ D 0 f'"h 0. (C3) dr? 2 Considerations of matching reveal that the boundary conditions for Equations (Cl) through (C3) are that g 0, g, and h 0 must approach zero as 77 -> ±. The method of solving Equations (Cl) through (C3) parallels that for Equation (B6). A first integral of Equation (Cl) can be found, and the boundary conditions for 17 -> ± can be applied to this integral to yield Use, in this expression, of the result G 0 = F 2 J dajda 0, obtained from Equations (10) and (BIO), reveals through integration by parts that 7 0 = 0 ; to lowest order the flame speed is unaffected by turbulence. With 7 0 = 0 it N can be seen that g 0 = ia 0 fjda 0 *da 0. <C4) A rather more complicated calculation, performed on Equation (C2), and using the result G, = rs/da*da 0 + T 2 /dajda,, obtained from Equations (10, (B5), (BIO) and (B13), reveals that 7, = f which is not zero, If we define a nondimensional variance A, of order unity for 8/1 <C 1, by A = (6//) 2 (u' + Y;)dr, (C5). then the 7, expression becomes 7, = (//6) 2 AA 2 f f"f 2 dy 0 = -(//6) 2 AA 0 I(n,a). (C6) J 0 dy 0 The value of 5 0, obtained from Equation (C3) by integrating from y 0 = 0 to y 0 =, is 6 0 = -i(//6) 3 BA 2 J f"'r 3 dyo = -(//6) 3 BA 0 J(n), (C7)

119 where the approach-flow skewness factor B, of order unity for 8/1 ^C 1, is defined by B = (6//) 3 (u' + Yi)dr. (C8)

120 TABLE 1 Turbulent Scales Symbol Name Meaning T z 7 aftve Reiati nshi p Approximate I Integral Scale Size of Energy- Containing Eddies 10 mm X Taylor Scale Length Characteristic of Dissipation Rate 1 mm /A/ R n Kolmogorov Scale Size of Smallest Fluctuation, at ' Viscous Cutoff 0.3 mm X/R 1/4 TABLE 2 Effects of Shear on Premixed Flames Stretch (f > 0) Compression Op < 0) steepening, thinning convection in downstream direction far upstream always a steady-state solution heat release (reactant consumption) per unit flame area less than normal heat release per unit initial flame area increases with time negative "burning velocity" (at heatrelease plane) for large lip I extinction of a hot spot by a simple mechanism for large lip I smoothing, thickening convection in upstream direction far upstream no steady-state solution heat release (reactant consumption) per unit flame area greater than normal heat release per unit initial flame area decreases with time approximate steady-state solution (with positive burning velocity) for small lip I corresponds to spherically expanding flame for small lip I

121 TABLE 3 Effects of Shear on Diffusion Flames Stretch (7 > 0) Compression (7 < 0) steepening, thinning always a steady-state solution increased reactant flux extinction at large 7T c j, = DJ" 1 smoothing, thickening no steady-state solution decreased reactant flux no extinction TABLE 4 Laminar Flame-Speed Integral n I n I

122 Unburnl Fig. 1 Coordinate system for investigation of effects of shear on premixed laminar flames Diffusion Zone v'y Turbulent Laminar Fig.2_ Profiles of mean reactant concentration Y, streamwise turbulent transport v'y', mean-square fluctuation Y' 2 and mean-cube fluctuation Y' 3, within the flame, as obtained from a composite expansion, for n = 1 and 0 = 5, in the limit of large-scale turbulence. For Y are shown the diffusion-zone solution, corresponding to 0 ->, the composite laminar solution, and the solution for the turbulent flame in the case ejdajda 0 = 10"'

123 Kn.a) 5 J(n) n Fig.3 The integrals I(n, a) and J(n), that appear in the formula for the turbulent flame speed. The integrals are plotted as functions of the reaction order n for various values of a, the nondimensional temperature rise across the flame 'L 8- Fig.4 Experimentally measured 9 influence of turbulence scale on the ratio of turbulent to laminar flame speeds, for stoichiometric propane-air mixtures in grid-produced turbulence

124 Fig.5 Profiles of temperature and concentration as functions of normalized inert concentration for turbulent diffusion flames 4V 2- Fig.6 Reaction-zone profiles for the laminar flame. The nondimensional distance 77 and the lowest-order concentration gradient F = -dy 0 /d77, as functions of the lowest-order normalized reactant concentration y 0, for a reaction order n = 1. Also, the first-order correction y, for reactant concentration, as a function of y 0 for n = 1, a = 1 and for n = 1, a = 0

125 DISCUSSION Prof. Classman: I am concerned with this and other papers which are dealing with chemical kinetics in particular about calculation of NO. From some recent work the role played by what people call "prompt" NO and by radicals appears of great importance; I don't see this anywhere in your calculations. It has been proved by German work that when one deals with fuel-rich flames these materials contribute at least half of the NO production. Author's reply: What you say is certainly true for premixed flames. For diffusion flames between fuel and air diluted with nitrogen, the lower flame temperature will reduce dissociation and radical concentrations. This seems to me to be a condition under which the Zeldovich mechanism may be quite good. Prof. Classman: I think that the Zeldovich mechanism may be even poorer under these conditions than for a premixed flame. In a diffusion flame there is always a fuel-rich side, and it is under fuel-rich conditions that there are difficulties connected with these compounds. I am not so sure that you can avoid the problem with a diffusion flame, since there is probably more "prompt" NO in a diffusion flame as compared with a premixed flame. Author's reply: But the decrease in temperature on progressing into the fuel-side (diffusion zone reduces production rates. I agree with you that we have not proven that for diffusion flames the Zeldovich mechanism is "the" mechanism. With presently available information, no other mechanism is sufficiently well defined to be used. Once a new mechanism is defined, it can be used in place of the Zeldovich mechanism in the turbulent diffusion-flame analysis described in the paper. I agree that further work directed toward defining NO production mechanisms in diffusion flames is needed. Prof.Lefebvre: You talked about the effect of turbulence on chemical reaction rates and I would like to know what evidence there is to support this hypothesis. I can appreciate that turbulence can increase chemical reaction rates by increasing transport phenomena, but I can also see that turbulence might have an inverse effect, say by dispersing some of the active chemical species into the cold gas. If we talk about flames of low turbulence intensity, which I believe is the subject of your paper, then what is wrong with a wrinkled laminar flame? If one looks at a flame in laminar flow one sees a smooth, unruffled surface. If turbulence is then introduced into the flow the flame surface remains essentially smooth but wrinkled by the effects of turbulence. In other words the increase in burning velocity or flame propagation is explained quite acceptably on the basis of an increase in flame surface area, so why do we need to invent new ideas such as the effect of turbulence on chemical reaction rates, which I agree could be important in high turbulence fields. Author's reply: That is an interesting point. The analysis I was referring to could be used either for large scale or for small scale; if the turbulence scale is smaller than or comparable with the laminar flame thickness, then the wrinkled laminar flame picture may be incorrect. However, I agree with you that at larger scale and at low enough intensities there may just be the wrinkling. The analysis neither excludes this possibility nor makes use of it. Yet a strong increase in flame speed with turbulence scale is predicted, in qualitative agreement with your experimental observations. As far as I know, such a prediction has not been obtained from wrinkled laminar flame models. So perhaps, even if it is a wrinkled laminar flame, you can get a useful result by by-passing that type of treatment. It would be interesting to introduce into the theory the assumption that the flame is composed of wrinkled laminar flames and to attempt to derive predictions of statistics of wrinkling geometries. This might lead to improved simplified wrinkled-flame models as well as to better understanding. Dr Barrere: I think it is important to consider that for diffusion flames the chemical reaction rates can decrease due to the effect of turbulence. For example if you have two species A and B it may be that, due to the turbulence, these two species do not react; in that case you have a reduction of the reaction rate. The situation is different with the premixed flames for which the turbulence tends to increase the chemical reaction rates; in the diffusion flames you might have just the opposite effects. Author's reply: In principle this is possible. However, so long as local strain rates are not so high as to cause extinction, the statistical predominance of stretch over compression means that on the average, turbulence increases overall reaction rates in diffusion flames.

126

127 112 KINETIC ENERGY OF TURBULENCE IN FLAMES by K.N.C.Bray University of Southampton United Kingdom

128 112 RESUME On emploie les equations exactes du jet reactif de turbulence, associe"es 3 une analyse des ordres de grandeur afin d'en de'duire la forme approximative que prendrait liquation d'equilibre de l'e"nergie cinetique de la turbulence pour des flammes pre'me'langees a deux dimensions, pour un nombre de Mach faible et e'leve' de Reynolds. On introduit ensuite des hypotheses plausibles de la fermeture afin d'obtenir une Equation qui se reduit a une forme connue de liquation de P^nergie cine'tique de la turbulence, dans le cas d'un jet nonr actif de densit^ constante. A cela s'ajoutent des Elements relatifs au de"gagement de chaleur et au defacement de masse, qui jouent un role important dans les flammes turbulentes. Les effects de turbulence observes au cours d'expe'riences faites sur des flammes turbulentes de configurations diverses sont lvalue's en fonction de cette Equation.

129 II2-1 KINETIC ENERGY OF TURBULENCE IN FLAMES K.N.C.Bray University of Southampton United Kingdom SUMMARY The exact equations of turbulent, reacting flow are used, together with an order of magnitude analysis, to derive an approximate form of the turbulence kinetic energy balance equation for premixed, two-dimensional, turbulent flames at low Mach number and high Reynolds number. Plausible closure hypotheses are then introduced, in order to obtain an equation which reduces to a familiar form of the turbulence kinetic energy equation, in the case of non-reacting flow of constant density. Additional terms, related to heat release and mass transport, become important in turbulent flames. Experimentally observed effects of turbulence on a variety of turbulent flame configurations are discussed in terms of this equation. LIST OF SYMBOLS a speed of sound a, constant a, constant a 2 a 3 a 4 q Cp DJ f f a 0 constant constant constant mass fraction of species i frozen specific heat at constant pressure for mixture molecular diffusion coefficient of species i a flow property frictional stress component F 0 Froude number, Uo 2 /g 0 L 0 g a flow property ga component of acceleration due to gravity in x a -direction h specific enthalpy <5/j heat of formation of species i per unit mass Jj tt k Kp component a of molecular diffusion mass flux of species i thermal conductivity constant / e reference value of turbulence energy length scale / 0 reference value of turbulent chemical reaction length scale / turbulence length scale L 0 characteristic length of flowfield in x-direction

130 II2-2 M 0 reference Mach number, 0 0 /a 0 n number of species p pressure q turbulence kinetic energy, iv^v^ q a R component a of molecular heat flux universal gas constant R L mean flow Reynolds number, P 0 Q 0 L 0 R T turbulence Reynolds number, Sf time-average flame speed t time T temperature u x-component of velocity v y-component of velocity v a w Wj W Wj x x a y z x a -component of velocity z-component of velocity mass rate of production of species i molecular weight of mixture molecular weight of species i distance in flow direction, in two-dimensional flow distance in direction a distance in direction perpendicular to x in two-dimensional flow distance in direction perpendicular to x and y 7 ratio of frozen specific heats for mixture 7i ratio of frozen specific heats for species i 5 0 characteristic width of shear layer e parameter 6 0 /L 0 e pi X 0 frozen specific heat at constant pressure for species i reference Taylor microscale of turbulence p. coefficient of viscosity t» T turbulent eddy kinematic viscosity p density Oq constant a p ^ <i> constant dissipation term in turbulence kinetic energy equation dissipation term in enthalpy balance equation Superscripts ( ) time average ( ) fluctuating component

131 II2-3 Subscripts i, j, k species m maximum value when 3u/3y = 0 a, fl cartesian component o reference quantity 1. INTRODUCTION The level of turbulence controls the transport of mass, momentum and energy within a turbulent flame. It can also influence the degree of inhomogeneity of the burning gases and hence the rates of reaction and heat release. Experiments designed to study the influence of an imposed turbulent flow on turbulent flame propagation have produced a rather confused picture, with open flames, ducted flames and nonstationary spherical flames all behaving differently. The objective of the present work is to discuss the growth, transport and decay of turbulence in flames, using the balance equation for the kinetic energy of the turbulent motion. An exact derivation of this equation for an arbitrary mixture of chemically reacting gases 1 *, and an estimate of the orders of magnitude of the terms 2 *, from the starting point for the present study. An approximate form of the turbulence kinetic energy equation is recommended for combustion modelling studies, and is also used to provide a qualitative explanation of some of the anomolies in the experimental literature. The turbulence kinetic energy approach, in which the kinetic energy of turbulence is regarded as an additional variable, governed by a balance equation, which must be solved together with the time-averaged balance equations for mass, momentum and energy, has been successfully applied to the modelling of several turbulent flows (e.g. References 3, 4, 5). Closure of the mathematical formulation is accomplished by using empirical equations to relate Reynolds stresses and other unknown terms to time-averaged properties, including the turbulent kinetic energy. This approach is popular as a basis for the mathematical modelling of turbulent flames 6 " 12. In some cases, other statistical properties of the turbulence are also calculated from solution of balance equations, in addition to the kinetic energy; these include fluctuation intensities for temperature 6 and composition 4, and a length scale of the turbulence 11. The hope is that turbulent kinetic energy models will provide a more accurate description of turbulent combustion phenomena than earlier eddy viscosity and mixing length models, because of their supposed ability to predict the non-equilibrium development of a turbulent flow field. At the present time experimental validation of the new models is far from complete. Calculations 9 ' 13 show that turbulent kinetic energy model predictions can be very sensitive to assumed initial profiles of turbulence quantities.. It follows that experiments designed to make a critical test of the kinetic energy models must include accurate initial profiles of turbulence quantities, in addition to measurements of velocity, temperature, composition, and turbulence intensities throughout the combustion zone. The limited comparisons between theory and experiment which have so far been published (e.g. References 8, 9, 10, 11) are reasonably encouraging, but fall far short of a complete validation of the models. Turbulence kinetic energy combustion models in the literature 7 " 11 have always employed a simplified form of the balance equation for turbulence energy, derived on the assumption of incompressible, non-reacting flow. It was shown in I that the exact equation for a chemically reacting gas mixture contains many additional terms which are not present in the incompressible flow case. Furthermore, an order-of-magnitude analysis reported in II suggested that some of these additional terms could become important in turbulent flame studies, whereas Bradshaw 14 found the incompressible flow equation to be adequate in compressible, turbulent boundary layer calculations at low supersonic Mach numbers. In the present work, mass-weighting of velocities, temperature, etc. is not employed. Thus, for example, a velocity component is written with Va = 0, rather than as recommended by Favre 15 and others, where v a = v a + v a. (1-1) v tt = v a + vi' (1.2) and v = v + - n i} v«v a+. (1.3) v a = - Ar 2 - * - (1-4) * References 1 and 2 are referred to as I and II, respectively, in the following discussion.

132 II2-4 In many cases, the statistical equations for turbulent flow are simpler in terms of v a and v^ rather than v a and \a. The reasons for the present choice are as follows. Firstly, promising experimental techniques for studying fluctuations in turbulent flames, such as laser anemometry and emission-absorption spectroscopy, are more likely to give measurements of fluctuating quantities without mass weighting. Experimental evidence is needed as a guide in formulating the empirical closure assumptions which must be made in order to obtain a closed set of equations. A second and more important reason is that these closure hypotheses appear more plausible, at least to the present author, in terms of unweighted rather than weighted velocities. All present day turbulent combustion modelling assumes, at least by implication, the validity of an extrapolation to the combustion situation of Morkovin's hypothesis 16 of independence of the vortical turbulence structure from density fluctuations. Thus, for example, it is useful in combustion modelling to assume that the ratio of Reynolds stress to turbulence kinetic energy is in some circumstances a universal constant, a,, as in incompressible flow. In terms of unweighted velocities, this assumption: -u'v" = ia,(u' 2 + v' 2 + w' 2 ) (1-5) may be interpreted as a description of a limiting form of anisotropy in a shear flow. The analogous assumption in terms of mass-weighted velocities, one form of which is - (p u)"v" = ia,(u" 2 + v" 2 + w" 2 ) (1-6) is less easily interpreted and, in the opinion of the present author, less plausible, particularly when it is recalled that v is non-zero. Experimental evidence is urgently required, to check the validity of either Equation (1.5) or Equation (1.6) under turbulent combustion conditions, where large density fluctuations are to be expected. Of course, if closure hypotheses are formulated in terms of unweighted velocities, v a and v^, it will still be possible to transform the resulting model equations into a form containing v a and v^, for the sake of computational convenience. 2. BALANCE EQUATION FOR TURBULENCE ENERGY In cartesian tensor notation, the instantaneous equation of motion in a viscous compressible flow may be written 3v a 3v a 3p 3f a 0 (2 n P + P v fl a - - 3t p 3x0 3x a 3x0 where f a n, which is the frictional part of the molecular stress tensor, is given by 2 9v <* s.l. 3v <* 9v A velocity component, v a, is divided into a time-averaged part, v a, and a fluctuating part, v^, so that v = v +v: a - (2-3) v a v a where vj = 0. All other dependent variables are similarly separated into time-averaged and fluctuating parts. If Equation (2.1) is multiplied by v^, time-averaged and re-arranged with the help of the continuity equation and Equation (2.2), the balance equation for turbulence energy is obtained in the following form, as shown in I»^+ at d) 3q (pv0 + p'vg)- = dx0 (2) (3) (4) (5) 3 3v' 3v,v 9 (6) (7) (8) (9) (10) 3x0 (Pq'v0-p'v0q + V0p'q' + p'q'v0 + p'v^) (2-4) (H)

133 112-5 where q = i v «v a is the kinetic energy of the turbulence, per unit mass, and q'f = ivavaf' where f is any variable. The viscous dissipation per unit volume is 2 3v a 3vn / 3v ' 3vn \ 2 2 3v ' 3vn L - ' - + ~ (2-5) 3x0 \3x0 3x a / 3 3x a 3x0 Terms in the kinetic energy balance Equation (2.4) may be identified as follows. Term (1) is the time rate of change of mean kinetic energy per unit volume, term (2) is the convective rate of change due to the mean velocities, V0, while term (3) is the convective rate of change resulting from the turbulent mass transport fluxes p' V0. The whole left-hand side of Equation (2.4) may be written Dq/Dt where D 3 3 = + vg (2.6) Dt 9t p 3x0 is the derivative following the time-average motion with velocity in which V0 is the mass-weighted, time-averaged velocity. Term (4) represents production of turbulence energy due to buoyancy effects; it is an increase in the total energy of the system, due to work done by gravity. The whole of term (5) may be interpreted as production of turbulence energy due to work done by turbulent stresses in velocity gradients, at the expense of the kinetic energy of the mean motion. Term (6) is a viscous diffusion term, while term (7) is the viscous dissipation of turbulence energy to heat. Term (8), is positive, represents a transfer of energy from the enthalpy of the gas to turbulence. If the signs are as shown, term (9) is a nonstationary transfer of energy from turbulence to the mean motion, while term (10) is part of the non-stationary gain of turbulent kinetic energy per unit volume of gas. Finally, the whole of term (11) represents turbulent diffusion of turbulence energy. 3. DIVERGENCE OF VELOCITY It was pointed out in I and II that the velocity divergence, 3v a /3x a, which is almost zero in low Mach number non-reacting flows, may become relatively large in turbulent flames as a result of reaction and heat release. Consequently, the part of term (5) of Equation (2.4) which is related to 3v a /3x a, the part of term (7) containing 3v^/3x a, and term (8) which contains dv' a /dx a, which would all be negligible in low Mach number non-reacting flow, may be significant in turbulent flames. These terms have all been neglected in turbulent kinetic energy combustion modes 6 " 11. As shown in I, the instantaneous continuity equation may be solved together with instantaneous balance equations for enthalpy and species, and equations of state, to given an expression for 3v a /9x a. Equations of state for a mixture of perfect gases are written p = P RT 2 Cj/ w (3.1) 1=1 h = SI Cj(er pi T + 5/j) (3.2) i=l where Cj is the mass fraction of species i, whose molecular weight is Wj, frozen specific heat at constant pressure, Cpj, and heat of formation per unit mass, 3(\. Both e p j and the ratio of frozen specific heats for species i, denoted by 7j, where e pi Wj/R = 7i/(7j- D (3-3) are assumed constant. The divergence of velocity may then be written

134 H2-6 j)v0 = _ J_/9P 3x p7 \9t + 9p' \ I P 9x / H 7 p 3x 7 P ; : J l^w/ _7-l 7j W ' rv ~7^TM Wj (3.4) where $ is the viscous dissipation term * = f ofl 3x0 in the enthalpy balance equation, q0 is the molecular and radiative energy flux in the x0-direction, Jj0 is the mass flux of species j in the x0-direction due to molecular diffusion and W: is the mass rate of formation of species j, per unit volume and time, due to chemical reactions. The molecular weight of the mixture W = (3.5) the frozen specific heat of the mixture Cp = t e pk c k, (3.6) and the ratio of frozen specific heats of the mixture, 7, given by C p W/R = 7/(7- D (3.7) are not assumed constant in the derivation of Equation (3.4). However, in many combustion situations, the variations in 7 and differences between 7 and 7; are not large. Therefore, following I, 7 will be assumed constant and the final set of terms in Equation (3.4) will be assumed zero, in the remainder of this analysis. The time-average of Equation (3.4) may then be written 9v0 = 1_ 9xfl P7 3p _ 9p + v, 3t 3x0 P V 0 3Xj< p P 2 3t 3 9x0 9x0 9x0 3x /7-l\ 2 1, 9q0 ^ a(-^. - * J ip\ - - j P * -p *- ~ 2_* ^i(p w i ~P, J \ 7 / P 2 L 3x0 ^ V 9x0/ (3.8) where fluctuations in 7 have been neglected and the approximation p p \ p has been employed on the assumption that pressure fluctuations are small. With similar assumptions, the term p' 9v^/3x a in the turbulence energy Equation (2.4) is given by 9v ' 1 P 9xa 7p 7 i i r 9 9t, 9 <l/3 1 7 P L 9x0 J 4. ORDER OF MAGNITUDE ANALYSIS V/3 a In ip T P V 0 3x0 7-1 l 3p 9, ^ va.. J P H 3x0 3x0 V i 7 p& [ / ajj',,\ t ' ' ' A H ^ I p Wj p i ' (3.9) A stationary, two-dimensional, premixed turbulent flame is considered. For this flow, an order of magnitude analysis has been carried out, and reported in II, for the terms in the exact, reacting, turbulent flow equations

135 listed in I. Here we shall report only the main assumptions of the order of magnitude analysis, and results relevant to the turbulence kinetic energy equation. Further details may be found in II. Cartesian coordinates (x,, x 2, x 3 ) are replaced by (x, y, z), with corresponding velocity components (u, v, w), x being in the direction of mean motion. A reference length L 0 is set equal to the distance x from the origin of the flame to the arbitrarily chosen station of interest, while 6 0 is a typical half-width of the shear layer at station x. The boundary layer approximation is employed so II2-7 - = e <SC 1. (4.1) LO For order of magnitude purposes, characteristic reference quantities are introduced, denoted by suffix zero. If the symbols f and g stand for any of the variables: u, v, p, T, h, q, the orders of magnitude of gradients are estimated as follows 3x L 0 ' 9y 5 0 9x L 0 ' 9y 6 0 For the x and y momentum equations while from continuity 3p p 0 u 2 9p p 0 u 2 (4.3) 3x L 0 3y L 0 ~ e. (4.4) u i Following a reivew of experimental data from turbulent flames, characteristic fluctuation magnitudes are estimated as 4- = e 1/2 (4.5) ID where f again denotes any of the variables: u, p, T, h, Cj. It is shown that the result is compatible with the approximation: production equals dissipation, in the appropriate fluctuation intensity balance equations. This is described in II as the "small fluctuation" case. A "large fluctuation" case, for which f 0 /fo ~ 1, is also analysed in II, but will not be discussed here. Pressure fluctuations are assumed to be of magnitude p- = K p p 0 u 0 2 (4.6) where the coefficient K p may be greater than unity. Two turbulence length scales are introduced. The first, / e, is characteristic of the energy-bearing scales of the turbulence while the second, X 0, is a Taylor micro-scale. Both are related to the dissipation term 0 in the turbulence kinetic energy balance, Equation (2.4), as follows It then follows that - (4.7) 'e *o A.. ; A. pl (4. 8) V R L e where R L is a characteristic Reynolds number of the mean flow = p " L. (4.9) Po Covariance magnitudes containing derivatives are then estimated as follows f' S-^2- (4.10) 9x a / e

136 II2-8 9f 9g' 3x ff 3x0 f ogo Xo f 0 go *o (4.11) However, the magnitude of covariances of the type f ' 3v^/3x a, which would be identically zero for incompressible flow, are estimated from the equations of Section 3. Finally, a distribution time-average chemical reaction zone is assumed, as indicated for example in the experiments of Shipman et al. 17 ' 18. A characteristic value, w 0, is introduced, typical of the time-average rate of production of major combustion product within the combustion zone. Numerous experimental and theoretical studies (e.g. References 7, 8, 17, 18, 19) make it clear that w 0 is generally very much less than the typical reaction rate within a premixed laminar flame in the same gas mixture, and also that w 0 often appears to be a function primarily of properties related to turbulent shear rather than to chemical kinetic factors. It is written Po a o,.\~ - ' where / 0 is a dimension characterising the length of the heat release zone. Consideration of orders of magnitude in the time-averaged species balance equation then shows (see II) that we must have / (4.13) if chemical reactions are to play a major role in determining the variation of composition within the flame. In view of wrinkled laminar flame and parcel models of turbulent flames 7 ' 19 ' 20 it is assumed that reaction rate fluctuations are large, so that w 0 /w 0 ~ 1. (4.14) Assume a low Mach number flow, such that M 2 <ic e (4.15) together with a high Reynolds number R (4.16) and a Froude number F 0 = 80 L o (4.17) where gg is a characteristic component of the acceleration due to gravity. Assume e <C 1, and treat the factor 15 in Equations (4.7) and (4.8) as 0(1). Then the order of magnitude of terms in the equations of Section 2 and 3 may be estimated as described above. Shown below are the larger terms in the relevant equations; the relative magnitude of each term is indicated directly below that term. The largest terms in each equation are given the relative magnitude of unity. The time-average divergence of velocity, Equation (3.8), becomes simply 9u 9v + =- 9x 9y n -! - -SZ 7 P i=l 1 (4.18) where neglected terms are all at most of order M or of order RLf" 3 ' 2. Equation (4.18) shows that the timeaverage divergence of velocity is due to chemical heat release. Similarly, Equation (3.9) becomes du' 9v' 9w' p' + p' - + p' - 3x 9y 3z T' -- k p' P I 9*0 e 1/2 /15 e 1/2 /15 1=1

137 where k is thermal conductivity, Dj is a diffusion coefficient for species i, and the factor 15 is regarded as being of order unity. The terms containing p'9 2 Cj'/3x^ result from combining the largest molecular diffusion contributions from the p' 3q0/9x0 and p' 3Jj'0/9x0 terms in Equation (3.9). Finally, consider the turbulence kinetic energy balance, Equation (2.4). Making use of Equation (4.19), and neglecting terms of relative magnitude e 3 ' 2, this becomes 9q pu + (pv+p'v') = (pu'u' + up'u') (p u'v' + vp'u'-f p'u'v') 9x 9v 9x 3y H2-9 9v PV'v' + p'u' g x + p'v' g y P 1=1 3 2 T' e e/f 0 e/f 0 ek n K p e 1/2 /15 9y (p v'q' q p'v' + p'v'q'+ p'v') K p e 1/2 /15-1/2 e 1/2 K,, 9x X0 0\ a (4.20) 1,1/2 It should be noted that the magnitudes proposed above are suggested as plausible approximate upper bounds. The variables concerned may be positive or negative and their magnitude may take any value less than the reference value, including zero. 5. MODEL BALANCE EQUATION FOR TURBULENCE KINETIC ENERGY Because of the closure problem, an exact representation of the statistical equations of turbulent flow cannot be used directly in numerical solution of flow problems. Some degree of empirical modelling is necessary. It is clear that, if the exact equations of I are to be replaced by empirical model equations, the first concern of the modeller must be to ensure that at least all the most important phenomena occurring in the exact equations are at least empirically represented in the appropriate model equations. The turbulence kinetic energy balance, Equation (4.20), will be considered from this point of view. In comparison with the corresponding equation for incompressible flow, Equation (4.20) contains terms representing effects of turbulent mass transport, buoyancy, divergence of velocity, chemical reaction, thermal conduction and diffusion of species. In attempting to incorporate the more important phenomena from this list into the turbulence kinetic energy equation, we use a formulation which reduces the Harsha's model equation 13 in the incompressible, non-reacting, single fluid case. This model is chosen because it has been successfully tested 13 against experimental data for incompressible flows and to some extent also for variable density flows. Harsha's model is used here as an example; it is not necessarily the most accurate possible representation of the kinetic energy balance. A basic assumption in Harsha's work is that the Reynolds stress is related to the turbulence energy by the equation where for two-dimensional flow -p' u'v' = a, pq (5.1) a, = a, 3u /I 30 9y 9y max /9Q '90 \ * o) ( 3y / a, = a, "97 (elsewhere) and a, = 0.3. A kinematic eddy viscosity is then introduced as

138 VT = p u' v' a, q p 3u/3y 30 ~9y where suffix m indicates that a maximum value is to be used when 90/9x «0. The turbulent diffusion flux is written m (5.2) j> T 9q -(pv'q' + p'v') = p (5.3) where a q is a turbulent Prandtl-Schmidt number for q. Finally, the dissipation function is written 9x a / i q 3 ' 2 (5.4) where / q is a dissipation length scale and a 2 is a constant, equal to 1.5 if / q is taken as the width of the mixing region. It is assumed that these expressions apply also in turbulent flames. We now consider empirical representation of the additional terms in Equation (4.20). By analogy with Equations (5.1) and (5.3) it is assumed that 3p (5.5) where a., is a turbulent Prandtl-Schmidt number for p. As wrinkled-laminar and parcel models of turbulent flames lead one to expect that density fluctuations will be highly correlated with temperature and enthalpy fluctuations, o p should be a number of order unity. The following approximation is employed to model the two normal stress terms in Equation (4.20): 90 3v ) +PV-V = an 9v (5.6) where a 4 is a constant of order unity. In considering the dissipation function, Equation (5.4), it should be noted that entry of a fluid element into a flame zone reduces the turbulence Reynolds number, since the density falls and the viscosity rises. More than an order of magnitude reduction in turbulence Reynolds number is to be expected. It is then quite possible that the dissipation will be considerably in excess of the value indicated in Equation (5.4). Finson 21 has found satisfactory agreement between a compilation of experimental data for incompressible flows and the following expression (see Rotta, Reference 22) where R T is a turbulence Reynolds number defined as 3x R (5.7) R T = P q'/2 (5.8) and a 2 and a 3 are constants*. If adjustment is made, for different definitions of the length f q, a and a 3 = 147. Using the above expressions and neglecting other terms, the turbulence kinetic energy Equation (4.20) becomes -.21 U 9x y ay m p 1=1 9y a, q 9q 90 9y 9y (D (2) /IL + _L^_!LJL n^. _ **? U + a oj 3Q 9y 9y /a 3y m (3) (4) (5) (5.9) (6) (7) * In view of the assumption in Section 4 of a high Reynolds number flow, Equation (5.7) must be regarded as a purely ad hoc correction.

139 This expression reduces to Harsha's incompressible flow equation if w; = 0, p = constant and R L -* <». Equation (4.18) has been used in term (4), in order to bring out the direct influence of chemical reaction rates on the kinetic energy balance. In computation it would probably be more convenient to calculate (3u/9x + 9v/9y) as shown in Equation (5.6). Note that term (4) is negative since W; < 0 in combustion. Terms from Equation (4.20) which have been neglected in Equation (5.9) are as follows. Buoyancy terms have been omitted on the assumption that the Froude number, F 0 3> 1. The group of terms in curly brackets containing the pressure fluctuation, p', has been neglected for the reasons explained in the Appendix. These reasons are admittedly speculative. Experiments show that, under suitable conditions, pressure fluctuations can interact strongly with flames, so it may be that these terms are more important than the present analysis would suggest. The term 9x0 9x a / is expected to be negligible because it involves a correlation between a scalar property and velocity gradients, which should be small 14 in the nearly isotropic turbulence responsible for dissipation. The remaining terms have been neglected because they are smaller than other terms, which appear to play a similar role in the kinetic energy balance. Equation (5.9) has been derived from the exact turbulence kinetic energy balance, Equation (2.4), from consideration of orders of magnitude in premixed flames. As noted in II, the balance between the terms is expected to be somewhat different in a diffusion flame. 6. VARIABLE DENSITY INERT JET MIXING A successful theory of turbulence in flames should be capable of accounting also for the observed effects of density variations in inert turbulent jets. Experiments show that a light gas jet spreads relatively quickly into a heavier gas (see for example the H 2 /air turbulent mixing experiments of Chriss and Paulk, Reference 23), in comparison with a jet whose density is the same as that of the surrounding medium. On the other hand, a heavy jet spreads relatively slowly (e.g. Reference 24). These differences in jet spreading rate with density ratio are not predicted by the classical eddy viscosity and mixing formulae, as has been clearly demonstrated by Harsha's comprehensive comparison 13 of experimental data with ^numerical solutions based on a range of turbulence models. In his critical evaluation, Harsha showed that, of all the turbulence models tested, the turbulence kinetic model was the only generally applicable model which was able to make successful predictions in both constant density and variable density flows, without arbitrary readjustment of parameters. Harsha's turbulence kinetic energy equation may be written 3q 9q PO +pv 3x 3y a, p q /90\ 2 3 ( ) +P \9y/ 9y ay ff q a, q 9q 3u ay m ay a, q 3n 3q 3p 9y 9y a 2 pq 3 (6.1) from Equation (5.9). This equation was solved numerically, together with a conventional form of the turbulent boundary layer equations for compressible flow. The boundary layer equations were of the same form in the turbulent kinetic energy calculations as in the relatively unsuccessful mixing length and eddy viscosity calculations. It follows that the physical explanation for the success of the turbulence kinetic energy model in predicting flows with variable density must lie in Equation (6.1) itself. A striking feature of this equation is that division by the density, p, removes p from all terms except for the second part of the diffusion term, which becomes a, q ay 1 9p 9q p 9y 3y

140 In the case of a light gas jet or a flame, this term has the effect of enhancing the diffusion of turbulence energy away from the centre towards the outer part, so increasing the spreading rate of the jet. In the case of a heavy gas jet, the term changes sign and opposite effect is produced. It is interesting to note from Equation (5.9) that the term derived frofn p' v' 3q/3y adds to the term discussed above, and enhances its effect, in term (6). Unfortunately, the agreement between Harsha's turbulence kinetic energy calculations 13 and Chriss and Pulk's H 2 /air mixing experiments 23 cannot be regarded as conclusive proof of the validity of Equation (6.1) in density gradients. Harsha found it necessary, in his H 2 /air comparisons, to commence integration some 5 or 6 diameters downstream of the jet exit. He noted (Reference 13, p.467) that an attempt to begin integration at the jet exit with an assumed shear stress profile gave wholly unsatisfactory results in H 2 /air comparisons, although the same method worked well in air/air cases. In later calculations, using similar numerical methods compared with the same experimental data, Rickeard 9 was able to obtain excellent agreement for both air/air and H 2 /air cases, starting his integration from assumed profiles at the jet exit. In order to achieve this agreement, Rickeard found it necessary to modify the quantity a, in Equation (6.1), so that it became an empirical function of Mach number and of mass flux ratio. In these circumstances, the validity of Equation (6.1) in the presence of density gradients, and hence the validity of the density-dependent terms in Equation (5.9), must be regarded as unproven. Experimental data is required, including jet exit profiles and turbulence kinetic energy measurements. 7. TURBULENCE IN FLAMES The influence of turbulence on the propagation and structure of turbulent, premixed flames has for many years been a subject of much controversy. Williams 20 has reviewed the literature up to Damkohler 25 and many others concluded, from experiments on open flames, that the response of a flame to sufficiently large scale turbulence is essentially passive, in the wrinkled laminar flame manner. The turbulent flame propagation speed, defined from the mean motion of the visible flame boundary, increases linearly with turbulent intensity. Spark-ignited, non-stationary, spherical flames in turbulent, premixed streams 26 ' 28 do not conform to this picture, since the corrected turbulent flame propagation speed is considerably lower than for open flames, and in some cases turbulence actually appears to reduce the flame speed. There are uncertainties regarding the method of analysis of this data 27 ' 28 Confined turbulent flames are different again, the effective flame speed being much greater than in corresponding open flames. Shear-generated turbulence provides a possible explanation, as noted by Scurlock and Grover 29 and others. On the other hand Westenberg's tracer dispersion experiments 30 ' 31 in ducted flames show only a slight increase in diffusivity in the flame region, and actually a decrease in diffusivity in some part of the burned gas. Recent experiments by Ballal and Lefebvre 32 suggest that the turbulent flame speed varies in a complex manner with both the intensity and the scale of grid-generated turbulence, in view of all the varied and complex phenomena mentioned above, the validity of the concept of a turbulent flame speed may be questioned. More detailed turbulence measurements are available from diffusion flames. Gunther and Lenze 33 ' 34 have shown that turbulence velocity, microscale and turbulent exchange coefficient are all larger in a town gas diffusion flame than at the same station in an air jet. The exchange coefficient is reasonably well represented by the product of the microscale and the turbulence velocity, as in the air jet. However, because of differences in the mean velocity, the relative turbulence intensity is much lower in the town gas flame than in the air jet. In a hydrogen flame on the other hand, its magnitude is similar to that in an air jet. Eickhoff 35 has linked these differences to the density gradient within the flame. This is positive for the hydrogen flame, but negative for the inner part of the town gas flame. Eickhoff also reports a sudden, steep drop in the relative turbulence intensity along the axis of a lifted, town gas flame, in the vicinity of the ignition point. Many of these experimental observations may be explained, at least qualitatively, in terms of the turbulence kinetic energy balance, Equation (5.9). This equation predicts that the following phenomena will occur under the influence of combustion. (i) Turbulence will be generated (term 3) as a result of shear produced within the flame, (ii) Turbulence energy will also be removed due to velocity divergence resulting from heat release (term 4). (iii) Diffusion of turbulence energy will be influenced by the sign and magnitude of the density gradient set up within the flame (term 6). (iv) A reduction in turbulence Reynolds number may cause a large increase in the viscous dissipation of turbulence energy (term 7). ' The interaction between turbulence and combustion depends upon the balance between these often opposing effects in each case. At one extreme is the non-stationary, spherical combustion wave, for which production of turbulence energy due to mean shear is negligible, whereas reduction in turbulence energy resulting from divergence of mean velocity may be significant. In coordinates fixed in the turbulent flame front, the convective and velocity

141 divergence terms give* p 3f 6q ~ p u' 2 6u neglecting flame front curvature effects (but see Reference 27), where Sf is the effective flame speed, 6Q = - P2 is the velocity change through the flame, p, and p 2 being the density of unburnt and burnt gas, respectively. Then the change in turbulence energy through the flame due to velocity divergence is of order In a plane combustion wave, this energy is removed from the u ' component of the turbulence. However, the irregular, wrinkled shape of the instantaneous flame front will in fact lead to removal of energy from all three components of the turbulence. It is concluded that the velocity divergence effect, plus the effects of enhanced viscous dissipation at the low turbulence Reynolds numbers in and behind the flame, will lead to a considerable damping of the turbulent motion, and hence a reduced effect of the turbulence on the propagation speed. At the other extreme are the high-speed, ducted, turbulent premixed flames, typified in the experiments of Shipman et al. 17 ' 18, and successfully modelled by Spalding 8 using an incompressible flow turbulence energy equation similar to Equation (6.1). Confinement in a ducted burner increases the mean shear, 90/9y, and hence the relative importance of the shear production of turbulence energy (term 3 in Equation (5.9)). As the heat release zone is distributed throughout the whole mixing region 17 ' 18, the effects of divergence of velocity (term 4 in Equation (5.9)) are less apparent. Also, at high velocity and with intense turbulence, the turbulence Reynolds number is expected to be large enough so that low Reynolds number corrections to the dissipation (term 7) may be neglected. Ducted premixed flame experiments at lower flow speeds and in larger ducts (e.g. Westenberg 30 ), and also open flame experiments, will be intermediate between these two extremes, because shear generation of turbulence will be relatively less important, and so the other terms on the right-hand side of Equation (5.9) will be relatively more important. The diffusion flame data of References also appears to be at least qualitatively in agreement with expectations based on Equation (5.9). In particular, Eickhoff s observations 35 regarding density gradient effects are compatible with term (6), and his measurements of a sharp reduction in relative intensity through a lifted flame are strongly suggestive of the effects of velocity divergence, term (4). Numerical calculations are required, in order to test whether inclusion of the new terms in Equation (5.9) does in fact improve agreement with experimental data over the wide range of conditions suggested above. 8. CONCLUSIONS (i) An exact turbulence kinetic energy balance, Equation (2.4), has been simplified using an order of magnitude analysis for a stationary, two-dimensional, premixed, turbulent flame at low Mach number and high Reynolds number. The simplified equation, Equation (4.20), contains several terms which have been neglected in earlier combustion studies. (ii) Plausible closure hypotheses have been used in order to derive from Equation (4.20) a model turbulence kinetic energy balance, Equation (5.9), which reduces to Harsha's equation 13, in the case of non-reacting flow of constant density at high turbulence Reynolds numbers. Additional terms represent effects of velocity divergence due to heat release in a turbulent flame, turbulent mass transport and viscous dissipation at low turbulence Reynolds number. (iii) It is recommended that this equation be employed in computer modelling' studies of turbulent flames. (iv) It is suggested that this equation can at least partially explain experimentally observed effects of turbulence on the propagation of open flames, ducted flames and spherical, non-stationary flames. The order of magnitude analysis of Section 4 is not directly applicable to non-stationary spherical flames. It is assumed here that convective and velocity divergence terms from the general equation, Equation 2.4, are important.

142 ACKNOWLEDGEMENTS The author is grateful to Dr J.B.Moss for reading and criticising the manuscript of this work and to Mrs P.V.Ayre for typing it. The present work is part of an experimental and theoretical study of fluctuating quantities in turbulent flames, supported by S.R.C. Research Grant B/RG/3552.

143 APPENDIX The magnitudes of three terms: 7-11 n X i ~ i _ v* 2., 7 P 1 = T' X, = _kp'- 7 p ax«x s = A Cj' T >. e pj T D J P P'TT 7 P fr? 3x 3 in the turbulence kinetic energy balance, Equation (4.20), will be considered. The point of view adopted is that the largest contributions to these terms are assumed to arise due to the appearance of discrete flamelets within the combustion zone. These flamelets are asumed to have a regular structure, so that the contributions to p' etc. arising from them are always correlated. The flamelets need not be laminar. Consider X, first. It is easily seen that this term will be exactly zero, if Wj' is symmetrical and p' is antisymmetrical about a plane in the flamelet. We assume that deviations from this symmetry are small enough so that p' Wj' <SC p' Wj'. The term X 2 is approximated as follows. Writing 9 2 T' 9 / 9T'\ 9p' 3T' P 'T~T = ^~I P 'T~/ ^ ~ ~" 3p' 3T' 9xs 3xo \ 3xo 3x0 / 9x0 3xo 9x0 3x0 9x0 we have 7-1 k 9p' 9T' X 2 = p 3x0 3x0 This is approximated by 7 1 k Apr ATf w X 2 = t 7 P If w f (Al) where If is the thickness of the reaction zone in a flamelet, where the reaction rate is Wf. On entering a flamelet, the temperature rises by an amount ATf, while the pressure rises by amount Ap f. The quantity w/wf is the ratio of the mean reaction rate to the flamelet reaction rate; it is an estimate of the probability of a flamelet existing at the measuring station. A momentum balance for the flamelet gives Ap /p, \ WS? - = - ( - 1 ] P \P 2 / RT, L (A.2) where p, and p 2 are the densities of unburnt and burnt gas, respectively, and Sf is the flame propagation speed. Assume Sf = -f^. k f f = ~^Tc7 (A.3) (A - 4) by analogy with laminar flame properties, where kf is the laminar or small-scale turbulent thermal conductivity controlling the flamelet structure. Finally, approximate AT 1 1

144 Substituting Equations (A.2) to (A.5) into (A.I) it is found that k W Wf ~ The order of magnitude of X 2 may now be estimated as follows, assuming that the molecular Prandtl number is unity, so that ic = C p ft 1 Wf X 2 ~ L-l_.. RL w o L o On the other hand the dissipation term in Equations (2.4), (4.20) and (5.9) is (A.7) Since R L = p 0 Q 0 L 0 /p 0 <^ 1, whereas experimental evidence 17 ' 18 shows that w 0 is a few percent of Wf, it follows that X 2 is very much smaller than <t>. Finally, if a binary mixture of reactant and product is considered, a similar analysis shows X 3 to be X 3 * IP -!lx 2 (A.9) where ep and e R represent the specific heats at constant pressure for product and reactant, respectively.

145 REFERENCES 1. Bray,K.N.C. 2. Bray, K.N.C. 3. Bradshaw, P. Ferriss, D.H. Atwell, N.P. 4. Spalding, D.B. 5. Lee, S.C. Harsha, P.T. 6. Spalding, D.B. 7. Spalding, D.B. 8. Mason, H.B. Spalding, D.B. 9. Rickeard, D.J. Equations of Turbulent Combustion. I, Fundamental Equations of Reacting Turbulent Flow. University of Southampton, AASU Report 330, Equations of Turbulent Combustion. II, Boundary Layer Approximation. University of Southampton AASU Report 331, Calculation of Boundary Layer Development using the Turbulent Energy Equation. J. Fluid Mech., Vol.28, p.593, Concentration Fluctuations in a Round Turbulent Free Jet. Chem. Eng Set., Vol.26, p.95, Use of Turbulent Kinetic Energy in Free Mixing Studies. AIAA J, Vol.8, p. 1026, Mathematical Models of Turbulent Flames, Lecture delivered at VDI Tagung, Verbrennung and Feuerungen, Karlsruhe, Also Imperial College, Mech. Eng, BL/TN/B/22, Mixing and Chemical Reaction in Steady Confined Turbulent Flames. Thirteenth Symposium on Combustion, Comb. Inst. p.649, Prediction of Reaction Rates in Turbulent Premixed Boundary Layer Flows. Combustion Inst. European Symposium (Ed. F.J.Weinberg) p.601, Academic Press, Performance of Air-Augmented Rockets: Turbulent Mixing and Chemical Reaction in the Secondary Combustion Chamber, Ph.D. thesis, University of Southampton, Kent, J.H. Bilger, R.W. The Prediction of Turbulent Jet Diffusion Rolling Res. Lab. TNF-47, Flames. University of Sydney, Charles 11. Lilley, D.G. 12. Borghi, R. 13. Harsha, P.T. 14. Bradshaw, P. Ferriss, D.H. 15. Favre, A. 16. Morkovin, M.V. 17. Howe, N.M. Shipman, C.W. Vranos, A. 18. Gushing, B.S. Faucher, J.E. Gandbhir, S. Shipman, C.W. 19. Howe, N.M. Shipman, C.W. Turbulent Swirling Flame Prediction. Presented at AIAA 6th Fluid and Plasma Dynamics Conf., Palm Springs, Calif., Chemical Reactions Calculations in Turbulent Flows. Application to a Co-Containing Turbojet Plume. ONERA T.P. No Presented at 2nd IUTAM-IUGG Symposium on Turbulent Diffusion in Environmental Pollution, Charlottesville, Virginia, Free Turbulent Mixing: a Critical Evaluation of Theory and Experiment. AEDC-TR , Calculation of Boundary Layer Development using the Turbulent energy Equation: Compressible Flow on Adiabatic Walls. J. Fluid Mech., Vol.46, p.83, Equations des Gaz Turbulents Compressibles. J. de Mechanique, Vol.4, p.361, p.392, Effects of Compressibility on Turbulent Flows in The Mechanics of Turbulence (Ed. A. Favre), Gordon and Breach, Turbulent Mass Transfer and Rates of Combustion in Confined Turbulent Flames. Ninth Symposium on Combustion, Academic Press, p.36, Turbulent Mass Transfer and Rates of Combustion in Confined, Turbulent Flames, II. Eleventh Symposium on Combustion, Comb. Inst. p.817, A Tentative Model for Rates of Combustion in Confined Turbulent Flames. Tenth Symposium on Combustion, Comb. Inst. p.1139, Williams, F.A. Combustion Theory. Addison-Wesley, 1965.

146 Finson, M.L. 22. Rotta, J. 23. Chriss, D.E. Paulk, R.A. 24. Abramovich, G.N. Bakulev, V.I. Makarov, S.I. Khudenko. B.C. 25. Damkohler, G. 26. Mickelsen, W.R. Ernstein, N.E. 27. Shchelkin, K.I. 28. Bolz, R.E. Burlage, H. 29. Scurlock, AC. Grover, J.H. Hypersonic Wake Aerodynamics at High Reynolds Numbers. AIAA J., Vol.11, p. 1137, Statistische Theorie nichthomogener Turbulenz. Z. fur Physik, Vol.129, p.547; Vol.131, p.51, Summary Report: An Experimental Investigation of Subsonic Coaxial Free Turbulent Mixing. AEDC-TR , Turbulent Submerged Jets of Real Gases in Turbulent Jets of Air, Plasma, and Real Gases (ed. G.N.Abramovich), p. 139, Consultants Bureau, N.Y., Z. Elektrochem, Vol.46, p.601, Growth Rates of Turbulent Free Flames. Sixth Symposium on Combustion, Reinhold Pub. Corpn., p.325, Measurement of Speed of Propagation of Turbulent Combustion. ARS J., Vol.30, p.76, Further Comments on the Measurement of the Speed of Propagation of Turbulent Combustion. ARS J., Vol.30, p.1032, Propagation of Turbulent Flames. Fourth Symposium on Combustion, Williams and Wilkins, p.645, Westenberg, A.A. Flame Turbulence Measurements by the Method of Helium Diffusion. Vol.22, p.814, J. Chem. Phys. 31. Westenberg, A.A. Rice, J.L. 32. Ballal, D.R. Lefebvre, A.H. 33. Gunther, R. Lenze, B. Further Measurements of Turbulence Intensity in Flame Zones, Combustion and Flame, Vol.3, p.459, Turbulence Effects on Enclosed Flames. Paper presented at Fourth Int. Coll. on Gas Dynamics of Explosions and Reactive Systems, Univ. of California, San Diego, Exchange Coefficients and Mathematical Models of Jet Diffusion Flames. Fourteenth Symposium on Combustion, Comb. Inst., p.675, Lenze, B. Turbulenzverhalten und Ungemischtheit von Strahlen und Strahlflammen. University of Karlsruhe, Dissertation, 35. Eickhoff, H.E. Experimental Investigation of the Influence of Combustion on Turbulent Transport in Jet Diffusion Flames. Combustion Inst. European Symposium (Ed. F.J.Weinberg), p.513, Academic Press, DISCUSSION Prof. Spalding: I would like to ask a question about the basis of the modelling and a general question. It seems to me that in modelling one has to make many choices and you gave reasons for them; but there was one choice for which I think you did not give an argument, and I will refer specifically to that choice. You started from the relationship between shear stresses and the kinetic energy of turbulence and put the shear stress proportional to the turbulence energy. Now there is one main alternative at the same level of complexity: i.e. the Prandtl-Kolmogorov proposal which assumes the shear stresses proportional to the velocity gradients and to the squre root of the turbulence energy. There are a number of objections to the model which you chose. In fact, as you said, at the center of a jet the kinetic energy is finite but the shear stress is zero; the same happens at the center of a pipe. So this model has basic defects. As soon as you start making computations, these defects become very severe. In order to avoid these complexities one has to introduce some dependence on the velocity gradient, eventually arriving (through the back door) at something similar to the Prandtl-Kolmogorov hypothesis.

147 Another objection is that the hypothesis from which you started has not been subjected to nearly as much testing as the Prandtl-Kolmogorov assumption. My questions therefore are: (1) why do you prefer that model? (2) do your results depend upon you having made that choice or not? Author's reply: I would like to thank Professor Spalding for this question, which raises an important point. My reason for choosing Harsha's adaptation of Townsend's assumption for the turbulent shear stress is that Harsha has carried out a very detailed comparison between predictions of this model and the experimental data of Chriss and Paulk for H 2 /air jet mixing. This was the most detailed comparison which I had seen of a turbulence kinetic energy calculation with experimental data for a variable density jet flow, and the results were fairly encouraging. On the other hand, I am aware that some of the data points the other way, as Professor Spalding rightly indicates. I do not wish to suggest that my model balance equation for kinetic energy of turbulence in flames cannot be improved. I have put it forward as an example of the terms required in such a balance equation, using as a basis Harsha's equation which in my opinion has been well tested. I think that we must now wait for experimental data from turbulent flames to guide us as to improvements in the modelling of this equation. The second part of Professor Spalding's question enquires whether my choice of Harsha's model influences the results of my study. Fortunately, it does not. Equation (5.1) of my paper may be replaced by -^ 30 - P U V = p!> T - 9y with the eddy viscosity coefficient V T given by a Prandtl-Kolmogorov assumption of the form V T = bq" 1/2 / q as recommended by Professor Spalding, where b is a constant and the length / q is defined in Equation (5.4). Applying Equations (5.3) to (5.8) as before, the turbulence kinetic energy equation becomes 9q 3q,, /9u\ n p u + P v = b q 1 ' 2 / 4 q p + a 4 p q XI <$; w; 9x 9y \9y/ 7 p i=i ' + + P 3 bq1/2 / a 3q 3y I a q 3y a q P 'q Terms in this equation have the same significance as the corresponding terms in Equation (5.9) of my paper. It can be used in place of Equation (5.9). Dr Chigjer: From the measurements which we have made in unconfined turbulent diffusion flames there are two results which have been bothering us for some time. The first one you discussed i.e. that the effect of the measured turbulence intensity in the open flame is either slightly less or not significantly different with respect to that in the corresponding isothermal case. The explanation you have given seems to me a valid one; obviously one has to check that by measuring the individual quantities. Incidentally we do find that when we make measurements in swirling flames then we do get an enhancement of the turbulence and so, apparently, some of the balance you were referring to for the non-swirling flames might be changed when we move to a swirling flame system. The other question which I would like to ask you is: when one has the type of open flame which you described, i.e. the lifted flame, one should expect to find acceleration of the mean velocity, simply because of the changes in temperature which one finds close to the flame front. We found from the measurements which we have made (and other people have made) that it is seldom that one measured this acceleration in the mean velocity and, as far as I know, one is not able to trace fully the effect of the acceleration and of the consequent changes in the pressure. Author's reply: I would be most interested to see the data mentioned by Dr Chigier in the first part of this question. The second part of the question concerns the acceleration of the fluid on passing through a lifted free turbulent flame. Our own observations seem to suggest that the main phenomenon on entering the lifted flame is a rapid transverse expansion of the fluid. Since most of the expansion takes place transversely we do not observe an acceleration 3u/3x. We have some data which could be compared with Dr Chigier's. Prof, libby: I have four brief comments. First I think we must thank Prof. Bray for doing a careful job in showing the enormous complexity of some reacting problems.

148 Secondly I want to amplify Prof. Spalding's comment about the turbulence model: not only are there some questions about the physics of that model but it also changes the equations to hyperbolic, so that one must deal with a different class of describing equations and different specification of initial and boundary data. Third point is that several years ago I was concerned on how to take the variable density effects into account and I came to the opposite conclusion about the utility of the Favre averaging. In fact the equations that result and the modelling which has to be done when one uses the Favre averaging are considerably simpler. Of course, as Prof. Bray quite rightly pointed out, once you make that decision and wish to compare prediction with an experiment, then of course you must get back to the presumably measured quantities. Now there are two aspects of that point: (1) it is not always clear whether the experiments give us a velocity or mass averaged velocity, (2) once you get the prediction on mass averaging quantities then of course you can make a limited further modelling to get to the prediction of the experimental data and in the meantime you have an enormously simpler analytic situation. My fourth point refers to the straight averaging versus Favre averaging; there will appear later this year, in the Physics of Fluids, some results by Stanford and myself about He-air mixtures. We have measured accurately a number of quantities such as: velocities, mass averaged velocities, intensity and any other quantities you would like to know; we have done both the straight and the Favre averaging so that one can see what the differences are. Also we have been able to decompose the differences and show that, in some case, they are due to the sum of two small numbers and in some other case they are due to the small difference of two relatively large terms. Author's reply: I look forward to seeing the experimental data mentioned by Professor Libby, but I do think that one might expect larger fluctuations in density in a turbulent flame than in a cold mixing situation, and possibly also a different coupling between the density field and the kinematics of the turbulence. I guess that one's preference in relation to Favre averaging as opposed to unweighted averaging is partly influenced by the experimental techniques available. Hot wire anemometry can be made to give Favre-averaged velocities, whereas we hope that the laser-doppler anemometry which we employ in turbulent flames gives us unweighted velocity components. I agree with Professor Libby that there are computational advantages in using Favre averaging, but nevertheless, if one's modelling is based on laser-doppler anemometry, one must first model the unweighted quantities and then transform to Favre-weighted equations before solving these equations. Prof. Monti: Did you ever feel embarassed in the evaluation of the order of magnitude of the different terms of your equation and in deciding which term has to be dropped out and which had to be retained? Author's reply: I felt embarassed quite often, because there are many terms in the exact equations whose orders of magnitude cannot be predicted with any confidence. I hope very much that people will check my order of magnitude estimates term by term, criticise my paper, and tell me where they disagree with my results. I suspect that this can happen since my order of magnitude estimates have been made on a purely ad hoc basis. Eventually we must rely upon experimental data to guide us as to which terms in the exact equations may be neglected.

149 113 A NUMERICAL SPECTROSCOPIC TECHNIQUE FOR ANALYZING COMBUSTOR FLOWFIELDS by Michael E.Neer Fluid Dynamics Facilities Research Laboratory Aerospace Research Laboratories Wright-Patterson Air Force Base, Ohio

150 113 RESUME Un programme permettant de calculer les spectres d'emission et d'absorption de OH dans 1'ultraviolet est presente avec pour optique la prediction des proprietes des ecoulements dans les foyers. Les distributions spatiales de la temperature et de la concentration de OH, resultant des calculs d'ecoulement sont utilisees comme donnees d'entree pour calculer les intensites absolues des spectres. L'interet particulier de cette technique est la possibilite de calculer les profits des intensites a partir de resultats donnes par un spectrometre a dimension de fentes conduisant a de faibles resolutions. Des comparaisons sont faites a partir de spectres obtenus avec des degres varies de resolution spectrale. Le programme de calcul est egalement employe dans une situation inverse relative a 1'analyse des spectres experimentaux. Un exemple est donne dans lequel cette technique est utilisee pour obtenir les temperatures moyennes et les concentrations le long de 1'axe d'une flamme de diffusion axisymetrique dans un ecoulement supersonique. Un autre exemple est presente pour demontrer la maniere permettant par une deuxieme inversion d'obtenir les profils radiaux de temperature du radical OH et sa concentration dans un plan perpendiculaire a 1'axe du foyer. Deux cas faisant intervenir une thermodynamique de non equilibre sont aussi discutes, 1'un deux porte sur bande vibrationnelle chaude et 1'autre sur un niveau electronique en non equilibre.

151 113-1 A NUMERICAL SPECTROSCOPIC TECHNIQUE FOR ANALYZING COMBUSTOR FLOWFIELDS Michael E.Neer Fluid Dynamics Facilities Research Laboratory Aerospace Research Laboratories Wright-Patterson Air Force Base, Ohio ABSTRACT A computer program which calculates the ultraviolet emission and absorption spectra of OH is presented for use in conjunction with numerical programs which predict combustor flowfield properties. Spatial distributions of OH number density and temperature, resulting from analytical flowfield calculations, are used as input data for calculating the absolute intensities of the spectra. Of particular interest is the ability to calculate the shapes of the intensity envelopes associated with the low resolution slit settings of a given spectrometer. Comparisons are made with actual spectral data obtained with various degrees of spectral resolution. The computer program is also used to generate graphical inversion techniques for analyzing experimental spectra. An example is given in which one such graphical technique is used to obtain average temperatures and number densities along the axis of an axisymmetric duct containing a>supersonic diffusion flame. Another example is presented to demonstrate the manner in which a second inversion technique can be used to obtain radial profiles of temperature and OH number density from radial scanning of an axisymmetric combustor flowfield. Two cases involving thermodynamic nonequilibrium are also discussed, one of which involves a hot vibrational band and the other an electronic nonequilibrium. NOMENCLATURE AJ b D G g I I 0 K k L N Q q I Re I 2 S T x Einstein transition probability Doppler half width apparatus function degeneracy spectral intensity spectral intensity before absorption absorption coefficient Boltzmann's constant optical path length OH number density partition function Franck-Condon factor electronic transition moment rotational line strength temperature axial distance from combustor inlet

152 II3-2 y a e X v distance along optical path spectrometer amplitude factor energy wavelength wave number Subscripts E eletronic state i layer of gas along optical path j rotational quantum number j spectral line / lower energy state R rotational state u upper energy level v vibrational state or quantum number INTRODUCTION Spectroscopic techniques are useful for analyzing flowfields in which chemical reactions are occurring because they generally do not interfere with the processes being studied. The hydroxyl or OH radical, which is present in the combustion products of many fuel-oxidizer mixtures, is particularly attractive because the radiation associated with the fl electronic transition (Figures 1 and 2) occurs in a region of the ultraviolet spectrum ( A) which is relatively free of extraneous radiation. A computer program has recently been developed at ARL which is designed to predict the ultraviolet emission and absorption spectra of OH associated with combustor flowfields. The computer program operates in four basic modes: emission, emission with self absorption, absorption of a discrete OH line source, and a combination of the last two. The spectra are determined for any degree of spectral resolution (Figures 3, 4 and 5), any apparatus function and any combination of vibrational bands (Figures 1 and 6) or rotational branches (Fig.2). Cases involving thermodynamic nonequilibrium can be accommodated by specifying the rotational, vibrational and electronic temperatures (Figs 23-26). While the intended purpose of the computer program is to facilitate the development of spectroscopic diagnostic techniques for specific application to the ARL supersonic mixing and combustion research effort 1 ' 2 ' 11, the program has a number of other uses. Of particular importance is the use of the program in conjunction with (or as a subroutine of) numerical combustor flowfield calculations in which OH number density and static temperature distributions are predicted on a point by point basis within the combustor. When the resulting distributions of temperature and OH number density (see solid lines in Figures 10 and 11) are used as input data to the spectroscopic computer program, the resulting absolute intensities of the associated OH emission (Fig. 12) or absorption (Fig.21) spectra can be produced along any optical path (Fig.20). Whether or not the expected absolute intensity of the emission spectra or the expected percentages of absorption are sufficiently high to obtain good experimental data can therefore be determined. Comparisons can be made of the emission spectra with and without self absorption (Fig. 12). Likewise, the intensity of the absorption source, necessary to dwarf the natural emission of the absorbing gas can also be determined. As shown in Figures 3, 4, and 5, the apparent increase in intensity and associated smearing together of spectral lines caused by opening up the slits on the spectrometer can also be determined. Spectroscopic techniques for inverting temperature and number density distributions from experimental spectra can be tested by applying the technique to theoretical spectra 1 ' 2 (Figures 21 and 22) generated by the computer program. The effectiveness of a given technique in inverting various types of number density and temperature distributions can thus be observed (Figures 10 and 11). Obviously, the validity of this-type of test or the validity of any spectroscopic technique depends on the accuracy of the mathematical models used to represent the radiating or absorbing gas. It has therefore been the intention here to represent the radiating and absorbing gas within the computer program using the most complete and accurate models which are consistent with the intended application of the resulting techniques. The mathematical modeling is presented in the next section. In addition to testing spectroscopic inversion techniques, the computer program can also be used to generate inversion techniques for application to specific problems. The sample techniques described in later sections have been generated in response to the specific problems associated with a recent series of supersonic mixing and combustion tests conducted at ARL. In planning these tests, numerical combustor flowfield calculations used in conjunction with the spectroscopic program indicated that insufficient intensity in emission would be produced to

153 obtain good experimental data. Furthermore, it was found that even if the conditions at which the experiments were to be carried out were adjusted to obtain sufficient intensity in emission, self-absorption would be a serious problem. Similar calculations, however, indicated that sufficient absorption could be expected for the original operating conditions as long as at least 10 to 20 percent of the hydrogen fuel were to be burned. From an experimental standpoint, it was necessary to record spectral data with a rapid scanning spectrometer because, of the short duration and quasi-steady nature of the burning process. Good high resolution spectral data could not be obtained in conjunction with rapid scanning, however, because of the turbulent and oscillatory nature of the combustion process. It was found, however, that low resolution spectral data in which the individual spectral lines blend together into an intensity envelope (Fig.5), could be used in conjunction with rapid scanning. The oscillations and turbulence merely resulted in a ripple appearing on the intensity envelope, but not a change in the basic shape. It was therefore desirable to generate a technique for inverting low resolution absorption spectra obtained from an axisymmetric combustor. Some conventional high resolution inversion techniques utilize mathematical models which have been greatly simplified for the purpose of obtaining closed form solutions for temperature and number density. It will be shown below that some of the assumptions made in these simplifications, at least as related to the present case, are invalid. The additional complication of the overlapping rotational lines, due to the low resolution, makes it nearly impossible to obtain closed form solutions even with these simplifying assumptions. Thus, it has been necessary to develop graphical inversion techniques by performing numerical parametric studies using the spectroscopic computer program. One advantage of the graphical techniques is that the ease with which they are applied is relatively independent of the sophistication of the mathematical model. It should be pointed out that, while the graphical techniques discussed below are general, they have some dependence on the specific apparatus used at ARL. The necessary graphs and charts presented here can easily be reproduced for any given apparatus by including the apparatus function and associated spectral half widths of the spectrometer and the -rotational and vibrational temperature of the source lamp within the computer program. The lamp temperatures themselves can be obtained by trial and error using the computer program in the emission mode. II3-3 MATHEMATICAL MODEL The ultraviolet bands 3 of OH result from energy transitions between the first excited electronic state 2 S + and the ground electronic state 2 II. The various bands result from vibrational quantum jumps between the two electronic states as shown in Figure 1. Each band is composed of a series of rotational lines, resulting from rotational quantum jumps, which occur in conjunction with the associated vibrational jump as shown in Figure 2. The OH radicals which emit and absorb radiation that enters the spectrometer are located along the optical path as shown in Figure 9. In order to predict the spectra resulting from distributions of temperature and OH number density along the optical path, the path is divided into any number of segments n of length Ay ; so that where S A Yi = L (1) 1=1 Ayj = Yi-yj-i (2) and L is the total length of the optical path through the test gas. The temperature and OH number density are assumed constant within each of these segments or layers of gas and equal to the average of the actual distribution. Since a spectrometer can only be focussed at one position at a time, most of the optical path will be viewed from an out-of-focus position. According to References 5 and 6, the effect of the increasing area viewed in the off-focus position is exactly compensated for by the loss of radiation due to off-slit focussing within the spectrometer. The total emitted and absorbed radiation may thus be calculated without regard to the cross sectional area of the actual optical path as it traverses the gas. The spectrometer is assumed to be located on the right side of Figure 9, while the absorption source lamp (if any) is on the left. The spectral intensity at wavelength X is defined as = SaG(X, Xj) lj(xj, 0) -f AIj(Xj, Ay ; )l (3) J j where: a represents the increased amount of light entering the spectrometer as the slits are opened up, G(X, X:) is the apparatus function for the particular spectrometer and slit setting, and Ij(X:, 0) is the intensity of radiation (due to the j 1 * 1 transition) entering the test gas from the absorption source lamp. If simple emission or emission with self-absorption is considered, Ij(X;, 0) is set equal to zero. The term AI:(Xj, Ay;) is the contribution of a given layer i to the intensity of the j"i spectral line and is defined as AIj(Xj, Ayj) = Ij(Xj, y;) Ij(Xj, y,_j). (4)

154 II3-4 For simple emission, Equation (4) can be written as AIj(Xj, Ay;) = NIJU> where the population, NJJ U of the upper state in the i* layer can be written as Ng E exp(-e E /kt E )g v exp(-e v /kt v )g R exp(-e R /kt R ) NJJU = - J QE Qv QR In the above expression, thermodynamic equilibrium has been assumed within the various energy modes, but not between the modes. The integrated transition probability is given by References 3 and 4 A; = -3- lre! 2 q v., v. S rr. (7) 1 3h g E gj The above equations for emission are used to determine both the emission spectra of radiating combustion gases as well as the emission spectra of the narrow OH line source. For absorption from a narrow OH line source, Equation (4) is written as -l) (8) where K(Xj, y t ) is the peak value of the absorption coefficient for the j th spectral line of the gas in the 1 th layer. The exact shape of an emission line has no influence on low resolution spectral intensities as long as the half width of the apparatus function is much greater than the half width of the emission line. The shape of the absorption lines on the other hand is very important because the amount of radiation absorbed depends both on the line shape of the absorbing radicals and the line shape of the incident radiation. When a microwave discharge cavity 7 containing water vapour is used as the discrete OH line source, the width of the source lines are narrower than the line widths of the absorbing gas 8. Therefore, using the peak value of the absorption coefficient is not a bad approximation. For emission with self absorption, the intensity is written as *" (9) where the average rather than the peak value of the absorption coefficient has been used since the shape of the emitting and absorbing lines are nearly the same. The expression for the case of absorption of a discrete OH line source plus emission with self-absorption is the same as Equation (9) except that the peak rather than the average value of the absorption coefficient is used, since the emission is assumed to be a secondary effect. Assuming Doppler broadening only, the peak absorption coefficient can be expressed in terms of the integrated transition coefficient as where b D is the Doppler half width and K(X, = u A 00) J J V IT. Nj 8Hc b D g, v] NJJ; is the population of the lower level. A complete description of the mathematical model, calibration and apparatus considerations, the computation scheme and operating instructions for the computer program are given in Reference 9. SAMPLE CALCULATIONS The effect of slit width is illustrated in Figures 3, 4, and 5. Figure 3 shows the high resolution emission spectra for a hypothetical spectral half width of.1 A. Figures 4 and 5 show effects of slit width for a spectrometer currently in use at ARL. For slit widths of.2 and 1.0 mm, the spectral half widths are 1.58 and A, respectively. The calculations were carried out for a temperature of 3000 K and an OH optical depth of 1.0 x radicals/ cm 2. The apparent increase in intensity shown in Figure 5 is due to the increased amount of radiation entering the spectrometer as well as the overlapping of many more spectral lines. Various vibrational bands in simple emission are shown in Figure 6, the same temperature, OH number density and slit width used in Figure 5. The 1-1 vibrational band and the 0-0 band can be seen to overlap beyond A.

155 Absorption calculations are shown in Figure 7 for two different temperatures and optical depths of the absorbing gas. The slit width is.5 mm for which the corresponding spectral half width is 5.22 A. The computation pf the intensities of the OH absorption source was carried out using a vibrational temperature of 2400 K and a rotational temperature of 1400 K. The electronic temperature and optical depth were chosen so that the absolute intensities were equal to those obtained experimentally. The numerically determined spectra of Figure 7 can be compared to similar experimental spectra shown in Figure 8 which were obtained during a supersonic combustion experiment in which thermal choking occurred. The absorption source is shown before the onset of combustion in the upper left corner of Figure 8. The absorption spectra obtained during supersonic and subsonic portions of the combustion experiment are shown in the upper right and lower right, respectively. During the transition from supersonic burning to subsonic burning (thermal choking), highly oscillatory spectra such as that shown in the lower left were occasionally observed. The four basic modes of operation of the computer program are illustrated below using the numerically predicted temperature and number density profiles 2 shown in Figures 10 and 11 for an axisymmetric duct and a slit width of 1.0 mm. The optical path was divided into 26 different isothermal layers to approximate these distributions. The results of the simple emission and emission with self-absorption modes of the program for viewing along the diameter of the flowfield are shown in Figure 12. As can be seen, the effects of self-absorption are strong even though the absolute intensity is quite low. The results of the absoption calculations of a discrete OH line source are illustrated in Figure 21 for the six optical paths shown in Figure 20. The associated transmittances are shown in Figure 22. The rotational and vibrational temperatures for the absorption source used in Figure 22 were both assumed to be 1500 K. Due to the low intensities of the emission spectra shown in Figure 12, the results of the fourth mode of the program, which includes the effects of emission with self-absorption in addition to the absorption of a discrete OH line source, were of negligible difference to those of Figure 21. It is often assumed in spectroscopic studies that the degree of absorption is independent of the absorption source. In other words, the transmittance I/I 0 is assumed to be the same regardless of the temperature of the absorption source. Figure 13 shows the results of numerical calculations in which two different test gases were specified in conjunction with three different absorption source lamp temperatures. The results show significant dependence on source lamp temperature for the region beyond 3110 A in which the 1-1 vibrational band is located (see Figure 6). It can therefore be seen that the correct rotational and vibrational temperature of the source lamp must be known if accurate spectra are to be predicted. Another assumption which is often made is that the shape, but not the magnitude, of the transmittance curves are independent of optical depth. Figure 14 shows the results of numerical calculations in which the transmittance of 16 test gases, using four different optical depths and four different temperatures, were determined for a given absorption source temperature. As can be seen from Figure 14, the shapes of the transmittance.curves for a given temperature definitely are dependent on the optical depth. II3-5 USE OF PROGRAM TO GENERATE DIAGNOSTIC TECHNIQUES As pointed out in the Introduction, a general diagnostic technique applicable to all cases can not be developed because the spectral half widths of a given spectrometer as well as the rotational and vibrational temperatures of the source lamp must be included in the development of the technique. It is the purpose of this section to give three examples of how the computer program can be used to generate diagnostic techniques for use in specific cases. The first technique which will be discussed is the two-wavelength technique which is similar to the two-line technique of high resolution spectroscopy. A numerical parametric study has been carried out for the absorption of radiation from a 1500 K equilibrium source through isothermal layers of gas with temperatures ranging between 1000 and 2400 K and optical depths between 1.0 x and 5.0 x radicals/cm 2. Figure 14.shows the results of only a portion of the total number of cases considered. As can be seen from Figure 14, the ratio I(X, L)/I(X, 0) is very nearly temperature independent at X = 3135 A. The degree of temperature independence at 3135 A is illustrated in Figure 15 where the transmittance at 3135 A is plotted as a function of optical depth for various temperatures. Thus, by measuring the transmittance at X = 3135 A, it is possible to determine the optical depth regardless of the temperature. Alternately, the widest variations in transmittance as a function of temperature as observed in Figure 14 occurs for wavelengths of 3090 A and 3160 A. The two-wavelength technique is based on determining the ratio of absorptivity at 3090 A or 3160 A to the absorptivity, at 3135 A. The absorptivity, defined as one minus the transmittance, is plotted as a function of optical depth for various temperatures in Figure 16 for the 3090 A case. As might be expected, the normalized absorptivities shown in Figures 16 and 17 are nearly independent of the optical depth for the lower values of optical depth. By observing the transmittance curves of Figure 14, it can be seen that, below an optical depth of 2.5 x radicals/cm 2, the absorptivity at 3160 A is too low to be measured accurately. Likewise, above 2.5 x radicals/cm 2 the absorptivity at 3090 A is not sensitive to increasingly higher optical depths. Thus, once the optical depth has been determined from Figure 15, the temperature of an isothermal gas can be determined from either Figures 16 or 17.

156 II3-6 The question arises as to what would happen if the two-wavelength technique were applied to the absorption spectra resulting from a non-isothermal distribution of temperature. As an example, if the transmittance curve marked with a six in Figure 22 (which corresponds to the 6th optical path in Figure 20 and the distributions of temperature and OH number density of Figures 10 and 11) is analyzed using Figures 15, 16 and 17, the resulting temperature is 1860 K and the optical depth is 1.0 x These numbers can be compared to the average temperature along the optical path 1740 K and the actual optical depth of 9.0 x radicals/cm 2. When the twowavelength technique is applied to the theoretical spectra resulting from the radial distributions of temperature and OH number density predicted to exist along the diameter at several other axial locations within the combustor, axial distributions of average temperature and OH number density can be determined. Figures 19 and 20 show the results of the two-wavelength technique as compared to the actual average temperatures and actual optical depths for three axial locations within the combustor when viewing is limited to optical paths traversing the diameter of the duct. Thus, it appears that the two wavelength technique can be used to obtain optical depths and average temperatures for optical paths along the diameters of an axisymmetric combustor flowfield. In those instances where off-diameter optical scanning is possible, the two-wavelength technique can be used to obtain the actual radial temperature and number density distributions at a given axial location. The entire procedure is too lengthy to discuss in detail here, but is discussed at length in Reference 9. Basically, the combustor flowfield is assumed to consist of several isothermal rings with corresponding constant OH number densities. There may be as many rings as there are off-diameter optical scans. Figure 20 shows a combustor cross section of 11.8 cm diameter which has been divided into six concentric rings with an equal number of optical paths. Numbers proportional to the areas scribed out by the intersection of the optical paths with the concentric rings are assumed equal to the optical path lengths through the various isothermal regions. The temperature and optical depth of the outer isothermal region are determined first using the two-wavelength technique as described above. The OH number density of the outer ring is then determined by dividing by the path length. The second optical path passes through three isothermal regions of which the properties of the outer two are already known. The absorption spectra obtained from the second optical path is adjusted so that the effects of the outer two regions are taken into account or "added back in", so to speak. The adjusted spectra is then evaluated according to the two-wavelength technique just as in the case of the outer optical path. In a similar fashion, the spectra from the third and subsequent optical paths are adjusted to take into account the effects of the outer layers for which the properties have already been determined. As the calculations proceed from the outer optical path inward, the errors are additive to some extent so that the properties determined for the inner rings are less accurate than those of the outer rings. The outer rings, however, represent a higher percentage of the total amount of gas for a given cross section than the inner rings. When the above procedure was carried out for the six transmittance curves of Figure 22, the results were as shown in Figures 10 and 11. The maximum number of rings which can be assumed depends on the amount of absorption. Experimentally, it is difficult to measure less than 10 percent absorption at any given wavelength. It is not necessary that a great deal of change occur in the transmittance from one optical path to the next, however, as witnessed by the results shown in Figures 10 and 11 from the 4th, 5th and 6th optical paths for which the transmittance curves are nearly identical. The second sample diagnostic technique is similar to the isointensity technique used in high resolution spectroscopy. Instead of finding two spectral lines of equal intensity, two wavelengths of equal transmittance are used to determine the temperature. For this second technique, different temperature conditions (T R = 1400 K, TV = 2400 K) have been assumed for the narrow line OH source lamp in an attempt to take into account the "hot" 1 1 vibrational band of the lamp used at ARL for the absorption work. Transmittance curves similar to those of Figure 14 are given in Figure 23 for these different source lamp temperatures. The principle on which the isotransmittance method is based is illustrated by the dotted line denoted as B in Figure 23. The dotted line B represents the wavelengths at which the transmittance is equal to the transmittance at 3080 A. As the temperature increases, the wavelength for which the transmittance is equal to that at 3080 A increases. The temperature dependence of this isotransmittance wavelength is shown in Figure 25 for temperatures ranging from 400 to 2800 K. In a similar fashion, the wavelengths for which the transmittance is equal to the transmittance at 3072 A are shown in Figure 26 as a function of temperature. The isotransmittance wavelengths are weakly dependent on optical depth as can be seen in Figures 25 and 26. The optical depth can be determined, once the approximate temperature and minimum transmittance of the spectra are known, by using Figure 24. In practice, several independent temperatures can be determined from Figures 25, 26 and others like them. The temperatures determined in this fashion are then averaged to reduce experimental error. The agreement between the individual temperatures before averaging is a good indication of the validity of the technique and the measurements. Experimental results for a supersonic diffusion flame using the isotransmittance technique appear in References 11 and 12. The most straight-forward technique for analyzing experimental data involves direct comparison with theoretical spectra by the trial-and-error process. Such a procedure was employed to determine the vibrational and rotational temperature of the narrow OH line source used at ARL. The direct comparison method was also used in the present study for a case in which chemiluminescence was observed to occur. During a supersonic mixing and combustion test (References 11 and 12), emission and absorption spectra were obtained. Once the emission spectra had been properly subtracted from the transmittance spectra, a rotational temperature of 1310 K and an optical

157 depth of 6.0 x was determined using the isotransmittance technique. When these values of temperature and optical depth were used as input to the computer program, the predicted absolute intensity in emission was orders of magnitude lower than what was actually measured. Only by assuming an electronic temperature of slightly more than 3000 K could the intensity predicted by the program be made to match the measured intensity. This high electronic temperature, characteristics of chemiluminescence, was therefore determined by a combination of the isotransmittance and directed comparison techniques. II3-7 SUMMARY A computer program has been developed which calculates low resolution ultraviolet spectra associated with the FI electronic transition of the OH radical. The program operates in four basic modes: emission, emission with self-absorption, absorption of a narrow OH line source and a combination of the last two. The program calculates absolute intensities over a wide range of spectral resolution resulting from any spatial distribution of OH number density and temperature. Thermodynamic nonequilibrium can also be included for which case the distributions of rotational, vibrational and electronic temperature are specified. The primary use of the computer program is to facilitate the development of diagnostic techniques for analyzing low resolution spectral data from quasi-steady and highly turbulent supersonic hydrogen-air flames which occur in the ARL supersonic mixing and combustion facility. Three examples are given of how the computer program can be used to generate diagnostic techniques. One technique which is called the two-wavelength technique is similar to the conventional two-line technique. An example is given to illustrate that the two-wavelength technique can be used to determine both the average centerline temperature along the axis of an axisymmetric duct or the radial distributions of OH number density and temperature for a given cross section. Another technique is known as the isointensity technique. The third technique is a trial-and-error method which is referred to as the direct comparison technique. In addition to developing diagnostic techniques, the computer program can also be used in conjunction with other numerical programs which predict combustor flowfield properties. The expected intensities in emission and percentages of absorption can therefore be predicted for various locations within the combustor flowfield. The effectiveness of a given inversion technique in inverting the specific types of temperature and number density distributions expected in a given experiment can be found by applying the technique to the predicted spectra as if it were experimental data. REFERENCES 1, Drewry, J.E. Dunn, R.G. Scaggs, N.E. 2. Drewry, J.E. 3. Dieke, G.H. Crosswhite, H.M. 4. Golden, D.M. Del Greco, P.P. Kaufman, F. 5. Nielsen, J.R. 6. Sawyer, R.A. 7. Fehsenfeld, F.C. Evenson, K.M. Broida, H.P. 8. Kostkowski, H.J. Broida, H.P. 9. Neer, M.E Supersonic Combustion Studies at Low Density Conditions, AIAA Paper No , presented at AIAA 5th Propulsion Joint Specialists Conference, US Air Force Academy, Colorado, 9-13 June Supersonic Mixing and Combustion of Coaxial Hydrogen-Air Streams in a Duct, ARL , December The Ultraviolet Bands of OH, Journal of Quantitative Spectroscopic Radiative Transfer, Vol.2, pp , April-June Experimental Oscillator Strength of OH, 2 S + 2 n, by a Chemical Method, Journal of Chemical Physics, Vol.39, No.ll, December Journal of the Optical Society of America, 20, 701, 1930: 37, 494, Experimental Spectroscopy, Third Edition, Dover Publications, Inc., New York, Microwave Discharge Cavities Operating at 2450 MHz, The Review of Scientific Instruments, Vol.36, No.3, March Spectral Absorption Method for Determining Population Temperatures in Hot Gases, Journal of the Optical Society of America, Vol.46, No.4, April Numerical Calculations of UV Emission and Absorption Spectra of OH, ARL Technical Report, to be published. Conference on Extremely High Temperatures, John Wiley & Sons, Inc., Library of Congress Catalog Card No , Boston, Massachusetts, March 1958.

158 Drewry, J.E. Supersonic Mixing and Combustion Studies of Ducted Hydrogen-Air Flows at an Inlet Neer, M.E. Air Mach Number of 2.6, AIAA Paper No , presented at AIAA/SAE 9th Scaggs, N.E. Propulsion Conference, Las Vegas, Nevada, 5-7 November Neer, M.E. Low Resolution OH Spectroscopy in a Supersonic Hydrogen Air Flame, ARL Drewry, J.E. Technical Report, to be published.

159 II3-9.-.CM -CM _ u. u. It U. J- b i-h ' H j (CM ICM to to -ICM ICM CM CM HCM-ICM -KM ; to CM - o t-'i 7 ij 7 (t) d (3; & d (l) c d ( ) 0 ( 1 ) O CM _ M K) C\ H l CM \ \l + (t.) a S-3\ Ivj (2) a y (E) 'd (3) 'd (1 ) 'd (E; o (3) '0 1 ) 'O HCM _, CM \ \ ^ IT> * \ tl CM r 1 H. i i \ i CM l~ \ \ ( M * \ \ \ + t t x> cv f CM HCM u CM -1 CM M - -lev. l ^0 CM «l - CM a ces 1.o ^-»2 JO u a I O U C 5. c c o p tu ed "S cd r^.q I > _0 A9W3N3

160 I _l b f UJ n Q_ O 0< to I 6 O.c TJ a WAVELENG CO 2 ICO C.o.ss u a o U 3 A1ISN31NI.? tu 0 CD to IO E E CM I. * I- o Q r c^ > Cj ^ '-' to t '" > n _l --, 1- in ^---~ EE^== ~ =gr- =- ^~" ".. ~^r ~*=^~^~~ "=- - "=" S!^_ ^. "' ~5i~ ^^. ^_. ' 0 2 q & 8 S S S. 9 g r ' ( uiu. NVIQId31S/ SllVMOdDIW ) A1ISN31NI o CM R CM to Q ^> ro 8 to 0 cd ro 0 CD O ro 0 o< 1 yuj 1 '? W 0 O rf o CO t o -a a. f «I " 3.y o u ^ z1 ^ m E

161 -o CO X) ~a e an a, o o 2 -*-* O II3-11,0 ' CO o <a 3 eh E ( 2uiui. NVIQIcJ31S/SllVMOdOIW ) A1ISN31NI 6 c> a co I I UJ 1 CO 2 o u a. CA C o u a o u 3 Z A1ISN31NI o U.

162 ABSORPTION SOURCE T R = 1400 K TV =2400 K NL= 5X I0 l5 cm" 2 T eq = 2000 K NL = 5 X I0 l6 cm~ WAVELENGTH A Fig.7 Numerical absorption spectra for a slit width of 0.5 mm

163 NO BURNING SUPERSONIC BURNING TRANSITION TO SUBSONIC BURNING Fig.8 Experimental absorption spectra obtained from combustor for a slit width of 0.5 mm' ABSORPTION SOURCE INTENSITY INDIVIDUAL LAYER OF EMITTING FINAL INTENSITY AT AT WAVELENGTH X; OR ABSORBING GAS WAVELENGTH Xj Fig.9 Schematic of optical path

164 CM 00 u X E E ot. a in 0. o e> z o UJ o E C co c "H to CM Iin o D. o u 1 ~: 10 9 _OI X N CM so E O UJ g en E o 1 co g. (D I.2Ti>_ W) in ro Q UJ O 10 o Q. O» 'S2 2> t-i co.5 a "3 If C4-H f O P o RI 8 CM O o o oco o o o o ocd oro o 0> o I CM 8 O o o o o(d en co S'i ^ ex E o O ao E

165 ao f r 5.6 GC UJ 4.8 EMISSION SPECTRA RESULTS FROM DISTRIBUTIONS OF FIGURES 10 AND II 4.0 cc o EMISSION WITH SELF ABSORPTION O WAVELENGTH A Fig. 12 Numerical emission spectra with and without self-absorption I.Or OPTICAL DEPTH 5 X IO 16 SOURCE TR- TV I500 K T R ' TV 1800 K g) TR TV 2250 K 306O 3O WAVELENGTH A Fig. 13 Numerically determined transmittance curves for various OH line source temperatures

166 o CO o< IO IO to a. O c o o C «C CO o m en 90 IL CD o ) Hld3Q 3 O O E I o I so U- ( 30NVllll/\ISNVyJL 39VlN30y3d ) 01 / I

167 cvi o X Q. UJ Q to Fig. 16 Intensity ratios used in two-line technique for 3090 A CM 'Eo CL UJ Q O Eo IO 16 : 1 i ' IABS,,,, / IABS,,., Fig. 17 Intensity ratios used in two-line technique for 3160 A

168 RESULT FROM GRAPHICAL INVERSION TECHNIQUE N U NUMERICAL CASE 10" CL UJ O < O Q. O IO 1,13 24 X( in) 36 Fig. 18 Optical depths determined from graphical inversion technique 2400r X e UJ CE ce tf RESULT FROM GRAPHICAL INVERSION TECHNIQUE 1 UJ I NUMERICAL CASE X( in) Fig. 19 Temperature profile obtained from viewing combustor along axis

169 SUBSCRIPTS \ OPTICAL PATHS I L l.59\ J 2 O/ X \3 J /1.23/ "NJ.45\I.23\ O/ \ \3 / /I.08/I.I5/I.3I 3.86 X. I.3I\I.I5\I.O8\ /4 \ \ \ / / 1.01 / 1.03/1.06/1.16 / 3.05 \I.I6\I.06\I.03\ 1.01 \ ( 5 Fig.20 Division of axisymmetric combustor flowfield into six isothermal rings ABSORPTION ' SOURCE UJ SEE FIGURE 20 FOR.1) LOCATIONS OF VARIOUS OPTICAL PATHS WAVELENGTH A Fig.21 Numerical absorption spectra from off-diameter optical paths of axisymmetric combustor flowfield

170 a o I '3 I o S 3 u a u< 30NV-U.IWSNVdl I ro O 10 ro O ro O CM to o< X r- Ul UJ 1 o c a il ta C 0. ca, H cd ^ o 7 *".S S o S 1 _>. -o (N r-i cb oi /I

171 TEMPERATURE K CM UJ O O to NARROW LINE SOURCE T R I400 K Tu 2400 K O MINIMUM TRANSMITTANCE I / I 0 Fig.24 Minimum transmittance as a function of temperature and optical depth NARROW LINE SOURCE T R I400 K T v = 2400 K TEMPERATURE K u I20O I UJ o < o 0. o i WAVELENGTH A Fig.25 Wavelengths for which transmittance equals transmittance at 3080 A

172 NARROW LINE SOURCE TR «1400 K Tu ' 2400 K TEMPERATURE K CM I Q_ \ 0. O IO 15 i 1,,,, 1,,, I, i, i WAVELENGTH A Fig.26 Wavelenths for which transmittance equals transmittance at 3072 A

173 114 METHODE ANALYTIQUE DE PREVISION DES TAUX DE REACTION CHIMIQUE EN PRESENCE D'UNE TURBULENCE NON HOMOGENE (APPLICATION A LA COMBUSTION TURBULENTE) par Roland Borghi ONERA Chatillon, France

174

175 114-1 METHODE ANALYTIQUE DE PREVISION DES TAUX DE REACTION CHIMIQUE EN PRESENCE D'UNE TURBULENCE NON HOMOGENE (APPLICATION A LA COMBUSTION TURBULENTE) Roland Borghi ONERA Chatillon, France RESUME Nombreux sont les problemes pratiques oil Ton tire parti de phenomenes chimiques en milieu ou ecoulement turbulent; on peut citer par exemple 1'etude de la composition chimique de la composition chimique de la stratosphere et de son evolution, les lasers chimiques, les reacteurs du genie chimique, et, ce qui nous interessera ici au premier chef, le developpement des flammes turbulentes. Dans de tels problemes, il se produit une influence rsciproque entre la turbulence (c'est-a-dire des fluctuations aleatoires de vitesse, concentration, temperature) et les reactions chimiques, influence souvent tres importante en pratique, et qui revet differents aspects difficiles a prevoire et a calculer. Nous presentons ici un etat d'avancement d'une approche theorique de ces phenomfines, basee sur une methode classique en turbulence non homogene d'ecoulements non reactifs, ou 1'evolution des fluctuations est suivie et calculee par 1'evolution de leurs moments d'ordre 2. Cette methode permet d'abord de distinguer les differents phenomenes et les differents aspects du probleme, mais aussi d'obtenir un premier calcul quantitatif de 1'influence de la turbulence sur les taux de reaction, en prolongeant 1'etude analytique par des calculs numeriques; elle pourra, et devra, etre amelioree dans ces approximations pour se rapprocher plus pres de la realite, dans certains domaines. SUMMARY Many are the practical problems taking advantage of chemical phenomena in turbulent media or flows; let us mention for instance the study of the stratospheric chemical composition and of its evolution, chemical lasers, chemical engineering reactors and especially, the development of turbulent flames, which is our prime interest. In such problems there is a reciprocal influence between turbulence (i.e. random fluctuations of velocity, concentration, temperature) and chemical reactions, an influence often very infportant in practice and which presentes various aspects difficult to predict and to calculate. In'the paper we describe the present situation of a theoretical approach to this phemonenon, based on a conventional method for studying turbulence in inhomogeneous, non-reactive flows, where the fluctuation evolution is followed and calculated by the evolution of their moments of order 2. This method permits one to first define the various phenomena and the various aspects of the problem, but also to obtain a first quantitative calculation of the influence of turbulence on reaction rates, by adding numerical calculation to analytical analysis; it can, and should be improved as regards its approximations in order to draw nearer to reality, in some domains.

176 II4-2 NOTATIONS B facteur de frequence Cj concentration (nombre de mole par unite de volume) du constituant i Cp capacite calorifique du melange, a pression constante d m coefficient (unique) de diffusion moleculaire f, g representent des fonctions aleatoires quelconques h enthalphie statique du melange k energie cinetique de la turbulence k r K vitesse sp6cifique d'une reaction (fonction de la temperature) represente le combustible /, L longueurs caracteristiques de la turbulence (respectivement longueur integrate et de dissipation) Oftj masse molaire du constituant i m debit O repr^sente 1'oxydant p pression statique du melange P represente les produits de combustion ou inertes PIT Qn coefficients fonction de T, definis par (20) R constante massique du melange (supposee constante) r s j coefficient stoechiometrique massique du corps i T temperature statique du melange T A temperature d'activation de la reaction chimique p. a composante de la vitesse dans la direction x a V volume total du reacteur a w taux de production (nombre de mole par unite de masse et de temps) de la reaction chimique YJ fraction massique du constituant i i Z fonction de Zeldovitch (definie par (12)). AQ (ou AH) chaleurs de reaction (definies par (1) ou (38)) 6j coefficients de diffusion turbulente (definis par (26) et (27)) 5 a 0 symbole de Kronecker (=lsi a = (3, =0 si c* = j3) 0 fonction "richesse" (definie par (11)) $ densite de probabilite d'une variable aleatoire 0, 6' constantes fonctions de distributions intiales X coefficient de conduction de chaleur moleculaire p. coefficient de viscosite moleculaire v coefficient stoechiometrique molaire p masse volumique T a 0 tenseur de frottement visqueux 1. INTRODUCTION Dans de nombreux systemes pratiques ou une combustion a lieu dans un ecoulement (flammes premelangees ou de diffusion), les reactions chimiques de combustion se produisent dans une couche de melange turbulente. Dans ces conditions, il se produit une influence couplee reciproque, souvent tres importante, entre les phenomenes chimiques et turbulents. Cette influence est tres difficule a prevoir et a calculer (car les phenomenes chimiques se produisent a 1'echelle moleculaire, alors que notre connaissance des caracteristiques de Fecoulement turbulent n'est jamais exacte a une echelle aussi petite, a cause des fluctuations turbulentes), et peut prendre differentes formes suivant les echelles et intensites respectives des phenomenes chimiques et turbulent. A 1'heure actuelle, et dans la majorite des cas interessants, les experiences ont montre la difficulte de prevoir theoriquement les caracteristiques de tels ecoulements, meme les carateristiques globales, telles, par exemple, que la longueur ou Pepaisseur des flammes turbulentes.

177 Nous presentons ici 1'etat d'avancement d'etudes sur ce sujet basees sur une approche particuliere de calcul des phenomenes aleatoires de combustion, s'appliquant dans des cas de flammes, qui seront detailles plus loin. D'autres approches aussi sont possibles et ont fait 1'objet de travaux, mais elles sont soit plus empiriques et done moins generates, soit plus generates mais moins facilement utilisables en pratique. L'approche que nous presentons ici nous semble constituer un bon compromis pour les ecoulements turbulents non homogenes; elle est basee sur 1'utilisation des moments de fluctuation et la modelisation des equations des moments d'ordre 2 (pour eliminer en particulier les moments d'ordre 3). II UN PEU DE BIBLIOGRAPHIE 2.1 Experiences de Base C'est sur des flammes a faible vitesse (bee Bunsen) que furent d'abord etudiees les flammes turbulentes premelangees, par Damkohler 1 et Summerfield 2 en particulier. De toute evidence les flammes turbulentes (reperees par un nombre de Reynolds calcule avec le diametre du bee Bunsen) etaient plus epaisses, se propageaient plus rapidement et quelquefois presentaient meme un aspect grossierement fluctuant (avec des "bouffees") contrairement a la finesse et la stabilite des flammes laminaires. Plus tard, furent etudiees les flammes turbulentes a grande vitesse surtout dans le cas de flammes premelangees se propageant dans un conduit a partir "d'accroche-flamme" inerte; les etudes les plus significatives sont celles de Talantov 3 ainsi que celle de reference 4. Les memes conclusions qualitatives ont ete tirees, et des courbes presentant 1'influence de divers parametres (la richesse, la temperature, la vitesse du melange frais) sur Failure globale de la flamme ont pu etre etabilies. Les figures 1, tirees de reference 4, montrent de telles courbes, ainsi que les figures 2, tirees de reference 3. Plus recemment des flammes turbulentes stabilisees par "flamme-pilote" ont ete etudiees, avec des moyens plus modernes, tels que la mesure de 1'intensite de la turbulence par anemometre a fils chauds ou la photographic ultra rapide (references 6 et 7). 2.2 Les Enseignements Tires de ces Experiences Toutes ces etudes experimentales ont eu pour resultats, d'abord de permettre une classification des differents types de flammes turbulentes, puis de tenter d'obtenir des formules reliant la vitesse de propagation de la flamme en regimes turbulent et laminaire (ceci pour les differents types), enfin de constater que, pour les types les plus interessants, le comportement etait difficilement previsible par une etude theorique, a 1'heure actuelle. Une classification claire des differents types de flammes turbulentes a ete recemment proposee par M. Barrere 8 d'apres celle de Damkohler, qui tient compte a la fois d'une longueur et d'une vitesse caracteristique de la turbulence (la vitesse caracteristique c'est la racine carree de 1'energie cinetique turbulente, la longueur une longueur integrate de turbulence / t ) et d'une longueur et une vitesse caracteristique des phenomenes chimiques (c'est 1'epaisseur de la zone de reaction laminaire /b et la vitesse de propagation de la flamme laminaire v^, ; en comparant trois temps caracteristiques (un temps turbulent t T, un temps chimique "laminaire" t c, et un temps mixte t m ) batis a partir de ces longueurs et vitesse, on obtient 6 domaines particuliers et 6 types de flammes, explicitees sur la figure 3; la separation entre ces domaines n'est, bien sur, pas aussi nette qu'il est indique. Pour la plupart de ces domaines, des formules reliant les vitesses de propagation laminaires et turbulentes ont ete proposees, d'apres les resultats experimentaux (voir par exemple references 1, 2, 3, 4, 5, 6), dont une justification theorique a quelquefois ete esquissee (refs. 3, 5, 9); en ce qui concerne les epaisseurs des flammes turbulentes, les formules sont cependant bien plus rares (voir seulement reference 3). De toutes facons, cependant, les differents types de flammes ne sont jamais bien clairement delimites dans une experience pratique, d'abord parce qu'ils peuvent apparaitre les uns apres les autres si les conditions d'entree de 1'ecoulement a enflammer varient (et qu'on ne connait pas bien les variations resultantes des grandeurs turbulentes, echelle de longueur et energie) et ensuite parce qu'ils peuvent coexister simultanement en differents endroits de la flamme (les echelles et energie turbulentes ne restant pas constantes en tout point d'un ecoulement). Aussi, a 1'heure actuelle, le calcul de la propagation d'une flamme dans un canal, probleme simple et realiste, dans des conditions realistes de vitesse, temperature, richesse d'entree, par des methodes de calcul assez raffmees sur le plan de la mecanique des fluides, est encore impossible. Les comparaisons entre calculs et resultats experimentaux ne sont pas entierement satisfaisantes, comme le montrent les figures 4 et 5, tirees respectivement des calculs de Spalding 5 et Moreau Les Approches Theoriques Surtout depuis deux ou trois annees, beaucoup d'etudes theoriques de ce probleme de calcul de flammes turbulentes ont ete entreprises. Elles peuvent se ranger en deux categories; dans la premiere categorie se trouvent

178 II4-4 des etudes tres reliees aux resultats experimentaux, done a caractere tres pratique, et ou la turbulence n'est considered que sous son seul aspect de fluctuations de vitesse; de ce type sont les etudes de Talantov et al 3 et Spalding 4 ; ces approches sont tres etroitement liees a la classification du paragraphe precedent et comportent une dose assez importante d'empirisme. Aussi, une seconde categorie d'etudes theoriques s'est developpee, qui devrait se passer de donndes experimentales dans une plus grande mesure; dans cette categorie il est apparu alors necessaire d'envisager la turbulence non plus seulement sous son aspect cinetique (fluctuations de vitesse), mais aussi comme comportant des fluctuations de temperature et de concentration des gaz. D'abord ont ete entreprises des etudes de reaction chimique du ler ou 2e ordre isotherme en presence d'une turbulence homogene et isotrope (Corrsin 10, O'Brien 11, Lin et O'Brien 12 ), puis dans une turbulence non homogene (refs. 9, 13, 14); ces approches sont serieusement limitees par la supposition de reaction isotherme, qui n'est valable que dans des cas pratiques peu nombreux (reactifs tres dilues ou reaction peu energetique), mais sont tres instructives, car elles ont mis en evidence les differents problemes qui se possent, et ont ete abordees avec la meme methode, celle des moments (methode classique dans les calculs des phenomenes aleatoires). Plus recemment, Dopazo et O'Brien 15 ont developpe une methode nouvelle qui parait fort prometteuse, mais qui n'est actuellement pas possible hors du cadre de la turbulence homogene et isotrope; elle s'affranchit des approximations inherentes a la methode des moments aussi bien que de la restriction de reaction isotherme. D'autre part, les reactions non isothermes en milieu turbulent non homogene ont aussi ete abordees, mais seulement dans le cas oil les processus chimiques sont tres rapides devant les processus turbulents (ce qui revient a se placer seulement dans certaines zones de la figure 3); c'est le cas de Libby 16, Toor 17 et ensuite Bush et Fendel 18, en ce qui concerne les flammes de diffusion. Les etudes n'ont cependant pas ete" menees, a notre connaissance, jusqu'au point de donner des resultats pratiques. Dans le cas plus general oil les "longueurs chimiques" peuvent etre du meme ordre (ou plus grandes) que les "longueurs turbulentes", le probleme est beaucoup plus complexe et peu d'etudes on ete effectu6es dans le domaine de la turbulence non homogene; on peut citer seulement Williams 19 et Borghi 20 et ces etudes sont pour le moment limitees a des intensites de turbulence faibles, encore que quantitativement le mot faible n'ait pas ete bien precise. C'est surtout ce cas que nous aliens developper dans le chapitre suivant, en montrant d'abord les problemes qui se posent en th6orie, puis 1'importance pratique des differents problemes, enfin en precisant une approche theorique qui, tout en ayant certaines limitations, peut se reveler tres utile dans de nombreux cas. 3. LA METHODE DES MOMENTS POUR UN ECOULEMENT TURBULENT NON HOMOGENE AVEC REACTIONS CHIMIQUES 3.1 Definitions Par souci de simplicite nous considererons ici un ecoulement oil une seule reaction chimique simple du 2e ordre peut se produire, du type: K + i»o -» P. Nous supposerons que la capacite calorifique du melange de gaz est constante, et que seul le combustible possede une chaleur de formation non nulle, AQ ; alors a chaque instant on peut ecrire: h = C p T + AQY K. (D Nous supposons aussi que la masse molaire du melange de gaz est constante p = RpT (2) constitue alors une deuxieme equation d'etat (instantanee) pour les gaz. Pour toute grandeur aleatoire, on peut poser: (a) soit oil T est la moyenne stochastique f = T + f ', (3) T = souvent confondue avec la moyenne sur un certain intervalle de temps

179 II4-5 (b) soit T f r = -f f(t)dt T J 0 est la densite de probabilite de f), f = r.+ f", (4) oil f est une moyenne stochastique de f ponderee par la masse, telle que: (pf) = pf (pt 7 ) = 0. (5) De meme on peut definir, les moments d'ordre n de la fonction f, ou de f' ou de f" par la definition generate:.4-oo Z n = J ; OO (6) et les moments couples de deux ou plusieurs variables aleatoires par des formules du genre: f+oo = J zfzjvtz,, z 2 )dz,dz 2. (7) Dans la suite on aura affaire surtout a des moments du type: (pu a )'u0 ou La connaissance de tous les moments, d'une ou d'un groupe de variables aleatoires revient a la connaissance de la densite de probabilite de ces variables. Moyennons les Aquation (1) et (2) grace a ces definitions; on obtient: a partir de (1); (2) entraine: ou h = C p T + AQY K h = C p T' + AQY K h' = CpT' + AQY K. h" = C p T"+ p = RpT = R(pT + p't') p'"" = R(p'T + pt' + p't' - pt) p = RpT p' = R(p'T + pt"). (8) (9a) (9b) On voit qu'il est plus interessant d'employer la ponderation par la masse, qui conserve a 1'equation d'etat moyennee une form semblable a celle instantannee; ces simplifications se retrouvent dans une certaine mesure dans les equations de bilan, mais pas dans le terme de reaction; aussi, nous continuerons pour le moment d'employer la notation classique, Bien sur, si 1'on neglige les fluctuations de p (ou si I'ecoulement est incompressible) les deux notations se confondent. La moyenne ponderee par la masse a ete introduite par A.Favre en Les Equations de Bilan pour les Variables Aleatoires Pour les variables aleatoires extensives d'une particule fluide, on peut ecrire les equations de bilan classique (en notation tensorielle): 3t a a / ay, \ PUc«Y i = -7 (d m r- L )+ r sipw O.J = K. O) 3x tt 3x a \ 3x a /

180 II4-6 at 3pT a at d\ 3 3p 3 / /3u a 3un \ 2p. 3u_, \ L pu OU^Un a u«= 1- U 1- ^- O n n 3x 3x a /x at\ u a ap p. au e /au a au«2 au-,\ i 3p AQ Sx^vCp 3xQ,/ Cp 3x a Cp 3x a \ax0 3x a 3 3x T / Cp at c p Pu a T = I -- I H I a n '- I (10) ou si nous definissons, a la place de Y 0 et de T, les fonctions: <t> = Y K/ r sk - Y o/r s0 (11) Z = T + AQ C p r sk Y K (12) on obtient un systeme plus simple, pourvu que 1'hypothese X dr (13) soit valable. En effet, en notant on arrive a 3x a (14) _3p at 3 3x a 3t a ap + 3x apv K at 3t + pu a Y K = 3x^ 'X 3Y K \ ] + r sk PW 3x a \Cp 3x a / 3 /X 30 \ = f -) +0 3x a \C p 3x a / (15) 3pZ 3t +., 3x 3 /X 3Z \ u a 3p 1 3p 3x a \Cp 3x a / Cp 3x a Cp 3t 3ufl 3x a r si est un coefficient stoechiometrique massique (r si = ^0??,), pw est le taux de production molaire (par unite de volume et de temps) du a la reaction chimique, de la forme pw = -BC K C 0 T (16a) ou pw = Bp 2 (16b) ou pw = B / T A p 2 Y K Y 0 T a exp( rv\/i c\f\a " "-* l T (16c) 3.3 Les Equations de Bilan pour les Moments A partir des equations (15), oil les variables u a, p, Y K, <j>, Z sont aleatoires, on peut trouver une suite de systemes pour les moments d'ordre 1, 2, 3... de ces variables; la methode est classique (voir par exemple reference 22); dans le cas qui nous occupe, ceci est fait en detail dans reference 21, en utilisant les moments des fluctuations du type f', de 1'equation (3); nous aliens donner ici les systsmes obtenus pour les plus importants moments, c'est-a-dire en negligeant les fluctuations de p et des coefficients X et p. pour ne pas alourdir 1'expose.

181 114-7 Pour les valeurs moyennes (moments d'ordre 1) on trouve 3p (pu a ) = 0 3t 3x a * 3 ~ 3t K. (PU a Y K ) 3x a 3 / X 3Y K 3x a Vc p 3x a \ pu a Y K J + r sk pw 3 ot 3 3x a 3p 3 3x0 3x a -. 0/3 PUaU(?) (17a) 3 _~ 3t f P2 H- 3t ou bien, avec les notations classiques, 3 3x a -(pu a Z). 3 /X 30 3x a \C p 3x a -i^sr P^tf*) \ pu;'z"j+s-.,. 3p 3 + (pu a ) = 0 3t 3x a 3pY K * 3pus 3t 3 3x a \ P Uj^ I j^ ) 3 axa(pu a u0) - i ( \>\ji \ i. a /X 3Y K pu a j I K 1 """ r skp w ox a \Cp ox a ' 3p 3 3 (Pu v a a )u0) 3x0 3x a ^ (17b) 3P0~! 3t 3pZ" 3 (pua 0; 3x a a CnTT 71 = 3 /X 30 1 ( a /x 3z \' P«a )'0'J \ in V7' 1 -(- o oil s est le dernier terme de 1'equation pour Z du systeme (15). On peut r6ecrire aussi (17b) en (17c) en utilisant la premiere equation de (17b): 3p 3 3t 3x n = 0 p at "^ax/- av Ir Sv ^ua^yk J-^sKPw ^Y K 3xa \Cp 3x a / 3r 30a 3 P lf +PUa 3x a^ 3p. 3 (_..,,. 3 -p/ = 3x0 + 3x a(^ (im *W *** ^ (17c) 30 3 p - + pu a - 0 at 3z "ax,, P +pu a - Z 3t 3x a a a /x 30 \ a / = ( - (pu a )>' } ow dx a \C p 3x a ^ a;v / i/ v a /x a/ \ a / = { (pu tt )'z' 1 + s /z', 3x a VCp 3x a ^ «' J 3/ J les termes s et pw sont aussi fonction des moments de fluctuations _ 3p 3p 3p U-v + Un, H 01 3x a a 3x Q 3t i r, ap 3p' c' n' in l n' Cp[ " 3x a a a 3X,, 3u«3ug 1 + '-ii +^J 3?', 3?',, 3Q0. 3u0 3ug 3u0 a a., ' T af}., ' r + a i a r a/i, T a(i., 3x a 3x a 3x K a 3x K a 3x K a 3x a (18)

182 II4-8 et une expression de pw, d'apres (16) peut etre trouvee en fonction des moments; on utilise les developpements en serie de T a et exp( T A /T) en fonction de T'/f, mais ce n'est valable que si T'/T < 1 ; on trouve 20 pw = p 2 Y () Y K T«exp ( T A (-T, F'Y; T'YK\ -=^- + -=*-]( TV I I n TV/ III/' Y 0 Y K P, + Q, + P.Q. 2 ' 2 T ' Y O Y 'K T:Y K.) ^ V + <P 2 + Q 2 -~- T2 I V 1 (P, + Q,)x (P 3 + Q 3 + Pj T-J (pw)' = ou T,\f T' YO YK --^- (P. + Q,) + ^ + =^ T i /L / T Y i - + P = Qn = a(a + 1) (a + n 1) n! (20) Alors dans les equations des moments d'ordre 1 interviennent des moments d'ordre superieur, de trois types: (a) u Q U0 u a 0', yu a Z' les termes de flux (b) u~ " 3xn 3ugV /3u0\/3ug\ - ], - - I dans les termes de sources d energie 3x a / Vax a /\ax a / (c) YgY^, T'YQ, T' 2,... etc, ou ce qui revient au meme tous les moments construits avec les fonctions Y K, 0 et Z (qui sont relies lineairements aux premiers, a cause de la definition de 0 et Z), dans le terme de source de masse. On peut remarquer deja, que s'il n'y a que des moments d'ordre 2 dans les termes de flux et de source d'energie, ce sont des moments de tout ordre qui interviennent dans le terme de source de masse, et ces moments auront d'autant plus d'importance que T A /f sera grand (done soit T A grand, soit T petit), a cause de 1'expression de Pn. Pour les moments d'ordre 2 du type (a) ou (c) on obtient des equations de la forme: p fy+pti a fv = - ((pu a )'f'g') + S(f, g), 3t 3x a 3x a (21) oil les differents termes sources sont donnes par. SOX r _ ua Yu' -, ^ 9u (3-1 u^, 9 3p - P' * S(u0, p ay) = - (pu a ) un - '- - (pu a ) u iu - + up ' M»v ^a ' 33x X a I [_ ' 33x0 Xfl AV 'ay S(u s, y,, -.3Y K,_ 30 R _., 3p' - V V' 3 _ 30 _._, 3Qn,3p', 3r a0, 3 /X 30'\ S(u0, 0) = - (pu a )'u (p^w E- - 0' + 0' ^ + u0 - ) + 0 p p 3x a 3x a 3x0 3x a p 3x a \Cp3x a / _ 3Z _ 300 W, a^afl, 3 /X 30' \ ^- r S(u0, Z) = - (pu a )'u (pu a )Z' E. - Z' - + Z' 2H. + U0 -( - ) + s'u0 p p 3x a 3x a 3x0 3x0 M 3x a \Cp 3x a / 3x a 3x a \Cp 3x a sk

183 II4-9 3Z 3 /X 3Z'\ S(Z,Z) = -2(pu a )'Z' + 2Z' (- )+2s'Z 3x a 3x a \C p 3x a /,_ 30 3 /X 30'\ S(0, 0) = -2(pu a )'0' +20' ( )+0 3x a 3x a \Cp 3x a / 3Z S(Y K,Z) = -(pu a )'Y^ 3x a 30 S(Y K, 0) = - 3x a 30 3x a /n.. \'7' (pua)z i(pu a.)<p /"mi V,V r3y K. 3 A 3YJA,_!\ 1 7 I I 1 3x a ' ~3x a Vc p 3x a y av,. Iv 3x a 3Z 3 /X 3Z'\ (pu a) 0 + 0' ( + 3x a \Cp 3x a /, 3 / x 9Y^\ 3x a \C p 3x a / a / ), ay'\ 0 /A OZ. \ ox a \*-p ox a / 3 /X 30'\ 1 Y 1 i l i K 3x a Vc p 3x a /, 3 / X 30' \ Z'- ( -!-)+0's' 3x a \Cp 3x a / Les moments d'ordre 2 du type (b), intervenant dans les equations d'ordre 1 par I'interm6diaire de s', et qui apparaissent aussi dans les termes sources (21), satisfont aussi a des equations de bilan, beaucoup plus compliquees, que nous n'ecrirons pas ici. On voit done, que si la possibilite d'obtenir des equations de bilan apparaltre toujours de nouvelles variables et deviennent rapidement extremement compliques, surtout dans le cas d'un ecoulement avec reactions chimiques, meme avec les hypotheses simplificatrices que nous avons prises. Aussi la technique de traitement de 'ces equations, jusqu'ici utilisee extensivement dans le cas d'un ecoulement inerte et incompressible par des nombreux auteurs, consiste a fermer le systeme d'equation en se donnant des expressions non differentielles approchees pour les moments d'ordre ou de type trop complique; on trouvera une revue de ces techniques de fermeture dans reference,23 et une description logique des methodes de fermeture dans reference Avant d'aborder dans notre cas ce probleme, commentons les differentes caracteristiques des equations obtenues, qui montrent deja, d'une fa?on qualitative comment 1'influence reciproque de la turbulence sur la reaction peut se produire.., 3.4 Discussion Qualitative des Equations de Bilan L'influence de la turbulence sur la reaction chimique se retrouve principalement dans le fait que pw n'est pas egal, en general a pw(y 0, Y K, p, T) ; en general, le long d'une ligne de courant, pw(~) a 1'allure de la figure 6, entre le debu^et la fin de la combustion (1'abscisse est un degre d'avancement de la reaction, par exemple: (Y Ko Y K )/Y Ko ; pw, d'apres la formule (19) comprend pw( ), mais aussi un crochet correctif pw pw peut done etre, soit plus grande que pw( ), soit plus faible, et ceci peut se produire suivant les signes de YO^K. TgY^, T'Y^, et les grandeurs relatives des differents termes de (22); en general P 2 + Q 2 + P, Q, est positif, T' 2 1'est toujours, done produit une augmentation de pw ; par contre:. YpY^, T'Yg, T'Y^ peuvent devenir negatifs (c'est justement 1'etude des equations de_bilan des moments du 26 ordre qui nous le dira) et done diminuer_ pw. On peut deja penser que 1'influence de T' 2 sera primordiale au debut de la cojnbustion (quand Y 0 et Y K sont encore assez forts), mais que c'est le contraire en fin de combustion, et que pw presente done une allure differente de pw( ), comme il est indique sur la figure 6. Ceci produirait un epaississement important de la zone de flamme, chose qui a ete constatee experimentalement. Remarquons a ce propos_que meme si pw( ) est tres grand, ce.qui conduirait en regime laminaire a une flamme infiniment mince, pw peut ne pas 1'etre si, par exemple Y 0 Yj c /Y 0 Y K = 1 et T trds grand, et produire une flamme epaisse; mais cette epaisseur ne serait qu'apparente, 1'enveloppe des fluctuations d'un front de flamme infiniment mince. La reaction chimique agit aussi sur la turbulence, de trois facons: (a) parce qu'elle accroit la temperature et done indirectement p et p, ce qui influe sur les termes sources S(u0, Oy), S(u0, Y K ), S(u0, 0), S(u0, Z).... etc, qui reglent la production ou la disparition des moments de fluctuations et particulierement des transferts turbulents de quantite de mouvement, de masse ou d'energie; (b) sur la diffusion de YK par des processus turbulents elle a en outre une action directe par la presence du terme source (pw)'u0. A cause de ce terme il serait possible d'ailleurs de casser la similitude generalement constatee entre la diffusion de masse et la diffusion de quantite de mouvement.' Williams 9 voit d'ailleurs la un facteur important regissant la vitesse de propagation des flammes turbulentes;

184 (c) enfin par la presence dans les termes sources S(Y K, Y K ), S(Z, Z). disparition des fluctuations de concentration et de temperature. reglant la production et 4. MODELISATION DES EQUATIONS DES MOMENTS Nous abordons maintenant le probleme de la fermeture des equations du chapitre precedent. Dans le cas d'ecoulements turbulents sans reactions chimiques, beaucoup d'auteurs se sont penches sur ce probleme et ont propose des approximations differentes plus ou moins compliquees; en general on s'est limite a des equations differentielles jusqu'aux moments du 2e ordre, en modelisant les moments incluant des derivees spatiales (type b), comme Font fait Lumley 24, Donaldson 25 par exemple; peu d'auteurs ont inclu des equations destinees a calculer les moments avec derivees spatiales (par 1'intermediaire d'une longueur) comme Rotta 26 et Spalding 27 ; plus rarement encore ont ete indues les equations pour des moments du 3e ordre 28. Dans le probleme qui nous preoccupe, qui est encore plus complexe, il nous semble, pour le moment du moins, plus realiste de se limiter aux equations que nous avons ecrites au chapitre precedent. II nous faut done trouver des approximations pour les moments du 2e ordre incluant des derivees (type b) et pour des moments en un point du 3e ordre ou d'ordre superieur, intervenant soit dans des flux de diffusion turbulente des moments du 2e ordre, soit dans les termes sources dues a la reaction chimique. 4.1 Moderation pour des Ecoulements Non Reactifs Au debut de ses etudes sur les reactions chimiques en milieu turbulent, Corrsin 10 a ddmontre que dans son cas (turbulence homogene et isotrope, reaction chimique simple isotherme du ler ordre) les longueurs caracteristiques qu'on est amene a detmir sont les memes que pour un ecoulement non reactif, car la reaction n'est pas spectralement selective. Ceci a ete demontre aussi pour des reactions simples, de facon approchee seulement, par O'Brien 29, toujours dans le cas de turbulence homogene et isotrope. On peut done penser que, dans notre cas, les precedes d'approximation pour la moderation dans le cas d'ecoulements non reactifs seront encore valables, ou du moins constitueront une bonne premiere approximation, de meme que les longueurs et les constantes definies au cours de ces approximations. Parmi les precedes de moderation, celui de Donaldson 25, peut se generaliser pour toutes nos equations; Donaldson definit les longueurs L et / et pose les approximations: -p/mx) 1 " OXa (23) 3xn On peut ecrire aussi: = -, Yu( u a u a p' 3x0 - (u a u a 'YU 0'Y")" 2 (u0y') (24) p^r = -P/ YU - 3x a 3x0 a -Yu avec Y' = Yj^ ou 0' ou Z' et aussi: 3Y' 3x v' v' (25) 2 Y l Y 2 -Y,Y 2

185 II4-1 1 oil Y', et Y 2 peuvent etre soit Yj^, #' ou Z'..'. Ces approximations permettent de fermer le systeme des equations de bilan (17c), (20), (21) a 1'exception des moments triples de fluctuations des grandeurs scalaires Y K, 0, Z. Un procede plus simple, directement derive de la theorie de longueur de melange de Prandtl, consiste a modeliser directement les flux turbulents: (pu a )'Y' sous la forme: _ 3Y (pu a )'Y' = pe Y - (26) 3x a et de prendre par exemple, dans le cas d'une couche de direction x, e,. = r ay -H = cste y Y. (27) On peut alors se passer des 4 premieres equations du type (20), mais il faut rajouter 1'hypothese (25) et negliger les termes cinetiques et acoustiques s' (autre approximation) pour que le systeme soit alors completement ferme (a part, toujours les moments triples des fluctuations des scalaires Y K, 0, Z). 4.2 Modelisation pour des Ecoulements Reactifs O - '. - Nous avons dit au paragraphe precedent que la moderation des ecoulements non reactifs pouvait rester valable pour des ecoulements reactifs. II reste done dans ce cas a donner une approximation des termes de moments triples des grandeurs scalaires qui interviennent dans pw et constituent le probleme primordial de la fermeture des ' equations, en presence de reactions chimiques. Pour bien montrer cette importance, placons-nous dans le cas plus simple de reaction dans un volume homogene (en moyenne) d'un fluide au repos (en moyenne), pour une reaction isotherme du type: A + A -» P. Si on suppose aussi que le champ des vitesses n'est pas modifie par la reaction (elle est isotherme), la seule equation du bilan a ecrire est 3C A 3C A 2 3 / 3C A \ TT *~ u a T ~ ~~ kr^a ~*~ T ( a m T ) > (^r ot 0x a ^x a \ ox a / = c ) (28) dont on peut tirer les equations des moments suivantes: dc 4 -jf- = -k r (Ci +C^2) i..... _ (29) dc 72 /"SC^2 c ov P 3 <"5n~> dt "M ^.. / «L A ^Kr^A ^u^ Pour un champ de turbulence isotrope, L 2 (t) c;2 (3D (c'est ici une relation exacte et non approchee, comme Test 25 la place de (30): dans le cas non isotrope) et finalement on obtient a dt 6a m, ^ C A k r -2k r C^3.. (32) On pourrait trouver aussi une equation pour C A 3, mais elle ferait intervenir C A, et d'autre part, on ne connait pas forcemeht tous les moments a 1'instant initial. Aussi, il nous faut tronquer le systeme, de la meme fa9on que nous 1'avons explique precedemment, par une hypothese suppiementaire. L'hypothese la plus simple est C A = 0 ou plus exactement < C 1; mais ceci revient plus ou moins a supposer que la densite de probabilite </>(c) (la probabilite pour que C A soit comprise entre c et c + dc est (^(c)dc)est

186 gaussienne; or C A etant une variable non negative, yj(c) ne peut etre gaussienne en theorie, et s'ecartera d'autant plus du profil gaussien que C A sera proche de zero. Dans ce cas, en effet, (f(c) a 1'allure de la figure 7, comme cela a deja ete mesure, meme en 1'absence de reactions chimiques 30. II existe de plus d'autres conditions a remplir pour que cette approximation de fermeture soit valable, dues aussi au fait que C A est une variable aleatoire non negative, en particulier: 0 < C^ < (33) (C?) 2 C; 3 > -ra- - C A 2 C A. (34) D'autres inegalites du meme type que (7) peuvent aussi etre etablies si on s'interesse aux moments d'ordre superieurs. Toutes les approximations de fermeture devront satisfaire ces inegalites (d'apres la terminologie d'orszag ce sont des IPCA: Inequality Preserving Closure Approximations 31 ) et d'autre part, ces approximations doivent etre aussi proches de la realite que possible, si on connait des methodes exactes de calculs. Aussi O'Brien a propose une relation de la forme 11 (35) oil A, et AO sont des constantes dependant seulement des conditions initiales. Plus generalement, dans reference 12, est proposee 1'approximation: CIC^CI = 0(c A +-g+-^+ C^ ^;-CAJ C A *Ci,+ -"-a, (36) dans le cas oil 2 especes A et B sont presentes (C A represente C A en un autre point). L'approche precedente peut directement se transposer au calcul des taux de reaction dans un "reacteur homogene" alimente en permanence par un debit de combustible et de comburant premeianges. En supposant que deux corps (un oxydant, un reducteur) reagissent suivant une reaction simple du deuxieme ordre, du type: K + vo -> P, dont le taux de reaction instantanee est: pw = BT<*exp(-^-jY 0 Y H = k r (T)Y Q Y H. (37) En supposant que la temperature ne fluctue pas (ce qui est faux, mais cet exemple est juste pour donner une idee de I'effet des fluctuations de concentration), et en negligeant la dissipation moleculaire (L 2 /D S> 1), on peut demontrer 32, que le systeme a resoudre s'ecrit Y _, Y) r ME> t ( YY +r I s'. T 0 I K^I( Y 0 ^O^e "~ r so _ P ~(Y K ) e k r (T) - - i sk _ k r (T) /'2 -L V Y' Y' -I- Y'2Y' "> '0 ^ Y0 Y 0 Y K ^ Y0 Y K^ _ = - 2r sk - t s (Y 0 Y^ + Y K P ts ( Y o ^ + Y K Y o Y k + Yj^) - r sk -- t, P AH T e = ((Y 0 ) e - Y 0 ) ; (38)

187 les indices e se referent a la valeur a 1'entree du reacteur; r so est le, coefficient stoechiometrique massique frso = " (^o); t s le temps de sejour moyen dans le reacteur (t s = m/v) ; on a suppose que la capacite calorifique des gaz dans le reacteur etait constante, ainsi que leur masse molaire; alors on a aussi: P = prt. (39) Le systeme (38), accompagne de (39), a ete resolu soit en negligeant les moments d'ordre 2 (solution classique negligeant la turbulence) soit en negligeant les moments d'ordre 3, soit en utilisant la formule (36) pour Jes calculer. Pour resoudre le systeme, on s'est ramene a 1'intersection de la droite (Y oe Y 0 )/t s et d^une courbe \V(Y p ), qui contient en elle les influences des fluctuations, s'il y en a. Pour.certaines valeurs initiales, (Yg 2 ),,, (YJ 2 ),, et (YgY' K ) e caracterisant la turbulence a 1'entree, W(Y 0 ) est tracee sur la figure 8, revolution de Yj, Y, YJ,Y' K en fonction de Y 0 (degre d'avancement de la reaction) sur les figures 9(a), (b), (c). On constate d'abord une influence nette de la turbulence sur les taux maximum de reaction W max (et aussi sur les limites d'extinction du foyer), mais surtout on obtient a cause de la turbulence une impossibilite de consommation totale de 1'oxydant (et combustible, car le calcul est fait a richesse 1); cet effet correspond au cas oil Y^Y^ = ~Y 0 Y K, c'est-a-dire au cas oil les fluctuations de Y n et Y K sont en opposition de phase totale, comme il est indique sur la figure 10; et cette tendance que YgY^ devienne negatif est provoquee par la combustion elle-meme, comme 1'indique la figure 9(c). L'importance de 1'hypothese de fermeture apparait aussi sur ces figures tres grande dans la domaine oil les concentrations moyennes Y 0 et Y K sont petites, et influe beaucoup sur le pourcentage de Y 0 restant a la fin de la reaction; par contre au debut de la reaction, cette influence est faible bien que les fluctuations soient cependant plus fortes. L'impossibilite de reaction totale que nous constatons ici ne se produit surement pas en pratique, car des fluctuations du type de la figure 10, entretenues, paraissent impossibles; en fait, si nous n'avions pas neglige la dissipation moleculaire cjes fluctuations, il est probable que la reaction aurait ete totale; il n'en reste pas moins que la forme de la courbe W au voisinage de Y 0 = 0 serait restee profondement modifiee par la turbulence, dans le sens que ces figures indiquent, c'est-a-dire avec une diminution de W. Remarquons enfin que cette diminution du taux de reaction par les fluctuations de concentration se traduit en realite par un temps plus long pour la combustion, done correspond, dans le cas d'une flamme et non d'un reacteur homogene, a un epaississement. Si nous comparons la courbe W tracee sur la figure 8 avec celle tracee sur la figure 6, on voit cependant que nous ne trouvons pas, dans le debut de la reaction, des vitesses beaucoup plus fortes; nous pensons que cela vient du fait que nous avons neglige ici les fluctuations de temperatures et que c'est justement T' 2, positif, qui produit un accroissement du taux de reaction au debut de celle-ci. 5. QUELQUES CALCULS D'ECOULEMENTS REACTIFS AVEC TURBULENCE NON HOMOGENE Par la methode des moments decrites au chapitre III, et la moderation dont nous avons pariee precedemment (chapitre IV), quelques calculs ont ete effectues, dont le but, pour 1'instant, a surtout ete d'avoir une idee qualitative des phenomenes intervenants, et non pas des resultats quantitatifs. D'ailleurs des experiences suffisamment precises et completes qui permettraient de justifier quantativement la methode, et d'ajuster en particulier certaines constantes, ne sont pas actuellement disponibles. 5.1 Influence des Fluctuations de Concentration sur la Reaction dans le cas de Flammes de Diffusion ou Pr6meiangees Dans ce premier exemple, tire de reference 33, les fluctuations de temperature ont ete negligees; la moderation simple (faisant intervenir un coefficient de viscosite turbulente) a ete utilisee, les correlations triples des concentrations ont ete negligees; on s'est attache a trouver les variations de: r OH = (ici K = H), Y 0 Y K a la traversee d'une zone de melange et de combustion. La figure 11 est relative a une flamme premelangee; la figure 12 a une flamme de diffusion. Le produit r OH presente une allure differente dans les deux cas: il est toujours negatif dans la zone centrale de la flamme (combustion vive), mais est positif avant et apres dans la flamme premelangee, ce qui n'est pas le cas dans la flamme de diffusion. Ceci est du aux termes de production de Y^Y^ autres que le terme du a la reaction: ce sont 2d rn (3Yp/3x a )(3Yj c /3x a ), dissipation moleculaire, et e y (3Y 0 /3x a )(3Y lc /3x a ), terme du aux gradients de concentration (e y est un coefficient de diffusion turbulente), qu'on peut retrouver dans le terme S(Y K, Y 0 ) du systeme (21), si 1 on remplace un Y K par Y 0.

188 En particulier, dans le cas de flammes de difusion, (3Y 0 /3y)(3Y K /3y) est negatif, alors qu'il est positif si le combustible et le comburant sont premelanges. _Avec les conditions initiales de turbulence choisies, lr OH I peut atteindre 10%; il n'a cependant sur les profils de Y 0 et Y H qu'une influence faible en pratique, n'agissant que lorsqu'il ne reste que quelques % de combustible. On pent done en conclure finalement que les fluctuations de temperature semblent d'importance preponderante, ainsi que, de toute fa^on, 1'effet de la turbulence sur 1'epaisseur d'une flamme peut etre plus accentue dans le cas d'une flamme de diffusion que dans le cas d'une flamme premelangee. 5.2 Calcul d'un Jet Reactif avec Turbulence Non Homogene Dans cet exemple, tire de reference 20, les fluctuations de temperature ne sont plus negligees; la meme moderation que precedemment a ete appliquee. II s'agit non plus du developpement d'une flamme a partir d'une stabilisation, mais au contraire de I'extinction d'un jet de gaz en reaction par un melange avec de Pair frais; la reaction simple qui s'y produit est representee globalement par CO + ^O 2 -» CO 2 et w co suit une loi semblable a (16a) oil T A = K et a = 0,5. A la traversee de la zone de melange a 5 cm du jet (c'est-a-dire a peu pres un rayon de jet), les profils moyens obtenus sont donnes figure 13. Les correlations turbulentes Y'^/T 2, Y^Y^,, T'Y^, \%, et k = lo? 2 + v 72 ) sont donnes figure 14. On voit que le melange provoque un figeage de la reaction, du a la decroissance de la temperature; mais, quand la temperature decroit, la reaction s'arrete plus rapidement que le melange croft, ce qui provoque ce pic dans le profil de Y co. Les profils de correlations sont tres compliques, mais on distingue deux zones: au centre une zone oil les fluctuations sont dues au jet, en une zone de melange a un rayon un peu plus grand, dans laquelle de nouvelles fluctuations sont produites par le melange lui-meme. L'influence globale de la turbulence sur la reaction d'oxydation du CO se voit tres clairement sur les figures 15 et 16. La figure 15 donne la decroissance du CO sur 1'axe du jet, due surtout a la reaction (dans les premieres longueurs, 1'air exterieur n'a pas diffuse jusqu'a 1'axe); la prise en compte des termes dejluctuations dans WCQ conduit d'abord a une consommation plus rapide de CO (due principalement au terme T' 2 ) puis a une diminution de cette consommation, qui va presque rendre nul w^ (due principalement aux termes T'Y^ et T'Y^g qui deviennent negatifs meme au centre du jet, comme 1'indique la figure 14 ils etaient nuls dans le plan initial). L'effet de T' 2, qui augmente w co au debut est meme responsable des fortes differences dans les profils transversaux de \ co (a 5 cm du debut), comme on le voit sur la figure CONCLUSIONS L'etude des influences reciproques de la turbulence, surtout non homogene, sur les reactions chimiques est d'un grand interet pratique, en particulier, comme nous 1'avons dit plus haut pour le developpement des flammes, mais aussi pour d'autres problemes, comme 1'etude des reactions photochimiques ou chimiques de la stratosphere, 1'etude de lasers chimiques ou des reacteurs du genie chimique. D'un point de vue purement scientifique, elle ne manque pas non plus d'interet; elle a d'ailleurs debute sur ces bases bien avant qu'une application se revele. Dans le cas de la turbulence non homogene, les methodes s'appuyant sur les moments de fluctuations nous semblent de 1'interet pratique le plus grand. Des etudes avec ces methodes ont seulement debute, et les resultats obtenus ne sont que partiels; ils montrent cependant que les fluctuations de temperature sont primordiales, done que le probleme est extremement complique, si on veut le prendre dans un cadre assez general. Si Ton se demande pour quels types de flammes turbulentes, dans quels domaines de la figure n 3, la methode des moments est valable, la reponse est: pour tous les domaines, en general; mais, pour etre plus precis, il faut tenir compte de: (a) (b) la moderation choisie qui peut ne pas etre valable pour certaines types de turbulence, 1'expression de w en fonction des moments qui n'est valable que pour T'/T < 1 done peut tomber en defaut dans certaines parties du domaine 3, oil des gros tourbillons et des grosses fluctuations coexistent. De toutes fagons, si Ton arrete les equations des moments au deuxieme ordre, la methode sera d'autant moins precise (bien que^toujours theoriquement valable) que 1'intensite des fluctuations sera grande. En fin de compte, le domaine de validite pratique de cette methode demande a etre etabli par une confrontation des resultats theoriques qu'elle donne avec des resultats experimentaux.

189 A 1'heur actuelle, ceux-ci sont_assez rares, ils concernent seulement des grandeurs globales, et non pas des caracteristiques locales telles que T' 2 ou T'Yg ou YgYj^ qui seraient d'un grand secours pour tester les moderations. C'est pour cette raison qu'il nous semble que cette etude necessitera encore beaucoup de travaux. REFERENCES 1. Damkohler, G. 2. Summerfield, M. 3. Talantov, A.V. Emorlaev, V.M. Zotin, V.K. Petrov, E.A. 4. Wright, F.H. Zukoski, E.E. 5. Spalding, D.B. 6. Ballal, D.R. Lefevre, A.H. 7. Moreau, J. 8. Barrere, M. 9. Williams, F.A. 10. Corrsin, S. 11. O'Brien, E.E. 12. Lin, C.H. O.Brien, E.E. 13. Donaldson, C. Du P. 14. Chung, P. 15. Dapozo, C. O'Brien, E.E. 16. Libby, P.A. 17. Toor, H.L. 18. Bush, W.B. Fendell, F.E. 19. Williams, F.A. 20. Borghi, R. 21. Bray, K.N.C. 22. Beran, M.J. 23. Mellor, G.L. Herring, H.J. NACA - TM 1112, April Jet Propulsion, Vol.25, No.8, August Combustion, Explosion, and Shock Waves, Vol.5, No.l, January th Symposium (International) on Combustion, pp. -933, 'h Symposium (International) on Combustion, pp , Communication 4th Int. Colloquium on Gas Dynamics of Explosions and Reactive Systems, San Diego, July La Recherche Aerospatiale, (a paraitre) Conference a la Societe franchises des Thermiciens, Paris, 4 avril Journal of Fluid Mechanics, Vol.40, Part 2, February The Physics of Fluids, Vol.1, No.l, January-February The Physics of Fluids, Vol.11, No.9, pp , Astronautica Acta, Vol.17, pp , AGARD Meeting Turbulent Shear Flows, London, 1971 The Physics of Fluids, Vol.13, No.5, pp , May Communication 4th Int. Coll. Gas Dynamics of Explosions and Reactive Systems, San Diego, 1973,. Sera publie dans Astronomica Acta. Combustion Science and Technology, Vol.6, pp.23-28, A.I. Ch. E. Journal, p.70, Vol.8, No.l, March T. Report TRW-7-PW, Project Squid, July T. Report UCSD-4-PU, Project Squid, July e Symposium IUTAM-IUGG on Turbulent Diffusion in Environmental Pollution, Sera publie dans Advances in Geophysics, Vol.18, Ed. F.N.Frenkiel, R.E.Mann Academic Press, N.Y. AASU Report No.330, University of Southampton, October Statistical Continuum Theories, Chapter 3, Monographs in Statistical Physics, Vol.9, Interscience publishers, 1968, N.Y. AIAA Journal, Vol.11, No.5, May 1973.

190 Lumley, J.L. Khajeh-Nouri, B. 25. Donaldson, C. DuP. Sullivan, R.D. Rosenbaum, M. 26. Rotta, J.C. 27. Spalding, D.B. 28. Kolovandin, B.A. Vatutin, LA. 29. O'Brien, b.e. 30. Libby, E.E. Communication presentee au 2e Symposium IUTAM-IUGG on Turbulent in Environmental Pollution, Charlottesville, Va., April AIAA Journal, Vol.10 No.2, February AGARD Meeting Turbulent Shear Flows, London, Chemical Engineering Science, Vol.26, p.95, Int. J. Heat Mass Transfer, Vol.15, pp , December The Physics of Fluids, Vol.12, No.10, p.1999, October Conference a 1'ONERA, Mai Diffusion 31. Orszag, S.A. 32. Cariou, R. Borghi, R. 33. Barrere, M. Borghi, R. The Theory of Turbulence, Ph.D. Thesis, Chapters 6 and 7, Princeton University, Rapport interne ONERA, CR. Ac. Sc. t.276, Serie C.139, DISCUSSION Prof. Williams: In your calculation of the mean reaction rate, in the case of a perfectly stirred reactor, you take into account only the concentration fluctuations; I think that temperature fluctuations would completely change the results. Author's reply: C'est seulement pour montrer 1'influence des fluctuations de concentration que nous avons neglige celles de temperature; toutefois je pense, et de nouveaux calculs fails depuis lors 1'ont montre, que les fluctuations de temperature ont une importance nette surtout dans le debut de la reaction, et pas vers la fin, oil alors ce sont les fluctuations de concentration qui sont importantes. Prof. Monti: Have you some experimental comparisons with the theoretical calculations? Author's reply: Actuellement, aucune comparaison detaillee avec des experiences precises n'a ete faite. Dr Barrere: Cependant, on a pu voir deja que, globalement, les resultats obtenus correspondaient qualitativement a des phenomenes observes experimentalement.

191 YF/Y TCK) 100 V(m/s) 1,2 1,1 1.0 YF/Y 1,2 1,0 0 foo %H 8 0,8 <.0 1,2 y> Fig.l Influence sur 1'epaisseur d'une flamme turbulente de <p 0, T 0, u 0, d'apres Reference 4 vitesse de propagation turbulente u = 60 m/s u = 100 m/s u = 60 m/s 1,4 1, ,4 f,6 Fig.2 Influence sur le developpement d'une flamme de T 0 d'apres Reference 3

192 flammes composers de petites poches de gar flammes compose'es de grosses poches de gaz (combustion distribute dans un volume ) flammes pi issues mais brisdes avec des poches de gaz tm= flammes quasi-laminaires flammes plisse"es Fig.3 Differents types de flammes turbulentes YF/Y, ' i 1,1 1,0 0,9 «^ *-HL / / f / S -v-. p V- 1,1 t T ( K) V{m/s) 0,8... } calculs de [5] experiences [4] Fig.4 Comparaison de calculs et experiences par Spalding 5

193 L (mm) essoi V (m/s) 500 Fig.5 Comparaison de calculs et experiences par Moreau 7

194 yow degre d'avancement de la reaction Fig.6 pw avec et sans turbulence Fig.7 Densite de probabilite des concentrations

195 mole m 3.s Fig.8 w(y 0 ) dans un foyer homogene

196 Fig. 9o Fig.9b Fig. 9c Fig.9 Y' 0 2, Y} 2, Y^YK dans un foyer homogene

197 Y' CO Y" J 09 ' co Y' "og Fig. 10 Cas de fluctuations empechant la reaction de se produire 0,04 0,03 0,02 0,01-0,001 3 y(cm) = 0,1 Fig.l 1 r OH a la traversee d'une flamme premelangee

198 "1 y(cm) Fig. 12 r OH a la traversee d'une flamme de diffusion ( K)T 1,5.10 -i % , ,5 % 0 L 300 L r(cm) Fig. 13 Profils moyens dans la zone de melange avec de 1'air frais

199 ior«t Kr«i 10 io / 10 Y * V ' 01 'CO K(J) '* L "' -2.10" Fig. 14 Profils des correlations turbulentes avec de 1'air frais YCO (%) x(cm) Fig. 15 Decroissement du CO sur 1'axe avec et sans effet de turbulence sur la reaction

200 r (err Fig. 16 Profils de CO transversal avec et sans effet de turbulence

201 115 STUDIES RELATED TO TURBULENT FLOWS INVOLVING FAST CHEMICAL REACTIONS by Paul A.Libby Department of Applied Mechanics and Engineering Sciences University of California, San Diego La Jolla, California 92037

202 115 RESUME Nous donnons les resultats d'une serie de recherches relatives a des reactions chimiques simples dans un ecoulement turbulent: reaction oxydant-combustible donnant un seul produit. Les conditions de 1'ecoulement sont supposees telles que a 1'echelle moleculaire les reactions sont infiniment rapides. Avec cette approximation limite les proprietes de la turbulence determinent revolution de la reaction chimique. La description physique des aspects chimiques de 1'ecoulement qui resulte de 1'hypothese d'une "chimie rapide" et 1'experience pour valider cette description sont soulignees. Les consequences mathematiques appropriees au cas de reactions fortement diluees sont alors developpees. On montre que le point crucial du probleme relatif a la description analytique du champ moyen des compositions reside dans la connaissance detailiee d'une quantite scalaire globale dont le comportement peut etre en relation avec un scalaire passif des ecoulements turbulents par exemple la temperature ou la concentration de rhelium dans des melanges air-helium. En particulier nous montrons qu'en chaque point de 1'ecoulement en question nous connaissons la fonction densite de probabilite de la variable quand la composition moyenne et le taux de creation moyen de chaque espece sont completement et parfaitement determines. Avec cette relation entre et le comportement des scalaires passifs nous analysons des donnees recentes sur la fonction densite de probabilite de la temperature et de la concentration en helium en vue d'indiquer les possibilites vari6es qui doivent etre prises en compte dans une analyse adequate. Le role de 1'intermittence est souligne de ce point de vue. Les difficultes theoriques d'un calcul "a priori" incorporant ces possibilites sont soulignees. Quoiqu'il en soit nous montrons les resultats de quelques calculs d'une couche de melange bidimensionnelle resultant de deux ecoulements 1'un combustible, 1'autre comburant. Les resultats font apparaitre la zone finie prevue de reaction. Nouse concluons en soulignant les problemes qui demandent une plus grande attention tout au moins tant que cette tres simple reaction chimique peut etre consideree comme une bonne approximation.

203 II5-1 STUDIES RELATED TO TURBULENT FLOWS INVOLVING FAST CHEMICAL REACTIONS By Paul A. Libby Department of Applied Mechanics and Engineering Sciences University of California, San Diego La Jolla, California SUMMARY We report on a series of investigations concerned with turbulent flows involving chemical reactions in the simplest chemical system, fuel-oxidizer resulting in a single product. The conditions of the flow are assumed to be such that at a molecular level the reactions are infinitely fast. In this limiting case the properties of the turbulence determine the extent of chemical reaction. The physical picture of the chemical aspects of the flow which results from the assumption of "fast chemistry" and the experimental evidence to support this picture are emphasized. The mathematical consequences appropriate for the case of highly dilute reactions is then developed; it is shown that the crux of the problem of describing analytically the mean composition field resides in knowledge of rather detailed properties of a synthetic scalar quantity whose behavior can be related to that of a passive scalar in turbulent flows, for example, temperature or the concentration of helium in helium-air mixtures. In particular, we show that if at each point in the flow in question we know the probability density function of, then the mean composition and the mean rate of creation of each species is completely and readily determined. With this relation between and the behavior of passive scalars in mind, we review recent data on the pdf's of temperature and of helium concentration in order to indicate the various possibilities which must be taken into account in an adequate analysis. The role of intermittency is emphasized in this regard. The theoretical difficulties of an_a priori calculation incorporating these possibilities are pointed out. Nevertheless, we show the results of some calculations of a two-dimensional mixing layer with fuel in one stream and with oxidizer in the second stream. The results show the expected finite reaction zone. We conclude by emphasizing the problems needing further attention before the analysis of even this simplest chemical can be considered to be in good order. 1. INTRODUCTION One of the largely unsolved problems connected with turbulent flows of interest to engineers relates to the effect to turbulence on chemical reactions. This problem arises in many practical applications from the wakes of reentering vehicles to chemical lasers to combustion chambers. The corresponding fluid dynamic aspects of these applications, although not completely treated by current phenomenology, can be considered to be better understood than the chemical aspects. From the point of view of the engineer concerned with making a prediction of turbulent flow properties when chemical reaction takes place, the crux of the difficulty concerns representation of the mean chemical creation terms, usually denoted w., the time mean rate of production of species i per unit volume per unit time. We can describe the instantaneous terms w. and in the usual fashion can take the time average of them. This process introduces difficulties when the molecular rate laws involve Arrenhius type expressions with high activation energies. In addition even in simpler cases there results a variety of correlations between the fluctuations of concentration of the various participating species. Given this situation and the need for some answer, it has been common practice for the engineer to ignore turbulence effects and to replace w. with mean concentrations and to insert mean fluid properties into the instantaneous relation. This practice appears to have been done, not out of conviction as to its correctness, but out of ignorance of a viable alternative. In recent years, there has been a quickening of activity devoted to the description of chemical reactions in turbulent flows with explicit effects of turbulence taken into account. However, the considerable modelling required and the several questionable assumptions which must be employed to obtain a closed set of equations suggest that it continues to be worthwhile to study a simple system in depth in order to establish a firm basis for its analysis. In this spirit we have been concerned with a particularly simple chemical system under a special set of circumstances. It consists of two reactants, possibly simulating fuel and oxidizer, leading to one product in a one-step, irreversible reaction. The conditions of the flow are assumed to be such that at a molecular level, the reactions are infinitely fast. We term this the case of "fast chemistry". It has been studied in the past presumeably because it should be the first problem properly handled in the full sequence leading to resolution of the problem of the effect of turbulence on chemical reactions in complex systems. Our efforts and point of view have been closely parallel to those of O'Brien and his coworkers [1, 2]. Here we shall first discuss the physical picture of the flow which derives from the assumption of infinitely fast chemical reactions and the experimental evidence to support it. This discussion

204 IIS-2 seems appropriate at the present time because much of the existing literature relating to turbulent flows with chemical reaction emphasizes the mathematical aspects at the expense of physical content. Next the mathematical consequences of fast chemistry are described. The interesting result which derives from these consequences is that the crux of the problem resides in the need to define at each point in the flow the probability density function for a passive scalar quantity which must be considered synthetic but whose behavior can be inferred from appropriate experimental data, e. g., knowledge of the pdf of concentration of a passive contaminant or of temperature can be applied directly. In this regard, several points are especially important to note relative to these introductory remarks. First, past studies of this case of fast chemistry have not benefitted from experimental data on the pdf's of scalars and have generally made assumptions concerning that function of an unrealistic nature. Recently the methods of digital analysis applied to experimental turbulence research have led to means for obtaining pdf's and have permitted more realistic considerations. Second, it is unusual in engineering problems involving turbulence to require as much detail as is implicit in a pdf, which in principle gives the mean value and all the moments of the fluctuations. In most problems the engineer is satisfied to know the mean and the intensity of the fluctuations. Third, although the point of view leading to the treatment of chemical effects by means of the probability density function is perhaps central to understanding the phenomenology involved, there are perhaps other approaches to the analysis of the problem. (Recently, in an as yet unpublished contribution. Professor Frank Marble has proposed a strain model for the case of "fast chemistry" discussed here in order to avoid the pdf approach. ) Finally, while experimental data on passive scalars can be employed by analogy to make predictions of chemical behavior in the flows studied experimentally, the main utility of such data is to assist the development of theories permitting a priori calculations of other flows. Finally, we describe some calculations which are clearly provisional but which indicate the main features of the flows with chemical reactions of the sort treated here. 2. THE PHYSICAL PICTURE To simplify the flow and thereby to expose essential aspects, we consider a turbulent shear flow with highly diluted reactants in a background gas of uniform properties. Two reactants lead by a one-step reaction to a single product according to and involve two atomic species. Thus #i and 7/1 may be thought of as hydrogen and oxygen and Tfl 3 as water reacting in air diluted with nitrogen. Under circumstances of pressure and temperature such that the equilibrium constant related to the reactions of Equation (1) is "large", we are naturally led in laminar flows to the flame sheet model, due originally to Burke and Schuman [3] in According to this model the product YiY 2 2= 0, where Y- is the mass fraction of species i, and a flame sheet separates the flow into a portion with Tfti, the product and diluent present and into another portion with Tl( 2, product and diluent present. In brief the two reactants do not coexist. In early calculations of turbulent shear layers [4, 5] this model is applied essentially with turbulent transport replacing laminar transport; as a result the flame sheet is sharp as in the laminar case and presumeably located at some mean position within the reaction zone of the real flow. Although such calculations have apparently been useful for engineering calculations, the resulting picture of the flow and chemical processes is not credible; for example, we expect the reaction and heat release to be diffused in a reaction zone as the turbulence causes the flame sheet to oscillate. One of the aims of the present research is to provide more realistic phenomenology. The notion of a large equilibrium constant with the consequence that Y^Y 2 0 can be applied to turbulent flow. The nature of the "flame sheet" depends on the flow. In some circumstances we can think of contiguous eddies with one reactant, product and diluent in one eddy and with the other reactant, product and diluent present in the adjacent eddy In this picture the product is formed at the sharp interfaces between such eddies as reactants diffuse to the boundaries of the eddies. This view of successive eddies with one reactant present may be the appropriate one for the transient experiment reported by Gibson and Libby [6]. A weak solution of acetic acid in a beaker was rotated by a magnetic stirring bar. Bursts of weak base solution were added. A single-electrode, conductivity probe whose output depends on the product of the acid-base reaction, a salt, ammonium acetate, was immersed in the beaker. If our ideas of the formation of product at the interface between contiguous eddies with one reactant present are correct, the signal from such a probe should be initially spikey and should indicate a gradual increase of the background level of salt as turbulent strain smears out the interfaces and as the reaction goes to completion. This is indeed found to be the case as shown in Figure 1 taken from Reference 6, which contains a detailed discussion of these results including the effects of spatial resolution of the probe. This simple experiment reinforces our notions of reacting surfaces and indicates that theoretical estimates of the thickness of the reaction zone given in Reference 7 are not unreasonable.

205 115-3 In flow situations of more practical interest, the physical picture to associate with the idea of non-coexistence of reactants may be that suggested in Figure 2. We consider a two-dimensional mixing layer with one reactant, perhaps 7H.\, in the faster moving stream with a x^ - velocity component of U, and with the second reactant in the slower moving stream with Uj = ru, 0 S r < 1. As shown, there are two interface surfaces between the turbulent fluid and the external, irrotational flow; these are the well-known interfaces reported by Corrsin and Kistler [7]. The intermittent nature of the turbulent flow plays an important role in our considerations so we discuss some of the features of intermittency, which in the present context implies the percent of time at a given space point the flow is turbulent. We denote the intermittency by y : y = 1 in a fully turbulent flow, y = 0 in the external potential flows. It is probably worth noting here that we have shown in Figure 2 the two interfaces to have "overhangs", i.e., cases wherein the interface position at a given XpX 3 is double valued in x 2. We do so because LaRue and Libby [8] found that in the downstream edges of the interface such overhangs occurred in roughly 40% of such edges whereas they occurred only 8% of the time on the upstream edges. In terms of our two-dimensional mixing layer we expect this result to imply that on the side of the faster moving stream overhangs are frequent on downstream edges and that on the side of the slower moving streams frequent on upstream edges. There appears to have been no direct experimental confirmation of this expectation. We also show in Figure 2 another interface; this is between two turbulent fluids in contrast to the other interfaces but between fluids with only one reactant present. This is the flame sheet now viewed as a convoluted oscillating surface. Our previous consideration concerning the production of product apply; in addition the theoretical estimation of Gibson and Libby [6] gives some indication of the structure of the interface in terms of the turbulent strain. The picture in terms of overhangs, i. e., the extent of multiple values of the flame sheet location in x 2 at a given xp x,, is unclear but overhangs are expected. It is the oscillation of this flame sheet interface across a finite zone of the mixing layer that results in a reaction zone. As a way of thinking of the phenomena we can consider the mean rate of destruction of species ^ at a given space point XpX 2,x 3 within the reaction zone to be represented by a pulse train in time corresponding to crossings by the flame sheet of the point in question and by an incremental destruction of #Jj by each crossing. Thus we could write w, = -w. 6(t-t ), t = t,, t_,..., t 1 In n n 12 N where t n is the sequence of crossing times and w^n is the incremental destruction of TTjj due to the crossing at t = t fl. Taking the usual time average, we would get (note that wj n is non-dimensional) J_ T _ * T T-co ^0 * T-o> Xi ' "" n=i N-" where fj is the crossing frequency of the flame sheet and Wj is the ensemble average of the incremental destruction of 7I\^. If wj n is roughly independent of location within the reaction zone, Equation (1) suggests that the quantity important for engineering calculations, w,, will have a bellshaped distribution across the reaction zone, will have a peak at the most probable location of the flame sheet, and will effectively vanish at the ends of the reaction zone seldom reached by the flame sheet. Although this way of thinking of the two-dimensional mixing layer applies to other flow situations and may be conceptually useful, it does not appear to lead to a strategy for calculations of the flow. The extent of the reaction zone and its position within the flow depends on the concentrations of the two reactants in the two streams. The increase of one reactant relative to the other tends to drive the reaction zone away from the increased reactant. It is conceivable that the flame sheet can be made essentially contiguous with the interfaces between the irrotational flow and the turbulent fluid. This fact may be useful in permitting information concerning the statistical behavior of interfaces to be interpreted in terms of flame sheet behavior. It will be important for further developments of our discussion to consider the probability density functions of the mass fractions of the two reactants at several points in the mixing layer. The probability density function (pdf) for species 77^ should be interpreted in the present context as giving the percentage of time that the concentration Yj is between Yj and Yj+ dyj =- Yj+dY, at a given point in the flow; mathematically we think of P^YpXp x 2, x 3 ). It may also be helpful to relate the pdf to the time history of the concentration which builds up the pdf after sufficient time. Accordingly, we show in Figure 3 the relevant pdf's and time histories in three distinct regions of the flow: external to the reaction zone near the faster-moving stream, PointA ; within the reaction zone, Point B ; and external to the reaction zone near the slower-moving stream, Point C. Several preliminaries are indicated; the concentration of reactant 5^ in the faster-moving stream is taken to be Y, that of reactant 1f( in the slower-moving stream to be Y. The Z 22

206 II5-4 concentrations Y u and Y 22 are arbitrarily taken to be such that the reaction zone is principally in the upper portion of the mixing layer but distinct from the upper interface. Qualitatively the same results apply when the reaction zone is in the lower portion of the layer. The flow is assumed to be intermittent at all three points under consideration. Thus at Points A and B a percent of time equal to I-VA and l-r B respectively, the concentration of 7n l equals Y U ; at Point C, a percent of time equal to \-y c the concentration of W 2 equals Y 22. Reactant #! 2 does not exist at Point A and ^j does not exist at Point C. Because /T^ and 5?I 2 are consumed in the reaction zone, their concentrations within the turbulent fluid will be less than Y U and Y 22. Although we are not particularly interested in the detailed behavior of the product, similar considerations apply to "ft 3 ; if we assume "ft 3 is absent in the two external streams, it will be present only in the turbulent portions of the flow. In Figure 3a we sketch the time history of reactant fn l at Point A and the related pdf in Figure 3c we do likewise for reactant 7n 2 at Point C. Attention is particularly drawn to the spikes in the pdf's at Y, = Y,, and Y 2 = Y 22 ; although we shall later for practical reasons idealize these spikes into delta functions, they have a finite width due to concentrations close to Y n and Y 22. These contributions are from the structure of the interface and possibly from recently engulfed external fluid not completely obliterated by the turbulent strain but within the turbulent fluid, i. e., not necessarily in the superlayer. In our subsequent discussion, we shall for simplicity, neglect the later contribution and associate the structure of the spikes near the bounds in concentration with the superlayer. Note that the pdf corresponding to concentrations within the turbulent fluid are not shown to be Gaussian or near-gaussian. As we shall see in detail below when we discuss related experimental data, the pdf's are known to be highly skewed. In this regard it is probably appropriate to emphasize the importance of the physical bounds on variables such as concentration and of the implications of those bounds on the pdf's of concentration; here 0 «Yj <: Y U, and 0 S Y Z * Y 22 and thus Pj = 0 for Yi < 0 Y! > Y n, P? = 0 for Y 2 < 0, Y Z > Y 22. The significance of these simple, physically obvious notions can be appreciated by the following consideration: At a point in the flow where the mean value of Y, is Y\, one can always find an intensity of fluctuation of m. l, Y{ 2, such that the bounds 0, Y n require a non-gaussian distribution of Y}. It is the neglect of this point which makes suspect the accuracy of some of the early calculations of the effect of turbulence on chemical reactions, based as they are on a Gaussian or two-sided Gaussian distribution of the concentration of reactants. O'Brien [1] has emphasized the necessity in an adequate theory of providing for highly skewed pdf's of the several concentrations. In the reaction zone, for example at Point B, the situation is more complex. Since we have assumed that the reaction zone is in the upper portion of the mixing layer, two types of values of the reactant % occur just as at Point A ; i. e., values equal to and close to Y U corresponding to the external flow and to the interface structure and a range of values clearly within the turbulent fluid. However, part of the time the flame sheet will be outside of Point B and the 1T( 2 will then exist there. The values of reactant ft! 2 will be in turbulent fluid and will thus be distributed across some range of values. Finally, if the pdf's of ft l and ff 2 correspond to the percent of total time that a value of Yj and Y, respectively prevails at a given point and if we recall the non-coexistence of 77^ and ^2, then whenever 5^ exists at Point B we must make entries at Y Z = 0 and vice versa. In addition, just as there is a structure to the pdf associated with the other two interfaces due to the superlayer, there will be a structure to the entries near Yp Y Z = 0 due to contributions from the structure of the reaction zone. These characteristics are shown schematically in Figure 3b. 3. THE PROBLEM OF PREDICTION With the physical aspects of turbulent shear flows involving fast chemistry set forth we now consider the quantitive treatment of such flows. (Our presentation is based on Reference [9]; similar treatments are given elsewhere, e. g., see Lin and O'Brien [2] but the details and exposition here are distinctive. ) The assumption of highly dilute reactants and uniform properties of the background fluid implies that attention may be focused on the conservation equations of the species and elements. In the usual notation of Cartesian coordinates the former equations for the mass fraction of species i are 3Y. 9 2 Y. 3t + (u, Y.) = v- -^ +w., i = 1,2,3 3x k i 1 k k (2) where we make the begnign assumption for present purposes that a single diffusion coefficient applies to the molecular transport of all quantities. If, as we have assumed, the reactants involve two elements, we introduce the element mass fractions (3) where W t is the molecular weight of species i. Related to the conservation of elements and in fact suggesting the definitions given by Equations (3) are relations among the creation terms

207 IIS-5 W l and az. t/az. T + (u k z > = JTT7-. i = i.2. (5) The mean of Equation (5) in the boundary layer approximation with the usual orientation of coordinates (cf., Figure 2) is where to close the equations some representation of, or additional equations for, the mean flux of the element mass fraction Z., i.e., u'z', is needed. i 2 i These relations do not explicitly account for the assumption of fast chemistry, i. e., of the noncoesistence of reactions /TJj and ft 2. To do so we define a new variable* C = Z 2 - (W 2 /W 1 )Z J. (6) From Equation (5) we have at axk which is the conservation equation for a passive scalar. The mean of Equation (7) subject to the same comments as those given above for Equation (5a) is We shall return to Equations (7) and (7a) below but we now focus on the utility of (x, x, x, t). From Equations (3) note that ' and we can now conveniently impose the assumption of fast chemistry; if at a particular point in space and time, > 0, then since Yj and Y 2 do not coexist at that point, = Y 2. Correspondingly, if at another point in space and time < 0, for the same reason, = -(W /W )Y. These features of the variable can be employed in the determination of the mean composition in a turbulent shear flow as follows: Suppose from an appropriate solution of the mean of Equations (5) for i = 1, 2 we know at a given point in space the mean element mass fractions, "Zp Z". Then the mean of Equations (3) permit the mean values of two of the three species to be determined in terms of the third; thus, e. g., Y 3 =w(z 1 -Y 1 ) (8) where w = W^/Wj is the ratio of molecular weight of the two reactants. Thus if we know Yj at the given point in space as well, we can determine the entire mean composition at that point. In addition if we consider the mean of Equation (2) for i = 1 and if we can estimate in that equation the x -derivative of the one correlation with velocity significant in turbulent shear flows of the boundary layer type, i. e., a/axj^fuj^yj), then we can determine Wj. This emphasizes that mean rate of destruction of reactant % is determined after the mean composition is obtained and not as part of the determination of the mean composition. The limiting behavior to be associated with "fast chemistry" as defined here should be noted; we can write for dilute reactants in general Wj = -ky Y where k is a rate constant. For fast *Mr. Peter Bradshaw has pointed out that there would be an appealing symmetry without alteration of the essentials if we had defined t^= (Z 2 /W 2 )-(Zp / W^; in order to preserve the development in [9] we retain the earlier asymmetric definition. In addition we note that if one works in terms of as defined in Equation (10), no difference arises.

208 II5-6 chemistry YpY 2 = 0 but k - co so that wj can take on any value, in particular it takes on the value consistent with the conservation requirements on Yj. To find Yj at the given point in question we employ the properties of by taking the mean of the values of when < 0 ; more precisely we have 0 -wy 1 (x r x 2, x 3 ) = E( < 0) = jj P C ( ;Xl, x 2, x 3 ) d (9) where. < 0 is the smallest possible value may take on, and where P^ is the pdf of at the space po'infin question. Equation (9) replaces the earlier statements YjY 2 = 0 which lead to a fixed flame sheet in turbulent shear flows and which disregard the effect of turbulence on the chemical behavior of the flow. Having established the utility of the variable and the need to know sufficiently about it to permit estimation of the conditioned expectation E( < 0), we can consider other aspects of. The variable is not a directly measurable physical quantity but because it is a passive scalar, measurements of such scalars as temperature, helium concentration in helium-air mixtures, etc. can be directly related to. This may be seen more clearly if we refer to two turbulent shear flows with identical velocity fields (cf. Figure 2); in one we have reactant % in concentration Y U in the faster moving stream and the reactant 77, in concentration Y 22 in the slower moving stream. In the other we have an absolute temperature Tj in the faster moving stream, T 2 in the slower. Two variables min - WY 11- Y 22 T - T, 9 ~ T - T 1 2 (10) will have identical initial and boundary values (0,x 2 >0,x 3,t) =e(0,x 2 >0,x 3,t) = (0,x 2 <0,x 3, t) = 6(0,x 2 <0,x 3,t) = (x 1,x 2 -oo,x 3,t) = etx^ x 2 -»,x 3,t) = 0. With identical velocity fields, implying identical behavior of the two interfaces, the behavior of 9 including its pdf will be identical with that of. * Before passing on to other matters it is perhaps worth noting that Equations (10) suggest ; _ C " ( /-wy ) +(Y /wy ) u 22 _ < Y 22 /WY 11 ) (ioa) Actually the parameter C = (Y 22 /wy u ) completely determines the mean composition in these flows involving highly dilute reactants; it is unnecessary to specify either the individual concentration in the two streams or the individual molecular weights. It is this parameter which determines the location of the reaction zone within the mixing layer. In this form the relevant conditioned expectation is E(C < C/(1+C)) so that if C «1 (Y u "large"), small fluctuations of from its value in the slowermoving stream, namely zero, involve the presence of reactant 77! 2 and chemical reaction. In^this case the reaction zone occurs on the side of the slower moving stream. If C» 1 (Y n "small"), only on the faster moving side of the mixing layer will % and fl! 2 be present at the same point at different times and the reaction zone is on the faster -moving side. *This point of view of the direct relevance of measureable scalar quantities to the determination of is due to Lin and O'Brien [Z]. Earlier the author had taken a more conservative but less imaginative, less constructive position (cf. [9]). In [2] the authors actually use data on the pdf of temperature in a round heated jet to calculate the behavior of such a jet if it contains one of the two reactants, the other being in the quiescent external region.

209 II5-7 In accord with our earlier discussion, we see that if C is sufficently large, the flame sheet and the lower interfacial surface may be coincident. 4. SOME RELEVANT EXPERIMENTAL RESULTS Having established the connection between the behavior of passive scalars and chemical reactions of the type considered here and the desirability of having the pdf's of scalars in turbulent shear flows, we give some recent pertinent results of temperature measurements in the wake of a heated cylinder from LaRue and Libby [8] in order to indicate the phenomena which must be handled by an adequate theory. We reinforce our earlier remark by stating that only in recent years as tape recording and the techniques of digital analysis have been employed in experimental turbulence research has it been readily possible to provide data on the pdf's of flow variables within turbulent flows. The absence of such data has clearly hindered the proper analysis of reacting flows even with simple chemistry. We show in Figure 4 experimentally determined pdf's of temperature relative to the ambient temperature in the wake of a heated cylinder. Because we are interested only in the qualitative nature of these functions, we present the pdf's directly as given in Reference [8] where a standard representation is used. * Figures 4a-c correspond to different transverse positions in the wake; the first is close to the wake axis where over 99% of the time the fluid is turbulent. There we see that the pdf is well represented for most purposes by a Gaussian distribution. Clearly the boundary 9 = 0 on the left is essentially ineffective. The next two figures correspond to decreasing values of intermittency and show increasing dominance of the spike corresponding to 9=0, i. e., to the external flow and to small temperatures presumeably associated with the interface. It is interesting to note in connection with Figure 4c that such a pdf can be reasonably represented by a delta function at (5=0 and an "equallyprobable" plateau within the turbulent fluid. Such a simple representation is shown in [9] to lead readily to predictions of the mean concentration distributions. This model may be considered appropriate when the reactant concentrations force the reaction zone to be near the limits of the turbulent fluid. In these cases the boundary value of 9 = 0 and the importance of intermittency and non-gaussian behavior are clearly of great physical significance. It should be emphasized that there may be circumstances in which the reaction is fully embedded in the middle of the turbulent fluid. In these cases intermittency and non-gaussian behavior may be unimportant. We now turn to a second set of data which involve a passive scalar and which are useful in clarifying our ideas. Mr. Richard Stanford and the present author are using hot-wire anemomstry and digital techniques with a sensor consisting of an extension of the Way-Libby helium probe [10]. Simultaneous measurements of two velocity components, Uj, and U 2, and of helium mass fraction c with good space and time resolution are obtained. The techniques of calibration, data collection, data reduction and accuracy assessment are rather elaborate and cannot be described here; however, they are presented in detail in References 10 and 11. It suffices for present purposes to state that there are obtained at each measuring station in the flow from these techniques three time series, one each for UpU 2 and c, from which the pdf's as well as mean values of each variable, moments of the fluctuations of each variable, and the various correlations of the fluctuations of the several variables can be obtained. A considerable effort is devoted to the assessment of accuracy; on the basis of the results thereof, we believe the instantaneous mass fraction of helium to be accurate to within ± We shall thus interpret as "air" any concentration of helium less than Our present experiments are being carried out in a porous tube 75 cm long and 7 cm in diameter; turbulent air is blown through one end and pure helium is injected through the porous cylindrical surface. The flow is shown schematically in Figure 5. We show there the interface between pure air and dilute helium. For the flow conditions of our experiment this interface is obliterated upstream of the end of the tube, i. e., all the air has been contaminated with some helium by the end of the tube. However, at x/l = 0. 5, 0. 75, two o t of three measuring stations, the interface can be clearly identified as we shall describe below. It is across this interface that pure air is entrained in fluid with some helium present, diluting the helium being injected in a pure state through the porous wall. We also show in Figure 5 the instantaneous position of another contour of constant concentration, say c = C Q. This contour can be interpreted to be analogous to a flame sheet since it directly corresponds to = 0 (cf. Equation (9) ). The dynamics of, and the velocity and strain field associated with this contour, would thus appear to be of interest in connection with the study of turbulent reacting flows. In what follows we focus on the air-dilute-helium contours; i. e., we let c = However, our data can be interpreted for larger values of C Q. Consider the experimental data at x/l = 0. 75; on the axis we find air 96% of the time. (An indication of the accuracy associated with these data is obtained by the results at this station on the axis, r/r = 0, at x/l = 0. 5; roughly a quarter of a million terms in the time series for c, obtained *The notation is as follows: 9 is the instantaneous temperature relative to the ambient temperature; is the mean temperature; 9' the root mean square of the temperature fluctuations. The dotted line on the left of these figures is the location of the gate in 9 used to discriminate potential from turbulent fluid.

210 II5-8 after eight hours of calibration and data collection give c = , c' 2 = (10" ). ) At radial positions approaching the wall the percentage of time we find fluid with helium present increases. At r/r = air is present 81% of the time while at r/r = 0. 5, 0. 75, air is present only 26% and 1% of the time respectively. Consider the time histories and pdf's of helium at the radial stations discussed above. In Figure 6 we show these data for the most interesting location r/r = Figure 6a shows the concentration history for roughly 0. 2 second of data, the unconditional mean concentration of , and the "gate" c = , which can be used to establish a zero-one discriminating function to identify fluid on each side of the interface. The spikey nature of the concentration is indicated; e.g., a maximum helium concentration six times the mean is noted. Figure 6b shows the corresponding pdf based on a quarter of a million data points, i. e., 500 traces such as that shown in Figure 6a. It is clearly of interest in understanding the nature of the flame sheet interface to investigate the characteristics of the contours of constant concentration; as an example, consider the air-dilute-helium interface and some preliminary results from the data shown in Figure 6. First, we develope a zeroone function as indicated above and then use this to obtain zone averages, crossing frequencies and point statistics * By these means it is found that the mean axial velocity in the air is 12% higher than m the dilute helium, m/sec versus m/sec; and the radial velocity in the air is 33% less than in the dilute helium, m/sec versus m/sec. These results are consistent with a priori ideas of the behavior of slower moving helium when injected into a faster-moving airstream near the axis of the pipe It is this sort of differential velocity field on the two sides of the flame sheet interface which determines the turbulent strain entering the estimates of the structure of the interface (cf. Reference 6.) By counting the number of times the.discriminating function changes from zero to one in a given period, we can find the crossing frequency fj and can determine a Strouhal number fpd/u, where D is the tube diameter and u the unconditioned mean axial velocity; we find fpd/u Finally we have determined the ensemble mean values of the axial and radial velocity components at the crossings; when the probe crosses a front from air to dilute helium, the axial and radial velocities are 8 86 and m/sec respectively. When the probe crosses from dilute helium to air, the corresponding velocities are and m/sec respectively. A little reflection indicates that these results are consistent with the expected rolling of large turbulent structures of dilute helium under the influence of the wall shear and the faster moving airstream near the axis. These same considerations and techniques can be applied to contours of constant concentration for a range of o It would be especially useful to obtain on such contours, as well as on the interface a measure of the velocity field from which the instantaneous turbulent strain can be estimated. Mr. Stanford and the present author have underway an experiment involving a multiple probe (two u-c probes, closely spaced) which hopefully will provide such data. 5. FURTHER REMARKS ON PREDICTION We now turn attention to the problem of estimating at each space point the pdf of so that we may compute the conditional expectation shown by Equation (9). Several preliminary remarks are indicated; if in the flow of interest, at a particular space point, data on the pdf of a passive scalar are actually available, e. g., as in Figures 4 or 6b, then following Lin and O'Brien [2], and Equations (10) or equivalent, we may use these data to compute the mean values ofjtj for a variety of values of the concentration parameter, C ; if, in addition, means for estimating Z l are employed, e. g., by use of Equation (5a), then at that space point all mean concentrations and the mean rate of creation may be computed. The nature of an a priori calculation of the mean concentrations in terms of the pdf of is clarified if we consider a simple model for such a pdf. There have been several such models. We suggested earlier in connection with Figure 4c that if the reaction zone is near the outer edges of the shear layer the pdf of can reasonably be represented by a delta function to describe the intermittency and a constant value P O to represent the values of within the turbulence. In [9] this is termed the "equally probable model" and is shown schematically in Figure 7 for the case when the reaction zone is near the high speed edge of a mixing layer where in the irrotational flow - -wy n. A similar analysis applies when the reaction zone is near the other edge. To exploit this model we assume that at the space point under consideration the intermittency I is known. In addition we assume that suitable phenomenology for passive scalars has been brought to bear so that at this point ~, ' 2 and Zj are known. These assumptions perhaps supplemented by others as well would generally have to be made for any pdf model. Now from the nature of *We use the standard verbiage of conditioned analysis. The reader may consult Kovasznay, et al. [12] for a detailed discussion of these techniques which are now in widespread use in experimental turbulence research.

211 II5-9 P ( ;x, x, x ) we readily compute P o =I/( o+ wy n ) (11) ' 2 = (!-!)( - + WY) 2 +_i C + ( w Y, - tcc-<wy) + C The first of these defines in terms of and I, namely. (12) If Q < 0, then at_the space point in question there never exists any reactant #! 2, no reaction takes place, and Yj = - /w. If o > 0, reactant 7^2 is present some of the time, in fact a fraction of the time equal to \ P d. In this latter case Y J r (wy ii> 2 Clearly the condition Q = 0 determines the edges of the reaction zone in this case. This simple "equally probably model" may be reasonably accurate for those cases in which the reaction zone is near the edge of shear layer. It is clearly inaccurate when the reaction zone is fully embedded within the turbulent fluid. More important for present purposes is the demonstration of how information on fluid mechanical quantities!.. ' form for the pdf of to calculate chemical behavior. can be employed along with an assumed Having indicated the situation concerning the quantitative treatment of the turbulent shear flows involving the simple reactions under consideration, we turn to approaches which may in due course provide appropriate procedures for a priori calculations and be realistic. An approach involving direct calculation of P»( ;xp x 2, x 3 ), perhaps along the lines pursued by Lundgren [13,14] for the pdf of velocity in turbulent flows, is appealing and may in due course be developed. Whether the complexity in the pdf of suggested above as being necessary for complete realism can ever by obtained by a direct calculation is, of course, unknown. For the present we follow a more modest path and after [9] take the following view: If we are able to calculate, with intermittency taken into account, the mean and first few moments of the fluctuations of, say ', ', ', etc., then we can estimate the pdf of yielding these quantities and can carry out the conditioned calculation associated with Equation (9). To exploit this view we neglect the structure associated with the spikes corresponding to limiting values of the concentrations and replace the spikes with delta functions. The assumed functional form for the pdf of must, on the one hand, be capable of describing Gaussian and near-gaussian distributions and on the other, highly skewed distributions. With these requirements in mind consider a point in the upper portion, x 2 > 0, of the mixing layer (either points A or B in Figure 2) so that = -wy... in the external flow; we take that If f \ P ( ;x r x 2,x 3 ) = (l-l")6( -HvY u ) + Ae (i+^k-^+a.,^) 2 *-" ) (14) where A, o, and as many a t coefficients as desired and needed are functions of x, x, x that the computed values of, ', etc. at each space point are in accord with Equation (14). A corresponding form applies to points in the lower portion, x 2 < 0, of the mixing layer; in particular the first term on the right side is replaced by (1-1 )6( -Y 22 ). Equation (14) provides the requisite flexibility in the pdf of. We see this as follows: if o =, i. e., if the peak and mean coincide, then depending on the intensity ' and the location of ~ relative to the bounds, -wyjj, Y 22, a full Gaussian, or a truncated Gaussian is obtained from such

212 Equation (14). If Q + C. and if, e. g., o > Y 22, then the pdf of within the turbulent fluid and within the bounds on will vary monotonically. In addition, Equation (14) provides for incorporation of additional information on the behavior of as it is available by the inclusion of additional a. ( coefficients. To indicate how Equation (14) may be used, we review some of the results given in [9]. Assume that only I, ~ and ' 2 are known, as we did earlier in the "equally probable model", and thus that j, i = 1, 2,... must be taken to be zero; determine A, a, and o such that at each point in the flow P is consistent with the known values of I,, ' 2 s 22 We have from the properties of the pdf we find 22 -wy, C' = - WY 11 2 _ 2 so that some calculation leads to I =fcair*(erf(a(c- o )) + erf (a(l + o >) J ~ ~ 2 ~ 2 ~ -a (C- ) -a (l+ ) A - - where to repeat C = (Y 22 /wy n ), and where A = (A/a), a = awy n, f o = ( o /wy u ). A corresponding set applies to the other side of the mixing layer.^ If the left sides of these equations are known for a given point in space, the values of K, a," and " o may be determined from Equations (15). With these values known the conditoned expectation yielding Y I can be carried out; we have which gives erf / ~ (a(l + C ~ -erf(ac 0 ) (16) The solution of Equations (15) for A, a., and o must be carried out numerically. It is readily possible to eliminate A' and to deal with two equations for a and ~ o We have solved these by forming an error measure and by employing either a direct search procedure in two dimensions or a steepest descent; the latter can be extended to cases where several aj coefficients are incorporated. We turn. now to the question of determining the quantities required to utilize this treatment; namely, Zj, ~ (cf. Equation (8) ) and '. In general we might expect that the first two of these are calculated by numerical treatment of Equations (5a) and (7a) with some suitable model or additional equations for the two transport terms, u 2 Z{ and u' 2 '. In addition, in general, we can follow the current developments in the new phenomenology of turbulent shear flows (cf., e.g., Har.jalic and Launder [15] for a widely available paper providing an appropriate entry into the extensive literature on these developments) in order to compute '. The appropriate equation therefore is

213 To close this equation modelling of the terms u' ', u' ' and of the dissipation term _ 1 l/(a Vax k )d '/ax k is required. Previous experience in modelling provides a guide here. However, it appears to be accepted widely by those working in the phenomenology of turbulent shear flows that additional attention must be devoted to the treatment of scalars and in particular to the modelling needed to close Equation (17). This may be true to some extent even for the transport terms u' 2 Z j and u 2. The need for additional attention to the modelling of scalars is related to the significance of intermittency in determining the unconditioned means appearing in the conservation equations; consider the case of a scalar 9 which has the constant value,_6 c, in the external flow. Then in the external flow the "fluctuation" of 9 has the constant value (9-9 c ); consequently, the flux term u' 2 9', which appears in the equivalent of Equation (14), would_be the result of fluctuations in the turbulent fluid alone and would thus decrease at least at the rate of I at the outer edges of a shear layer. Whether the usual modelling properly accounts for such behavior appears doubtful. Accordingly, in [9] a heuristic assumption for ' is developed by adding to the intensity due to intermittency a contribution which results from balancing in Equation (14) the production term, u '0 /dx 2 ), and the dissipation term on the right side. Thus for the upper portions of the mixing layer (18) where K is a constant and where the quotient yu/(3u/3x) _ introduces a length scale required by the dissipation term. If in Equation (17) K =0, ' 2 is due only to intermittency, having the value, (-wy^), (1-1) of the time and f + (1+1 )wy,,)/!. the remainder of the term, so as to yield the specified mean *. On the other hand, if I = 1, i. e., the flow is fully turbulent. Equation (18) corresponds to the modelling based on the local balance indicated above. Thus Equation (18) contains in a crude and probably inaccurate manner the requisite physics of the intensity of the fluctuations of The applications of this theory in [9] are to the two-dimensional mixing layer since it provides a simple test flow. Distributions of both interrn_ittency_and mean velocity are assumed. Crocco-type relations between the streamwise velocity and and Zj and Equation (18) is employed to determine at a given value of the similarity variable TJ = x 2 /x^ the left sides of Equations (12). In Figures 8a and 8b we show typical results for C = 0. 5, i. e., the case when the concentration of reactant TTjj in stream 1 is relatively larger than that of reactant 77J 2 in stream 2 so that the reaction zone tends to be on the lower portion of the mixing layer (T) < 0) and for two velocity ratios in the two streams: r = 0, corresponding to one stream relatively quiscent, and r =0.3. The regime in which mean values of both reactants coexist is the reaction zone; according to our previous discussion it is across this zone that the flame sheet interface oscillates. Additional work needs to be devoted to the phenomenology of scalars before calculations of the sort presented here can be considered convincing. However, the present formulation, based as it is on a rather general, flexible description of the pdf of, would appear to be capable of incorporating as much information on the fluctuations of as we are likely to be able to predict. The accuracy of a given number of terms in the representation of P must be compared with experimental data on the pdf's of passive scalars. However, a formalism would appear to be available for the treatment of the rather special but basic case of turbulent reacting flows considered here. 5. CONCLUSIONS We have discussed with primary emphasis in the applicable physical aspects a case of a simple chemical reaction,?7ij+#! 2 -«#; 3, occurring in a turbulent shear flow under conditions such that chemical kinetics are infinitely fast and the reactants highly diluted. For shear flows there is considered a flame sheet which oscillates across the reaction zone and which can be thought of as an interface with turbulent fluid on both sides. The problem of describing analytically the mean composition and mean rate of creation at each point in these turbulent flows is shown to be related to the computation of a conditioned expectation of a passive scalar variable,. Accordingly, recent experimental data on the probability density function of temperature in the wake of a heated cylinder is reviewed in order to emphasize the *The "edge" of the reaction zone in this case is associated with the location in the flow where this second value equals zero; until this value is positive there is no reactant T!\ present.

214 importance of intermittency; of the physical bounds on the permissible values of a variable such as concentration; and of highly skewed, non-gaussian pdf's. Experimental data from turbulent flows of helium-air mixtures are shown to have direct relevance to reacting flows in providing pdf's of passive scalars and in providing information on the dynamics of, and the velocity field associated with, the oscillating "flame sheet". A formal procedure permitting an a priori calculation of the mean composition and mean rate of creation in turbulent shear flows is suggested. To be exploited this procedure requires knowledge of the fluctuations of with intermittency taken into account. Further work on the phenomenology of such fluctuations appears necessary before satisfactory calculations using this procedure can be considered to prevail. There remains much additional experimental and theoretical work on turbulent reacting flows before even relatively simple cases can confidently be computed. Experimental data on the characteristics of interfaces separating two turbulent fluids distinguished by two scalar quantities would assist in the development of the theory. In addition flows involving reactants in sufficient concentrations so that significant density fluctuations arise must be treated theoretically. 6. REFERENCES 1. O'Brien, E. E., "Turbulent Diffusion of Rapidly Reacting Chemical Species ", Vol. 18, Advances in Geophysics, Academic Press. 2. Lin, C.H. and O'Brien, E. E., "Turbulent Shear Flow Mixing and Rapid Chemical Reactions: An Analogy". JFM, Vol. 64, 1974, p Burke, S. P. and Schumann, T.E.W., "Diffusion Flames", Ind. and Engineering Chemistry, 20, (1928). 4. Libby, P. A., "Theoretical Analysis of Turbulent Mixing of Reactive Gases with Application to Supersonic Combustion of Hydrogen", ARS J., 32, (1962). 5. Ferri, A., "Mixing Controlled Supersonic Combustion", Annual Review of Fluid Mechanics, Vol. 5, 1973, Annual Reviews, Inc. Palo Alto, California, pp Gibson, C.H. and Libby, P. A., "On Turbulent Flows with Fast Chemical Reactions, Part II: The Distribution of Reactants and Products Near a Reacting Surface", Combustion Science and Technology, 6, (1973). 7. Corrsin, S. and Kistler, A., "Free-Stream Boundaries of Turbulent Flows, " NACA TR (1955). 8. LaRue, J. and Libby, P. A., "Temperature Fluctuations in the Plane Turbulent Wake", Physics of Fluids (accepted). 9. Libby, P. A., "On Turbulent Flows with Fast Chemical Reactions, Part III: Two-Dimensional Mixing with Highly Dilute Reactants", Combustion Science and Technology (submitted). 10. Way, J. and Libby, P. A., "Application of Hot-Wire Anenometry and Digital Techniques to Measurements in a Turbulent Helium Jet. " AIAA J., 9, (1971). 11. Stanford, R. and Libby, P. A., "Further Applications of Hot-Wire Anemometry to Turbulence Measurements in Helium-Air Mixtures", Physics of Fluids (accepted). 12. Kovasznay, L.S.G., Kibens, V., and Blackwelder, R. F., "Large-Scale Motions in the Intermittent Region of a Turbulent Boundary Layer", J. Fluid Mech., 41, pt. 2, (1970). 13. Lundgren, T.S., "Distribution Functions in the Statistical Theory of Turbulence", Physics of Fluids, Vol. 10, No. 5, (1967). 14. Lundgren, T.S., "Model Equation for Nonhomogeneous Turbulence", Physics of Fluids, Vol. 12, No. 3, (1969). 15. Hanjalic, K. and Launder, B. E., "A Reynolds Stress Model of Turbulence and Its Application to Thin Shear Flow", J. Fluid Mech., Vol. 52, Pt. 4, (1972). ACKNOWLEDGMENTS The research reported here was supported by the.office of Naval Research under Contract N (Subcontract No ) as part of Project SQUID. The author gratefully acknowledges the assistance of Dr. John La Rue and Mr. Richard Stanford.

215 LIST OF FIGURES 1. The production of salt from a weak-acid, weak-base reaction (From Reference 6). 2. The two-dimensional mixing layer between reacting streams. 3. The schematic representation of the probability density functions and related time histories of concentration at selected points in the two-dimensional mixing layer. a) Near the edge of faster-moving stream. b) In the reaction zone. c) Near the edge of the slower-moving stream. 4. The probability density function for the non-dimensionalized and scaled temperature in the wake of a heated cylinder (From Reference 8). a) On the wake axis. b) Off-axis where the intermittency is c) Off-axis where the intermittency is Helium-air mixing in a porous tube. 6. Experimental results for the time history of helium concentration and probability density function: x/l = 0. 75, r/r = a) Time history of concentration. b) The probability density function. 7. A highly idealized model for the probability density function of. a) External to the reaction zone. / b) Within the reaction zone. 8. Predicted distributions of mean concentration in the two dimensional mixing layer (From Reference 9 ): C = 0. 5, K = a) r = 0. b) r = /i PROBE DIAMETER Calibration Bubble INITIAL VALUE 0.5cm -J 0.05 U- I sec I \/ Possibly Reacting Surfaces Figure 1

216 i K> CM g) CM CM CM ~\ r -i i i r O in h- O CD O N- 0 O CM O N. O «> 2 ^ o o o> o d ~. d d " ri ~ Q to p CJ o_ g, h l 5 O ' "o d ul Q I CM Q V ID d in d ro d c\j d {e/( -0)} 9dr,e o ' d

217 T I I I CM cr> oj N ro <n oo o to oo oo o ro in CM CM CM * o o o o - c\i r-' d N u u n u n it o in CONCENTRATED HELIUM ' I I 00 (D {,e/(0-0)} e d;e 1 - T ~I - 1 ' O ' S DILUTE HELIUM C=C 0 = O.OI O> ID CM CM h- <fr odd CM m o> ^ in o d CM d n n n 8 C 0 = I. I <* ID TURBULENT AIR Figure 5 Q to u> in to CM o '

218 MEAN VALUE LEVEL of GATE O SflMPLE NUMBER Figure 6a CONCENTRATION Figure 6b *IO,-2

219 AJ1 A CM [^ <U g. s *../> CO 00 Figure 8a Figure 8b

220 DISCUSSION Dr Barrere: Since your experiments are made in a tube, are you sure that there is (pressure) wave propagation? In similar experiments we had troubles with combustion. Author's reply: I must say that these experiments are performed with extreme care; both the temperature and the speeds are monitored carefully. Our speed is low and we are sure that the temperature is uniform; on the other hand our conditions are far from a direct simulation of combustion.

221 METHODE DE QUASI-EQUILIBRE POUR L'ETUDE DES ECOULEMENT RELAXES par R.Prud'Homme Laboratoire d'aerothermique du C.N.R.S., Meudon

222

223 METHODE DE QUASI-EQUILIBRE POUR L'ETUDE DES ECOULEMENTS RELAXES R.Prud'Homme Laboratoire d'aerothermique du C.N.R.S., Meudon RESUME La methode du quasi-equilibre, presentee dans cet article est un procede d'integration permettant de calculer un ecoulement relaxe stationnaire dans 1'hypothese monodimensionnelle. II s'agit d'une methode pas a pas utilisant un sous-programme d'equilibre. Le quasi-e'quilibre est d'abord presente a partir de considerations sur I'affinite chimique et 1'entropie. Les equations de 1'ecoulement sont ensuite ecrites dans le cas de plusieurs reactions chimiques. La methode de resolution est ensuite etudiee et les resultats numeriques sont exposes pour deux melanges reagissants: le melange H-H 2 et le melange H H 2 HF F 2. L'article se termine par un examen des problemes importants qui se posent lors de la resolution: stabilite de 1'integration, determination du debit et zone transsonique. SUMMARY The quasi-equilibrium method, presented in this paper, is an integration technique which permits to compute the evolution of a steady unidimensional relaxed flow. It is a step to step method using an equilibrium subroutine. The quasi-equilibrium is first presented from considerations on entropy and affinity. The flow conservation equations are then written in the case of a multireaction system. Numerical results are presented in the case of mixture H-H 2 and mixture H-H 2 -HF-F 2. In the final part the paper underlines the most important problems which arise in the resolution: integration stability, mass flow rate determination, transonic zone. INTRODUCTION Les ecoulements a grande vitesse et a haute temperature font intervenir des phenomenes de relaxation. C'est le cas des tuyeres de fusees ou se manifestent des desequilibres de differentes sortes mais surtout chimiques. Les generateurs de laser a gaz (CO 2 et N 2 par exemple) utilisent la relaxation de vibration pour recuperer 1'energie resultant des "inversions de population". Dans tous les cas de relaxation (chimique, de vibration, de rotation, etc) on a affaire au meme type de probleme lies a la resolution du systeme d'equations differentielles decrivant 1'ecoulement. De nombreuses methodes de calcul ont ete utilisees ces dernieres annees 1 ' 2 ' 3. Ces methodes ont toutes leurs avantages. Parmi les methodes d'integration pas a pas nous presentons ici celle du quasi-equilibre basee sur 1'utilisation d'un sous-programme de calcul d'equilibre 4. La methode presentee est done interessante avant tout si Ton possede un tel sous-programme. Elle represente un compromis valable entre le temps de calcul et la precision des resultats obtenus. Les inconvenients sont ceux de la plupart des methodes d'integration pas a pas lorsque les equations a resoudre presentent une singularite, un probleme de valeur propre se posant alors. Nous definirons d'abord ce que nous entendons par quasi-equilibre a 1'aide de considerations sur I'affinite chimique et 1'entropie. Les equations de 1'ecoulement relaxe seront alors ecrites et la methode numerique presentee. Les problemes poses par cette integration seront etudies ensuite, notamment ceux qui concernent la zone transsonique et la stabilite du procede.

224 HI CONSIDERATIONS GENERALES SUR L'AFFINITE CHIMIQUE ET L'ENTROPIE Envisageons un melange de plusieurs fluides susceptibles de reagir chimiquement entre eux et decrivons son etat thermodynamique a 1'aide des R + 2 parametres extensifs independants: s, 7, 1, - r, IR. representant respectivement Pentropie, le volume et les degres d'avancement specifiques. L'energie interne est une fonction homogene de degre 1 de ces variables et nous pouvons ecrire: e = e(s, 7, Hi,. SR) W p de = Tds-pd7+ A r d r r=l (relation de Gibbs)., (2) Dans le cas devolutions infmitesimales mecaniquement reversibles, dw = pd7 (3) represente le travial recu par 1'unite de masse de fluide, et si 1'evolution est de plus adiabatique, nous avons: ou: de = dw (4) de = -pd7. (5) II s'ensuit que la relation de Gibbs devient: Tds + S A r d r = 0. (6) r=l Les cas ou 1'entropie est invariante correspondent a la reversibilite totale de la transformation infinitesimale considered. Ces cas sont caracterises par la relation: A r d r = 0. (7) r=l On distingue pour ces transformations isentropiques des evolutions a composition figee avec: des evolutions en equilibre chimique avec: d r = 0, (8) A r = 0, (9) et des cas mixtes ou certains degres de libertss r sont figss et les affinites chimiques correspondant aux autres degres de liberte sont nulles, en classant ces degres de maniere a ce que les Q premiers soient figes, on a: d? r = 0 r = 1,... Q, A r = 0 r = Q + 1,... R. (10) Dans les cas decrits ci-dessus, il est possible de caracteriser une evolution fine et reversible par 1'egalite: s = c*, le comportement interne du melange obeissant a des equations du type: (HO ou: r = c5? (12) A r (s,7, r ) = 0- (13) II existe cependant d'autres cas moins reels que les precedents ou la simple reversibilite mecanique permet d'integrer 1'equation d'adiabaticite (6). Les evolutions correspondantes ne sont pas totalement reversible et aboutissent done a une production d'entropie consecutive a la relaxation chimique. Pour ces transformations appelees evolutions en quasi-equilibre ou encore "pseudo-equilibre", nous avons:

225 1-3 R. (H) La variation d'entropie d'une evolution en quasi-equilibre est obtenue en integrant 1'equation (6), ce qui donne: s+v^ = < r=l * (15) On peut envisager la realisation pratique de telles transformations mais ce n'est pas notre propos. L'interet de ces transformations reside dans la simplicite des equations bbtenues qui permettent d'assimiler pour le calcul ces evolutions a des equilibres chimiques fictifs. En effet, en affectant du symbole (") les nouvelles grandeurs introduites, nous pouvons poser: Ar A r a r =, a r etant constant (16) pour r = 1,... R, et: s + 5H a r r = s ; r=l les equations (14) et (15) se ramenent alors a: A r = 0, r = 1,... R, (17) s = c. (18) Mais les consequences de ce changement de variables ne s'arretent pas la car la forme de 1'equation de Gibbs est elle-meme invariante et toutes les proprietes de stabilite thermodynamique permettant de de"duire des cele"rites caracteristiques, des chaleurs specifiques, etc, restent valables. On obtient ainsi pour I'^nergie interne: e(s, 7, r ) = e(s, 7, ) (19) de = Tds-pd7+ A r d r. r=l (20) Nous verrons dans ce qui suit, qu'avec les variables usuelles T et p (au lieu de s et v ), la resolution du quasi-squilibre est identique a celle de 1'equilibre chimique. 2. EQUATIONS D'UN ECOULEMENT RELAXE STATIONNAIRE MONODIMENSIONNEL 5 L'ecoulement d'un melange chimique dans une tuyere adiabatique est g^neralement bien represente par une approximation monodimensionnelle stationnaire. Les Equations obtenues sont classiques: pvff)(x) = m (masse) (21) ^s R dg,. dx ^ rdx = 0 (entropie) (22) V 2 h + = h 0 (energie) (23) pv-^= w r, dx r = R (production des especes) (24) et ne tiennent pas compte des differents phenomenes de transfert. Les notations employees sont les suivantes: p h masse volumique (egale a 1/v), enthalpie massique,

226 HI 1-4 V vitesse,,<%kx) aire de la section droite a 1'abscisse x, w r taux de production du degre de liberte r, rh et h 0 constantes d'integration correspondant au debit massique et a 1'enthalpie generatrice respectivement. A ces equations, il faut ajouter les diverses equations d'etat exprimant h, A r, etc... en fonction des variables fondamentales choisies, ainsi que 1'equation de la cinetique chimique donnant les taux de production w r en fonction de ces memes variables. Pour les demonstrations generates, nous conserverons les parametres s, 7,,,... R comme variables fondamentales, mais pour les applications nous utiliserons de preference T, p ainsi que R compositions chimiques independantes, ce qui donne les equations suivantes: pv.0(x) = m (masse) (25) ds ^-> dni T + > MJ L = 0 (entropie) (26) dx < f ' dx V 2 h + = h 0 (energie) (27) dnj K pv i = ^ a: J r w,, (production d'especes) (28) dx r=l auxquelles il faut ajouter les diverses definitions de thermodynamique et de cinetique chimiques ci-dessous: h =.S nj(h»)j (29) ij + RTlogg-^p (30) N S nj (31) J J=i (32) p = pnrt (loi d'etat) (33) N. N» w r = k DrTT (phj)"* - k Rr TJ (pnj)"jr. (34) On obtient R compositions chimiques independantes en classant de 1 a L les elements chimiques contribuant au melange (et que Ton suppose presentes comme especes du melange) et de L + 1 a N les autres especes. Les N L dernieres compositions chimiques sont alors independantes en Ton a: N - L = R. (35) Les compositions chimiques nj (nombre de moles d'espece j par unite de masse de melange) sont reliees par les L equations de conservation des elements: N 3E a/j rij = Tj;, / = 1,... L. (36) Les N L dernieres valeurs nj forment un vecteur f et les L premieres valeurs n; s'expriment en fonction de.

227 HIl-5 On obtient ainsi: n = II T?, + U, avec Uj = 1 - S a/j, (37) L'equation de 1'entropie devient done: L h = S f?/(h )/ + r-, avec r, J = (H ); J T*. a/itm?)/ (38) /=! /=i ' > ' N i S Mj dnj = A d, avec A: = MJ ~ 2C a/j M/ (39) j = i /= 1 N Cpf = S n: Cp : est appele chaleur specifique figee. (40) j=i dt 1 dp d Cpf- F ~ +r '~ = (entropie). (41) dx p dx - dx D'autre part, seules les N L dernieres equations de production des especes sont necessaires. En posant: k N \ = ~T7 TT (PHjA J (42) PV j=i L on obtient: kn N B r = 1 - ~ TT (pnj) a Jr, a jr = ^ - v\ t, (43) K Dr j =1 ^ = M r. (44) La grandeur sans dimension B r etant reliee a I'affinite chimique de la reaction A r = S a jrmj par 1'identite: B r s 1 - e-ar/ R T, (45) I'affinite de la reaction A r etant elle-meme reliee aux affinites des degres de liberte A: par: A r = ^^ a jr Aj. (46) Les coefficients stoechiometriques 3j r, j = L N, r = 1... K, formant une matrice N, on a done: Mi 1 dx = N (production des especes). (47) On peut aussi faire intervenir les coefficients Bj associes a chaque degre de liberte tels que: Bj = 1 - e-v RT. (48) Ainsi, toutes les grandeurs intervenant dans 1'ecoulement relaxe peuvent s'exprimer a 1'aide des variables T P et f. 3. PRINCIPE DE LA METHODE DU QUASI-EQUILIBRE La methode du quasi-equilibre est une "methode pas a pas" qui consiste a calculer le point n + 1 a partir des conditions existant au point n.

228 III1-6 Pour cela, on calcule une premiere approximation en admettant que 1'evolution a lieu en quasi-equilibre. Puis, on effectue une correction sur les valeurs obtenues, ce qui conduit a la seconde approximation du point n -t- 1. Seule cette seconde approximation fait intervenir la cinetique chimique. Presentons cette methode a 1'aide des parametres s, 7,, la signification de s et de A etant celle du point x n, c'est-a-dire: La premiere approximation verifie les equations: A = A- A n (49) Tn S = s + -LA-. (50) T n - - = - V n%n) (51) (53) = 0. (54) La seconde approximation correspond a: -(2) _ -d) _ ; (55) s n+l s n+l s n pour les corrections sur s et j[, et a: _e =-e + [!r(f) 1 (Xn+ '- Xn) _ P ' v \ ux /qe J n (56) (58) ces deux dernieres equations permettant de deduire la valeur de 7^ connaissant 7^ et les corrections effectuees sur s et _ respectivement. Le point suivant (n + 2) est calcule de la meme maniere en prenant cette fois pour s et A les grandeurs: A = A L An+i (59) 'n+l s = s + - A n+1 -!. (60) 'n + l La definition meme du quasi-equilibre change done a chaque pas. Dans la pratique, les variables T, p et _ sont utilisees et on calcule les corrections a apporter a T^, et a p^, a partir de J^, - J^, = A grace aux formules approchees: avec: Alog e T = Y n -A (61) A log e p = Z n AJ (62)

229 RU jl/nrt T \ V RT; Ill 1-7 (63) Cpf RT? (64) La valeur de d /dx pour 1'evolution en quasi-equilibre est egale a: '^-} =,dx/ qe d log e dx (65) avec, pour la matrice J2 : /nrt U + Ue : V v 2 " " """ <66> La matrice carree F_ ayant pour terme general: q/jq/j + 5ji_ (67) Le point courant (n+1) est done calculable a partir du point (n) a condition qu'en ce point (n) la quantite: nrt (68) ne soit pas nulle, et que la matrice fi ne soit pas singuliere. On peut demontrer que ces conditions ne sont pas remplies en deux points de 1'ecoulement: - le point E ou la vitesse V de 1'ecoulement est egale a la cele"rite" caracteristique du quasi-equilibre: le point F ou la vitesse V est egale a la ceierite caracteristique figee: af =\/[(3p/9p) s j]. Ces points E et F sont situes de part et d'autre du col geometrique de la tuyere et sont tres voisins de celui-ci. 4. RESULTATS DE CALCUL La methode precedente a ete appliquee au cas d'un ecoulement d'hydrogsne et en cas d'un melange d'hydrogene et de fluor. Ils sont presentes sur les figures 1,2,3, 4, ainsi que les resultats de 1'equilibre chimique. Si Ton voulait representer une evolution en quasi-equilibre, la meilleure fa9on serait de tracer les courbes B r (x) ou Bj(x). Ces courbes seraient des paralleles a 1'axe des x. Pour 1'equilibre chimique, elles deviennent confondues avec 1'axe des x, pour 1'ecoulement a composition figee, on obtient: si I'affinite Aj est positive (fig. 5, 6). La figure 7 montre ce qui se passe dans un pas pour les fonctions Bj(x). On voit que plus on est pres de 1'equilibre, plus la correction a faire pour obtenir la seconde approximation est faible. Cette correction sur Bj(x) est egalement faible pour les grandes valeurs de x. On pourra cependant utiliser dans cette seconde zone une variante de la methode du quasi-equilibre utilisant comme premiere approximation une evolution a compositions constantes, mais ce changement de methode n'est pas necessaire et complique le programme de calcul. Lors de 1'integration numerique, plusieurs probtemes se posent. Tout d'abord, un probleme de stabilite et de demarrage du calcul, ensuite un probleme transsonique, au voisinage des points E et F, qui, s'il est resolu, doit nous fournir le debit de 1'ecoulement ou le profil de tuyere. (69)

230 III STABILITE DE L'INTEGRATION NUMERIQUE La stabilite des solutions a ete etudiee 4. Mathematiquement, les solutions sont stables en ce sens qu'une petite perturbation par rapport aux conditions de demarrage tend a se resorber dans 1'evolution ulterieure si Ton suppose que 1'ecoulement reste stationnaire et de meme enthalpie generatrice et debit. Cependant, cette stabilite est locale car le point F, ou la vitesse V est egale a la ceierite caracteristique figee, est un point singulier pour les equations. II s'ensuit qu'une perturbation creee en amont de F tend a se resorber jusqu'a la zone transsonique ou elle tend au contraire a se developper, une perturbation creee en aval du point F se resorbe. Cette stabilite est aussi relative, c'est-a-dire que 1'ecart a la solution initiate se reduit d'autant plus fortement que la cinetique chimique est rapide. Les criteres de proche equilibre donnent done des indications utiles au sujet de la stabilite 4. Dans le domaine du proche equilibre, ou les solutions sont stables, nous venons de le voir, la stabilite numerique n'est cependant pas assuree si Ton n'a pas pris soin de limiter superieurement les pas d'integration. Pour un pas trop grand, la solution oscille. Cette limite superieure du pas augmente considerablement lorsque Ton s'eloigne de 1'equilibre de sorte que le probleme ne se pose plus dans le domaine proprement relaxe. Ces particularites de 1'integration se reproduisent pour toutes les methodes pas a pas et s'expliquent parfaitement a 1'aide d'un modele d'equations etudie par Emanuel 6. Les limitations de pas dependent de la methode numerique utilisee; avec la methode du quasi-equilibre le pas maximum est egal au plus petit des temps chimiques. 6. LA ZONE TRANSSONIQUE Nous avons vu que la methode du quasi-equilibre etait en defaut en deux points E et F voisins du col geometrique de la tuyere. L'etude des ecoulement transsoniques 7 et celle de la propagation du son 8 montrent bien que ces deux points ont une signification physique precise. La ceierite du son est en effet comprise entre les et valeurs a e = N/[OP/9p) s,al a f = \/K a P/ 9 P)s, 1 1 ui en constituent les limites extremes. Lorsque ces ceterites thermodynamiques a e et af sont voisines 1'une de 1'autre, les points E et F sont voisins, ce qui etait le cas dans les exemples etudies. Numeriquement parlant, si Ton se fixe les constantes m, h 0 et I'affinite chimique initiate ainsi que 1'entropie initiate, la solution obtenue diverge au voisinage du col. Si Ton considere le debit rh comme une inconnue du probleme, on peut toujours en trouver une valeur donnant une solution continue satisfaisante pour 1'ecoulement relaxe. Inversement, si Ton considere 1'aire de la section droite au col comme une inconnue, on peut aussi determiner un profil de col de tuyere satisfaisant, le reste du profil de la tuyere etant fixe. Le probleme n'est pas fondamentalement different du cas des ecoulements figes ou en equilibre, les conditions au point singulier determinant le debit ou le diametre du col de la tuyere. Mais la difference fondamentale est que pour les ecoulements relaxes la position du point singulier F n'est pas connue alors qu'il s'agissait toujours du col geometrique pour les ecoulements isentropiques (voir figure 8). Du point de vue numerique, si le profil de tuyere est donne, il faut recommencer 1'integration des equations avec plusieurs debits et determiner par interpolation le debit convenable. Une approximation lineaire pour 1'une des grandeurs (p, V ou M f, etc) en fonction de 1'abscisse pouvant etre utilisee a 1'extreme voisinage du point singulier afin de depasser cette zone sans inconvenient. La methode qui consiste a choisir le debit (en principe celui de I'ecoulement en equilibre calcute au prealable) et a modifier le profil de tuyere pour obtenir une solution continue, est moins satisfaisante mais plus simple a realiser. Elle n'est valable que si la modification geometrique ainsi realisee n'est pas trop importante. Le comportement de I'ecoulement dans la zone transsonique peut etre precise grace a la methode des developpements asymptotiques. Nous appliquerons cette methode au cas d'un melange gazeux monoreactif en proche equilibre en traitant tout d'abord le probleme monodimensionnel. En rapportant les differentes grandeurs a des valeurs de reference convenables, les equations de 1'ecoulement se reduisent a: v ) = rii7 (continuite) (70)

231 III1-9 v 2 h + = HO (energie) (71) ds d T A = 0 (entropie) (72) dx dx v = LA (production des especes). (73) dx En considerant un profil de tuyere de section droite faiblement variable, nous introduisons le nombre positif e <SC 1 tel que I3(x) = 1 + el3,(x), (74) et nous admettons que les parameires (sans dimension) de 1'ecoulement admettent des developpements asymptotiques en puissances de e au voisinage d'un ecoulement de base (indice o) en equilibre chimique (s 0 = 7 0 = 1, 0 > AO = 0). Pour un parametre quelconque de I'ecoulement nous posons done: f (x; e) = f 0 + e n f, (x) + e 2n f 2 (x) +, (75) les terms successifs f 0, f,, f 2,.... etant d'ordre 1 dans la zone de validite du developpement. Le nombre n devant etre choisi positif lors de 1'analyse des equations resultantes. On fixe 1'echelle d'analyse en abscisse a 1'aide d'un nombre 0 tel que: x = e0 n x. (76) L'abscisse reduite x etant, par definition, d'ordre 1 (autrement dit, x est d'ordre e~p n ). Compte tenu de ces hypotheses, on obtient les equations suivantes: ds, = 0, on choisit s. = 0 (entropie) (77) dx v, -7, - m, + e 1 -"^, + e n (v rh 2-7,m,) + = 0 (masse) (78) ^ * ro 9 h = 0 (energie) (79) d i\. _ i )+ -e-0 n {L 0 A, + e n (L 0 A, + L,A,) + }= 0. (80) dx \dx dx / Nous avons suppose que les ceierites du son etaient voisines, ce que Ton peut traduire par: oil Z est une donnee pour n fixe. a fo -r-1 = Ze" (81) a /o Pour simplifier 1'expression du probleme, il faut faire un choix judicieux de certaines grandeurs de reference (indice r), dont nous donnons sans demonstration les valeurs dimensionnelles (exposant +) : (82) Nous posons ensuite F = h^, %. Avec n = 1 on obtient des equations lineaires correspondant aux domaines subsonique et supersonique.

232 Le choix n = \ nous fournit les equations de la zone transsonique qui, en prenant vj = vj = afg, deviennent: u -1 0 x ~~ X* ~% (D ' (ID TABLEAU I Equations F+ 1 Z, vj 13, + iii 2, = c^, 7, = v,, s, = 0, r + i ^ Zj, o v, i/j, 1 m 2 -r"+si+7i = 0, 7, = v,, s, = 0 +1 el/2 Jil 'c (III) r + i 7 t "' u" 7 (25 1 ril S, + 7i = 0, 7, = v,, s, = 0 Avec 0 = 1 et (3 = 1 on retrouve 1'ecoulement fige et I'ecoulement en equilibre respectivement. Les echelles d'analyse correspondant a x + <C / dans le premier cas et x + ~^> I* dans le second cas. Seul le cas (3 = 0, x + = 0(lj.) est a proprement parler relaxe transsonique, les dimensions de la zone etudiee y sont de 1'ordre de grandeur de la longueur chimique. Etudions la solution du systeme (II) lorsque le profil de tuyere est donne par:.(3, = 0,01(x 4 + 4x 3-5x 2-60x). (83) On admet que 1'equilibre chimique est realise en x = 0 ce qui implique qu'en ce point on ait: A[ = 0 ou, + 7, = 0. On pose F = 1 et Z = 2,1. Le systeme (II) admet alors une solution continue a v, croissant: v, = 0,l(x 2 + 2x- 15), (84) la variable chimique etant egale a: g, = -0,l(x 2-15). (85) La perturbation de-debit rh 2 est egale a 0,9. Si, partant des memes valeurs en x = 0, on suppose que 1'evolution a lieu a 1'equilibre chimique, la solution obtenue (exposant *) n'est plus continue car le debit rh 2 = 0,9 ne convient pas. Si on change rh 2, la solution relaxee obtenue n'est plus continue. Pour rii 2 = 0,185, la solution d'equilibre est continue. Enfin, le point critique F a pour abscisse x,,- = 3 et correspond a v, = 0. Ces resultats, resumes sur la figure 9, correspondent bien au comportement que nous avions decrit au debut de ce paragraphe. L'analyse par developpements asymptotiques s'applique aussi aux cas ou les ceierites caracteristiques a e et af ne sont plus voisines 4, mais son interet est encore plus grand pour les ecoulements bidimensionnels. En effet, si ' I'approximation monodimensionnelle est souvent suffisante dans le convergent d'une tuyere, le calcul de 1'ecoulement aval par la methode des caracteristiques peut presenter un interet certain. La zone transsonique doit alors etre connue avec suffisamment de precision. L'analyse par developpements asymptotiques appliquee a la zone transsonique dans le cas de ceierites caracteristiques voisines conduit aux equations:

233 TABLEAU II p x y Vjf«arto«s -1 e-2/3^1 /C -1/3-^1 /c (D z«, = (r + i)* tt * ttf -* w Si = c^, 7, = v, = 4>, x, s, = 0 0 X + "^ y + el/3 JL ''. (ID z«, = (r +!)* *, -* w ^ + 5, +7, = 0, 7, = v, = * lx, s, = 0 1 e2/3 X + 'c + g 'c f z? = (p + 1)4).$.._<!, 1X 1XX lyy (Ill) ' (?, +7, = 0, 7, = v, = * lx, s, = 0 Dans ces equations les grandeurs de reference sont les memes que pour I'ecoulement monodimensionnel. Les developpement asymptotiques sont aussi en e n, e etant positif et tres petit devant 1'unite, n etant un nombre positif, Les composantes de la vitesse sont: u = u 0 + e n u, + e 2n u 2 + v = e(v, + e n v 2 + ) (86) L'isentropie de la premiere approximation permet d'introduire le potentiel 4>,(x, y) tel que: "i = *ix. v i = *iy (87) L'echelle d'analyse est definie par les coordonnees reduites x et y : x = e(3n X) y = e(/3-l)n+ly ; les variables sans dimension x et y etant rapportees a la longueur chimique a n = 2/3. (88) F c. La zone transsonique correspond Comme precedemment, seul le systeme II est caracteristique de la relaxation transsonique. On connait a ces equations une "solution tuyere" donnant pour un polynome de degre quatre en x et y. Les lignes particulieres de 1'ecoulement correspondant sont representees sur la figure 10. Une autre solution est egalement connue et fait intervenir des exponentielles complexes et des polynomes de x et y, mais il ne s'agit pas d'une "solution tuyere" CONCLUSION La methode du quasi-equilibre convient a 1'etude des ecoulements relaxes stationnaires ou Ton fait 1'hypothese monodimensionnelle. L'interet de cette methode est assez evident lorsque Ton n'est pas tres eioigne de 1'equilbre chimique car I'affinite chimique A est alors pratiquement constante et voisine de zero. Lorsque les reactions chimiques sont plus lentes et que I'affinite varie de maniere importante, la methode du quasi-equilibre donne des resultats corrects mais est moins simple qu'une methode classique (Runge-Kutta, etc). Cependant, tout procede numerique doit permettre de resoudre les problemes de proche equilibre. Si Ton dispose d'un programme de calcul d'equilibre utilisable comme sous-programme du calcul relaxe, la mise en oeuvre de ce precede est alors tres simple. L'analyse de la zone transsonique a montre par ailleurs 1'interet de la methode des developpements asymptotiques dans le domaine du proche equilibre. Lorsque 1'ecart a 1'equilibre devient plus important cette analyse reste valable mais il faut alors envisager les termes d'ordre superieurs au premier.

234 BIBLIOGRAPHIE 1. Bray, K.N.C. 2.. Moretti, G. 3. Anderson, J.D., Jr 4. Prud'Homme, R. 5. Barrere, M. Prud'Homme, R. 6. Emanuel, G. 7. Knesser, H.D. 8. Napolitano, L.G. Atomic Recombination in a Hypersonic Wind Tunnel Nozzle. Journal of Fluid Mechanics, Vol.6, Part 1, July A New Technique for the Numerical Analysis of Non-Equilibrium Flows. AIAA Journal, Vol.3, No.2, A Time Dependent Analysis for Vibrational and Chemical Non-Equilibrium Nozzle Flows. AIAA Journal, Vol.8, No.3, Ecoulements Relaxes dans les Tuyeres. These de Doctoral d'etat, Faculte des Sciences de Paris, Equations Fondamentales de I'Aerothermochimie, Masson, Problems Underlying the Numerical Integration of the Chemical and Vibrational Rate Equations in a Near-Equilibrium Flow. AEDC TDR 63-82, March Relaxation Processes in Gases. Physical Acoustics, Vol.11, Part A, Warren P.Mason, Academic Press, Transonic Approximations for Reacting Mixtures. Israel Journal of Technology, Vol.4, No.l, 1966.

235 Fig.l TO = K - PO = 10 atm. 1 equilibre 2 relaxe 3 fige a partir du foyer TK \\ AS \ \3 \ X CM ATM Fig.2 TO = K - PO = 10 atm. 1 equilibre 2 relaxe 3 fige 0 10 x CM Fig.3 TO = K - PO = 10 atm. 1 equilibre 2 relaxe 3 fige

236 fl/mo/e/g. 0,04 < 0,02 H- 0,01 10 Fig.4 Evolutions des compositions chimiques Equilibre Relaxe Fig.5 TO = K - p 0 = IO atm. Evolution du parametre B en ecoulement relaxe

237 Xf/» Fig.6 Evolution des B: (Melange H - F) Resultats de calcul Courbes de resultats de calcul Evaluation (proche equilibre: Bj (Bj) e ; zone du cd) B Fig.7 Schema simplifie du precede d integration Fig.8 Solutions pour un ecoulement isentropique dt m.. = m rh., = rh > 0 dx dt rh. = m, < 0 dx

238 Fig.9 Evolution de la vitesse v,, du degre d'avancement de la reaction, et de la section droite de tuyere A, (premier terme du developpement asymptotique) Relaxe v,,, Equilibre rh 2 = 0,9 vf, f Equilibre rh 2 = 0,185 Vj

239 mi-17 Fig.lO Lignes particulieres pour une "solution tuyere" (S e ): ligne sonique d'equilibre (Sf): ligne sonique figee (H): ligne des cols (C,), (C 2 ): caracteristiques transsoniques en F

240

241 III2 CALCULATION OF THE EFFECT OF AFTERBURNING IN EXTERNAL SUPERSONIC FLOW BY MEANS OF A METHOD OF CHARACTERISTICS WITH HEAT ADDITION AND MIXING LAYER ANALYSIS by P.Mittelbach Messerschmitt-Bolkow-Blohm GmbH - Munich Space Division

242 III2 RESUME Une methode est donnee permettant de calculer 1'effet d'une combustion externe dans un ecoulement supersonique sur la distribution de la pression le long de profil au voisinage duquel la combustion se poursuit. Elle a pour base la methode des caracteristiques avec apport local de chaleur. Moyennant une distribution des sources de chaleur il est possible de d6crire 1'ecoulement turbulent par 1'analyse de la couche limite reactionnelle en appliquant le programme de couche limite de Patankar/Spalding. Quelques exemples montrent la valeur de cette approche.

243 III2-1 CALCULATION OF THE EFFECT OF AFTERBURNING IN EXTERNAL SUPERSONIC FLOW BY MEANS OF A METHOD OF CHARACTERISTICS WITH HEAT ADDITION AND MIXING LAYER ANALYSIS P.Mittelbach Messerschmitt-Bolkow-Blohm GmbH - Munich - Space Division SUMMARY A method is described for the calculation of the effect of afterburning in supersonic flow in the vicinity of a base body on the pressure distribution along this body. The basis for it is a method of characteristics, where the heat addition is prescribed. Information on the distribution of heat sources is gained by an analysis of the turbulent reacting mixing layer applying the Patankar/Spalding boundary'layer program. A few examples show the usefulness of this approach. NOTATION (MKS units) Cj, C K C C R C v, Cp h mass fraction of a chemical component element mass fraction mass fraction of rocket exhaust gas specific heat at constant volume (pressure) "specific enthalpy H IT *L m R M P PT q(x, r) q' = pq Q(x) Qm r r 0 (x) ), r E (x) ), r 2 (x) total enthalpy flight altitude turbulent Lewis number diffusor air flowrate nozzle (rocket gas) mass flowrate Mach number pressure turbulent Prandtl number energy released (added) per unit mass and time energy released per unit volume and time energy released per unit time within the region with distances less than x from the nozzle exit plane total energy released per unit tune at complete combustion distance from axis of symmetry boundaries of mixing layer base body contour boundaries of the region of energy addition in the method of characteristics program /.IO(XB) \ SB aerodynamic force acting on the base I = I 2;rprdr I \ r 0 (0) /

244 III2-2 S R T u w W D jff x p p. Machangle Subscripts rocket thrust temperature velocity component in x direction amount of velocity pressure drag of diffusor distance from nozzle exit plane density oo L R free stream condition airgap resp. air rocket 1. INTRODUCTION AND PROBLEM FORMULATION The subject of this report is a problem of external combustion in supersonic flow. The special arrangement considered is shown in Figure 1. A central rotational fuselage is surrounded by a row of rockets (approximated by a coaxial slot-nozzle) with a finite gap between. By this arrangement the rocket exhaust gases which contain a certain fraction of unburnt fuel (assume hydrogen) are mixed with two coaxial airstreams in a turbulent mixing zone, where the remaining fuel can react with the free oxygen of the air (afterburning). By the heat release the pressure level in the region around the base body is increased in particular at the base body itself. The result is a reduction of the base drag or equivalently an augmentation of the primary rocket thrust. Our task was to estimate the possible thrust augmentation of such a system, which is sometimes called a partially ducted one, since the mixing and combustion zone is confined only at one (the inner) side by a rigid wall, which consists of a part of the vehicle itself. It is clear that the exact consideration of the complicated interdependence of turbulent mixing, chemical reactions and pressure variation is extremely difficult. Our approach to this problem is based on a certain decoupling of the mixing layer problem and the effect of energy release on the pressure field. The basis of our method is a method of characteristics with prescribed distribution of released respectively added heat. So in frictionless approximation the effect of heat addition on the pressure distribution along the base body is calculated. Beyond that the mixing zone is analyzed on the basis of boundary layer theory with the pressure prescribed in order to get information about possible distributions of energy released by reaction, where a diffusion controlled mechanism is assumed. For this problem we applied the well-known boundary layer program of Patankar and Spalding 1 based on an implicit finite difference scheme. In the following the method is described in short and is illustrated by a few applications. A more detailed analysis is given in References 2 and ANALYSIS 2.1 Method of Characteristics with Heat Addition The basic equations of inviscid nonisoenergetic flow differ from that in the isoenergetic case by the fact that the streamwise variation of the total enthalpy - coupled with a continuous entropy production - is no longer zero but given by ±(h + iw2) = Tj i = ± (1) at at w with q the added heat per kg and sec. Under condition of supersonic flow the characteristic equations can be derived from the basic equation of continuity and momentum in formal analogy to the isoenergetic case (see e.g. Reference 4) by treating the density in the continuity equation as function of pressure p and entropy s and considering Equation (1). The result is (see Reference 2 for details)

245 III p de I sin 6 1 /3p\ q 1 cot p. + + sin ji j = 0 * pw I r p\3s/ptw] (2) with = tan (6 ± p.) (characteristic direction) 3x and p. = arcsin (Machangle) M In the ideal gas case we have (^) = (3) pw p Cp It is not intended to go into details of the method of characteristics itself but a few remarks must be made regarding the prescription of heat sources within the field. (1) The region of heat addition is assumed to lie between given boundaries TI(X) < r(x) < r 2 (x). (2) The distribution within this region is described by a profile in x-direction and a profile in r-direction. The more important variation in x-direction is represented by the function Q(x) the heat added (released) per second in the total region of points with distances less than x from the nozzle exit plane or by its derivative which is the heat released in a strip of unit width in a distance x from the nozzle exit plane. According to a certain flowrate of unburnt fuel the maximum energy Om, assuming complete combustion, is known. Then assuming complete combustion again O(x) will approach Om with increasing x. The coordinate x Q at which Q( X Q) * Qm with practically sufficient accuracy can be termed a length of combustion. When prescribing the lateral heat distribution the following condition must be met 2,J r2(x) J r,(x) dq(x) pqrdr =. (4) dx If the function q is prescribed this condition can only be approximately fulfilled with help of a calculation without energy addition (see Reference 2 for details) because of the unknown density. It seems more appropriate to the fact that the region of heat addition is fixed a priori, that the heat added per unit volume in unit time is prescribed. q' = PQ (5) The profile itself we chose generally as a sin n x-function (with n = 1 in most of the cases run) with q = 0 at the boundaries r,, r 2 and one maximum between. Regarding the choice of the lateral boundaries r,(x) and r 2 (x) the following applies: Since the flowfield is calculated in a finite number of meshpoints by the method of characteristics the region of heat addition must not be too small in any direction in order to prevent that the prescribed heat sources are not "felt" sufficiently by the discrete meshpoints. Therefore we generally chose nearly the whole field between the body contour at the inner side and the shock starting from the outer nozzle-lip (because p ex j t > p<x>) at the outer side as region of heat addition. j = 0 plane flow, j = 1 axisymmetric flow.

246 II Mixing Layer Program For the analysis of the mixing layer around the rocket jet, we used the Patankar/Spalding boundary layer program GENMIX4, which has been developed for mixing and combustion problems in fully ducted flows. The flow system, for which this program is applied here is shown in Figure 2. It is a special feature of the program that the region of computation is restricted to the mixing layer itself. Only the first core of pure rocket jet is formally included in the procedure. Since the basis of the method is first order boundary layer theory a mean pressure depending only on X has to be prescribed, which follows as a first approximation from a method of characteristics calculation without heat addition. The turbulent reacting boundary-layer equations are found elsewhere (see References 1 and 2) and are omitted here. To evaluate the turbulent shear stress the Prandtl-mixing length model is used. In applying the program the dependent function to be integrated along x have to be specified with initial and boundary conditions. Beyond that a model for the chemical reaction has to be chosen, by means of which the density can be determined. Because we assume chemical equilibrium as the easiest possibility and because we have an effective binary mixture of rocket exhaust gas and air (uniform turbulent Lewis number) the appropriate variables are u C R velocity component in x mass fraction of rocket exhaust gas H total enthalpy l*= h H 1. The initial conditions are represented by a constant profile across the nozzle exit with two boundary points given by the flow state of the air streams (box profile). As long as the mixing layer does not reach the body wall the conditions for the variables at the inner (I) and outer (E) boundary are given by C RJ (x) = C RE (x) = 0 H,(x) = H,(0) = const. H E (x) = H E (0) = const, while the velocities follow from the pressure gradient, e.g. du[ 1 dp dx P U, dx Under condition of equilibrium density p and temperature T of a H, O, N-System can be calculated from the variables u 2 P, C R, h = H- because with C R all element mass fractions are given due to the binary mixture approximation. The considered species'were H 2, O 2, H 2 0, N 2, where N 2 has been treated as inert component. For a simplified calculation it was assumed that with the product H 2 O only O 2 or H 2 is present. The chemical composition is then immediately determined and the temperature follows from 2c K h K (T) = h with the enthalpy of a chemical component given by h K = h KO The full equilibrium calculation is described in Reference 2. The density then follows from the equation of state. Once the flow variables and the chemical composition is known the energy O(x) released per sec in the total mixing layer within distances lower equal x is given by Q(x) = 2vf E X pu[c R 2C K)R h KO + (1 - C R )2C K>L h KO - S K h KO ]rdr ri(x) where the subscript R respectively L denotes pure rocket gas respectively air (Luft). More directly one has with h KO =0 for 0 2, H,, N 2

247 III2-5 Q(X) = with the heat of formation 3. EXAMPLES OF APPLICATIONS 3.1 Parameters of the Computation As an example of application we consider the following case taken from an earlier study 5 flight altitude flight Mach number conical base body with length 15 km Moo = 3.5 r 0 (0) = 2.1 m dr 0 /dx = X B = 5 m nozzle mixing ratio chamber pressure flowrate Unburnt hydrogen flowrate heat release per sec at perfect O/H = «100 Atm m R = 400 kg/sec, thrust S R = N kg/sec combustion Q m = 4.22 IO 9 joule/sec profile of flow variables see table below. air gap air flowrate rh L = kg/sec (this flowrate is sufficient for stoichiometric combustion of the remaining hydrogen) The flow variables in the air gap are calculated assuming a plane two-shock in-take according to Reference 6 prescribing the pressure after compression (1 Atm in our case). The following table contains the flow variables in the nozzle exit plane. free stream nozzle air gap pressure (N/m 2 ) density (kg/m 3 ) temperature ( K) stream velocity (m/sec) Mach number c p /c v IO s IO From the air flowrates and the mass fluxes pu the positions of the nozzle lips are calculated to be r,(0) = , r E (0) = m. 3.2 Free Variation of the Heat Function Q(x) First of all we wanted to see the effect of a free variation of the heat function Q(x) subject to the condition Q(x) < Q m = const.. For this purpose we chose a linearly decreasing function ; dx with the combustion length Xn = 2Q n (dq/dx) x=0 (6)

248 III2-6 Figure 3 shows the pressure distributions for different lengths X Q and Figure 4 the development of the force r dr S B (x) = -27T pr dx. "o (7) The effect of the thrust augmentation is approximately* measured by the expression = SB - W Diff SR where Wpjff is the drag generated by the diffusor, which is proportional to rh L and S R is the primary thrust proportional to rh R. With Woifr = N according to Reference 6 the augmentation factor f is given by Figure 5 as function of the combustion length XQ. In order to make a definite prediction of the additional thrust we need definite information on the Q(x) function. 3.3 Mixing Zone Results The development of a mixing zone at constant pressure (1 Atm) and with unity turbulent Prandtl- and Lewisnumber is shown in Figure 6. The variation of Q(x) has been calculated by means of the profiles as shown in Figures 7 and 8 (mass fraction of rocket exhaust gas and temperature). The u-profile is similar to the C R -profile because of P T = Lp = 1. Influence of pressure on Q(x) : As shown in Figure 9 as a result of a method of characteristic calculation there is a strong pressure variation in x-direction. With the simplified pressure function p(x) in the upper half of Figure 10 one gets the mixing layer beneath and the corresponding Q(x) variation. According to Figures 10 and 11 there is a remarkable influence of the pressure on the combustion length, which increases strongly when the pressure decreases. 3.4 Variation of Rocket Mass Flowrate An example of combined mixing layer and energy addition analysis is given in the following examples where we changed the rocket mass flow retaining a constant air flowrate. Applying the same pressure distribution the Q(x) function for various rh R are shown in Figure 12. Since the first slope of Q(x) is nearly independent of m R the resulting combustion lengths increase strongly with the rocket mass flowrate. Correspondingly the increase of the base force S B with rh R weakens with increasing rocket mass flowrate (Fig. 1 3). The augmentation factor f in the upper half of Figure 13 therefore decreases for larger rh R 's. The decrease in regions of very small m R is due to the predomination of the constant term W D jff. The maximum value of f to be gained is Similar results are shown in the Figures 14 and 15 for the case of a doubled diameter. Here shorter combustion lengths lead to distincly higher values of the augmentation factor (to 0.35). These few examples may be sufficient to show the applicability of the programs, although we are aware of the fact that they can only represent an intermediate step forward a uniform analysis. 4. REFERENCES 1. Patankar, S.V. Heat and Mass Transfer in Boundary Layers. Intertext Books, 2nd edition, Spalding, D.B. 2. Mittelbach, P. Schuberhohung durch Nachverbrennung bei Teilummantelung im Uberschall. MBB- Bericht UR A, Mittelbach, P. Ermittlung des Zusatzschubs durch Nachverbrennung bei einem teilummantelten Triebwerks mittels Charakteristikenverfahren und Mischzonenanalyse. MBB-Bericht UR , Liepmann, H.W. Elements of Gasdynamics. Wiley, Roshko, A. * It is assumed that the base force of the fuselage (in absence of the diffusor) is comparatively low.

249 5. Nitsch, E. Berechnung von Triebwerkskernfeldern fur Raketenmotoren mit Luftbeimischung bei Polzer, K. Teilummantelung. MBB-Studie, Dez., Schneider, K. 6. Geuenich, W. Bemerkung zur Erstellung der Anfangsbedingungen fiir die Berechnung des sich bei Mischung von Warmeschaltstrahlen und Nachverbrennung ergebenden Schubs. MBB-TN.RE /71. III2-7

250 III2-8 Fig. 1 Partially ducted system (schematic) air central body Fig.2 Flow system (schematic) for mixing layer calculation with a given mean pressure distribution p(x). Region of computation is shaded

251 III2-9 do. dx Q» 4.22 < IO 9 Joule/sac p [lo 5 N/m 2 Fig.3 Pressure distributions along base body for a given total heat addition Q m and linear dq/dx variation with different combustion lengths X Q dq_ dx Q m m k.22 Joule/sec 10 s«( 8,[-] C0(q. 0) n 5 x L n J Fig.4 Development of base force S B (x) with x for different combustion lengths X Q

252 S B" Dlff. lt.22 «10 9 Joule/sec. 5B. r 0 (o).2.1ii, r 0 (x B )-0.2m Fig. 5 Variation of augmentation factor with the combustion length u., 3800, U T 800, u_ «1030 n/sec ~" JC A * * T n , T T U20, T., K H J. at OlH r[m], Q l0 9 Joul«/8ecl p 10 5 M/» con»t Fig.6 Example of the development of the turbulent mixing layer at constant pressure and of the energy release function Q(x)

253 Fig.7 Profile of the rocket mass fraction at various x (same example as Figure 6) 2500 ( 2000 H t»o A " Fig. 8 Temperature profile at various x (same example as Figure 6)

254 Po> 15000m, MO, - 3.5, *,, - 400kg/»ec kg/sec, Q« x 10 5 N/m 2, X B - 10m Fig.9 Pressure distributions according to the method of characteristics (no heat addition!) along (a) shock at the outer nozzle lip- (b) outer boundary of rocket jet (c) inner boundary of rocket jet (d) contour of base body 1.0 :.[At.] r[»], t MO 9 Joule/secJ «[ ] Fig. 10 Development of mixing layer and Q(x) function for the above pressure distribution. Initial profile see text

255 S.KA 0.2 Q [l0 9 Joule/eecl [Atm] ~ ~ thcor. p.0.2 JAtrn] 2 L Fig.l 1 Q(x) variations at various pressure distributions. Initial profile see text 7 6 Q [lo'joule/secl r 0 (o). 2. I L = kg/aec p[atb] 5 60Okg/sec Fig. 12 Q(x) variations at various rocket mass flowrates and constant air gap flowrate

256 H«15000m, J^-3.5, r_(0)=2.1i 100 ZOO " kg/sec) N [kg/sec] Fig. 13 Base force and augmentation factor as a function of rocket mass flowrate as derived from the method of characteristics program with the Q(x)-variations of Figure 12

257 Q [l0 9 Joule/sec] Ap.800 kg/sac H m Moj-3.5 th«or. r n (o) kff/aec Fig. 14 Q(x) variations at another diameter of base body and air gap mass flowrate and various rocket mass flowrate

258 S B" V Diff r (0)=4.2 D, 1^= kg/sec r Q (o)=2.1 m, * L = kg/sec 0.1 ^[kg/sec] [' '» 10 W Dlff (* L = kg/sec) = N H = m, M00 = Fig. 15 Base force and augmentation factor as a function of rocket mass flowrate calculated with Q(x) variations of Figure 14

259 III3 SUPERSONIC MIXING AND COMBUSTION IN PARALLEL INJECTION FLOW FIELDS by John S.Evans Aerospace Technologist and Griffin Y.Anderson Head, Combustion Section NASA Langley Research Center Hampton, Virginia, 23665, USA

260 III3 RESUME Les techniques permettant de predire correctement les caracteristiques des ecoulements supersoniques avec melange et reactions chimiques sont d'une grande importance dans 1'analyse des performances des systemes de propulsion du type statoreacteur. On dispose a 1'heure actuelle de programmes analytiques relatifs a des ecoulements avec injection parallele, reactions chimiques et melange turbulent correspondant a des jets multiples. L'application de cette analyse a des geometries d'ecoulement simples est discutee et des comparaisons sont faites avec des donnees relatives a des cas plus complexes correspondant a des jets multiples en milieu reactif. Une revue est donnee des recherches effectuees au Langley Laboratory d'ecoulements avec injections paralteles. Parmi celles-ci 1'etude d'un seul jet du melange turbulent non reactif (H 2 dans 1'air et H 2 dans N 2 ) et de jets turbulents reactifs (H 2 dans 1'air) avec un seul jet ou des jets multiples. Ces implications des resultats a 1'etude des injecteurs de combustible utilises dans les statofusees supersoniques sont discutees.

261 III3-1 SUPERSONIC MIXING AND COMBUSTION IN PARALLEL INJECTION FLOW FIELDS John S.Evans Aerospace Technologist and Griffin Y.Anderson Head, Combustion Section NASA Langley Research Center Hampton, Virginia, 23665, USA SUMMARY Adequate prediction techniques for supersonic, mixing, reacting flows are of great importance in the design and performance analysis of supersonic combustion ramjet (scramjet) engines. Analytical programs for parallel injection flow fields with chemical reaction and turbulent mixing are now available for both single and multiple-jet flows. The application of these analyses to simple flow geometries is discussed, and comparisons also are made with data on the more complex case of multiple-jet, reacting flows. A review is given of Langley investigations of parallel injection flow fields. Among these are single-jet studies of nonreacting, turbulent mixing (H 2 in air and H 2 in N 2 ), and of reacting, turbulent mixing (H 2 in air) with both single and multiple jets. Implications of the results of the studies for scramjet fuel injector design are discussed. LIST OF SYMBOLS d; diameter of jet f r h fraction reacted, defined as mass fraction of reacted hydrogen divided by the mass fraction which would have reacted if the mixture were in equilibrium; f r = f r (x, r) distance of duct wall from center line k proportionality constant; see Equation (1) p p s p t r s static pressure stagnation pressure pitot pressure radial distance from center line jet spacing u flow speed x z e t T7 m axial distance from injection point mixing length scale turbulent kinematic viscosity mixing efficiency parameter, defined as the fuel that would react if complete reaction occurred without further mixing divided by the amount of fuel that would react if the mixture were uniform; Tj m = fj m (x) M t turbulent dynamic viscosity p Subscripts density e o free-stream value center-line value

262 III3-2 INTRODUCTION Several researchers in recent years have pointed out the advantages of using hydrogen-fueled, supersonic combustion ramjet (scramjet) engines for hypersonic air-breathing aircraft 1 " 4. One of the more important of these is the capability for active cooling of not only the engine but also the airframe by using the heat sink provided by the liquid hydrogen fuel 2. Good design will be necessary to insure that engine cooling requirements are kept low while thrust and/or efficiency are maximized. Among the principal factors to be considered are the distribution and quantity of heat release in the combustor. The research reported in this paper is aimed at improved understanding of this part of the design problem. Experimental testing at every stage of the design process is indispensable, but progress toward the ultimate design is expedited by using mathematical analysis to unify and extrapolate experimental results. With the aid of modern computers, it is possible to calculate fuel-air mixing and the distribution of heat release. Though such predictions are quite valuable, the limitations always present in theoretical calculations must be kept in mind, and the calculated results checked by experiment as completely as possible. For preliminary engine design studies a one-dimensional analysis has often been useful for estimating quantities such as wall pressure distribution and heat transfer. Cross-sectional area of the duct and chemical composition of the flow are specified functions of distance, and the conservation equations plus the equation of state are used to determine temperature, density, pressure, and flow velocity. For analysis of a single jet mixing with a parallel airstream, two-dimensional programs 5 ' 6 of greater complexity are used. The boundary-layer equations are cast into parabolic form and are solved by marching downstream from an initial plane perpendicular to the flow direction. Turbulence in the flow is modeled so that mixing takes place by diffusion as in laminar flow but at a much faster rate. When more than one jet is present, the mixing analysis must account for interaction between mixing regions after some point in the flow field. This interaction requires a three-dimensional analysis such as the three-dimensional parabolic treatment of Patankar and Spalding 7 or the quasi-three-dimensional approach of Alzner 8. The phenomenon of turbulent mixing is far too complex to represent in all respects by a model based on average values and inserted into the diffusion term of a laminar equation. An important aspect of turbulent flow which is often neglected by modeling approaches based on mean values is "unmixedness" a term used to describe the condition in which eddies of fuel and oxidant have swirled together but in which the molecular mixing necessary for burning has not taken place. In such conditions gas sample measurements yield results in which unreacted fuel and oxygen are both obtained at the same point in space. It may be possible to include "unmixedness" effects in turbulence modeling by calculating root-mean-square values of concentration fluctuations, as has been done by Spalding 6 and his associates. The two principal modes of fuel injection are injection normal to the flow or parallel to the flow. Normal injection gives more rapid mixing, leading to short combustors and rapid heat release. However, normal injection also causes some blockage of the airflow and creates relatively strong pressure disturbances. The necessity to avoid thermal choking by tailoring heat release to the expansion of flow area was discussed in Reference 4, where combinations of normal and parallel injection of fuel from struts were proposed to control Keat release. The mathematical task of calculating mixing and reacting flow is more difficult for normal injection than for parallel injection because it is necessary to solve the boundary-layer equations in three dimensions instead of two and because recirculation in the stream direction is important in part of the flow field. For these reasons practical results achieved thus far are empirical 9 and efforts toward developing the capability for computing normal injection flow fields are only now beginning to be feasible 7 ' 10. The remainder of this paper will be concerned with a review of recent NASA work on supersonic, parallel, mixing flows - some reacting and some not. This will include some work on multijet injection as well as injection with single jets. Implications of the results will be discussed in relation to design of the scramjet engine. REVIEW OF EXPERIMENTAL INVESTIGATIONS OF PARALLEL INJECTION Since both parallel and normal injection of fuel are needed to achieve proper distribution of heat release in a combustor, it is necessary to investigate the dependence of mixing characteristics on the parameters of the flow. For parallel injection the principal problem is to achieve sufficiently rapid mixing to keep combustor length within reasonable limits. The mixing is turbulent and, in the present state of the art for engineering calculations, can be calculated only by means of models which put the equations into the same form they have in laminar flow. Experiments are necessary to check the calculated results and to adjust empirical constants in the transport models. The following paragraphs are a brief review of some of the work on parallel injection either funded by NASA or performed in a NASA laboratory. In an investigation which looked at mixing between two airstreams in a coaxial configuration, Eggers and Torrence 11 tested an eddy viscosity model proposed by Zakkay 12 in the form p. t = pe t = kz(pu) 0, (D

263 where z is defined to be distance from the center line to the point where u = l/2(u e u 0 ). Since Equation (1) is valid only in the far field of a jet, they re-defined z to be the distance between the points where u takes the values IH3-3 u, = u e (u 0 - Uj), and (2) u 2 = u e (u 0 -u e ) ' (3) and found that this permitted center-line decay of velocity and concentration to be predicted in both near and far fields. Later Eggers 13 reported data taken on a coaxial hydrogen jet mixing with a supersonic stream of air without reaction. He attempted to predict the new data using Equation (1), with much poorer results for the H 2 -air tests than in the previous air-air tests. He noted that the H 2 -air data could be predicted using a more complicated model developed by Cohen and Guile 14 in which kinematic viscosity, e t, was modeled rather than dynamic viscosity, Mt = P^t This led him to omit density from Equation (1), so that it became e, = kzu 0. (4) The viscosity, jlt t, now modeled by M t = pe t = pkzu 0 (5) has proved to be highly successful for axisymmetric flows both with and without large radial density gradients and in reacting as well as nonreacting flow. A large part of the success of this model can be attributed to the fact that u t varies radially (through p ) as well as axially (through u 0 ). The possibility that heat release from reacting gases would influence the mixing process and thereby invalidate the use of nonreacting studies for prediction of mixing in combustors was examined by Anderson, Agnone, and Russin 15 in an investigation of a Mach 2 airstream and also by Cohen and Guile 14 in another coaxial mixing experiment with and without reaction. Both found that reaction had little effect on mixing. This conclusion is probably generally true for supersonic mixing, but too few experiments have been performed to be sure. Measured composition profiles reported in Reference 15 indicated unreacted hydrogen and oxygen were collected in the same sample for samples taken near the flame. This can be explained in terms of "unmixedness", which means that turbulent mixing of eddies of fuel and oxidant is not enough to produce complete burning of the fuel, since burning takes place only at the surfaces of these macroscopic regions. When the authors of Reference 15 modified the measured compositions by completing the reaction of any unburned hydrogen and oxygen present at a given point, excellent agreement was obtained with a composition profile computed with the equilibrium chemistry analysis of Reference 5. Comparison of the degree of completion of reaction in the measured compositions with calculations including unmixedness will be given later in this paper. Beach 16 measured wall static pressures and pitot pressure profiles in a coaxial burner to study the influence on mixing of using different gases to simulate air in the external flow. He used the same coaxial hydrogen burner with heated air, with heated nitrogen (to simulate nonreacting air), and with vitiated air. (Vitiated air consists of the combustion products of hydrogen burned in air plus enough added oxygen to restore oxygen content to that of air.) Beach found that calculations using Eggers' viscosity model [Eq. (5)] matched all the data from his experiments with one value of the experimental constant. Single-jet studies are necessary for investigation of basic factors in the mixing process, but it is also important to look at the results of bringing a number of jets into close proximity with each other, since it is obvious that better distribution of fuel across a combustor cross section can be obtained by injecting fuel from an array of small jets than by injecting it at the same rate using a single large jet. Anderson and Gooderum 17 recently reported an investigation of mixing and combustion downstream of a strut from which five H 2 jets issued parallel to the airstream. Good agreement was noted between measured wall pressures and the pressure distribution calculated using a onedimensional theory in which fuel was reacted in direct proportion to distance from the injector. ANALYTICAL APPROACHES AND COMPARISON WITH MEASUREMENTS The three computer programs currently in use for predicting mixing and reaction in parallel injection flows are identified by the names CHARNAL 6, JETMIX 8, and A2330 (Ref.5). CHARNAL and A2330 predict the properties downstream of a single injector for plane or axisymmetric geometry. JETMIX treats the merging of multiple jets by assuming symmetry at midplanes between jets. Input to JETMIX is generated by one of the other two programs (for either plane or axisymmetric geometry), and the equations are solved in terms of a rectangular grid. In all the programs, values are given to all variables in an initial plane and the solution proceeds by marching downstream. In A2330 effective values of turbulent viscosity are obtained from Eggers' model [Eq. (5)]. A constant value of

264 III3-4 viscosity roughly equal to that in A2330 at the point where the solution switches from A2330 to JETMIX is used in JETMIX, CHARNAL computes turbulent kinetic energy and its dissipation rate from differential equations, but the initial distributions of these quantities are obtained from the Prandtl mixing-length model. In the following pages flow properties calculated with the tools described are compared with data from some of the experiments mentioned earlier. Pitot Pressures in a Coaxial Burner In Figure 1, pitot pressure measurements from Beach's coaxial mixing experiments 16 are compared with those calculated using A a single-jet, diffusion-controlled, mixing program using the Eggers' viscosity model. The nozzle, injector, and combustion duct are shown in Figure 2. Agreement between theory and experiment is good Not only do the data clearly show the radial positions of the mixing regions and the large variations in pitot pressure there, but they also show the evolution of profile shapes and the variation of center-line values with distance from the injection point. The fidelity with which the theory predicts these fairly complex curves is remarkable. Also, it was found that the same empirical constant in the viscosity model matched data in air, N 2, and vitiated air. Since pitot pressure is a good indicator for showing where mixing and reaction are taking place, these results lend confidence to the use of A2330 and Eggers' model for predicting mixing of circular jets with a parallel airstream. Composition Profiles in Multijet Mixing The comparison of theory and experiment for multiple-jet injection from a strut in a rectangular duct 17 is especially interesting. Figure 3 shows a schematic of the experimental apparatus used. Both the jets and the surrounding vitiated airstream were at Mach 2.2; hydrogen and air stagnation temperatures were 300 K and 2000 K, respectively. The flow from any one of the five injectors was treated (using A2330) as a single jet mixing with an unbounded coaxial airstream until the edge of the mixing region approached the plane midway between two neighboring jets. Conditions along a radius at this station were used as input to JETMIX and the solution was continued. As fuel and air mixed they were considered to react instantaneously. The flow region in which JETMIX calculates is illustrated in Figure 4. Symmetry of the flow permits the bottom and sides of the rectangle shown to be treated as frictionless, impermeable walls. For convenience, the duct wall at the top is treated in the same way. The dimension s is the distance between injector holes. The dimension h is determined by the mass flow of hydrogen, the mass flow of air, and the constraint that the equivalence ratio would be 0.6 if mixing were complete. The wake flow behind the strut was ignored. Step profiles of all properties were used as input to A2330, and two ways of defining initial conditions at the injection point were tried. (The strut blocked 27% of the duct cross section.) In the first approach, both H 2 and air flows were expanded isentropically from an area defined as duct cross section minus strut cross section to the full duct area. Conditions in this expanded flow were used as initial conditions for A2330. In the second approach, H 2 jet exit conditions and air free-stream conditions were used for A2330 and duct height was reduced by the strut thickness to maintain the overall equivalence ratio at 0 = 0.6. Comments on the results from the two approaches are given later. The calculated amount of mixing achieved at a given station, x, is plotted in Figure 5, where results are shown for both A2330 and for JETMIX. A reduced rate of mixing is evident after JETMIX picks up the solution from A2330. The overal amount of mixing achieved at a given x does not differ greatly between calculations using the two different approaches to account for strut blockage. Both reacting and nonreacting runs were made during the tests. To avoid reaction for the hydrogen while nearly duplicating the conditions of mixing in a hot flow, oxygen replenishment was omitted in the production of the vitiated air for nonreacting runs. From the plots of measured wall pressure shown in Figure 6, the rise in pressure due to burning of the fuel is easily visible, and the calculated variation of pressure (one-dimensional theory) is seen to agree well with the data. Profiles-of total hydrogen concentration measured at a point 0.77 m downstream from the strut by nine probes equally spaced across the center line of the duct exit are shown in Figure 7 and are compared with 'calculated profiles. All samples collected were essentially completely reacted, and total hydrogen concentration is used so that the spread of injected fluid for both reacting and nonreacting cases can be compared directly. As noted earlier, reaction seems to have little effect on mixing, since the measured profiles differ little between the reacting and nonreacting flows. The data are plotted against distance from the nearest jet center line with corrections applied as indicated to account for shifts in streamline positions caused by the duct exit shock. Note that the corrections are large only for probes 2 and 8 and that they are larger for the nonreacting case than for the reacting case, because the duct flow is more overexpanded for nonreacting flow. Prediction of profile shape is good, but the distance downstream of the injector at which best agreement is found does not match well between theory and experiment. This is attributed to unsatisfactory modeling of the turbulent transport in JETMIX, since by varying the level of viscosity the calculated distance from the injector can be made to agree with experiment. Assuming local chemical equilibrium, the calculated fuel distribution profiles in Figure 7 correspond to reacting between 68% and 79% of the injected fuel. This compares well with the amount of reaction (66%) required at the probe location to match the measured duct wall pressures with the one-dimensional calculation shown in Figure 6.

265 III3-5 Unmixedness Due to Turbulence As was mentioned earlier, measurements of composition by Anderson, Agnone, and Russin 15 revealed the presence of unreacted oxygen and hydrogen in the same sample for samples taken near the flame. In Figure 8 values of a quantity which represents.the degree of completion of reaction for each sample has been reproduced from one of their figures in order to compare with a theoretically predicted variation of the same quantity. The experimental points represent the ratio of hydrogen present in the form of water to total hydrogen present. The theoretical curve is based on the mean value and the root-mean-square fluctuation of total hydrogen concentration as calculated by CHARNAL. A simple step function variation with time for instantaneous concentration is postulated in which total hydrogen concentration has the mean value plus the root-mean-square fluctuation for one-half the time and the mean value minus the root-mean-square fluctuation for the other half. When the mean value is near stochiometric, both unbumed fuel and oxidizer can be present on the average. This model, which gives a reasonable representation of the observed data, may be valuable because of the potential importance of unmixedness in modeling turbulent flows in the near field. APPLICATION TO SCRAMJET DESIGN In the preceding sections several examples have been presented of the ability of current mixing calculations to model experimental data in supersonic combustion flow fields. Further developments, such as improved transport models and three-dimensional calculations, are becoming available and are expected to provide additional accuracy and flexibility in predictive capability. Still, problems remain which will require much additional effort before some phenomena can be satisfactorily predicted. (Examples are: ignition behavior, recirculating flows, and finite reaction rates in turbulent flows.) In designing a supersonic combustion engine, it is necessary to provide the structural engine with numerical estimates of heat transfer, drag forces, and pressure distribution, and to be able to specify the location and nature of fuel injectors to achieve the desired distribution of heat release. Maximum use is made of predictions based on theoretical analysis and on tests of separate engine components before going to tests on a complete engine. The fuel injection strut results shown in Figures 5 to 7 are an example of the combined theoretical and experimental approach. The variation shown in Figure 5 of accomplished mixing with distance from the injector can be used as input to a one-dimensional analysis for predicting wall static pressures (see Figure 6) and other flow properties in the supersonic combustor. These, in turn, are available to calculate combustor cooling requirements and wall pressure forces to allow rational evaluation of scramjet performance pontential from conventional engine cycle calculations. CONCLUDING REMARKS A brief review has been presented of some of the work carried out or supported by the NASA to develop and apply mixing analyses for the calculation of parallel-injection, supersonic-combustion flow fields. Satisfactory agreement was found between data and calculations using a simple mixing length turbulent transport model for single axisymmetric jets. A multiple-jet flow-field calculation using constant turbulent viscosity gave composition profiles of the correct shape, but the variation with distance from the injection point was not adequately predicted. Newer turbulent transport models now becoming available promise improved predictive ability in all calculations and offer additional capability to model features observed in real flows (i.e., unmixedness). However, a number of problems important for the application of mixing analysis to scramjet design will require considerable additional research. REFERENCES 1. Ferri, Antonio Mixing-Controlled Supersonic Combustion. Annual Review of Fluid Mechanics, Vol.5, pp , Becker, John V. New Approaches to Hypersonic Aircraft. Presented at the Seventh Congress of the International Council of the Aeronautical Sciences, Rome, Italy, September (Also in Aeronautics and Astronautics, Vol.9, No.8, pp.32-39, August 1971.) 3. Henry, John R. Hypersonic Air Breathing Propulsion Systems. Paper No.8, NASA SP-292, Beach, Harry L. November Henry, John R. Design Considerations for the Airframe-Integrated Scramjet. Presented at the First Anderson, Griffin Y. International Symposium on Air Breathing Engines, Marseille, France, June (Available as NASA TM X-2895, December 1973.)

266 III Edelman, R. 6. Spalding, D.B. Launder, B.E. Morse, A.P. Maples, G. 7. Patankar, S.V. Spalding, D.B. 8. Alzner, Edgar 9. Rogers, R.C. 10. Baker, A.J. Zelazny, S.W. 11. Eggers, James M. Torrence, Marvin G. 12. Zakkay, V. Krause, E. Woo, S.D.L. 13. Eggers, James M. 14. Cohen, Leonard S. Guile, Roy N. 15. Anderson, Griffin Y. Agnone, Anthony M. Russin, William Roger 16. Beach, Harry L., Jr 17. Anderson, Griffin Y. Gooderum, Paul B. Diffusion Controlled Combustion for Scramjet Application. Part I - Analysis and Results of Calculations. TR 569, General Applied Science Laboratories, Inc., December (Available as NASA CR ) Combustion of Hydrogen-Air Jets in Local Chemical Equilibrium. Report 73-1, Fluid Mechanics and Thermal Systems, Inc., Waverly, Alabama, November (Available as NASA CR-2407.) A Calculation Procedure for Heat, Mass, and Momentum Transfer in Three-Dimensional Parabolic Flows. International Journal of Heat Mass Transfer, Vol.15, pp , Pergamon Press, Three-Dimensional Mixing of Jets. ATL TR150, Advanced Technology Laboratories, Inc., Jericho, New York, July (Also available as NASA CR ) Mixing of Hydrogen Injected from Multiple Injectors Normal to a Supersonic Air Stream. NASA TN D-6476, September A Theoretical Study of Mixing Downstream of Transverse Injection into a Supersonic Boundary Layer. Report , Bell Aerospace Company, (Also available as NASA CR ) An Experimental Investigation of the Mixing of Compressible Air Jets in a Coaxial Configuration. NASA TN D-5315, Turbulent Transport Properties for Axisymmetric Heterogeneous Mixing. AIAA Journal, Vol.2, No.ll, November Turbulent Mixing of Coaxial Compressible Hydrogen-Air Jets. NASA TN D-6487,. September Investigation of the Mixing and Combustion of Turbulent, Compressible Free Jets. NASA CR-1473, December Composition Distribution and Equivalent Body Shape for a Reacting, Coaxial, Supersonic Hydrogen-Air Flow. NASA TN D-6123, January Supersonic Mixing and Combustion of a Hydrogen Jet in a Coaxial High Temperature Test Gas. Presented at the AIAA/SAE Eighth Propulsion Joint Specialist Conference, New Orleans, Louisiana, Paper No , November Exploratory Tests of Two Strut Fuel Injectors for Supersonic Combustion. NASA TN D-7581, 1974.

267 II13-7 HIGH TEMPERATURE (2220 K) TEST CONDITIONS CALCULATION GRID DUCT WALL D O NITROGEN DATA NITROGEN THEORY, AIR DATA AIR THEORY / /-DUCT (j) A_ - 6 "s -4 Figure k. Flow region for JETMIX calculations. Figure 1. Pitot pressure profiles for coaxial H 2 -air and Hg-Ng mixing. 1.0 MULTIPLE JET SINGLE JET O PROFILE LOCATIONS (SEE FIG. 6) WITHOUT INITIAL EXPANSION.6 WITH INITIAL EXPANSION cm.2 COMBUSTION DUCT j i i i x, m Figure 5- Mixing achieved as a function of distance from injector. Figure 2. Nozzle, injector, and combustion duct for coaxial mixing experiment. OMBUSTION DUCTi O REACTION D NO REACTION ONE-DIMENSIONAL CALCULATION -PARALLR INJECTION STRUT Figure 3. Supersonic combustion duct with parallel fuel injection strut. Figure 6. Duct wall pressure for parallel injection (equivalence ratio = 0.6).

268 III {.08f O REACT ING FLOW D NON-REACTING FLOW WITHOUT INITIAL EXPANSION WITH INITIAL EXPANSION 1.0 p MASS FRACTION m 0.68m THEORY (REF. 6) O DATA (REF. 15) r/d, Figure 8. Effect of unmixedness on fraction reacted; x/d. = 15-7, d. = m. <J 0 Figure 7. Total hydrogen profiles. (Horizontal bars show streamline shifts caused by exit shocks; numbers by symbols identify probes.)

269 IV! TURBULENT BOUNDARY LAYER IN HYBRID PROPELLANTS COMBUSTION by R.Monti University of Naples Institute of Aerodynamics

270

271 IV1-1 TURBULENT BOUNDARY LAYER IN HYBRID PROPELLANTS COMBUSTION R.Monti University of Naples Institute of Aerodynamics ABSTRACT A review is made of the different hybrid propellant combustion models (solid fuel and liquid oxidizer). The general equations for turbulent boundary layer combustion, together with the interface boundary conditions, are written from which several proposed models based on a number of assumptions can be derived. Most of the combustion theories and working formulae for solid fuel regression rates assume different orders of magnitude for the characteristic time ratios of the relevant processes (i.e. fuel vaporization, gas-phase chemical reactions and oxidizer diffusion). It can be shown that the models based on finite characteristic time for diffusion and chemical reaction can explain the experimentally observed regression rate dependence on both mass flux (pu) and combustion pressure. RESUME Ce travail a 1'intention de presenter une review de plusieurs modeles proposes pour la combustion de propellents hybrides (combustible solide et oxydant liquide). On e"crit les equations plus generates pour la combustion dans le strate limite turbulent et d'elles descendent, dans une maniere unitaire, les modeles proposes base's sur differentes hipotheses. La plupart des theories de la combustion et des formules proposees pour la vitesse de rstrogradation est base sur hypotheses environ 1'ordre de grandeur des rapports entre les temps caracteristiques des proces qui arrivent pendant la combustion (vaporization du combustible solide, reactions chimies dans la phase gaseuse et diffusion gazeuse de 1'oxydant). On demontre comme les modeles bases sur temps caracteristiques finis du proces de diffusion et des reactions chimies peuvent deployer la dependence (observee experimentalement) de la vitesse de retrogradation du flux de masse (pu) et de la pression de combustion. NOMENCLATURE c Cf Cp D E number of chemical elements in the mixture specific heat of the solid fuel specific heat at constant pressure diffusion coefficient activation energy F blowing rate (pv) w /(pu) e h specific enthalpy (chemical + sensible) H total specific enthalpy K) k thermal conductivity Le Lewis number m, molecular weight of species i rhj rate of mass transfer of species i at the fuel surface

272 IV1-2 p pressure Pr Prandtl number q heat flux R gas constant r number of gas phase chemical reactions f regression rate Re Reynolds number s number of species in the mixture St Stanton number t, characteristic time for i process T temperature u velocity component along x v velocity component along y Wj chemical reaction species production x,y system coordinates Oj concentration of species i ft, mass fraction of atomic component j in the molecule of species i 6 boundary layer thickness Ahf specific heat of formation X parameter defined in equation (15) p. viscosity v oxidizer/fuel ratio p density 0, diffusive mass flux of species i Tjj characteristic time ratios t,/tj SI parameter defined in equation (50) Subscripts c combustion e external f fuel h heterogeneous ox oxidizer s sensible T turbulent v vaporization w wall 1. INTRODUCTION This paper attempts to review some of the relevant existing models of hybrid propellant combustion. The general problem of the combustion of hybrid propellants is too complex to be treated analytically and numerically; thus attention will be confined to characteristics hybrid propellants exhibiting typical Hybrid combinations are of two fundamental types: 1) Solid fuel and liquid oxidizer; 2) Solid oxidizer and liquid fuel. We will mainly refer to the first, more "classical", combination and will examine a solid fuel exposed to a hot gaseous oxidizer flowing on its surface (i.e. we assume that vaporization of the oxidizer has been accomplished upstream of the combustion chamber). The burning process of the solid depends on the properties and on the conditions of 'both the fuel and the oxidizer; it may include a number of the following steps: melting, vaporization, ablation, solid, liquid and gas thermal decomposition, heterogeneous and homogeneous chemical reactions, diffusion of chemical species and heat transfer. We will focus attention on typical classes of ablating hybrid propellants and will, therefore, make a few assumptions regarding the fuel composition, its behaviour under high heat fluxes and at high temperatures. First of

273 all we will assume a homogeneous solid fuel (i.e. non-metallized fuel); this will allow us to neglect the presence of solid combustion products and therefore the effect of radiative heat transfer (Ref.l). Furthermore we assume that the fuel ablates (i.e. vaporizes without melting) so that no liquid phase will be present in the combustion process; this assumption somewhat simplifies the mathematical treatment of combustion. (*) The case under consideration is that depicted in Figure 1 where the gaseous oxidizer flows on the solid fuel surface; a combustion zone exists in the boundary layer where the diffused oxidizer and the gasified fuel burn. An important point here is to establish beforehand if the boundary layer type of analysis is applicable to the present situation. In fact it has been stated that the rather high mass blowing rates at the fuel surface may invalidate some of the basic assumptions of the boundary layer theory. This point is raised in References 3, 4 where a Rayleigh type analysis is carried out to avoid the boundary layer limitations (When the sublimation rate exceeds a critical value the boundary layer is blown off and a b.l. solution ceases to exist). At any rate the simpler Rayleigh analysis was motivated also by the difficulty associated with the non-equilibrium vaporization boundary condition at the fuel surface (Ref.3). However if attention is confined to normal bydrid propellants (i.e. solid fuel and gaseous oxidizer), which exhibit rather low values of regression rates, it can be shown (Ref. 1) that the applicability of boundary layer theory is valid within a rather broad range of experimental conditions. Excellent reviews have been made by Barrere (Ref.5) and Marxman (Ref.l) about the state of the art and the first theories on the combustion mechanism of hybrid propellants; these reviews were mainly concerned with diffusion controlled theories. Since then a number of works have been published which consider other phenomena not accounted for in the earliest theories. In order to put all the works and theories to be reviewed in the proper perspective, we will first of all state the general problem and write the governing equations for the gas and for the solid fuel. From the complete set of equations it is possible to derive approximate equations which are valid when proper assumptions are made regarding the different processes; in this way it will be possible to identify a number of categories of combustion model. Subsequently, within each category, the available combustion models will be discussed. IV STATEMENT OF THE PROBLEM With reference to Figure 1 let us consider the case of a steady hot oxidizer gas flowing parallel to the fuel surface (y = 0). A boundary layer is formed in the gas on the solid surface; at typical combustion conditions transition occurs at a Reynolds number of the order of 10 4 (low values of the critical Reynolds number are due to the combined effects of combustion and mass addition at the fuel surface) whereas values of Reynolds numbers of order 10 s + IO 6 are achieved in practical propulsive devices; this suggests we restrict our attention to turbulent flows (Refs. 1,6,7). The gas mixture (oxidizer, combustion products and vaporized fuel) is confined inside the duct (generally axysimmetric) formed by the solid fuel itself. Ordinary fully developed turbulent flow in a circular pipe is achieved in 40 to 100 diameters (Ref.6): the blowing effect should reduce this entry length but not in a very substantial way (the boundary layer thickening due to blowing diminishes downstream). Since typical length/diameter ratios for propulsive application are about 30, one may conclude that the boundary layer is not generally fully developed at the end of the combustion chamber. It is, therefore, of prime importance to study the entry length of the duct (pure oxidizer flowing at the outer edge of the boundary layer). Fuel consumption causes the duct diameter to change with time; even if the oxidizer mass flux were held constant at the outer edge of the boundary layer, the enlargement of the cross sectional area is responsible for rather substantial variations of the temperature, pressure, composition and burning rate with time. However, if the burning rate is small compared with the duct radius divided by the time scale of the combustion process, the problem under examination can be considered as quasi-stationary; this is indeed the case since the values of f are of the order of mm/sec and the fuel grain inner radius is several centimetres. In conclusion, not considering transient phenomena such as ignition, unsteady combustion and flame extinguishing (some of these phenomena are analysed in Reference 7), we may focus our attention on the problem of a turbulent, two dimensional, steady boundary layer in a reacting gas. The processes which may be present and govern the problem under consideration are: - The conductive heat transfer inside the solid fuel - The convective heat transfer from the flame zone to the solid fuel The thermal decomposition of solid fuel into other solid products and the vaporization of fuel at the gas-solid interface (*) In Reference 2 the case of hydrid propellant forming a flowing liquid layer on the solid fuel surface is considered with reference to a class of organic propellants with low fusion temperature additives or metallic hybrides utilized at ONERA.

274 IV1-4 The heterogeneous reactions at the fuel surface between the solid fuel and gaseous species diffusing through the boundary layer - The convection and diffusion of gasified fuel and/or of the heterogeneous chemical reaction products through the boundary layer - The diffusion of oxidizer through the boundary layer. The number and complexity of these processes pose a formidable task for both theoretical and numerical solutions of the problem. In general the available hybrid combustion models make some drastic assumptions, consider only one (or two) rate controlling processes and neglect the remaining ones (either absent or not important or in equilibrium). In the next section two sets of equations are written for the kinetic and thermal boundary layer in the gas, and for conductive heat transfer in the solid fuel; these equations are linked through the fuel-gas interface boundary conditions (at y = 0). From the general equations, from the boundary conditions and from the kinetic description of the processes it is possible to derive the simplified combustion models proposed in the literature. 3. GOVERNING EQUATIONS 3.1 Boundary Layer Equations The equations for the steady 2-D plane (*) boundary layer for a reacting gas mixture composed of s species are written with the usual notations as follows: Continuity: Momentum: Energy: f- (pu) + ^- (pv) = 0 (D ox 9y 9u 9u _ 9p 9T 9x 9y 9x 9y 9h 9h 9P 9u 9q pu 1- pv = u T 9x 9y 9x 9y 9y Equivalently, in terms of total enthalpy (H = h + v 2 /2), Equation (3) reads: 3H 9H 9 9q pu + pv = - (UT) - 9x 9y 9y 9y Species: 9o!j 9otj 9$;... pu + pv = + Wj W 9x 9y 9y ' Where: the species production rates, the tangential stress, the species diffusive fluxes and the heat flux are expressed respectively by: Wj = w 1 cr,«i,p); r = -G/ + M T ):r (5) 9«i Pi = -p(di + D T i)ttq = (6) (*) More appropriately the b.l. equations, for the usual geometries of hybrid propulsive systems, should be written for the axisymmetric case. However when the non-dimensional b.l. thickness (8/n) is much less than one, the solutions for the two cases practically coincide.

275 We recall that the species concentrations («j) and the diffusive mass fluxes (0j) must satisfy the conditions: IV1-5 S «i = 1 ; Jb 0j = 0 (7) i=l i=l The "effective" Prandtl and Lewis numbers are introduced: "-<,^ -> = < C PT^ and equation (6) reads: (9, If one may assume that the diffusion coefficients are all equal, then the effective Lewis numbers are also equal: In this case the energy equation (3a) reads: Le ( = Le (10) 9H 9H pu + pv = 9x 9y (11) ay The enthalpy (h) of the mixture is: s s r.t 1 h =. Ojhj = S en J Cpj dt + Ahfil (12) where Ahg is the specific heat of formation for the species (i) at the reference temperature T r. Assuming Dalton's law is valid for the partial pressures (p;) the perfect gas equation reads: The equations, (1), (2), (3), (4),and (7) together with the definitions (5), (9) and the state equations (12) and (13) constitute a set of s + 4 equations in the s + 4 unknowns (a,, T, p, u, v). 3.2 Solid Fuel Equation The solid fuel is exposed to the hot combustion flame and heats up. It will be assumed that the solid fuel-gas interface is flat and parallel to the x axis and that the gas-solid interface regresses into the solid with a steady velocity t (x). It is convenient to refer to a moving coordinate system for which the plane y = 0 coincides with the gassolid interface. Even with reference to the moving coordinate system, in practice true steady state conditions will never be reached for the solid fuel in the axisymmetric case (or even in the plane case, when the unburnt fuel thickness is of the same order (or less) than the heat penetration depth). At any rate we will consider that a quasi-steady state is reached for temperatures within the solid fuel so that, in the heat conduction equation, the time derivative term may be neglected. The energy equation for the solid fuel, in terms of the solid temperature Tf (x, y) reads: 9T f 9 / 9T f \ 9 / 9T\ PfCf f L = k fx M + (kfy j (14) 9y 9x \ 9x / 9y \ y 9y / where pf and Cf are the density and the specific heat of the solid fuel.

276 IV Generally the conductive heat flux along x can be neglected Tf = 0 and the fuel can be assumed 9x to be thermally isotropic (kf x = kf u = kf). Furthermore the solid fuel properties (pf, Cf, kf) are assumed to be constant with temperature, Tf. With these assumptions equation (14) may be simplified into: where: k f 1 X = - = const. p f C f f If the thermal solid-solid decomposition is neglected (so that the thermo-physical properties can be assumed to be independent of Tf) equation (15) can be integrated to give: which satisfies the boundary condition T f = T 0 -CXexp(y/X) (16) y = - oo ; Tf = TQ The constant C is found by imposing the boundary condition at y = 0 (which can be different according to the assumptions on the vaporization process). 3.3 Boundary Conditions The boundary conditions for the gas mixture are written at the edges of the kinetic and thermalboundary layers and at the solid-gas interface where they are shared with the solid fuel. (1) Edge of boundary layers Kinetic y = 5 k ; u[x, 8(x)] = u e (x) (17.1) v[x, 8(x)] = 0 Concentration y = 5 C ; «j[x, 6(x)] = a ie (1 < i < s) Thermal y = 6 T ; H[x,5 T (x)] = Hg (17.2) (2) Gas-solid interface (y = 0) u(x, 0) = 0 (18.1) p(x,0)v(x,0) = (p«0 w = pff(x) (18.2) q(x,0) = k f + p t f(x) [h(x,0) - hf(x,0)] (18.3) \0y/y=o S %mj(x,0) = p f f(x)0jf (18.4) where: ftj is the mass fraction of atomic component "j" in the molecule of species i I S 0ji = 0), rhj(x,0 J is the rate of mass transfer of species i at the -fuel surface (net flux) due to convection and diffusion: rhj (x,0) = p f r(x) aj(x,0) - < j(x,0) (19) In our problem the most significant boundary condition is equation (18.3) which expresses the energy balance at the solid-gas interface: the heat flux out of the gas q (x,0) must balance, at steady state, the conductive heat flux entering the solid Ikf (9T/9y) y= o I plus the energy necessary to transform the mass of fuel Ipf f(x) I from

277 IV1-7 solid to gas at the interface temperature T (x,0) = Tf (x,0). The expression of q (x,0) is given by equation (9) and is a sum of the conductive and of the diffusive heat transfer. 3.4 Chemistry In order to complete the set of equations for the boundary layer and the solid fuel, the expressions for the production terms of the gas phase reactions, the surface vaporization and the heterogeneous reactions must be given as function of the field variables Gas phase reactions (a) Global chemistry Let us assume that r chemical reactions, among the s gaseous species (Aj), take place in the mixture; the j reaction reads: fe V ij A i «"?, V ij A i < 20 > kjr (T) ' where kjpco, k ir (T) are the direct and reverse reaction rates. The time variation of the molar density (nj), due to all r chemical reactions, is given by: dn ; W: 1 = m; ' dt = mj. (Vjj' - Vjj) k kj - k jd (T) i n (21) where the molar densities are related to the mass fractions («;), to density (p) and to the molecular weights (m,) by: When the gas mixture can be assumed to be in chemical equilibrium then the differential equations (21) may be replaced by the algebraic equilibrium equations which, for the general j chemical reaction, read: fj (c*j, T, p) = TT nk^lcj - v kj) - Kj(T) (22) where Kj(T) kj R (T)/kj D (T) is the equilibrium constant, which can be expressed in terms of thermodynamic functions and of the stoichiometric coefficients of the j chemical reaction. Equation (22), which can be written for (s-c) independent chemical reactions, together with the (c) atom balance equations, determine the mixture composition in the field. (b) Simple chemistry Often the species balance equations are written only for Fuel (Fu), Oxidizer (Ox) and reaction products (Pr). This allows a large reduction in the species variables (from s to 3) (References 3, 8, 9, 12). The only chemical reaction to be considered is: Fu + r ox Ox *" Pr (23)

278 IV1-8 The expressions for the reaction rates are: nrr - MT) nfu n^* (24) w ox _ r ox m ox w Fu m Fu A fictitious mass fraction is often defined as: which, from equation (24), has a zero production term: m ox Q = «ox - r ox «FU (25) m FU = w ox - rox ± *W FU = 0 (26) m Fu Mass fraction <p can be conveniently chosen as a variable for the problem (together with a ox or «Fu ) to define the mixture composition. Again the condition for chemical equilibrium is immediately obtained from equation (24) putting Wpy = W ox = Surface vaporization In some instances the surface pyrolysis (vaporization) of the fuel can be one of the controlling mechanisms for the hybrid combustion (Ref.l3). The surface pyrolysis law is of the form (Ref.3): Differentiating equation (27) yields: Pff = w v exp(-e v /RT w ) (27) dr / E v \ dt w When a small value of dt w /T w causes large values of df/f one may say that, in practice, the vaporization process can provide any mass flow rate of the fuel vapor with a small adjustment in the surface temperature (T w ). The conditions under which this "vaporization equilibrium" is realized is Ev/RT w»1 i.e. when the required vaporization rates are low. Therefore when the vaporization characteristic time is much less than the other times involved in the process one may replace equation (27) by the equilibrium condition: TW = T ab, (28) In this case vaporization will not be the rate controlling process since it is capable of providing the fuel vapor at any mass flow rate. To check the vaporization equilibrium condition, the computed mass flow rate (pff). which is found (as an eigenvalue) from the set of the governing equations, must be less than that given by equation (27), computed at the fuel interface temperature (T w ) Heterogeneous reactions The surface reactions, between the oxidizer diffused through the boundary layer and the solid fuel, can be described by a reaction rate law of the form (Ref.7): k h = w h exp(-er/rtw) (29) In a non equilibrium situation (kinetically limited) the effect of surface reactions is to yield a surface heat flux proportional to k^ and to the heat of reaction. The amount of oxidizer consumed at the surface and the heat flux due to heterogeneous surface reactions depend on the surface temperature (T w ) through equation (29). When surface reactions are so fast as be considered in equilibrium, all the reactive species, diffused or convected onto the fuel surface, react to give the heterogeneous reaction products.

279 The hybrid propulsion mechanism strongly depends on the characteristic times of the gas phase reactions, fuel vaporization and heterogeneous reactions. These phenomena are somehow coupled; for instance the importance of the heterogeneous reactions which take place depends also on the gas-phase reactions: when the latter are in equilibrium no oxidizer is able to reach the surface and experience heterogeneous reactions (Ref.7).. At the same time ablation and heterogeneous reactions are coupled because part of the heat released by the heterogeneous reactions serves to vaporize the solid fuel. IV RATE CONTROLLING PROCESSES A number of studies have been made on hybrid combustion with the main task of deriving semiempirical formulae for the regression rate (f(x)) as functions of the thermophysical characteristics of the propellants and the flow conditions in the combustion zone. The motivation for these studies is that the functional dependence of f(x) on rocket parameters is necessary for hybrid rocket design. With reference to Figure 2, the relevant processes involved are: 1) solid fuel decomposition, 2) vaporization (and/or heterogeneous reactions), 3) diffusion of oxidizer and fuel vapor (and/or heterogeneous reaction products) through the boundary layer, 4) and gas phase reactions (fed by the oxidizer and the fuel diffused through the boundary layer). For the sake of simplicity and only to perform an order of magnitude analysis we will refer to local film conditions (i.e. we neglect variations in the flow direction compared with variation across the flow); in this case the above processes can be considered as steps in series. The controlling process is therefore the slowest one (which acts as a bottleneck). If tj is the characteristic time for the general process (i) one may have a number of simplified combustion models depending on the order of magnitude of the tj's ratios. The relative orders of magnitude of these times depend on the propellants and on the burning conditions. In general the solid-solid decomposition is not a very relevant process and it can be neglected due to the relatively small decomposition energies involved in typical polymeric hybrid propellants (Refs.13,14). Even if present, this process could be considered as the first step of the vaporization process and be confined at the fuelsolid interface. Assuming furthermore that heterogeneous chemical reactions are either absent or not important, the relevant characteristic times involved reduce to: - t 2 for the vaporization process t 3 for the diffusion processes - t 4 for the gas phase reactions Let us take the diffusion process characteristic time t 3 (of the same order of the heat transfer time) as reference and let us denote by: _ t 2 t 4 T 23 ~ 7~ > l3 T 43 ~ 7~ T 3 The orders of magnitude of T 23 and r 43 may give an indication of the rate controlling processes and of the possible simplifications to make in the governing equations. In Table I the limiting cases are reported with the indication of the controlling processes. Let us see what are the mathematical implications reported in Table I (row A and Column I), (a) Equilibrium of the vaporization process. If the interface temperature is at the fixed temperature T abl the constant C in the equation (16) may be immediately found and the temperature distribution in the solid fuel reads: T Tf ~! = exp(y/x) (3D T abl ~ T o The temperature gradient at the interface reads: dt f \ - T 0

280 IV1-10 i.e. the conductive heat flux warms up the ablating mass of the fuel from the initial (ambient) temperature (To) to the ablation temperature (T ab ) The boundary condition (18.3) at the interface may be written now, neglecting the heat conduction inside the solid fuel along x, as: where Ah v = h(x,0) hf(t 0 ) is the fuel effective heat of vaporization i.e. the heat necessary to bring the solid fuel from the initial (ambient) temperature to the gaseous state at the ablation temperature. (b) Gas phase chemical reactions in equilibrium. If the gas mixture is in local chemical equilibrium, the mixture composition is a function of the temperature, of the chemical components concentration (cp and of the density (or of the pressure). The s species balance equations can be replaced by the diffusion equations for the c chemical components (Ref.l 5, 16) which take the form (1 < j < c) ; where c; = i: feoi (34) J i=l J 4>cj = i: % 0i (35) When the c: values are known at each field point the mixture composition (i.e. the mass fractions a, ) may be found by the s - c equilibrium equations (22) plus the c atom balance equations (34). 5. USUAL SIMPLIFICATIONS Various authors adopt in their analysis a number of simplifying assumptions which are often the same; these mainly refer to the turbulence model and to the effect of mass blowing at the wall. 5.1 Turbulence Models A usual assumption made by the majority of authors is that the effective Prandtl and Lewis numbers are unity outside the thin reaction zone (Refs. 6,10,17); in this case equation (11) simplifies considerably. When the pressure gradient term can be neglected a further simplification is possible and the momentum and energy equations reduce to the identical form: where f is one of the variables u or H. 9f 9f 9 [ 9f I,,.. pu + pv = (M + %) r (jo) 9x 9y 9y I 9yJ Equation (36) states that the Reynolds analogy could be extended to the present problem (i.e. that there is a similarity in the non-dimensional velocity and total enthalpy "y" profiles); this is true also for the case of blowing at the surface (Ref.l7). The similarity allows one to express the Reynolds analogy as: a... 90/9y Outside the reaction zone w ; = 0 and also the mass fractions satisfy equation (36) so that the similarity is extended to 0!j(y). In the present situation the only effective (turbulent + laminar) transport coefficient is (p. + p. T ).

281 The eddy viscosity expression may be based on the classical formulations (Prandtl mixing length, Von Karman similarity law, Van Driest) or the more recent ones (Patankar-Spalding, Cebeci and Smith, Reichardt etc.) which are examined in detail in References 15 and 18. Turbulent "simple" flow empirical formulae (flat plate, constant properties, no blowing) have been extended to the hybrid propellant boundary layer. For instance velocity profiles and friction coefficient expressions IV1-11 = (y/5)" 2 (38) u e 4 Cfo = S to =.03 R; x 2 (39) have been used in References 7, 17, Blowing effect Many authors, instead of going into a numerical approach, which would have obscured the role of the various factors involved in the problem, have tried to separate the effect of blowing (analyzed with film theories) from that of two dimensionality (analyzed with the more classical approaches of non-reacting boundary layers with no mass addition). A classical Couette flow analysis performed by Spalding (Ref.27) examines the effect of the mass transfer parameter B = (pv) w /(pu) e ^- on the friction coefficient and on the Stanton number under the assumption that variations in the flow direction are negligible as compared to those in the y direction. For laminar, constant property flow the blocking effect caused by the surface mass addition is expressed by: where Cf 0 is the "zero blowing" friction coefficient. C f In (1 + B) ~- = ^ B (40) Cfo This simple formula has been derived under the assumption that blowing does not affect the boundary layer thickness and is therefore confined to small values of B. An analysis is performed by Marxman (Ref.l9) who finds a correction factor for equation 40 for turbulent flow (mixing length), taking into account the boundary layer thickening for the blowing conditions. For a range of 5 < B < 100 he suggests that the friction coefficient ratio can be conveniently approximated by: - = 1.2 B"" (41) 6. COMBUSTION MODELS Existing combustion models have been derived by assuming different rate controlling processes. The first models refer to the simplest case where only one process is assumed to be the rate controlling one (for instance diffusion or vaporization of the fuel). These simple models have sometimes been corrected to include the effect of other processes (when some of the characteristic time ratios tend to approach unity.) For instance, since the diffusion limited models fail to explain the regression rate pressure dependence at low pressures (Ref.28) (because at these conditions both r 23 or r 43 cannot be assumed much less than one) a pressure dependence term may be included as a correction when kinetic processes begin to become important (Ref.27). We will refer to specific papers for details of the derivations of the different models, the main features of which will be only briefly summarized here. We will try to label, whenever possible, each combustion model with a letter (A, B, C) and a Roman number (I, II, III) according to the cases illustrated in Table I.

282 IV Diffusion Controlled Combustion (A-I) The basic formula is equation (32) which, in terms of the blowing mass flux reads: Pf t = (PV) W = «±» (42) Ah v The main contribution of the different authors is to suggest explicit expressions for q(x, 0) at chemical equilibrium conditions. Several simple expressions for the q(x, 0) dependence on flow conditions and on the solid fuel characteristics have been proposed. These formulae, which are listed here for convenience, are fully reported in Reference 5. Barrere and Moutet q(x,0y = 0- (T C -T W ) (43) x (pv) w where 0 is a constant and k is an average thermal conductivity for the gas. Classman and Fineman: q(x, 0) = PT' In l +-- Ah v (44) where: = Q + C D (T c - T w ) Ah v and (pv) w Ah v is the radiation incident on the surface emitted from solid particles at temperature, T p ; The expression for the boundary layer thickness depends on the particular geometry considered. Spalding: where q(x, 0) =.0296^- f-^-y (Re x )- 8 Ah v B u * B " '' (45) x u = H c - h (x, 0) Ah v and P/Pe Y x /2 Af d(u/u i J "' * e ) 1 + B u u/u e Marxman and Gilbert (Ref.6) suggest the following expression: q(x, 0) = S t p c u c [H c - h(x, 0)] (46) where subscript c denotes conditions at the combustion layer, and the Stanton number is written as function of a non-blowing Stanton number St 0 = icfo = CRe x -->

283 IV1-13 u St St = C Re-- 2 (pu) e - u c St 0 The velocity profile and the position of the flame in the turbulent boundary layer are found by means of the empirical turbulent law equation (38) and the mixing length theory as function of the oxidizer/fuel mixture ratio and of the temperature at the flame; the St/St 0 ratio is found as function of the blowing rate F = (pv) w /(pu) e Marxman in subsequent works (Ref.7,19) presents a refinement of his model defining the heat flux as: q(x, 0) = -Cfo I-M p e u e -*- Ah s (47) 2 \ c fo/ u c where Ah s is the gas sensible enthalpy difference between the flame and the wall Ah s = (C p T) c (C p T) w, the friction coefficient ratio is given by equation (41). The effect of variable fluid properties is taken into account in Reference 19 by a coordinate transformation of the Howarth-Dorodnytsin type (Ref.23) where the reference incompressible state has to be found in a semiempirical way by measurement of the boundary layer thickness. The position of the flame as function of the O/F ratio (i>) is determined by the condition that the free stream reactant is carried to the flame by turbulent diffusion: Kpv) w = 0ox (y c ) where the diffusion flux (0 0 x) i g given by (5.3). Denison (Ref. 16) considered chemically active ablating surfaces for rather low values of B (graphite ablation) and equilibrium chemistry. Expressions for the turbulent skin friction are derived for both the Prandtl mixing length theory and the von Karman similarity law, under the assumption that turbulent Prandtl and Lewis numbers are unity. 6.2 Laminar Models As pointed out in the introduction the experimental hybrid propulsion conditions are such that a turbulent boundary-layer is established at a short distance from the leading edge of the fuel grain; laminar flows are therefore not very important in practical applications. However, theoretical analyses for laminar flows are much simpler so that in them it is possible to account for non-equilibrium conditions that are not easily analyzed in turbulent flows. Thus it may be worthwhile to review the few works on laminar boundary layers available in the literature. Reference 20 reports some experimental data obtained in laminar flows of O 2 /N 2 mixtures flowing on PMMA slabs in a range of pressures below atmospheric. These data are rather satisfactorily correlated by the classical heat transfer controlled regression rate: p f f = c /-^ L (48) V x No pressure dependence was found, probably because at these very low regression rates the surface temperatures are in equilibrium (case A.I). In Reference 3 an analysis is presented which is based on the Rayleigh analogy instead of on the boundary layer. This work is relevant because it is able to predict, at least qualitatively, the regression rate pressure dependence (Fig.3). The two cases of equilibrium surface temperature (diffusion controlled, case A-I) and of non-equilibrium pyrolysis at the interface (case B-I), have been examined in connection with Plexiglas-Oxygen propellants. The set of equations is solved with the sublimation rate obtained as the eigenvalue from the boundary conditions at the interface. The analysis points out two important conclusions: (1) there exists a lower limit of chamber pressure below which the regression rate is no longer given as a real eigenvalue (at surface equilibrium condition with a Burke-Shumann diffusion flame); (2) when the surface non-equilibrium is accounted for, the tendency to saturation of the regression rate with increase in the flow rate can be predicted.

284 IV1-14 In Reference 4 a number of computation have been made using the method of Reference 3 at conditions such that the characteristic time for evaporation of the fuel, from the fuel surface, is comparable with the diffusion time (i.e. r 32 = O[l), case B-I). The fuel surface temperature is calculated by the equations and the regression rate is shown to depend on the ratio between the partial pressure of the oxidizer and the square root of the total pressure (p ox /\/P) (Fig. 4). 6.3 Gas Kinetics and Diffusion Controlled Models Reference 10 examines the case of a diffusion controlled combustion (case A-I) and the case where both the diffusion and the gas phase kinetics are of importance (case A-II). The authors divide the boundary layer along x into three regions : 1) 0 < x < x m where there is no combustion. At x = xm combustion begins since concentration gradients and diffusion flows become equal to the limiting values. 2) x m < x < x' m where the burning rate is affected by both diffusional and kinetic factors; 3) x' m < x where the boundary layer is rather thick and therefore the concentration gradients and diffusion flows become relatively small. In this zone the chemical reactions can be considered in equilibrium and the process is diffusion controlled. For purpose of qualitative estimates the effect of chemical kinetics was assumed to be of the form: w ox = K p" c& agj exp (- E/RT) (49) The main parameter, which measures the relative importance of the diffusional and kinetic factors in terms of the external conditions, is: where 5 is the boundary layer thickness. The expression for the burning rate has the usual form : where the value of: p f t = C r S to (pu) e C r = B! ^to depends on the structure of the reacting boundary layer and is a function of the parameter SI. It was found that for T 43 «1 (gas reactions in equilibrium, small SI) C r is almost independent of SI ; for increasing SI, C r decreases (i.e. when r 43 tends toward values of order unity) which apparently corresponds to the diffusion-kinetic combustion regime. The critical values of SI (Sl cr ) controlled law: corresponds to the departure of the regression rate from the diffusion f = (pu)-» Figure 5, taken from Reference 27, shows qualitative agreement for the points of departure (higher mass flow rates for higher pressures) where (pu) e = 8p v Sl ct. Results of calculations are presented in Figure 6 together with the experimental data obtained in References 27,28. The satisfactory correlation for the average burning rates of the Oxygen-Butyl rubber propellants (average over the entire fuel grain length) should be noted. 7. CONCLUSION It is clear that the solution of the reacting boundary layer on hybrid propellant is a very difficult problem. Complex numerical calculations are required when the number of the involved processes is large; they should mainly be intended for the analysis of particular propulsive conditions. On the other hand it is necessary to make available simple calculation procedures for use in the design of hybrid rockets. The need for working formulae for the regression rate dependence on the design parameters has resulted in the formulation of a number of simple models, with rather narrow range of applicability.

285 IV1-15 One purpose of the present paper has been to examine a number of combustion models and to identify their range of validity with reference to the rate-controlling processes. It would be convenient to know, beforehand, what should be the most appropriate model to adopt for given experimental conditions. However, it is not easy to quantify the order of magnitude of the ratios of the characteristic times of the relevant processes; in fact these times strongly depend on the local conditions in the boundary layer. It is therefore essential that the chosen model be checked with experimental data. It is important here to warn against the application of the boundary-layer solution to particular zones in the combustion chamber (e.g. at the liquid injector end) where the ideal model can be very far from the local flow field conditions. Similarly it must be pointed out that the models described and the equations written refer to "short burning" times (the fuel surface is assumed to be flat and parallel to the axial direction) and to steady state (which are somehow conflicting conditions). One of the main concerns of the rocket designer, apart from the achievement of maximum specific impulse, is maximum solid propellant utilization). In this respect some of the results from "short burning" analysis are sometimes too pessimistic if applied to long runs. In fact, non-uniform burning rates along x (exhibiting a minimum or a maximum), will in practice be more uniform due to the effect of a local increase (or decrease) of the mass flow rate. This may be one of the reasons for the usually observed uniform consumption of fuel grains in long runs (Ref.25,26). The b.l. type of analysis is still however the fundamental line of attack for hybrid combustion problems and work along these lines will eventually produce more accurate semiempirical formulae with extended validity range. REFERENCES 1. Marxman, G.A. 2. Lieberherr, J.F. 3. Tsuge, & Fujiwara, Y. 4. Omori, S. 5. Barrere, M. Moutet, A. 6. Marxman, G.A. Gilbert, M. Research on the Combustion Mechanism of Hybrid Rockets. AGARD Phenomenes de fusion et reactions de surface dans la combustion des lithergols. ONERA Fine Structure of Hybrid Combustion. Astronautica Acta Vol. 14 pp Gasdynamic Aspects of Hybrid Combustion with Non-Equilibrium Surface Pyrolysis. Astronautica Acta Vol. 15 pp La propulsion par fuses hybrides. International Astronautical Congress Turbulent Boundary-Layer Combustion in the Hybrid Rocket. 9th Symposium on Combustion Marxman, G.A. Boundary-Layer Combustion in Propulsion. pp th Symposium on Combustion 8. Hermance, Shinnar, Summerfield 9. Patankar, S.V. Spalding, D.B. 10. Kustov, YU.A. Rybanin, S.S. 11. Kashiwagi, T. Summerfield, M. 12. Borghi, R. Ignition of an Evaporating Fuel in a Hot Oxidizing Gas. Astronautica Acta Vol. 12, No Heat and Mass Transfer in Boundary-Layers. Intertext books London Effect of Chemical Kinetics on the Burning Rate of a Propellant Slab in a Turbulent Oxidizer Flow. Combustion, Explosion and Shock Waves Vol. 6, No. 1, pp Ignition and Flame Spreading over a Solid Fuel: Non-similar Theory for a Hot Oxidizing Boundary-Layer. Fire and Explosion pp Chemical Reactions Calculations in Turbulent Flow - Application to a Co-containing Turbojet Plume. ONERA Rabinovitch, B. Regression Rates and the Kinetics of Polymer Degradation. Combustion th Symposium on 14. Houser, T.J. Peck, M.V. Research in Hybrid Combustion. Progress in Astronautics and Aeronautics, Vol. 15 Heterogeneous Combustion, Academic Press, New York, pp

286 IV Anderson, B.C. Lewis, C.H. Laminar or Turbulent Boundary-Layer Flows of Perfect Gases of Reacting Gas Mixtures in Chemical Equilibrium. NASA CR Denison, M.R. The Turbulent Boundary Layer on Chemically Active Ablating Surfaces. of the Aerospace Sciences. Giugno Journal 17. Wooldridge, C.E. Muzzy, R.J. 18. Launder, B.E. Spalding, D.B. Measurements in a Turbulent Boundary Layer with Porous Wall Injection and Combustion. 10th Symposium on Combustion Mathematical Models of Turbulence. Academic Press, London and New York, Marxman, G.A. Combustion in the Turbulent Boundary Layer on a Vaporizing Surface. Symposium on Combustion th 20. Krishnamurthy, L. Williams, F.A. 21. Brian, P.L.T. Reid, R.C. Laminar Combustion of Polymethylmethacrylate in O 2 /N 2 Mixtures. Fire and Explosion, pp Heat Transfer with Simultaneous Chemical Reaction: Film Theory for a Finite Reaction Rate. A.I.Ch.E. Journal p Luglio, Blottner, F.G. Finite Difference Methods of Solution of the Boundary-Layer Equations. Journal, Vol. 8 No AIAA 23. Crocco, L. 24. Jaffe N.A. 25. Monti, R. Sorge, S. Recchi, V. 26. Gany, A. Manheimer, Timnat, Y. Transformation of the Compressible Turbulent Boundary-Layer with Heat Exchange. AIAA Journal, Vol. 1. No. 12 p The Numerical Solution of the Non-Similar Laminar Boundary-Layer Equations Including the Effects of Non-Equilibrium Dissociation. Lecture Series 12 AGARD-VKI Short Course on Hypersonic Boundary Layers Von Karman Institute for Fluid Dynamics Progettazione e prove preliminari su un motors a razzo a propellenti ibridi. Aerotecnica, Missili e Spazio, No Parametric Study of a Hybrid Rocket Motor. Israel Journal of Technology, Vol. 10, No. 1-2, pp Smoot, L.D. Price, C.F. Regression Rate of Non-metalized Hybrid Fuel System AIAA Journal, Vol. 3 p Smoot, L.D. Price, C.F. Pressure Dependence of Hybrid Fuel Regression Rates. AIAA Journal, Vol 5 p

287 IV1-17 TABLE I I II III T 43 «1 T 43 =0[1] r 43» 1 t 3»t 2,t 4 t 3,t 4»t 2 t 4» t 3» t 2 Vaporization in A T 23 «1 Diffusion equilibrium Controlled T w = Tf(x,0) = T abl B TV = 0[1] t 3, t 2»t 4 0[t 2 ] =0[t 3 ] =0[t 4 ] t 4»t 3,t 2 C T 23» 1 t 2» t 3» t 4 t 2» t 3, t 4 t 2,t 4»t 3 Kinetically Vaporization Kinetics Controlled Controlled Gas phase Gas phase reactions in Kinetics equilibrium Controlled «i = a ie (T,P) _ t 2 _ Vaporization time t 3 Diffusion time t 4 T 43 = = t 3 Gas reaction time Diffusion time

288 IV1-18 LAMINAR TRANS. TURBULENT OXIDIZER BOUNDARY _ REACTION ZONE HETEROGENEOUS REACTIONS SOLID FUEL Fig. 1 Boundary layer combustion model li DIFFUSION OF OXIDIZER GAS PHASE CHEMICAL REACTIONS DIFFUSION OF FUEL VAPORIZATION HET. REACTIONS SOLID FUEL DECOMPOSITION Fig.2 Relevant processes in boundary layer combustion model

289 IV1-19 surface equil. P(atm)= r </g/cm37sec) Fig.3 Saturation characteristics and pressure dependence of the regression rate in Plexiglass-Oxygen (Ref.3) polyethylene - 90%H / V I / X cm ~ n y i-u.^u 10 CO [.gr/(cm 2 -secjj i i i i ? 1/2? Q Afi (kg/cnr)' Fig.4 Influence of the oxidizer partial pressure P 0 on the regression rate (Ref.4)

290 IV1-20 REGRESSION LOWFL DWI RATE 1 fregiohi l i r NOT DEPENOE:NTL ON PRESSUREf y / HEAf TRANSFER CONTROLLED r=ag - 8 RATE ~T /LOW F'RESSURE- / I INTERMEDIATE- -HIGH FLOW- REGION / RATE REGION i i DN / f NOT DEPENC)EN1 / ON F,LOW JRE/ K INETICAL.LY ' CONTROLL Et) f=b pn LOCIJS OF MAXIMUM ri t ola)pe REGSESSION RATES HEAT TRANSFER CONTROLLED V X 1 HIGH- PRESSUR E x^ ^«- X x > /» ^^, / A ^^~ / IN- FE Rr-IEL[ / PIREss II R )l A F / (y^ ^^s* &_^ ^* ^ / i ii DEPENDENT' BOTH FLOW i AND PRESSL fl SPECIFIC MASS FLOW RATE TE Fig.5 Pressure and flow rate dependence of typical nonmetalized hybrid systems (Ref.27) MO 2 (cm/sec) J ? e u e (g/cm 2 sec) Fig.6 Average burning rate versus mass flux at constant pressures (Ref.10) Numbers near the experimental points denote pressures in atmospheres.

291 IV1-21 DISCUSSION Professor Williams: There is one aspect of your time ratios that I find a little confusing. You identify a vaporization time and a diffusion time, yet in the steady state these two times must be equal. To say that there are conditions under which one of these times is small compared to the other seems to me to violate a steady state condition. Author: Certainly under steady state conditions the vaporization rate must equal the rate of consumption in the reaction zone. But in some instances vaporization may be rate controlling and thereby influence the surface temperature. What we may do, for instance, is to assume that vaporization is not rate controlling and calculate the burning rate together with the surface temperature and then check to see whether or not this temperature is capable of providing the required rate of evaporation. Professor Williams: That is a useful way to look at the problem. The vaporization rate is strongly dependent on temperature and therefore you can, as a first approximation, find a reasonable temperature and calculate the diffusion rate. That should be a good approximation to the overall rate and you can then go back to the vaporization expression and make any necessary corrections to temperature. F.Jaarsma: I would like to make a comment based on some experimental work we carried out in this field many years ago. We boosted up the diffusion rate so that the diffusion time diminished considerably and we still found that diffusion was the limiting factor. M.Barrere: I would like to draw attention to the importance of the eddies which are formed at the surface and which are visible in photographs of the boundary layer. If these become significant the exponent of the flow could decrease from 0.8 to perhaps 0.5. In regard to the re-entry problem, the situation at the surface is very complicated because we have chemical reactions and gas emissions which are very difficult to model. It is more complicated than the hybrid system where there is a firm solid surface.

292

293 IV2 SOME PROBLEMS AND ASPECTS IN COMBUSTOR MODELLING.by F.Suttrop DFVLR Porz-Wahn, West Germany

294

295 IV2-1 SOME PROBLEMS AND ASPECTS IN COMBUSTOR MODELLING. F.Suttrop At DFVLR (Porz- Wahn, W.-Germany) some activities have been initiated which are aimed at developing a theoretical combustor model that allows prediction of CO- and NO - emissions and some of the more important burner characteristics with satisfactory reliability. The model should not be confined to gas turbine application but should also be applicable to industrial burners etc., where relative low temperatures and pressures can be observed, and which are gaining more and more attention from a pollution point of view. Thus the considerations are not restricted to the special case of equilibrium composition which can be assumed only at rather high temperature and pressure levels. A basic element of the considered model will be the Homogeneous Reactor. The program used is that developed by Dr.JONES (SHELL-Thornton). The kinetics introduced into the homogeneous reactor are simplified insofar as instantaneous decomposition of the hydrocarbon to CO and H 2 O is assumed, following the proposals of HAMMOND and MELLOR, EDELMAN and others. As a first step, a sequence of well stirred reactors was programmed which after ESSENHIGH gives an approximation of the one-dimensional or plug flow reactor. So this procedure is a simple method for solving ordinary differential equations. At this state, there is a possibility of comparing the theoretical results with experimental data, which were obtained in the quasi one-dimensional post reaction zone of a pre-mixed burner flame at atmospheric pressure using propane as fuel. This should be a very sensitive check of the validity of the assumptions before extending the well stirred reactor concept to more complicated models. The work done so -far has disclosed a number of difficulties which have to be considered even in such a simple system, before a satisfactory agreement with experiment will be achieved. Figure 1 shows NO-fractions plotted vs.flow time in the post-reaction zone considered. One problem is the proper determination of the amount of prompt-no which is formed in the flame front. So far, experimental values have been taken and introduced as initial conditions into the calculations. This problem becomes more important for fuel rich mixtures. The next question is to choose the correct rate constant for the reaction which controls NO-formation, denoted as k 20 in Figure 1. Shown are calculated curves using the rate values given by BAULCH and others and NEWHALL respectively. Experimental data lie between those curves as indicated by the dashed line. The third problem is the temperature drop caused by radiation. In the present case estimations indicated a value close to 1 IK/ms. Figure 2 shows the influence of radiation on NO-formation. The temperature drop was varied between zero and 11 K/ms. One can see that for residence times between 5 and 10 ms this influence is considerable. A more fundamental problem is raised by the question of what initial composition can be assumed for the plug flow reactor, in other words, what amounts of the free radials O, OH and H are leaving the flame front and entering the reactor. In order to obtain a feeling for the real conditions two extreme cases were considered in one of which zero concentrations of the radicals were assumed (which is certainly too low) whereas 10% dissociation of O 2 and water was assumed in the other case (which is probably too high). Figure 3 shows the mole fractions for the first case, plotted against time. In the left hand section of Figure 3, which was calculated with a sequence of extremely small homogeneous reactors, one can recognize that a partial equilibrium of the radicals is established very quickly, followed by a moderate decrease farther downstream. Figure 4 presents the case with rather high initial amounts of radicals. The partial equilibrium is achieved still more rapidly in this case. However, the equilibrium concentrations differ from the previous case. Note that the temperatures are much lower due to the energy loss caused by dissocation. So the differences between the two considered cases in the downstream decay of the radical concentrations are considerable thoughout the plug flow reactor, as shown in Figure 5 (presenting the O-concentrations as an example). The effect of the initial amount of radicals on the NO-formation is important, as demonstrated by Figure 6. Though higher initial O- concentrations should favour the NO- formation, the effect of the lower temperature

296 IV 2-2 in this case is dominant and yields lower net rates for the NO- formation. Though there seems to be a closer numerical agreement between theory and experiment in the case of the higher initial radical concentrations, the characteristics of the curves differ appreciably. The author tends to prefer the case where no initial radicals are present, but the question is still open to discussion. However, it seems clear that all these effects should be taken into account when using simplified combustor modelling techniques.

297 Z5D n dtnoj/dt i 1.36 IV2-3 c PRO PANE-AIR FLAME (BAULCH \l o o SU. i o Z EXPERIMENTAL PROMPT-NO-LEVEL EXPERIMENTAL DATA 7x 0' 3 exp (75500/RT) NEWHALL) TEMP. DROP pue TO RADIATION «II I TIME, ms Fig. 1 NO-formation downstream of flame front position P = 1 atm, T E = 298 K, A/F-ratio =.99 a. Q_ I I dtnch/dt kjo - 7xlO xexp ( / RT) (NEWMALL) 2 o ISO -II K/ms i o TEMP. DROP DUE TO RADIATION TIME, ms Fig.2 NO-formation downstream of flame front position P = 1 atm, T E = 298 K, A/F-ratio =.99

298 IV24-2 I: Z 4 6 TIME.m* J I 8 10 IZ TIME, *H«10' Fig.3 Free radical mole-fractions (zero initial dissociation of O 2 and H 2 O) t i l l I T-2032K I o S u. I I o T-1905 K PART.EQUIL., K JO' 1 I I ) 0 TIME, ws j I I I I I TIME, io- Fig.4 Free radical mole-fractions for 10% initial dissociation of O 2 and H 2 0

299 IV I U. O o % INITIAL DISSOCIATION OF Q^ AND HiO, T$ 2032 K u İ ZERO INITIAL DISSOCIATION, ' 2054 K ID'- > IZ TIME.O.S Fig.5 Mole-fractions of O for different initial conditions 1.4 a a..1.2 Z O 11.0 < a uḟ' I I i r ZERO INITIAL DISSOCIATION OF 0 2 AND HfcO, T K E a. a. o i.6 U. o -4 o g.2 10% INITIAL DISSOCIATION, T«1830 K 10% IN IT. DISS o.0! TIME, ms I I I I I IO TIME, ms Fig.6 NO-formation for different initial amounts of free radicals O, OH and H

300 IV2-6 DISCUSSION In reply to questions posed by Professor Classman on the kinetic data employed for CO + OH = CO 2 + H reaction, the author confirmed that two different values of activation energy were employed, depending on the level of temperature.

301 IV3 MEASUREMENTS IN TURBULENT FLOWS WITH CHEMICAL REACTION by N.A.Chigier Department of Chemical Engineering and Fuel Technology University of Sheffield

302 IV3 RESUME Get expose traite de mesures effectuees au sein d'ecoulements turbulents et en particulier de leurs possibilites d'application a la prediction des ecoulements s'accompagnant d'une combustion. L'auteur mentionne la precision relative des instruments utilises pour les mesures de vitesse, de temperature, et de concentration solide et gazeuse, et presente des exemples des modifications quantitatives qui ont marque les proprietes des flammes au fur et a mesure de 1'amelioration des techniques. II etudie le principe qui consiste a separer les grandeurs a dependence temporelle en valeurs moyennes par moyenne de temps et valeurs des moyennes quadratiques dans le cas ou Tin tensile des fluctuations turbulentes varie considerablement. II souligne la necessite de faire varier la periode de temps pour effectuer la moyenne des donnees suivant les conditions locales et montre que, dans certains cas, le processus d'etablissement de moyennes peut masquer la nature physique des phenomenes que Ton mesure. L'utilisation de tubes de Pitot a refroidissement par eau pour les mesures de vitesse, de pyrometres a aspiration pour les mesures de temperature et de sondes a aspiration de dimensions relativement importantes, a refroidissement par eau, pour 1'analyse des gaz et des particules perturbe I'ecoulement et constitue un facteur d'erreurs. Pour toutes mesures a effectuer au sein d'un milieu en combustion, il serait opportun d'utiliser a 1'avenir des sondes a laser, et d'avoir recours a des methodes optiques qui eviteraient 1'introduction de sondes physiques. L'auteur fait le bilan des developpements dans le domaine des andmometres a laser et dans celui de la spectroscopie Raman a laser. A litre d'exemple, il rend compte de 1'utilisation de sondes a laser pour les mesures de vitesses dans des flammes affectees de mouvements tourbillonnaires, et pour les mesures de temperature et de concentration de composes chimiques au sein de flammes de diffusion turbulentes. II souligne enfin le role important que joue la precision des mesures dans la formulation et la verification des methodes theoriques de prediction, tant analytiques que numeriques.

303 IV3-1 MEASUREMENTS IN TURBULENT FLOWS WITH CHEMICAL REACTION N.A.Chigier Department of Chemical Engineering and Fuel Technology, University of Sheffield. ABSTRACT Measurements in turbulent flows are considered particularly for their relevance to the prediction of flows with combustion. The relative accuracy of instruments used for the measurement of velocity, temperature, gas and solid concentrations is discussed and examples are given of changes in the magnitudes of flame properties as measuring techniques have improved. The concept of separating time-dependent quantities into timemean average values and rms values is examined for conditions in which the intensity of the turbulent fluctuations varies greatly. The requirement of varying the time period for averaging according to the local conditions is stressed and it is shown that, under certain conditions, the averaging procedure can conceal the physical nature of the phenomena that is being measured. The disturbance to the flow and the errors introduced by using water-cooled pitot tubes for velocity, suction pyrometers for temperature and relatively large water-cooled suction probes for particle and gas analysis are discussed. It is argued that future measurements in flames should be made with laser probes and that all measurements should be made optically, without the introduction of physical probes. Developments in laser anemometry and laser Raman spectroscopy are reviewed. Examples are given of measurement by laser probes of velocity in flames with swirl and of temperature and specie concentration in turbulent diffusion flames. The important role of accurate measurement in the formulation and testing of analytical and numerical prediction theories is stressed. NOMENCLATURE a = jet nozzle radius D = jet axial diameter f = measured frequency F = mean frequency f s = applied frequency h = Planck's constant r = radius R = radial distance R H = radius to Y H =iy H(, U e = mean velocity of external flow Uj = mean velocity of jet at exit plane u' = fluctuating axial velocity component u = mean velocity v' = fluctuating radial velocity component Vp = velocity of particle x = axial distance y = transverse distance Y H = mass fraction of hydrogen YH c = mass fraction pf hydrogen on the centreline Z = distance along flame centreline AE = rotational energy change for the rotational Raman spectrum

304 IV3-2 X L = laser beam wavelength t> d = difference beat frequency V L = laser beam frequency v 0 = frequency of photon V/ = half angle of laser beam intersection w = frequency INTRODUCTION The measurement of physical and chemical properties plays an important role in the development of analytical and numerical methods applied to flows with chemical reaction. Combustion in turbulent flows is such a complex phenomena that the formulation of the problem, with its appropriate boundary conditions, requires the specification of a physical model of the flow. The basic equations used in the analysis require to be shown to be relevant and correct for the system being analysed. When approximations are made on the basis of neglecting terms in the equations that are considered of lower order of magnitude than the dominant terms in the equations, these assumptions require verification. Semi-empirical modelling laws formulating relationships for shear stress, mixing length, kinetic energy of turbulence and correlation functions are used extensively. Finally, the accuracy of the analytical and numerical predictions is tested by comparison with experiment. Many of the papers which compare experiment with theory use phrases such as "good agreement" and "close relationship". In many of these cases the predicted results are adjusted or made to conform with the experimental results on the assumption that the experimental measurements are correct. In order that a set of experimental data can satisfactorily meet the above requirements it is necessary for the measurements to be accurate and sufficiently comprehensive in both the time and space domains. The purpose of this paper is to demonstrate how measurements made in turbulent flames contribute towards the understanding of the physical phenomena. Experimental studies are examined critically and measurements have been selected on the basis of their ability to provide new information as well as their suitability for comparison with theoretical predictions. In the past much of this experimental information was dependent upon probes introduced into the flames and these suffer from the disadvantage of causing physical and chemical disturbances to the system. The recent developments which have taken place in laser technology allow the measurement of instantaneous values of velocity, temperature, particle and specie concentrations at points within the flow system. It has become feasible to measure correlations of the turbulent fluctuating components as formulated in the more comprehensive theories of turbulent flows at high temperatures with chemical reaction. TIME-MEAN AVERAGING IN TURBULENT FLOWS Time-mean averaging is generally adopted in theoretical analyses in order to remove the time-dependency and consider the quasi-steady state equations. All measuring instruments have frequency response limitations so that measurements need to be made over a finite time period. The sheer bulk of information of data from instruments with high frequency response requires some form of averaging in order that general conclusions can be derived rather than reporting measurements which are only specific at a particular time to a particular flow configuration. The process of reducing instantaneous measurements into time-mean average components and rms fluctuating components requires to be used with considerable care and discrimination. The criteria used for selecting the time period during which the averaging procedure is carried out is that this time period must be sufficiently long that the time average value would not change if the time duration for averaging was increased. In making measurements in shear flows where the intensity of turbulent fluctuations is high and for intermittent flows, the time period for adequate averaging can be of the order of several minutes. When such long time-mean average periods are being used, alternative means have to be used in order to demonstrate the structure of the flow. Figure 1 shows a photograph of a turbulent diffusion flame impinging on a flat surface 1. Under the special conditions of this experiment the cellular structure of turbulent eddies has been visualized. The structure of turbulent jet diffusion flames is generally inhomogeneous. When samples are withdrawn from the flame brush and rapidly quenched the co-existence of fuel, oxidant and combustion products are shown in the sample. The chemical reaction times are so short in the high temperature reaction zones that it is not feasible for the fuel and oxidant to co-exist at the same location. The samples withdrawn do not come from a homogeneously mixed mixture but rather from a flow at the point which is fluctuating in composition between a fuel-rich and fuel-lean mixture. At the interface, between the fuel-rich and the fuel-lean parts of the flow, chemical reaction will occur with fuel and oxidant diffusion towards this interface in a manner similar to that occurring in a laminar diffusion flame. This phenomena was first reported by Hawthorne 2 and has been studied more recently by Bilger 3. Hawthorne et al 2 coined the term "unmixedness" to describe this phenomena and demonstrated the relationship between unmixedness and the intensity of turbulent fluctuations.

305 The concept of a flame front is quite clear for the case of a laminar diffusion flame. In turbulent flames reaction zones occur where mixtures are within the limits of inflammability and temperatures are sufficiently high for reaction to take place. A flame front can be considered to exist in the region of the temperature maximum where the mixture ratio is generally close to stoichiometric. On the basis of measurements made by Kent and Bilger 4 of a hydrogen turbulent diffusion flame in a co-flowing stream of air, flame contours are shown in Figure 2. The H 2 limit is for a mole fraction of 0.1% and the O 2 limit for J%. The reaction taking place in the flame would then need to be confined between the fuel and oxidant limit surfaces. The analyses of Barrere 5 and Bray 6 show the importance of measuring correlations between velocity, temperature and concentration fluctuating components but, as yet, little progress has been made in the measurement of such quantities. The majority of measurements which have been reported in the literature have been time-mean average quantities. IV3-3 PHYSICAL PROBES A large number of measurements reported in the literature have been made with physical probes. These probes need to be sufficiently robust to withstand the high temperature conditions, with the presence of dust and liquid particles, found in flames. The range of water-cooled pitot tubes for velocity measurement, suction pyrometers for temperature measurement and water-cooled suction probes for gas and solid sampling have recently been reviewed by Beer and Chigier 7. Probes for laboratory scale flames are discussed comprehensively by Fristrom and Westenberg 8. At the time that these probes were used no alternative probes were known or considered feasible for use by the experimenters. Though attempts were made to minimize the interference effects it was generally not easy to assess the degrees of inaccuracy introduced due to the flow, cooling and catalytic reaction disturbing effects of the probes. One of the most comprehensive experimental studies of turbulent diffusion flames has recently been made by Kent and Bilger 3 ' 4 ' 9. They examined a hydrogen turbulent diffusion flame in a co-flowing stream of air. There were considerable density gradients in the system, due both to temperature variations and differences in the molecular weight of species. Water-cooled pitot and static pressure probes were used for velocity measurements. Temperature was measured with a Pt/Rh thermocouple covered with a non-catalytic coating and gas samples were withdrawn from the flow isokinetically, using a hot water-cooled probe. Examples of the measurements they obtained are shown in Figures 3, 4 and 5. It is interesting to note in Figure 4 that there is an overlap in the hydrogen and oxygen concentrations, in a region of the flame where temperatures were at their maximum, in the vicinity of 2000 K. The hydrogen and oxygen could not co-exist at these high temperatures without a reaction taking place. Because of the turbulent conditions the probe, situated in the flame, withdrew samples from the fuel-rich and fuel-lean sides of the flame front as the flame front fluctuated across the probe. The quantities of hydrogen and oxygen measured in the reaction zone can, therefore, be used in order to get an indication of the local turbulence level. Hawthorne et al 2 used this "unmixedness" in order to explain the fact that the observed timemean average flame lengths were approximately 25% greater than the length along the axis to the stoichiometric mixture line based on mean concentration measurements. Peak concentrations of nitric oxide occurred for both axial and radial profiles on the rich side of stoichiometry (Fig.5). The rate of production of NO was derived from the experimental data through the use of the species balance equation with the turbulent mass diffusivity derived from the hydrogen element species balance. Production rates for NO were also computed theoretically on the basis of the Zeldovich equations, assuming steady state, main species equilibrium and adiabatic temperature conditions. With these assumptions the production rate is a function of equivalence ratio alone with a peak at low fractions of equilibrium close to the stoichiometric concentration. TURBULENCE MEASUREMENTS A number of attempts have been made at measuring turbulence characteristics in flames. The hot wire anemometer cannot, in general, withstand the high temperature conditions of a flame. Rao and Brzustowski 10 have made turbulence measurements in a plume using a platinum-iridium wire which can be operated with a mean wire temperature up to 900 C. Parker" has made measurements of turbulence intensities in a flame using an iridium wire but this could only be used in regions of the flame where there was a reducing atmosphere. The probe could not withstand the high temperatures in the oxidizing atmosphere regions of the flame. Hypodermic glass tubes with internal flows of high pressure cooling water and surrounded by a thin film of platinum have been manufactured for turbulence measurements in flames but have generally been found to be too fragile for use in turbulent diffusion flames. Considerable progress in the measurement of turbulence in flames has been made by Gunther and co-workers 12 ' 13. Gunther reported on turbulence measurements made by Ebrahimi 14 using a water-cooled condenser microphone probe and Eickhoff 15 using a water-cooled disc static probe. The results are presented in Figure 6 for measurements of time-mean fluctuating axial, and normal components of velocity measured in flames compared with isothermal jet measurements. These results showed, surprisingly, that turbulence intensities in flames are lower than corresponding measurements in isothermal jets, and thus no evidence was found of flame generated turbulence.

306 IV3-4 PHOTOGRAPHIC MEASUREMENTS When there are solid or liquid particles in the flow, velocities can be measured by using high-speed photography. Local gas stream velocities were measured in a diffusion flame in the laminar boundary layer over a flat plate by Hirano and Kanno 16. Magnesium oxide particles were suspended in the flow and, by using a repetitive illumination system, velocities were measured from the particle tracks. These velocity measurements are shown in Figure 7. The velocity distribution across the boundary layer with fuel injection indicated that the thickness of the boundary layer with a diffusion flame was large compared with that without a flame, while the velocity gradient normal to the plate, at the plate surface in the presence of a diffusion flame, was found to be steep compared with that without a flame. A marked feature of the velocity distributions across the boundary layer with a diffusion flame was the appearance of a maximum velocity, which exceeded the air stream velocity. Even^at a point only a few millimetres from the leading edge of the porous plate a velocity over-shoot was observed near the flame zone. Once a diffusion flame was established the streamline and velocity profiles changed considerably compared with those without a flame. These aerodynamic changes were attributed by Hirano and Kanno 16 to the pressure distribution changes due to the flame reaction. Pressure distributions were calculated from the measured velocity and temperature distributions. A higher pressure region was found at the air stream side of the leading edge of the flame zone. At the same time, a lower pressure region appeared at the fuel side of the flame zone. The stream near the lower wall was decelerated and turned away from the lower wall until it attained the higher pressure region. After passing through the higher pressure region the stream was accelerated and turned towards the lower wall. The aerodynamic flow fields of the air stream side of the flame zone were found to be distorted, as if an object had been placed on the porous plate instead of a diffusion flame. The effects of pressure changes, acceleration and increase of the width of the flow fields as a direct consequence of combustion has not yet been clearly established in turbulent diffusion flames. Measurements made recently at the University of Sheffield using a laser anemometer in flames have shown clear indications of accelerations in the vicinity of flame fronts. Measurements of flame properties in flames containing particulate matter present additional problems, due to the impaction of particles on any physical probes introduced into the flame. A series of studies has been carried out at Sheffield University 17 ' 18 ' 19 ' 20 ' 21 ' on spray flames burning liquid fuel particles injected into the spray by liquid atomizers. Much of this work has concentrated on the measurement of particle sizes of the liquid droplets in the flames and the particle dynamics. A double-spark high-speed photographic system was developed in which two sparks are fired consecutively at electronically controlled times. A narrow depth of field of < 1 mm thickness is photographed. The droplet images are magnified 100 times and particle sizes are determined by direct measurement of the diameter of drop images. Measurement of the distance between the two images, coupled with the measurement of the time interval between the two sparks, gives the droplet velocity, and the angle of flight of the droplets is also determined from analysis of the photographs. The optical system, photographic equipment and the electronic control system, together with the method of analysis of photographic plates, is described more fully in Reference 17. Changes in mass of liquid droplets, as measured in a liquid spray burning in the wake of a stabilizer disc, 18 are shown in Figure 8. The mass of droplets < 50 p.m as a proportion of a total mass of droplets is plotted as a function of axial distance along the spray. The effects of variation of the annular air velocity from 8-10 m/s are shown in Figure 8 and the differences in drop size are compared between the unignited cold conditions and the hot burning conditions. Measurements of drop velocities are shown in Figure 9 as a function of axial distance along the spray for drop size ranges < 50 urn and > 100 urn. When the velocities of drops are compared with measurements of air velocity it can be seen that there are very significant deviations between the particles and the air. There are also significant differences between velocities of droplets in the hot burning spray as compared to those in the cold unignited spray. For the prediction of flames with particulate matter, such as liquid sprays, the spatial distribution of drops and their size distribution needs to be known. The rates of evaporation and burning of particles are a function of local temperature, vapour pressure, heat transfer and mass transfer conditions. The spray studies at Sheffield have shown that for both hollow-cone pressure jets and for twin fluid atomized jet spray flames, temperature conditions are too low and mixtures are too rich to allow burning to take place within the spray cloud. Instead of the classical concept of envelope flames surrounding individual droplets the main reaction zones were found at the outer periphery of the sprays. These flames are mixing controlled, as in diffusion flames, and not controlled by the rate of vaporization of the droplets. Theoretical predictions made on the assumption of envelope flames surrounding individual droplets assume that rates of heat transfer are governed by molecular conduction and rates of mass transfer are governed by molecular diffusion between the flame front and the droplet surface. For the cases of the spray flames studies at Sheffield, with flame fronts at the outer periphery of the spray cloud, rates of evaporation of droplets become dependent upon the turbulent transfer of heat and mass between the droplet surface and a flame relatively far removed from the drop surface. These studies have shown the important role that management has and must continue to play in providing the appropriate physical model and boundary conditions, which are a necessary prerequisite for the correct formulation of the theoretical problem.

307 IV3-5 LASER PROBES Lasers provide high intensity, monochromatic light sources which allow measurements to be made with spatial resolutions of < 1 mm 3. The developments in laser technology have been rapid and measurements have now been made of velocity, temperature and specie concentrations in flames. The laser anemometer has proved to be particulalary useful for studies in turbulent flames and measurements have been made under high intensity turbulent swirling conditions where measurements of turbulence characteristics can be made by no other means. Prior to the advent of lasers the signals in Raman spectroscopy were extremely weak and intensities were not sufficiently high for the instrument to be used for engineering purposes. The use of lasers, tuneable lasers, high resolution Raman spectrometers, double monochromators and high pressure cells have made it possible to make accurate point measurements of temperature and specie concentration in flames. Laser anemometers depend upon the measurement of frequency and frequency meters allow accurate measurements of velocity to be made in the ranges between fractions of a millimeter per second and supersonic velocities. The measuring volume is determined by the region of inter-section of two incident laser beams with wavelength X L and frequency V L. The scattered light is detected by a photomultiplier and, after data processing, the power spectrum is divided into a d c term, contributed by simultaneous scattering of both beams from a single particle moving through the volume with a velocity v p, a term corresponding to the beat frequency produced by scattering of the two laser beams from other particles, and noise. Analysis of the power spectrum for interfering light beams yields frequency shifts and amplitude functions that can be evaluated in terms of the properties of the scattering medium and of the light source intensity. For the determination of velocity components we simply utilize Doppler's Law, giving the relation between the velocity components and frequency shifts. Penner and Jerskey 22 have reviewed developments in laser probes. In earlier laser anemometers the laser beam was divided by a beam splitter with one beam serving as a reference beam and the other as the source of the scattered radiation. Durst and Whitelaw 23 have shown the advantages of using the fringe mode, sometimes known as the double Doppler configuration, and this mode is now being generally used. Seeding of the flame with particles is required and magnesium oxide particles have proved to be very effective for light scattering. This velocity is determined from the equation. p 2 sin \1/ For signal diagnostics, spectrum analysers, frequency trackers and single particle counters have been used. In the single particle counting method the number of fringes crossed by the particle passing through the measuring volume is determined and the time duration is measured by a nano-second clock. By this means instantaneous velocities are measured for each particle that passes through the control volume. When the output from the counter is linked to a computer, time-mean average and rms values of velocity can be determined. Since the laser anemometer cannot distinguish between the direction of the measured velocity components, a frequency shifting device needs to be used in systems with turbulence intensities above 30% and, when the velocity magnitudes are in the vicinity of zero, a rotating disc diffraction grating is particularly effective as a frequency shifting device and measurements are currently being made at Sheffield in swirling jet flame systems with recirculation zones. Baker, Hutchinson and Whitelaw 24 have used an electro-optic modulator to shift the frequency of one of the light beams when making measurements of the three components of instantaneous velocity in the highly turbulent recirculating zone of a flame with swirl. The burner was operated at a swirl number of 0.5 with methane as fuel and seeding the air supply with titanium dioxide particles. The fringes within the measuring volume are made to move with a finite velocity by means of the light frequency shifting device. When a particle crosses this moving set of fringes it now becomes possible to determine both the direction and the magnitude of the velocity. Signals from the photomultiplier were analysed by a computing counter which measures the frequency of the burst within a pre-set measurement of time, rms velocities are derived from rms frequencies as follows: n (F- f s ) The measured profiles of mean velocity and turbulence intensity are shown in Figure 10. By comparing these results with data obtained previously without frequency shifting the authors showed that, with frequency shifting, mean velocities were 100% higher and rms turbulence levels were 300% lower. Local turbulence intensities are seen to reach very high levels above 300%. It is not possible to make measurements under conditions of such high turbulence intensity by any alternative means. A description of the Raman effect is given by Penner 22, who compares Raman scattering with elastic Rayleigh scattering. When a photon of laser light interacts with a molecule, the Raman Stokes line returns to an energy level AE above the ground molecular energy level, while the Raman anti-stokes line is at a value AE above the virtual energy level. The virtual energy level is the one obtained when the energy of the photon hc 0 is added to the ground molecular energy level of the molecule. These energy levels are associated with molecules

308 IV3-6 in the excited state. For an equilibrium system the intensity ratio of the Stokes to anti-stokes lines is dependent upon the temperature of the molecule. The laser frequency V L must be well removed from the molecular absorption frequency. For non-resonant scattering the intensity of a Raman line is proportional to the frequency change, the number density of molecules, and a transition probability for the energy change. The number density is determined from a calibration experiment carried out under the same conditions of temperature and pressure as those in the test experiment. High laser intensities are required with the exciting radiation at high frequencies. The relatively much more intense scattered Rayleigh radiation at the exciting frequency is removed by filtering. Filtering can be acomplished with a high resolution Raman spectrometer using double monochromators. Filtering can also be accomplished with high resolution interference filters or by polarizing the exciting frequency. Substantial progress in the measurement of temperature by laser Raman scattering has been made by Lapp and co-workers 25 ' 26. Observation is made of temperature dependent effects in the spectral distribution of the Stokes Q-branch vibrational scattering. These effects arise predominantly from the vibration-rotation interaction and from significant population of excited vibrational levels. Upper-state bands originate from these excited levels and these are usually shifted towards the blue region of the spectrum. Observations were made using an argon ion laser, operated at 1.5 watts with 4880-A. The scattered light was analysed by a double monochromator with 5000-A blazed gratings. Scattering data for H 2 O and O 2 were obtained from lean H 2 -O 2 flames and data for N 2 were obtained from a lean H 2 -air flame. For the diatomic molecules considered in these experiments the Stokes Q-branch fundamental series profiles are calculated. These calculated profiles are used to fit experimental profiles in order to determine the scattering gas temperature. Figure 11 shows calculated Stokes Q-branch fundamental intensities for nitrogen over a range of temperatures from 300 K to 3500 K. Vibrational temperatures are proportional to the integral of intensity for particular bands, while rotational temperatures are proportional to the profile on the short wavelength side of each band via the influence of the vibration-rotation interaction. For calculations made of nitrogen profiles, the spectral width can be seen to increase relatively with increase in temperature (Fig. 11). The N 2 temperature was determined by fitting theoretical profiles to experimental profiles obtained from the flame. An example of this fitting procedure is shown in Figure 12. The Raman scattering signatures are direct measurements of the relative populations of the molecular internal modes and, for equilibrium situations, these relative populations correspond to the fundamental definition of temperature. This form of temperature diagnostics has the potential for becoming the most fundamentally accurate scheme for non-perturbing three-dimensional measurements. Temperatures measured by Raman spectroscopy are shown to agree with independently measured temperatures utilizing a fine wire thermocouple to within 2 per cent. Relatively few measurements have been made of gas concentration in flames but those reported recently by Regnier et al 27 have demonstrated the feasibility of making gas concentration measurements with laser probes. In order to improve the sensitivity they used a stimulated scattering process called "parametric four-wave mixing" or "coherent Raman anti-stokes scattering". This scattering is much more intense than spontaneous Raman scattering. Two co-linear light beams of frequencies u 1 and to 2 generate a co-linear (anti-stokes) wave at frequency 2coi - w 2 vibrational frequency co v = to, co 2. The intensity of the new wave is proportional to the square of the number density of resident molecules. In the experiments of Ragnier et al 27 a single mode ruby laser was used, together with a stimulated Raman scattering oscillator. This oscillator consisted of a high pressure cell filled with a mixture of 80% H 2 and 20% He at a total pressure of 30 atm. This pressure cell converts a fraction of the punp pulse at o^ and produces a pulse at the required frequency co 2. The high pressure cell contains the same gas that is being detected. The detection capability of the instrument was shown by making measurements on H 2 diluted in N 2 at concentrations ranging from 10 ppm to 100%. Measurements were made in a flame of natural gas containing 75% methane and 25% ethane, pre-mixed with air. The results of the H 2 distributions in a horizontal gas flame are shown in Figure 13. The peaks of H 2 concentration are in the vicinity of the reaction zone. The results shown in Figure 13 demonstrate the type of new information that can be obtained using laser optical concentration methods. The H 2 concentrations would not normally be detected by a gas sampling probe. H 2 is generated as a result of the cracking of hydrocarbons as they pass through the high temperature gradients immediately prior to the main reaction zone. Developments in tunable dye-lasers will allow measurements to be made of a number of species in the flames and will probably eventually replace the high pressure cell used by Regnier et al 27. CONCLUSIONS Measurements in turbulent flows with chemical reaction need to be examined critically, taking into account the instruments used for making the measurement. The detailed flow structure can be complex, as illustrated by the photograph in Figure 1. Measuring instruments cannot give a full description of the physical phenomena and the process of time-mean averaging can conceal the detail structure. In turbulent flow systems instruments with frequency response in the khz range are required in order to supplement time-mean average measurements with rms values of the turbulent fluctuating components of velocity,

309 temperature and specie concentration. Some information can be obtained from the use of water-cooled microphone and static pressure probes for velocity fluctuation and with bare thermocouples with wires in the range of 25 p. in diameter. No way has yet been found to measure concentration fluctuations using physical probes. Optical means are preferable for making measurements in flames. High-speed photographic techniques can be used for measuring size and velocity of liquid and solid particles present in the flame. Developments in laser anemometry and laser Raman spectroscopy have reached a state where it is possible to measure "instantaneous" values of velocity, temperature and specie concentration at "points" with a volume of the order of 1 mm 3. Measurements made in recirculation zones of flames with swirl by laser anemometer could not have been obtained by alternative means. Measurements of hydrogen concentration obtained by laser Raman spectroscopy show the formation of hydrogen as a result of cracking of hydrocarbons in the vicinity of the flame front. This information could not be obtained by suction sampling probes. Measurements using laser probes have many advantages over those obtained by physical probes and it is recommended that laser probe techniques should be adopted for measurements in flame wherever possible. IV3-7 REFERENCES 1. Milson, A Chigier, N.A. 2. Hawthorne, W.R. Weddel, D.S. Hottel, H.C. 3. Bilger, R.W. Kent, J.H. 4. Kent, J.H. Bilger, R.W. 5. Barrere, M. Prud'Homme, R. 6. Bray, K.N.C. 7. Beer, J.M. Chigier, N.A. 8. Fristrom, R.M. Westenberg, A.A. 9. Kent, J.H. Bilger, R.W., 10. Rao, U.K. Brzustowski, T.A. 11. Parker, K.H. Guillon, O. 12. Gunther, R. Simon, H. 13. Gunther, R. Lenze, B. 14. Ebrahimi, I. 15. Eickhoff, H. Studies of Methane and Methane-Air Flames Impinging on a Cold Plate. Combustion and Flame, (in press). Mixing and Combustion in Turbulent Gas Jets. 3rd Symposium on Combustion Flames and Explosive Phenomena, pp , Williams and Wilkins, Baltimore (1951). Concentration Fluctuations in Turbulent Jet Diffusion Flames. Technical note F-46, Department of Mechanical Engineering, University of Sydney (November 1972). Turbulent Diffusion Flames. 14th Symposium (International) on Combustion, pp , The Combustion Institute, Pittsburgh (1973). Equations fundamentales de I'aerothermochimie, Masson et Cie, Paris (1973). Equations of Turbulent Combustion I, Fundamental Equations of Reacting Turbulent Flow, University of Southampton, Department of Aeronautics and Astronautics, Report no. 330 (October 1973). Combustion Aerodynamics, Applied Science Publishers, London and Halsted-Wiley, New York, (1972). Flame Structure, McGraw-Hill, New York (1965). Measurements in Turbulent Jet Diffusion Flames. Technical Note F-41, Department of Mechanical Engineering, University of Sydney (October 1972). Combustion Science and Technology, 1, p. 171 (1969). Local Measurements in a Turbulent Flame by Hot-wire Anemometry. 13th Symposium (International) on Combustion, pp , The Combustion Institute (1971). Turbulence Intensity, Spectral Density Functions and Eulerian Scales of Emission in Turbulent Diffusion Flames, pp th Symposium (International) on Combustion, The Combustion Institute (1969). Exchange Coefficients and Mathematical Modes of Jet Diffusion Flames. 14th Symposium (International) on Combustion, pp , The Combustion Institute (1973). Chem. Ing. Techn., 40, pp (1968). Chem. Ing. Techn., 40, pp (1968).

310 1V Hirano, T. Kanno, Y. 17. McCreath, C.G. Roett, M.F. Chigier, N.A. 18. McCreath, C.G. Chigier, N.A. 19. Chigier, N.A. Roett, M.F. 20. Chigier, N.A. Makepeace, R.W. McCreath, C.G. Aerodynamic and Thermal Structure of the Laminar Boundary Layer Over a Flat Plate with a Diffusion Flame. 14th Symposium (International) on Combustion, pp , The Combustion Institute (1973). A Technique for Measurement of Velocities and Size of Particles in Flames. Journal of Physics E: Scientific Instruments, Vol. 5, pp (June 1972). Liquid Spray Burning in the Wake of a Stabilizer Disc. 14th Symposium (International) on Combustion, pp , The Combustion Institute (1973). Twin Fluid Atomizer Spray Combustion, American Society of Mechanical Engineers, Winter Annual Conference, New York, ASME paper no. 72 WA/HT-25 (November 1972). Aerodynamic Interaction between Burning Sprays and Recirculation Zones. Combustion Institute European Symposium, 1973, Academic Press, pp (1973). 21. Chigier, N.A. McCreath, C.G. Combustion of Droplets in Sprays. Astronautica Acta (in press). 22. Penner, S.S. Jerskey, T. 23. Durst, F. Whitelaw, J.H. 24. Baker, R.J. Hutchinson, P. Whitelaw, J.H. 25. Lapp, M. Goldman, L.M. Penney, C.M. 26. Lapp, M. Penney, C.M. St.Peters, R.L. 27. Regnier, P.R. Moya, F. Taan, J.P.E. Use of Lasers for Local Measurement of Velocity Components, Species Densities and Temperatures. Annual Review of Fluid Mechanics, Vol. 5 (1973). Optimization of Optical Anemometers. Proc. Roy. Soc. London, Ser. A324, pp (1971). Velocity Measurements in the Recirculation Region of an Industrial Burner Flame by Laser Anemometry with Light Frequency Shifting. Atomic Energy Research Establishment, Harwell, Report no (1973). Raman Scattering from Flames. Science, Vol. 175, pp (1972). Laser Raman Probe for Flame Temperature, Project SQUID. Tech. rep. no. GE-l-PU, Thermal Science and Propulsion Center, Purdue University (April 1973). Gas Concentration Measurement by Coherent Raman Anti-Stokes Scattering. American Institute of Aeronautics and Astronautics, paper no (1973).

311 IV3-9 Fig.l Photograph of a turbulent methane diffusion flame impinging on a flat plate, showing the cellular structure of the turbulent eddies (Ref.l) 20 -.' STOICHIOMETRIC 160 Fig.2 Flame contours for a hydrogen turbulent diffusion flame in a co-flowing stream of air, U;/U e = 10 (Ref.4)

312 IV UJ KO Fig.3 Axial compositions and temperature distributions in a hydrogen turbulent diffusion flame with a co-flowing air stream, Dj/U e = 10 (Ref.4) LU z g u 0-3 cr U_ uj 0-2 o OJ r /n 8 10 Fig.4 Profiles of temperature and composition for a hydrogen turbulent diffusion flame with a co-flowing air stream (Ref.4)

313 IV l20 d. z o I o 80 < or u. UJ _i o Fig.5 Nitric oxide concentration profiles measured in a hydrogen turbulent diffusion flame in a co-flowing air stream as a function of axial distance and jet to external velocity ratio (Ref.4)

314 1V r ISOTHERMAL V.15 TURBULENCE INTENSITY l Fig.6 Turbulence intensity measurement in flames. (Ref.l2) (1) Axial velocity, time-mean values, Re = (2) Microphone measurement, axial velocity fluctuations, Re = 8000 (3) Static pressure probe, normal fluctuating velocity, Re = (4) Isothermal jet, normal fluctuating velocity, Re = WITH FLAME WITHOUT FLAME FLAME ZONE/o VELOCITY COMPONENT IN x DIRECTION, u, cm/sec Fig.7 Velocity profiles for a diffusion flame in a laminar boundary layer, demonstrating the phenomena of velocity over-shoot (Ref. 16)

315 IV HOT SPRAY COLD SPRAY ANNULAR AIR VELOCITY AXIAL DISTANCE, mm 150 Fig.8 Mass fractions determined from size analysis of droplets photographed in liquid spray flames. The effects of variation in annular air velocity and differences between burning and non-burning conditions (Ref.20) VELOCITY, m/s Fig.9 Velocities of droplets measured photographically in burning and non-burning liquid fuel sprays in the wake of a. stabilizer disc (Ref.20)

316 IV u 4 C/.) U U,m/s Fig. 10 Time-mean average and fluctuating velocity components of velocity measured by laser anemometer in the recirculation zone of a flame with swirl (Ref.24)

317 IV3-15 l l I ' I ' ' I ' I I L l WAVELENGTH (A) Fig. 11 Calculated Stokes Q-branch fundamental intensities for nitrogen, used for the determination of temperature by laser Raman scattering (Ref.26)

318 IV POINT DATA AVERAGE WAVELENGTH (A) Fig. 12 Determination of temperature using laser Raman scattering. Comparison of measured and theoretically calculated profile for nitrogen at 1546 K (Ref.26) E q. OL < cr 24-5 o zo o Fig. 13 Concentration distributions of hydrogen in a natural gas flame measured by laser coherent Raman anti- Stokes scattering (Ref.27)

319 IV3-17 DISCUSSION Professor Ferri: I am not questioning your conclusions, but I believe I could explain the results on the assumption that the chemical reaction time is long compared with the mixing time without having to introduce a time dependent fluctuation. Author: The point I wish to emphasize is that we cannot use a simple model of a very thin reaction zone separating the fuel and the oxidant. Dr Winterfeld: Is it possible for small particles to agglomerate into clusters and thereby impair the measurements? Author: This question is one that is discussed at great length among those working in laser anemometry. In principle the extent to which a solid particle will follow the fluid flow is dependent on its drag to momentum ratio which increases as the size of the particle is reduced. Ideally one should use particles of 0.1 micron or less but we have not yet succeded in grinding magnesium oxide so finely, and we are using particles of the order of 1 or 2 microns which is certainly small enough for the measurement of time-mean average velocities. The real question is to what extent can these particles follow the fluctuations in the flow and particularly the higher frequency fluctuations. Clearly for any given particle size there must be a critical frequency above which the particle can no longer follow the flow. However, despite this limitation, we feel the method has considerable merit and we shall continue to use it until some better alternative can be found. Professor Bray: We have measured experimentally the fidelity with which particles follow a fluid flow using a cross-beam correlation technique in which the laser anemometer is crossed with another laser beam which is insensitive to density fluctuations. It is a schlieren beam and is therefore measuring a property of the flow, while the laser anemometer is measuring a property of the particles. If we carry out a Fourier analysis of the crosscorrelation of the two signals and obtain the phase relationship, we then find a phase lag which increases with frequency up to a cut-off value beyond which there is no phase relationship between the particle and the gas. This, I believe, gives a direct indication of the fidelity of the particles. Professor Ferri: We have also studied this problem and we believe there is no simple relationship between particle size and cut-off frequency owing to the large number of parameters involved. For example the drag of a particle depends on its shape as well as its size. However, our analysis suggests that one micron particles will follow all large scale fluctuations which are of most interest to us. Professor Williams: I am very interested in the first photograph you showed because it appears to illustrate the wrinkled laminar diffusion flame sheets that, as I pointed out yesterday from theoretical considerations, must exist in a turbulent diffusion flame. Could you say more about the photographic technique and the geometrical arrangement of the burner? Author: The apparatus we used is described in detail in Combustion and Flame. Basically it was a natural gas diffusion flame emerging from a round nozzle. The flame came upwards and impinged on a flat, cooled plate which cause it to spread outwards. Essentially it was an open, gaseous diffusion flame entraining ambient air. The photographic technique was very simple. What you saw was a direct photograph taken with an ordinary camera. I cannot recall the exposure time. Professor Williams: I think the plate has something to do with the flame structure. It is very analogous to data shown in a previous paper which was also concerned with a flame stabilized by a plate. Author: We have not seen this effect in ordinary gaseous diffusion flames so the presence of the plate must somehow have made visible the separation of the eddies. However, I believe this type of structure exists in most turbulent diffusion flames although it is seldom clearly evident. Major Schumaker: Is the picture identical to what the eye would see? If so it is time resolved. Have you taken any photographs with much shorter exposure times?

320 IV3-18 Author: The only photograph we have is the one you have seen. It was taken with a simple camera merely to obtain a photographic record of the apparatus.

321 IV4 SOME MEASUREMENTS AND NUMERICAL CALCULATION ON TURBULENT DIFFUSION FLAMES by Th.T.A.Paauw Laboratory for Thermal Power and Nuclear Engineering Department of Mechanical Engineering Delft University of Technology Rotterdamseweg 139A, Delft, The Netherlands

322 IV4 RESUME L'auteur presente les resultats de mesures et de calculs portant sur un champ d'ecoulement de combustion a I'interieur d'un four conique construit de telle maniere que les approximations relatives a la couche limite soient valables. Deux types de flammes ont et< observes: les flammes collees et les flammes decollees. Des mesures detaillees de temperatures, de vitesses et de fractions massiques de N 2, O 2 et CH 4 ont ete effectue'es pour les profils radiaux a differentes distances a I'inteSrieur du cone. La concentration de NO, telle qu'elle a ete donne"e par les mesures, s'est rsvelee etre sensible au type de flamme. Les resultats respectifs des mesures et des calculs portant sur la flamme de'collee, en particulier, concordent de fa9on raisonable.

323 IV4-1 SOME MEASUREMENTS AND NUMERICAL CALCULATIONS ON TURBULENT DIFFUSION FLAMES Th.T.A.Paauw Laboratory for Thermal Power and Nuclear Engineering Department of Mechanical Engineering Delft University of Technology Rotterdamsewfig 139A, Delft, The Netherlands ABSTRACT Measurements and the result of calculations are presented for the combustion flow field in a conical furnace, constructed so that boundary layer approximations were valid. Two types of flames were observed, an attached and a lifted flame. Detailed measurements of temperature, velocity and the mass fractions of N 2, O 2, CH 4 and NO were obtained r for the radial profiles at different distances along the cone. It appeared that the measured concentration of NO was sensitive to the flame type. The agreement between the calculated and the measured profiles of the lifted flame was especially satisfactory. INTRODUCTION In the last decade the analytical modelling of combustion systems has received growing attention. This is partly due to the development of numerical analysis to predict flow patterns to which a chemical reaction is added. Furthermore incentive to modelling has been the necessity to predict the levels of pollutant emissions. The production of pollutants is strongly influenced by combined action of the temperature field and the residence times in the flow system, directly coupled to the rate of mixing and chemical reaction. The most simple flow system in which mixing can be described is the boundary layer flow. Combustion and mixing in boundary layers have been numerically investigated by Edelman 1 et al. for a detailed kinetic system, and by Spalding 2 using a one-step chemical reaction. The numerical technique employed there for the parabolic system was a forward step finite difference technique. In most practical combustion systems, however, the boundary layer approximation is not valid because of the existence of recirculating zones. The numerical solution of the elliptic Navier Stokes equations describing a two dimensional flow field with recirculation and a one-step chemical reaction is given by the Spalding group 3, while Kennedy 4 et al. extended the method to obtain the solutions of the coupled mixing and detailed chemical kinetic process using a numerical successive over relaxation technique. The last method mentioned is, compared with a forward step method, rather time-consuming. Moreover for the case of turbulent flow the effective viscosity cannot be defined by simple laws such as the Prandtl mixing length theory. So an additional set of differential equations has to be solved 5. Also our knowledge of the reaction rate in turbulent flames is very restricted. As a rule its value is based on the time mean concentrations and temperatures, but in practice it is necessary to account for the covariance of the fluctuating concentrations and the fluctuations in the temperature 6. Moreover, when the Damkohler number, giving the ratio of the characteristic time for the reaction and the fluctuation time of the eddies which characteristizes the turbulent dissipation, is large, then the chemical reaction becomes controlled by the molecular diffusion process between the eddies 7. On this phenomena Spalding 8 based his eddy break up model, which was supported by the calculations of Lilley 9 for premixed swirling flames and Mason 10 for reactions in premixed boundary layer flows. The present paper gives the results of similar boundary flames in a conical geometry, specially constructed so that recirculation could not occur. Also some numerical calculations are presented using the forward step method for boundary layer flows. EXPERIMENTAL SETUP AND MEASUREMENTS A flame without recirculation was realised by designing a conical furnace (Fig.l). The smallest diameter of the cone was 300 mm. The conical angle was 11 (tg a =.2). Furthermore methane was supplied centrally

324 IV4-2 through a fuel pipe of 44 mm inner diameter and 89 mm outer diameter. Air flowed through the remaining part of the upstream surface. Fireproof cement was used as a lining and measurements were made through four holes in the wall at heights of respectively.2,.4,.6 and.8 m. To obtain flat velocity profiles at the upstream end use was made of settling chambers, while fine-mesh gauzes were also used to equalize the profiles. The burner was constructed so that an angular momentum could be imparted to the air flow by means of a vane-type swirler. During the measurements it seemed that at low swirl levels the flame lifted off. To establish a stable flame near the entrance of the furnace a ring of stabiles steel with a diameter of 80 mm was used as a flameholder halfway across the first hole at the upstream end of the conus. In this paper we will restrict ourselves to measurements without swirl and with a load of 100 m 3 /hr Slochteren gas (76% methane). The air flow was chosen to give an excess air of 5%. This corresponds to an inlet gas velocity of about 28 m/s and an air velocity of 5 m/s. Measurements were performed to determine the velocity the concentration and the temperature profiles. The velocity profiles were obtained with a five hole pitot probe. This method measures the velocity vector. From the measurements it appeared that the velocity vector was mainly directed perpendicularly to the crosssection of the furnace, so the boundary layer approximation was validated. The temperature was measured by a suction thermocouple pyrometer. The pyrometer was shielded to prevent heat losses by radiation. As for the velocity measurements the errors were less than 5%. For the gas analysis a Becker gas chromatograph was used, which gave the concentrations of N 2, O 2, CH 4, H 2, CO 2 and CO. Additional measurements for the concentrations of NO and NO 2 were made with a specially developed chemiluminiscent analyser. The error was several percent and the measurements reproduced well. The gas samples for the gas chromatograph were also taken by suction probes of 30 mm outer diameter. The diameter of the probes influenced the measured profiles at the lowest traverse for points passing the flameholder. For this reason only measurements were made between the flameholder and the measurement opening in the lining. Furthermore the measured wall temperature of the cone was below 400 C and measurements with a Shell heat flux radiation pyrometer indicated that the heat flux by radiation was lower than 15 kw/m 2, i.e. less than five percent of the total heat release in the furnace. Thus the measured temperatures were some 5% lower than if radiation were absent. The measurements of the flow and the concentration field were performed for both an attached flame as well as for a lifted flame. The profiles of concentration, temperature and velocity are respectively given in Figures 2, 3 and 4. NUMERICAL CALCULATIONS As already mentioned, the numerical method developed by Patankar-Spalding 2 was used to solve the boundary layer approximation of the Navier Stokes equations in confined flows. The differential equations can be generalised in the form shown in Figure 5. The quantity Q gives the source term, and the quantity <t> stands for respectively velocity, stagnation enthalpy, fuel mass fraction, the mixture fraction and nitrogen oxide mass fraction. For velocity the source term is the pressure gradient. This gradient is supposed to be constant over the cross section and is updated each forward step of the numerical procedure in the x-direction. It is given such a value that the integral balance of momentum and continuity over a cross section are satisfied. The source term for the stagnation enthalpy is due to kinetic heating, while the source term for the mixture fraction is zero. For the burning of methane we used the one-step global reaction system proposed by Hammond and Mellor 11 : CH 4 + 3/2 O 2 -» CO + 2 H 2 O. For reaction rate we used the empirical value for such a one-step reaction systems for hydrocarbons formulated by Edelman 12 (Fig.6). From early calculations with this reaction system it appeared that the temperature broadened after igriition much faster than was observed in the experiments. This could be understood by means of the eddy break up model, which implied that the reaction rate inside the mixing region could be completely determined by the rate of diffusion between the eddies and not by the chemical reaction rate. In our calculations we did not use an extended turbulence model. Because the concentration and velocity gradients in the mixing part of the flow were too low to give a reactionable eddy break up reaction rate, the following approximation was used. An upper limit (1.5) and a lower limit (.5) for the equivalence ratio was assumed between which ignition and burning could take place. This kind of inflammability limits forced the burning to take place where the gradient of the fuel mass fraction was steepest as illustrated in Figure 7. Of course this method was quite crude and should be seen as a first step. Furthermore we let the mixture ignite in the same place as ignition was observed in the experiment (.4m downstream for the lifted flame and. 1 m for the attached flame). This procedure was not only a consequence of the

325 fact that axial diffusion in the model was neglected but also because the effects of inter eddy diffusion on the effective reaction rate were not well known. The reaction rate for methane disappearance in the ignition zone was supposed to be the same as the reaction rate determined by an ignition temperature of 980 C. Because of our interest in NO formation an additional equation for NO was used. For the source term, the reaction rate put forward by Zeldovich, was used (Fig.6). It is based on the Zeldovich mechanism and the assumption that the O 2 molecules are in equilibrium with the oxygen radicals. It is dubious if equilibrium could be reached in one forward axial step of the numerical method. However, detailed kinetic studies 13 show that equilibrium could be reached in a few milliseconds, so for low velocities, as in our experiment the assumption seems justified. Furthermore the fluctuations in the mass fractions and the correlated temperature were calculated by the method Spalding 14 suggests for a turbulent flame. The fluctuations in the mass fractions in this study were taken as the mass fraction gradient times a characteristic length. These fluctuations gave the root mean square deviation of the distribution of the mentioned quantities. The variation of the quantities in time was presumed to be of castellated form so the distribution consisted of two peaks. Given this distribution the effective reaction rates and the mean density were calculated. IV4-3 DISCUSSION Some results of the computations are shown in Figure 8. Here the profiles are given for the two axial distances at.6 and.8 m from the entrance of the furnace. In the case of the lifted flame ignition occurred at.4 m axial distance; for the attached flame the ignition took place at.1 m where the flame holder was placed. For both cases the calculated mixing pattern was more or less the same. The only significant difference was that in the central region the O 2 mass fraction was lower for the attached flame but still present. Comparing the measured profiles with the calculations, one notes that the features of the lifted flame were rather well predicted. In the attached flame the influence of mixing was substantially greater. This was in the first place expressed by the disappearnace of O 2 in the central zone. Much more interesting, however, was the excessive production of nitrogen oxide. In Figure 9 the measured values of nitrogen oxide are given for both flames, together with the calculated ones. The calculated values of the attached were somewhat higher than those for the lifted flame, so there was a greater discrepancy between the predicted and the measured NO values for the attached flametype. A reason for this might be the assumption that the actual distribution of the temperature was possible much broader than predicted by the gradient method. This meant that small, very hot spots were generated next to large relatively cold spots. It is necessary to predict this distribution before a good estimate can be made of the reaction rate. Another explanation may be that the turbulent diffusion in the attached flame was small. If the characteristic time for diffusion is large compared with the characteristic time for the chemical reaction, the growth of the mass fraction of NO with the covered distance in the furnace is larger than for a small diffusion time. This idea is supported by the fact that the broadening of the measured temperature profiles is very slow. In this light the flatterning of the velocity profile in the central region caused by the flame holder is significant, because this means that the production of turbulent energy is negligible. It is clear that the numerical method used has to be improved to account for this effect. A first attempt to describe the diffusion process and the distribution of quantities like temperature more satisfactorily would be by the introduction of a turbulence model. Secondly, it may be necessary for describing ignition to introduce the covariance between the concentrations of the reacting species 70 2 TCH 4 /ro 2 rch 4» wl^h 7 as fluctuations of F, to predict the influence of concentration fluctuations on the reaction rate, by describing the evolution of this quantity as an additional differential equation. A third important point could be the choice of the form of the distribution function for the various quantities with time, which we have supposed to be of a castellated shape. ACKNOWLEDGEMENTS A substantial part of the numerical work was done by ir.h.bartelds and the assistance of ing.e.arnold Bik and W.van der Burgh was very valuable in obtaining the measurements. REFERENCES 1. Edelman, R. A.I.A.A. paper (1971) Economos, C.

326 IV Patankar, S.V. Spalding, D.B. 3. Gosman, A.D. Pun, W.M. Runchal, A.K. Spalding, D.B. Wolfshtein, M. 4. Kennedy, L.A. Scaccia, C. 5. Launder, B.E. Spalding, D.B. 6. Donaldson, C. du P. 7. Chung, P.M. 8. Spalding, D.B. 9. Lilley, D.G. 10. Mason, H.B. Spalding, D.B. 11. Hammond, D.C. Mellor, A.M. 12. Edelman, R Fortune, O. 13. Marteney, P.J. 14. Spalding, D.B. Heat and Mass Transfer in Boundary Layers, (Intertext Book, London, 1970) Heat and Mass Transfer in Recirculating Flows, (Academic Press, New York, 1969). Paper at the Int. Conf. on Num. Meth. in Fluid Dyn. (1973) Mathematical Models of Turbulence, (Academic Press, London, 1972) AGARD publ. CP3 paper B-l (1971) Phys. of Fluids (1972). 13th Symp. on Comb. 649 (1971) A.I.A.A. paper (1973) Eur. Symp. on Comb. 601 (1974) A.I.A.A. paper (1971) A.I.A.A. paper (1969) Comb. Sc. and Techn (1970) Rep. RF/TN/A/4. (Dept. of Mechn. Eng., Imperial College, London, 1971)

327 IV4-5 Conical furnace A: attached flame B: lifted flame conus angle: 22 length: 1.4 m entrance diam.: 0.3 m S: swirler f lame-holder M: methane A: air Fig. 1 Schematic drawing of the furnace VS/10 Differential equations: d0 d0 1 d with 0 = u velocity h stagnation enthalpy ^,:.JQ fuel mass fraction mixture fraction nitrogen oxide mass fraction Fig. 5 Generalised differential equations

328 IV4-6 Source terms: - velocity -(ap/»x) - stagnation enthalpy fc [r "eff WV 1 ) *T - fuel mass fraction p* * T n T exp(-12200/t) VS/11 - NO mass fraction p r M I exp(-67500/t) Fig. 6 Source terms Fig.7 Graphical description of reaction zone

329 IV4-7 3 o o 00 ^O ^r CM -r- II II II II x a < x o

330 IV4-8 5 o oh 2 o o o. u ro oi b

331 ^ I E E to co K- IV4-9 <U +» 1 2 o ca 6 -O V 3 C E E VO CO o S 2 73 <u i OO oil

332 IV4-10 l i lifted flame Tl in ppm x in m =.8 calculated =.6 D =.8 measured A = r in cm Fig.9 Calculated and measured NO massfractions

333 IV4-11 DISCUSSION In reply to questions from Major Schumaker, the author stated that on the graphs the open symbols represented experimental data while the closed symbols were based on calculations.

334

335 App.A- APPENDIX A LIST OF PARTICIPANTS REGISTERED AT 43rd PEP MEETING, 1-5 APRIL 1974 AUTHORS ANDERSON, Mr Y. Griffin BORGHI, Mr Roland BRAY, Prof. K.N.C. CHIGIER, Dr N.A. HUFFENUS, Mr J.P. LEUCHTER, Mr O. LIBBY, Prof. Paul A. MITTELBACH, Dr Phil P. NEER, Dr Michael PAAUW, Th T.A. PRUD'HOMME, Mr R. SPALDING, Prof. D.B. WILLIAMS, Prof. F.A. NASA Langley Research Center - M/S 168, Hampton, Virginia ONERA, 29 Ave de la Division Leclerc, Chatillon-sous-Bagneux Dept of Aeronautics and Astronautics, University of Southampton, Southampton 5NH S09 Department of Chemical Engineering and Fuel Technology, University of Sheffield, Mappin Street, S13JD Alsthom, Techniques des Fluides, 75 rue General Mangin, Grenoble ONERA, 29 Ave de la Division Leclerc, Chatillon-sous-Bagneux University of California at San Diego, PO Box 109, La Jolla, California Messerschmidt-Bolkow-Blohm GmbH, UR 8 Miinchen 80, Box Technology Incorporated, 3821 Colonel Glenn Highway, Dayton, Ohio Laboratorium voor Energievoorziening en K'einreactoren, Technische Hogeschool, Rotterdamseweg 139a, Delft Laboratoire d'aerothermique du CNRS, Meudon Imperial College of Science and Technology, Department of Mechanical Engineering, Exhibition Road, London SW7 2BX AMES Department, University of California San Diego, La Jolla, California USA FRANCE UK UK FRANCE FRANCE USA GERMANY USA NETHERLANDS FRANCE UK USA OBSERVERS BECKO, Ing. Y.B. BELLET, Mr J.C. CHAMPION, Dr GASTEBOIS, Mr P.G. HAMANN, ir R.J. HENNINGSEN, Ing. Swend KRAUSE, Prof. Dr E. Research Laboratory, ACEC, BP 4, Charleroi ENSMA, rue Guillaume VII, Poitiers ENSMA, rue Guillaume VII, Poitiers SNECMA, Centre de Villaroche, Moissy-Cramayel Department of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, Delft Laboratoriet for Energiteknik, Polyteknisk Laereanstalt Lundtoftevej, 100 Bygn 403, 2800 Lyngby Rhein-Westf. Techn. Hochschule Aachen, 55 Aachen, Templergraben 55 BELGIUM FRANCE FRANCE FRANCE NETHERLANDS DENMARK GERMANY

336 App.A-2 OBSERVERS (Continued) LEDUC, Ing. Bernard LINARES-DIAZ, Mme S. SAINT-CLOUD, Prof. J.P. SCHOYER, ir H.F.R. SCHUMAKER, Major Karl H. VALENTIN, Prof. P. WILKINS, Dr Mark L. COEVERT, Mr K. Institut de Mecanique Applique'e, Universite Libre de Bruxelles, 50 Avenue Franklin Roosevelt, 1050 Bruxelles SNPE, Centre de Recherches du Bouchet, Vert-le-Petit Laboratoire d'energetique, ENSMA, Poitiers Department of Aerospace Engineering, Delft University of Technology, Kluyverweg, 1, Delft BOARD, Old Marylebone Road, London NW1 5TH, England Faculte des Sciences de Rouen, Place Emile Blondel, Mont-St-Aignan Lawrence Livermore Laboratory, University of California, PO Box 808, Livermore, California Technologisch Laboratorium RVO-TNO, Lange Kleiweg 137, Rijswijk (ZH) 2100 BELGIUM FRANCE FRANCE NETHERLANDS USA FRANCE USA NETHERLANDS

337 1. Recipient's Reference 2. Originator's Reference s.unginator 6. Title REPORT DOCUMENTATION PAGE AGARD-CP Further Reference 4. Security Classification of Document Advisory Group for Aerospace Research and Development North Atlantic Treaty Organization 7 rue Ancelle, Neuilly sur Seine, France UNCLASSIFIED Analytical and Numerical Methods for Investigation of Flow Fields with Chemical Reactions, Especially Related to Combustion 7. Presented at Liege, Belgium 8. Authors) 10. Author's Address Various Various 9. Date May Pages Distribution Statement This document is distributed in accordance with AGARD policies and regulations, which are outlined on the Outside Back Covers of all AGARD publications. 13. Keywords/Descriptors Combustion Applications of mathematics Chemical reactions Flow distribution 15. Abstract Laminar flow Proceedings Pollution 14.UDC :536.46: \J 14 These Proceedings consist of the 4^ papers presented and analysis and discussion of same for the Specialists Meetingsron "Analytical and Numerical Methods for Investigation of f Flow Fields with Chemical Reactions, Especially Related to Combustion" during the 43rd Meeting of the Propulsion and Energetics Panel held on 1, 2 April 1974 at Liege, Belgium. L-Four one half day sessions were conducted on Classical Methods for Numerical Computation of Laminar Flow and Mean Flow with Chemical Reactions, Modern Methods of Analysis of Turbulent Flames, General Methods of Analysis of Flows with Numerous Chemical Reactions, and Numerical and Analytical Methods Applied to Combustors and to the Study of Pollution. P - P

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