(c)l999 American Institute.of Aeronautics & Astronautics

A99-33556 AIAA-99-3367

VALIDATIQN OF SUB-GRID WALE FINITE-RATE CHEMISTRY MODEiS FOR TURBULENT DIFFUSION FLAMES

Emad Amin* and lsmail Celik+ Mechanical and Aerospace Engineering Department, West Virginia University, Morgantown WV 26506-6106

~ Abstract /

The purpose of this investigation is to validate finite-rate ~ chemistry models to be used for turbulent combustion ! predictions. The first of these models is the constrained

equilibrium and the second is the strained laminar diffusion j flamelet. The flow-field and thermal-field predictions using I these two approaches have been validated against / published experimental data in the literature. -This paper

discusses the detailed implementation techniques for these two approaches in multi-dimensional CFD

I simulations. The turbulence/chemistry interaction is modeled by the conserved scalar/assumed probability

/ density function (pdf) description for turbulent diffusion ! flame. The reacting flow-field is modeled by a 2-D elliptic / finite volume CFD solver. The predictions show ; satisfactory”agreement with the published measurements

computations. This will enable. us to examine their potential validity as Sub-Grid Scale (SGS) chemistry models for Large Eddy Simulation (LES) of turbulent reacting flows. The different thermo-chemical models investigated here are the Full Equilibrium (FE), Constrained Equilibrium (CE), and Strained Laminar Diffusion Flamelets (SLDF). The flow-field and thermal- field predictions using these chemical sub-models are compared to the published experimental data in literature.

Full Eauilibrium Chemistry (FE)

: given the known limitations of the used isotropic turbulence model (k-c model) and the assumption of a fixed shape for

c the conserved scalar probability density functionacross all I mixing stages. The predictions suggest that the two : models can be used as attractive alternatives for Sub-Grid I Scale, (SGS) combustion modeling. in Large Eddy

Simulation (LES) studies of turbulent reacting flows.

Introduction

; The present study is concerned with the CFD validation of ; two different finite-rate chemistry models. Our obecvtive

here is to test the performance of theses sub-models in ! the frame of Reynolds Averaged Navier Stokes (RANS)

In this model the thermo-chemical state variables (e.g. temperature, density, and concentrations) are known from the initial mixing conditions and by assuming total chemical equilibrium for the reacting mixture. The average or filtered values of the thermo-chemical variables are obtained by the probability density function (pdf) convolution integral over the mixture fraction space. The pdf parameters are found from the solution of modeled transport equations for the mixture fraction and the concentration fluctuations (variance). This technique is considered to be an extension to the variable density mixing case (e.g. Bilger, 1976; Libby and Williams, 1983; Faeth & Samuelsen, 1986) and for this reason this model is classified here as a single-variable formaksm. The transport equation for the time-dependant mixture fraction can be written as;

, 1

* Research Assistant Professor, Member ’ Professor

Copyright O,l999 The American Institute of Aeronautics and Astronautics Inc. All rights reserved

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Where v and x are the velocity and position vectors in j space coordinate direction (j=1,2,3). In the context of RANS computations we solve for the Favre-averaged form of the mixture fraction transport equation. While In LES implementation we should solve for the instantaneous 5 instead. The transport equation of the square of concentration fluctuations (g = 5’“) can be written as (Spalding, 1971);

In the context of LES the above equation can be written for the filtered (volume averaged) g using modeled values for the sub-grid kinetic energy of turbulence (k) and sub-grid scalar dissipation rate (s). In the present RANS validation, the above equations for 5 and g are solved with the set of the governing flow-field equations. The numerical values of the constants in the above equations are listed in (Amin & Celik 1999). These model constants are similar to those used in previous flow modeling (e.g. Jancika et al., 1978, Amin et al., 1995a, Amin, 1995). The computed values of Favre mean 5 and g are then used to calculate the local pdf (P(Q). The pdf is usually assumed to follow a specific functional distribution. Many functional forms have been suggested in the literature among which is the Beta function (e.g. Rhodes et al., 1974; Janicka & Kollmann, 1978) and Clipped Gaussian distribution (Lockwood & Naguib, 1975). Beta function is chosen in this work because of its ability represent the different stages of mixing including the bimodal distribution. It has also the potential for extension to multiple scalar mixing (Girimaji, 1991) and has been used in many LES studies (e.g. Cook and Riley, 1994). The p (5) is given by;

The exponents a and 6 are related to the mixture fraction and the variance of the turbulent part of the composite pdf as follows;

(5)

p Ay& (6)

If the above model is used as a sub-grid turbulence/interaction model then the instantaneous mixture fraction should be used instead of the mean value. The mean (filtered) density is thus given by,

Constrained Equilibrium Chemistry ICE) In this approach a second constraining parameter is added to the single-variable formalism. The treatment of this variable differs from one implementation of the model to the other. One implementation has been suggested by Jancika and Kollman (1978), and used for the H2/Air flame (Jancika and Kollman, 1982), and H2/Argon-Air flame (Chen and Kollman, 1990). In this approach the degree of departure from total equilibrium is measured by the change in the number of total moles/kg in the system. This approach has been extended and used in hydrocarbon flames -by Correa et al., (1985). A general theory for the constrained equilibrium modeling has been formulated by Keck, .(I 990). To account for the turbulent fluctuations in the progress variable a pdf for the reaction progress variable (f-t) is usually assumed (Jancika and Kollman, 1982). Similar to the p(k) the p(q) may take the form of a one of the standard probability distributions. In the present study an assumed Delta function ‘pdf shape have been proposed for the fluctuations in rl. However other options like 3-Delta, and Beta functions have also been used in the literature (Jancika and Kollman, 1982). As we neglect here the fluctuations in the progress variable its marginal pdf is totally fixed by the calculation of Its mean value (or filtered value in case of LES). The averaged or filtered value of any thermo-chemical quantity, o, can be found from a 2-D convolution integral over 5-f-t space as;

In the above integral it is commonly assumed that the joint density weighted pdf can .be written as

P(r7d3 = Pmm (9) Which implies statistical independency between the mixture fraction and the progress variable. Pope and Correa (1986) have performed Monte-Carlo simulation for the joint pdf of 5 and n. They showed that the covariance and the correlation coefficient do not always vanish (i.e. statistically dependent variables). This might be a

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potential important area for LES researchers to investigate.

Strained Laminar Diffusion Flamelet (SLDFJ The third, and perhaps one of the most important modeling approaches is the strained laminar flamelet approach (Liew et al.; 1981, 1984, and Peters; 1984). The approach provides an alternative way of incorporating finite rate chemistry effects in turbulent combustion modeling. The turbulent flame is assumed to be consisting of an ensemble of laminar flamelets. Each individual flamelet is stretched and distorted by the hydrodynamic flowfield. The detailed structure of the turbulent and laminar flamelets is assumed to be similar. The state-relationships between the mixture fraction and the reactive scalars obtainable from laminar flame calculations or alternatively from experimental measurements, can be used for turbulent combustion modeling (Gore and Feath, 1986). The different flamelets in the ensemble respond differently to the mixing field. The difference in the turbulent stretching leads to different reaction-diffusion responses of the flamelets. This is the main difference between the flamelet and flame-sheet models were the internal structure of the flame front is not important. As a result of this fluid dynamics straining on the laminar flamelet the scalar gradient is increased in the reaction zone. This gradient is usually expressed as the scalar dissipation rate (Liew et al., 1984). At very small values of strain rate the laminar flamelet relationship approaches that of a total chemical equilibrium. With increasing strain the finite rate chemistry effects increase until flame extinction occurs. A necessary condition for the strained laminar flamelet concept to be applied in turbulent diffusion flame modeling is to have a laminar flame thickness smaller than Kolmogorov Length scale hk which is defined as

(‘0) Where v is the kinematics viscosity and E is the kinetic energy of turbulence dissipation rate. This length scale is also called the “dissipation scale” as it represents the lower cut-off of length scales within a turbulent flow-field. Thus the necessary condition for applying the laminar flamelet concept can be formulated as;

L,<., & -

(‘1)

Where Lc is the laminar reaction zone thickness in mixture fraction space. As argued by Bilger (1988) the above condition may be justifiable near the burner zone. However, further downstream and away from the intense mixing condition the reaction zone become broad and distributed in which the above condition is not met. These

distributed reaction zones contain. the turbulence structure within them. The distributed combustion images have been measured and experimentally observed by Bilger (1988) and Schefer et al., (1994).

Numerical Modeling The CO/H2/N2 bluff body stabilized flame of Correa and Gulati (1992) is chosen here as a test case to investigate the finite-rate chemistry effects. This represents a good case for the present validation for several reasons. First the fuel used, a CO/H2/N2 mixture, is of known detailed reaction mechanism. Second, the axisymmetric bluff body stabilized flames can provide regions of high scalar dissipation rate far downstream of the near burner zone. This is considered to be an important advantage since in jet flames the inlet boundary conditions can affect the scalar dissipation rates near the burner exit. Thus the precise description of the inlet turbulence boundary conditions will not have a significant impact on the solution in this case. Third, in this type of flames there is no need for pilot flame stabilization and the associated problems in modeling a secondary heat input (Bilger, 1985). The axisymmetric bluff-body flame-holder has an outer diameter of 38.1 mm with a concentric fuel jet of diameter 3.18 mm. The flame is stabilized in the recirculation zone provided by the bluff body, whose blockage ratio in the tunnel is approximately 5%. The ratio of the bluff body diameter to the central jet diameter is 12:i. The fuel mixture consists of 27.5% CO, 32.2% H2, and 40.2% N2. The experimental data includes the instantaneous and spatially resolved measurements of major species (e.g. 4 029 co, CQ!, H20 and N2), density, and temperature. The numerical model is solved using a 2-D axisymmetric finite volume code. The grid used is 52x120 in the radial and axial directions respectively. The set of boundary conditions used is similar to that used by Correa and Gulati (1992).

Thermo-Chemical Tables Before performing the CFD simulations for the reactive flowfield the database (look-up tables) relating the thermo- chemical variables to the instantaneous mixture fraction is prepared. The code is modified to accept the variables listed to any size of the instantaneous mixture fraction array. This flexibility in mixture fraction array size is found necessary to enable the CFD code to communicate easily with the computations obtained from different pre- processors. The pre-processors used in the present work are the chemical equilibrium code EQULIB and the laminar opposed diffusion flame code OPPDIF of SANDIA National Laboratories. The output calculations of these codes are tabulated in the look-up table. The tabulation

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methodadopted in the present work lists the laminar state- relationship as a function of the instantaneous mixture fraction. This has the advantage of easily incorporation of additional thermo-chemical quantities (i.e. species or laminar rates of formation). Another important advantage of the present tabulation strategy is that the convolution integral can be evaluated as accurately as required by simply adding additional rows in the table. The calculations of the thermo-chemical library are performed for the three different chemistry models. The first was the adiabatic equilibrium in which the free energy for the mixture of products was minimized. The only inputs required for those calculations were the initial concentrations, initial temperature and pressure. The second is the constrained equilibrium case in which the equilibrium computations were done for a fixed mole fraction of CO. The CO has been chosen as a constraining parameter here because its oxidation step can be quite important in this type of recirculating flames. However, it should be noted that other reactive species can be chosen for the same purpose. The mole fraction of CO was fixed at each mixture condition by assuming a value of the reactedness. The reactedness in the present computations was defined as;

(12)

The subscripts u and b stand-for the unburned and burned conditions respectively and X is the mole fraction. In this way the value of the reactedness is bounded and varies between zero and one similar to the mixture fraction. The value of the reactedness is fixed in the present computations at 0.71 for two reasons. First, this is value at the stoichiomertic conditions (i.e. flame sheet) which gave a similar CO concentration in laminar flame calculations by the constrained equilibrium and flamelet methods. Secondly, by this mean we avoid solving an additional equation for the mean value of the. reactedness and possibly its fluctuations. With this simplification the code will need only one input- table instead of having different tables as per CO concentration of the flame (similar to having a constant strain rate throughout the flame in flamelet models). In the third model, the laminar diffusion flame of CO/H2/N2 is calculated using the opposed diffusion flame code (OPPDIF) and CHEMKIN library. The species and a detailed reaction mechanism for COIH2/N2 oxidation is taken from Drake and Blint (1988). Different computations are performed in order to adjust the strain rate on both sides of the flame (air and fuel sides) to a fixed value of strain rate. Similar to the choice of a fixed value of the reactedness the laminar flame calculations are made for a

specified rate of strain (120 s-l). This choice is somewhat arbitrary but is excepted to represent a major portion of the flow. The solution for.a range of strain rates was not ,attempted in this simulation. ‘However the effect of variable strajn rate computations on the flow-field will be considered in more detail in future investigations. Thus in all the above assumptions a one dimensional thermo-chemical data library was built and linked to the code. All the relevant thermo-chemical properties to the computations of the mixing and reacting flow-field are listed in the table as-column entries. The CFD predictions are compared with the experimental measurements in the next section for the three different models.

Laminar flame comrwtations The full equilibrium relationships are computed using the codes; STANGAN and CHEMKIN (Kee at al., 1987). For the CO/H2/N2 diffusion flame the laminar flame state- relationships obtained by there different assumption are shown in Fig. 1 .l. As could be seen the peak value of specific volume (l/p) occurs in the vicinity of the stoichiometric mixture fraction &.t = 0.323). The SLDF profile showsa slight shift to the lean part of the flame. While the constrained equilibrium solution shows a shift towards the rich side. This might be due to the effects of preferential diffusion. It should be noted here that these effects are accounted for in the opposed diffusion flame computations but not in the equilibrium computations. The temperature profiles are shown in Fig. 1.2. Again the shift of the profiles are reflecting mainly the assumptions involved in both computations. However, the peak temperatures are almost the same in all of the three cases. The species mass fraction profiles are seen next in Figs 1.3-1.6. For CO mass fraction, In the lean region of the flame the flamelet profile shows an .intermediate value between equilibrium and constrained equilibrium solutions. While in the rich part of the flame (5 > 0.6) the flamelet calculations show a higher CO levels. This is because we included the effects of finite rate chemistry by using a detailed reaction mechanism for predicting CO oxidation at rich conditions. The over-predictions of CO oxidation to CO2 in the equilibrium and constrained equilibrium assumptions are also noticed in Fig; 1.4. Again this is a result of incorporating a detailed reaction mechanism for oxidation of the CO. In most -of the lean region of the flame and over the finite reaction tone thickness in mixture fraction space the water vapor (H20) is in excess concentration in comparison to the equilibrium and constrained equilibrium solution (Fig. 1.5). The failure of the constrained equilibrium solution to provide a more realistic estimate for H20 in comparison to flamelet

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solution might be due to the fact the reactedness parameter used for the constrained equilibrium calculations was chosen a function of CO concentration only. As it was discussed in Keck (1990) the success of the constrained equilibrium assumption relies mainly on the proper choice of the constraining parameter which is highly problem dependent. The H2 mass fraction (Fig. 1.6) profiles show underprediction in the rich side of the flame for the equilibrium solution in comparison to the constrained equilibrium and flamelet computations. Next we present the model validation in the reacting flow- field. The mean temperatures and mean species mole fractions were calculated in post-processing after a converged solution for the variable density mixing field has been obtained. The density weighted average of relevant quantities were calculated from

(13)

Where $I stands for temperature and major species concentration. The mean mixture fraction axial decay is shown in Fig. 2.1. The predictions show fairly close agreement among the three thermo-chemical models and the experimental data. The comparison of the rms of mixture fraction in Fig. 2.2 indicates that the turbulence modeling is underestimating the mixture fraction fluctuations at axial distance of X/D = 10 (where X is the axial distance and D is the burner diameter). The predictions of the thermo-chemical quantities (temperature and mass fractions) along the flame centerline and at different axial locations are discussed next. The predictions of the mean temperature at X/D = 10 is shown in Fig. 2.3. As we can see the centerline temperature is overpredicted by 15% at X/D=lO. The predicted temperature profile shows a small peak at R/D = 2.5 (R is the radial distance from the flame centerline) while the measurements show a continuous decrease in temperature across the jet width. It should be noticed that the overprediction of the jet mixture fraction by 1.4% at the flame centerline is the main reason for the overprediction in temperature. The predictions from the three thermo- chemical sub-models are almost identical in the lean region of the flame. The temperature is plausibly reproduced on the rich side by the constrained equilibrium approach. Also ,the rate of temperature decay is in agreement with the measurements. It should also be noted i:hat the maximum temperature at this axial location is almost the same between the three sub-models and very close to the measurement (Tmax = 1550 K). The predicted radial location of the peak temperature is shifted slightly to the lean part of the mixture fraction space. The mean predicted CO mass fraction in Fig 2.4 shows a satisfactory agreement with the experimental data. The

flamelet approach shows an expected slight variation due to the finite rate chemistry effects. However for the mass fraction of H2 the three sub-models (Fig. 2.5) underpredicted the radial decay. The constrained equilibrium approach gives a better agreement. At radial locations greater that R/D = 1, the experimental data and the predictions collapsed almost on a single curve. In predicting CO2 concentration (Fig. 2.6) the constrained equilibrium and flamelet model showed a closer agreement than full equilibrium model. However, the three sub- models predicted the maximum CO2 mass fraction rather poorly. The peak value is predicted to be of twice the experimental data (in case of flamelet predictions). Better predictions of CO2 are expected if a flamelet profile with a lower strain rate be selected for the present computations. The flamelet approach overpredicted the location and the value of the maximum H20 concentration at X/D = IO (Fig. 2.7). This may be, again, explained by the selection of a highly stretched flamelet (a = 120 s-l) for the present computations. The equilibrium and constrained equilibrium predictions show closer agreements than the flamelet approach for H20 at this location. Further downstream (at X/D = 20) the predictions and measurements start to deviate considerably The mean temperature is overpredicted at the centerline by more than 50% (Fig. 2.8). At radial locations higher than R/D = 1 the predicted temperatures show better agreement to the measurements. The flamelet solution seems to provide a better representation for the flame at those locations. Figures 2.9-2.10 show the mass fractions of CO and H2 are seriously underpredicted at X/D = 20. While CO2 and H20 are overpredicted in the lean part of the flame and underpredicted in rich part (Figs. 2.11-2.12). The calculated location of the peak in case of CO2 mass fraction is almost half of the measured one indicating a slower jet expansion. The overprediction of CO oxidation to CO2 and H2 oxidation to H20 is consistent with the underprediction of CO and H2 mass fractions in the rich part of the flame. This indicates that a possible reason for the disagreement is due to the chemistry model.

Summarv and Conclusions

In this paper the validation of different approaches for modeling the finite-rate chemistry process in a bluff-body stabilized diffusion flame have been presented. The RANS computations show that both models can reproduce the macro features of the flame (i.e. chemical flame length, mean temperature and concentration distribution) fairly

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well. Suggestions to extend the applicability of these finite- rate chemistry models to LES have been made. In general, there are acceptable agreements. in mean values for these sub-models. The simplified turbulence chemistry interaction model in the present computations shows that the turbulence level is underpredicted at the jet boundary. The use-of these models as sub-grid chemistry models in LES of turbulent combustion is justified on the bases that small-scale turbulence is of more isotropic nature compared to large-scale structure. Future LES studies will help to resolve some important issues in turbulence modeling and will enable us to investigate the reasons ,for some of the observed discrepancies between predictions and measured data. The choice of a specific thermo-chemical model for modeling the turbulent diffusion combustion must be based on an accurate determination of important variables like average strain rate in the flame or average reactedness. The difference in the performance among the different thermo-chemical models is a strong function of those parameters, which have to be, determined accurately and prior to the CFD modeling.

Acknowledament

Part of this investigation has been conducted under the sponsorship of US DOD Army Research Office through the EPSCoR program (Grant No.: DAAH04-96-1-0196).

References

Amin, E. M. (1995). CFD Predictions of Ultra Low NOx High Intensity Combustion, Ph.D. Thesis, University of Leeds, Leeds, UK.

-Amin, E. M., and Celik, I. (1999) A Validation Study of A Turbulent Mixing Model Based on the Probability Density Function Approach” Paper no. 1999-01-0231, SAE International Congress and Exposition Detroit, Ml.

Amin, E. M., Andrews, G. E., and Pourkashanian, M., Williams, A. and Yetter, R. A. (1995) A Computational Study of Pressure Effects on Pollutants Generation in Gas Turbine Combustors, Presented at the international Gas Turbine Congress & Exposition Houston, Texas - June 5- 8, 1995.

Bilger, R. W. (1975). A Note on Favre Averaging in Variable Density Flow. Cornbust. Sci. Techno/. 11, 215- 217.

Bilger, R. W. (1976). The Structure of Diffusion Flames. Combust. Sci. Tedhnol, 13, 155-70

Bilger, R. W. (1980). Turbulent flows with Non-Premixed Reactants. In turbulent reacting flows ed. P. A. Libby, F. A. Williams, pp. 65-113. Berlin: Springer-Verlag.

Bilger, R. W. (1988). The Structure of Turbulent Nonpremixed Flames. Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 475-488.

Bilger, R. W. and Starrier, S. H. (1983). A Simple Model for Carbon Monoxide in Laminar and Turbulent Hydrocarbon Diffusion Flames. Combust. Flame. 51, 155- 176.

Chen, J. Y, and Kollman, W. (1992). Combust Flame 397:412-88.

Chen, J. Y. and Kollman, W. (1988). PDF Modelling of Chemical Nonequilibrium effects in Turbulent Nonpremixed Hydrocarbon Flames. Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 645-653.

Chen, J. Y. and Kollman, W. (j990b). Chemical Models for Pdf Modelling of Hydrogen-Air Nonpremixed Turbulent Flames, Combust. Flame, 79, 75-99

Chen, J. Y. and Kollmann, W. (1988) Pdf Modelling of Chemical Nonequilibrium effects in turbulent Nonpremixed Hydrocarbon -Flames, Twenty-Second Symposium (International) on Combustion, The Combustion institute p. 645.

Chen, J. Y., Kollmann, W. and Dibble R. W., Pdf Modelling of .Turbulent Nonpremixed Methane Jet Flames, Combust. Sci. and Tech., 64:315-346.

Correa, ,S: M. (1992), and Pope, S. B. (1992) Comparison of A Monte Carlo PDF/Finite-Volume Mean Flow With Bluff-Body Raman Data, TwentyiFourth Symposium (International) on Combustion, The Combustion institute pp. 279-285.

Correa, S. M., Drake, M. c., Pitz, R. W., Shyy, W. (1984). Twenties Symposium (International) on Combustion, The Combustion institute p. 337.

Drake, M. C. and Blint, d. J. (1988) Structure of laminar opposed-flow diffusion flames with COIH21N2, Combust. Sci. Technol. 61 :I 87-224.

Faeth, G. M. and Samuelsen G. S. (1986) Fast Reaction Nonpremixed Combustion Prog. Energy Combust. Sci. 12:305-372.

Frankel, S. H., Adumitroaie, V., Madina, C. K., and Givi, P., (1993) ,Large Eddy Simulation of Turbulent Reacting Flows by Assumed PDF Methods, in Engineering Applications of Large Eddy Simulations (Ragab, S. A.and

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Piomelli, U. ed., pp. 81-101, New York, NY: ASME, FED- Vol.

Girimaji, S S (1991). Assumed P-pdf Model for Turbulent Mixing: Validation and Extension to Multiple Scalar Mixing Combusi!. Sci. Technol. 783 77-I 96.

Gore, J. P., and Faeth, G. M., (1986) Structure and Spectral Radiation Properties of Turbulent Ethylene-Air Diffusion Flames Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 1521-1531.

Jancika, J., Kolbe, W. and Kollman, W. (1978). Closure of the Transport Equation for the Probability Density Function of Turbulent Scalar Fields. Journal of Non-equilibrium Thermo-Dynamics 4, 47.

Janicka , J. and Kollmann, W., (1978) Seventeenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 421-429.

Janicka , J. and Peters, N. (1982) Prediction of turbulent Jet Diffusion Flame Lift-Off Using A Pdf Transport Equation. Nineteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 367-374.

Janicka , J., Kolbe, W., and Kollmann, W. (1978). The solution of a PDF-transport equation for turbulent diffusion flames. Proc. Heat Trans. Fluid Mech. Inst., Stanford University.

Keck, J. C. (1990). Rate-Controlled Constrained- Equilibrium Theory of Chemical Reaction in Complex Systems. Prog. Energy Combust. Sci. 125:154-l 990.

Kee, R. J., Ruply, F. M. and Miller, J. A. (1987). The Chemkin Thermodynamic Data Base. Sandia Technical Report SAND87-8215.

Libby, P. A. and Williams, F. A., ed., (1980). Turbulent Reacting Flows, Springer-Verlag, New York.

Liew, S. K., Bray, K. N. C., Moss, J. B., (1984) A Stretched Laminar Flamelet Model of Turbulent Nonpremixed Combustion, Combust. Flame 56:’ 99-213.

Lockwood, F. C., and Naguib, A. S. (1975). Cornbust. Flame, 24, 109-I 24,.

Peters, N. and Kee R. J., (1987). The Computation of Stretched Laminar Methane-Air Diffusion Flames Using a Reduced Four-Step Mechanism. Combust. Flame 68:17- 29.

Pope, S. B. (1976). The probability approach to the modelling of turbulent reacting flows. Combustion and Flame, 27, 299.

Pope, S. B. (1985). Pdf Methods For Turbulent Reactive Flows. Prog. Enrg. Combust. Sci. 11 :I 19-l 92.

Pope, S. B. and Correa, S. M., (1986). Joint PDF Calculation of a Non-equilibrium Turbulent Diffusion Flame. Twenty-First Symposium (International) on Combustion, The Combustion Institute p. 1341.

Rhodes, R. P., Harsha, P. T., and Peters, C. E. (1974) Turbulent Kinetic Energy Analysis of hydrogen -air diffusion Flames. Acta Astronautica, 1, 443.

Schefer, R. W., Johnston, S. C., Dibble, R. W., Gouldin, F. C. and Kollman, W., Non reacting turbulent mixing flows: a literature survey and data base. Sandia Rep. SAND86- 8217, Sandia National Laboratories, Livermore, CA (1986).

Schefer, R. W., Namazian, M. and Kelly, J. (1994) Stabilisation of Lifted Turbulent-Jet Flames, Combust. Sci. Technol. 99:75-86.

Spalding, D. B. (1971). Concentration fluctuations in a round turbulent free jet, J. Chem. Engng. Sci. 26, pp. 95- 107.

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Figure 1 .I volume.

The laminar state-relationships for specific

0x5 (1.73

Figure 1.2 The laminar temperature state- relationships.

0.6

0.55

0.5

0.46

6 0.~

$j 0.35 Ii al a.3

H 0.25

8 03

cl.15

0.1

0.05

II

Figure 1.3 The laminar CO mass fraction state- relationships.

Figure 1.4 The laminar CO2 mass fraction state- relationships.

Figure 1.5 The laminar H,O mass fraction state- relationships.

0.05 -.-.-.- BP

El .- _ &q

o.o+ - - ST.oE

Figure 1.6 The laminar H mass fraction state- relationships.

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1.1

0.2

0.1

0

Figure 2.1 Centerline decay of mean mixture fraction.

Figure 2.2 Radial profiles of RMS of mixture fraction at X/D =: 10.

Figure 2.3 X/D = 110.

Radial profiles of mean temperature at

Figure 2.4 Radial profiles of mean CO mass fraction at X/D = 10.

Figure 2.5 Radial profiles of mean H2 mass fraction at X/D = IO.

0.15

O.lC

0.13

0.12

6 0.11

'E 0.1

", 0.w

83 0.m

5 0.07

3 O.(lB

k 'J.05

z 0.w

0.03

o.a2

0.01

D

Figure 2.6 Radial profiles of mean CO2 mass fraction at WD = 10.

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Figure 2.7 Radial profiles of mean H20 mass fraction Figure 2.10 Radial profiles of mean H2 mass fraction atX/D= IO. at X/D = 20.

Figure 2.8 Radial profiles of mean temperature XID = 20. . c

I a.o+ 0.03 %Kn o.a,

FVD

at Figure 2.11 Radial profiles-of mean CO2 mass fraction it X/D = 20.

Figure 2.9 at X/D = 20.

Radial profiles of mean CO mass fraction Figure 2.12 Radial profiles of mean H20 mass fraction at X/D = 20. ,

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