AIAA JOURNALVol. 30, No. 2, February 1992

Modeling Ideally Expanded Supersonic Turbulent JetFlows with Nonpremixed H2-Air Combustion

R. Villasenor*Vanderbilt University, Nashville, Tennessee 37235

J.-Y. GhentSandia National Laboratories, Livermore, California 94550

andR. W. Pitzt

Vanderbilt University, Nashville, Tennessee 37235

A prediction model for a turbulent axisymmetric diffusion flame with equilibrium and nonequilibriumhydrogen-air kinetics is presented for supersonic jet flows. For the first time, turbulence-chemistry interactionsare included in a supersonic combustion model. In the finite rate chemistry analysis, the fast shuffle reactionsin the H2-air reaction system are assumed to be in partial equilibrium and three-body recombination reactionsare treated kinetically. The turbulence model consists of the Navier-Stokes equations in Favre-averaged formwith a modified two-equation turbulence closure adapted for supersonic flows. The model uses probabilitydensity functions to describe fluctuations in both the mixture fraction and the reaction progress variable. Otherrelevant features of the supersonic code include a local convective Mach number that quantifies the compress-ibility effects and a thermodynamic table technique that speeds up the numerical computation. The modelpredictions are compared to probe measurements in a supersonic Hi-air jet flame by Beach. Comparison of themixture fraction deduced from Beach's data to the model results indicates that the model predicts the turbulentmixing reasonably well. Predictions of the major species concentrations are nearly identical for the equilibriumand partial equilibrium models. The partial equilibrium model predicts substantial superequilibrium of theminor species and more closely predicts Beach's data than an earlier chemistry model based on eddy breakup.

NomenclatureC£, Cg = turbulence model constants/ = mixture fractionfcc = correction factorg = variance of mixture fractionk = turbulent kinetic energyMc = convective Mach numberp = variance of progress variableq = progress variabler = radial coordinateSc = Schmidt numberSq = source termu = longitudinal velocity componentv = radial velocity componentx = longitudinal coordinate7 = specific heat ratioe = turbulence dissipation rateH = dynamic viscosityp = densitya = Prandtl-Schmidt numberT = time scale ratio\2 = second mixing term of q equation

Received Feb. 19,1990; revision received July 16,1991; accepted forpublication Aug. 1,1991. Copyright © 1991 by the American Instituteof Aeronautics and Astronautics, Inc. All rights reserved.

*Graduate Student, Mechanical Engineering. Member AIAA.tMember of Technical Staff. Member AIAA.% Associate Professor, Mechanical Engineering. Senior Member

AIAA.

Introduction

A VITAL part of the effort to develop advanced propul-sion systems capable of sustaining hypersonic flight

within the atmosphere is the ability for theoretical predictionof flow properties in scramjet combustors where hydrogen isinjected and burned in a supersonic air stream. For completegenerality, the prediction method should be able to calculateturbulent fuel-air mixing and kinetically controlled combus-tion in a combustor for arrays of hydrogen injectors at variousangles and configurations to the air stream. Other aspects suchas the effects of the combustor walls on the flow and theability to predict the effects of ignition sources are also impor-tant. Practical limitations severely restrict the inclusion of allof these capabilities in a single program. In the present work,we consider a simple flow geometry that consists of an axisym-metric diffusion flame of hydrogen in vitiated air. To furthersimplify the numerical simulation, the supersonic jet is as-sumed to expand ideally into the coflow stream.

An important concern in supersonic reacting flows is theincomplete combustion due to slow chemical kinetics. Thereare various approaches in modeling chemical kinetics, namely,frozen flow, complete reaction, equilibrium kinetics, andfinite rate kinetics. Frozen flow and complete reaction modelsrepresent the two extremes of the modeling. The equilibriumreacting flow approach assumes that the chemical reactiontime is much shorter than the time required for mixing andflow residence time. However, in supersonic flows, the gasresidence time and chemical reaction time can be comparable.Therefore, the local equilibrium assumption may not be validfor the entire flowfield. This paper is intended to address theproblem of finite rate chemistry by simulating the chemicalreactions in a free shear layer.

In previous work1 on axisymmetric supersonic jet flames,the mixing of fuel and oxidant was computed for parallelinjection of H2 into coflowing vitiated air using a two-equa-tion (k - e) turbulence model. The extent of the chemical

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396 VILLASENOR, CHEN, AND PITZ: EXPANDED SUPERSONIC TURBULENT JET

reaction was modeled by using three different assumptions:1) no reaction, 2) complete burning of all fuel mixed withoxygen, and 3) finite rate burning based on the rate of decayof large turbulent eddies into small ones. The last assumptionis known as the eddy-breakup (EBU) model. In later work,2

the numerical model was extended to include the effect ofunmixedness.

Recently, Eklund et al.3 investigated the same flow but withseveral algebraic eddy viscosity turbulence models. The chem-ical reaction mechanism included a seven-reaction model in-volving seven chemical species. However, none of the previouswork included the effect of turbulence-chemistry interactions.

For high Reynolds numbers turbulent flows, the interactionbetween turbulence and chemistry can greatly influence boththe mixing and reactive scalar fields in the computation. Al-though these effects have been studied extensively in the sub-sonic regime by Janicka and Kollmann,4 Janicka,5 Chen,6Kollmann and Janicka,7 and others,8"10 no comparable studieshave been done for supersonic flows. In view of that, wecalculate the mean values of the thermochemical variables[i.e., concentrations (mass fractions), temperature, and den-sity] for the scalar field through a probability density function(PDF) approach. An assumed shape for the PDF is used todescribe fluctuations in both the mixture fraction and thereaction progress variable.

To account for compressibility effects on the turbulent mix-ing, which reduce mixing rates between the two streams atsupersonic speeds, we have incorporated an empirical correla-tion that relates the turbulent viscosity to the convective Machnumber.

A common practice among the numerical predictions donein the past is that the thermochemical properties are evaluatedon the fly when the marching-downstream integration is per-formed. Such an approach is computationally expensive andbecomes undesirable for flows in which the number of chemi-cal species is very large or if more complex turbulence orchemical models are to be tested. A viable alternative used inthe present work is a lookup table that contains all of thethermodynamic quantities for all possible states. As a result,the amount of computational time is reduced considerably,but the memory requirements increase due to the table.

This paper consists of four parts. In the first part, theturbulent model is introduced. In addition to the usual set ofequations for the velocity field (i.e., A: - e - g), an extra gov-erning equation (stagnation enthalpy) is solved so that themixture enthalpy can be determined from the mixture frac-tion, stagnation enthalpy, and velocities. Next, the combustionmodel is discussed, and thermochemical tables are constructedfor both chemical models. The equilibrium table has the mix-ture fraction and the mixture enthalpy as the independentvariables, whereas the nonequilibrium table is augmented bythe reaction progress variable. The statistical informationabout the distribution of the thermodynamic states is providedby assuming a certain shape for the joint PDF among thescalars. Finally, the results of the calculations are discussedand compared to measurements in a reacting supersonicaxisymmetric jet.

unity so that the energy flux caused by interdiffusion does notappear in this equation. For the momentum equation, onlygravity forces are neglected, and the pressure gradient is as-sumed to be a function of the axial coordinate only. In thepresent parabolic flows, the static pressure is further assumedto be constant throughout the flowfield and, hence, the pres-sure gradient is zero. The standard form for the turbulencekinetic energy k and its dissipation rate e in a free shear layercan be found elsewhere.1'4'10'11

The corresponding scalar field is described by the mean andvariance of the mixture fraction/, which is the normalizedmass fraction of an atomic element originating from one ofthe input streams, usually the fuel stream. The mixture frac-tion is chosen to be 1 in the fuel stream and 0 in the outerco flowing stream. The transport equations for the mean /andthe variance g of the mixture fraction have the following form:

»{\ df]/ ad

(2)At Mach numbers above 1, the growth rate of mixing layers

and jets is reduced. A correction for the turbulent viscositysimilar to that derived by Dash et al.12 for axisymmetric jets isapplied. A factor whose magnitude varies from 1.0 atM = 0.0-0.2 for large Mach number is multiplied by the tur-bulent viscosity as computed from the k~e model. The correc-tion factor used by Dash et al.12 is an empirical expressionderived from experimental observations and is formulated asfollows:

•fee = 0.25 + 0.75/ {1.0 + exp[24.73 (P - 0.2)]} (3)

where the parameter P is kl/2 divided by the local speed ofsound.

In recent experimental studies,13'14 it has been observed thatthe thickness of two-dimensional turbulent shear layers isdecreased significantly for Mach numbers larger than the

, sonic value. The reduction of the shear-layer growth is directlyrelated to a compressible mechanism that occurs between themixing streams. In order to provide an explanation for suchphenomena, Bogdanoff15 concluded that the convective Machnumber could be correlated with the reduction of the shear-layer growth rate. To account for the reduction in turbulentmixing rates due to compressibility effects, an empirical corre-lation based on Papamoschou's measurements13 is expressedas a function of the convective Mach number. Therefore, theturbulent viscosity is modified according to the followingrelationship,

0.801.0807 - 0.4728)]J (4)

Turbulence ModelThe turbulence model is an extension. of the widely used

k - e - g combustion model for subsonic flows.4'11 For super-sonic flows, effects of significant levels of kinetic energy mustbe considered; that is, the mixture enthalpy depends not onlyon the mixture fraction but also on the Mach number.

The Favre-averaged transport equations for mass, momen-tum, and energy in a free axisymmetric jet are given by Evanset al.1 For the energy equation, it is assumed that there isneither radiant energy transfer nor work by body forces. Fur-thermore, the Soret-Dufour effect is neglected and the un-steady pressure term is not considered. The Lewis number forthe laminar and turbulent contributions is taken to be equal to

and the convective Mach number is defined as

— \u) (5)

where \u = «2/«i» \> = p2/pi, \ = 72/7i» and MI = u\/a\.Subscripts 1 and 2 refer to gases or conditions in thefreestreams on the high- and low-speed sides of the shearlayer, respectively. The new correlation reduces the mixingrates between two streams at high convective Mach numbersand retains subsonic mixing rates at low convective Machnumbers. Finally, the model constants in the equations for thevelocity field are standard values that can be found in theliterature.1'4'10'11'16

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VILLASENOR, CHEN, AND PITZ: EXPANDED SUPERSONIC TURBULENT JET 397

Combustion ModelTo describe the chemical reactions in turbulent flows, the

conservation equation for individual species needed to besolved. Because of the highly nonlinear nature of chemicalreaction rates, time-averaged expansions around mean valuesproduce scores of correlations involving species and tempera-ture, and the expansion may not converge as the temperatureand species fluctuations become significant compared to themean values. Simulation of the effects of turbulence of com-bustion are primarily through these correlations and ne-glecting them can severely impair the modeling of turbulentcombustion. One common approach for overcoming thisproblem is to assume an analytical function for the joint PDFamong the scalars and to compute the mean chemical sourceterms by integrating over the allowable domain.4'5'10'11 In whatfollows, we will extend this approach to supersonic combus-tion flows, where two chemical models, one with equilibriumand the other with simplified finite rate chemical kinetics, areconsidered.

In the equilibrium model, all of the chemical reactions areassumed to be in equilibrium and the thermochemical proper-ties can be determined from the mixture fraction / and themixture enthalpy h provided that the pressure is given.Although one could solve for the equilibrium compositionsby minimizing the Gibbs function when the calculations arecarried out, it would require considerable computing time. Analternative approach is to generate a lookup table of thermo-chemical properties in terms of/and h for all possible condi-tions. During the calculations, all of the thermochemicalquantities can be determined by a multilinear interpolationscheme using the table, avoiding repeated calculations forequilibrium compositions. This approach has been extensivelyemployed for predictions of subsonic turbulent reactingflows,4'8'11 with significant saving of computing time. How-ever, this computationally efficient scheme is feasible onlywhen the thermochemical states are functions of a small num-ber of scalars.

In the partial equilibrium (or nonequilibrium) model, weassume that the two-body shuffle reactions are equilibrated.Therefore, the progress of chemical reactions is determinedentirely by the slow three-body recombination reactions:

H + OH + M = H2O + M

follows,

..dqdxdq „ dq 1 a fYj i / fit\ dq] 1j* +pv_* = (T+ )rir +:;dx dr r drl\Sc aq/ dr] 2

r\\

£

I'

(6)

(7)

The only new model constants introduced in Eqs. (6) and (7)are the turbulent Prandtl numbers oq and ap. The parameter ris the ratio of time scales of velocity and scalar fluctuations.The values assigned to these constants were as follows: Sc = 1,aq = 0.85, Op = 0.90, and r = 2.0.

The equilibrium table for the thermochemical variables at agiven (f — h) is calculated with the Chemkin computer codedeveloped by Kee et al.19 Similarly, the nonequilibrium tablefor the thermodynamic state of the gas for a given (/ - h - q)is also calculated with the Chemkin routines prior to theprediction of the flowfield.

Results and DiscussionIn this section, the parabolic code is applied to the predic-

tion of a reacting supersonic axisymmetric jet of hydrogenmixing with vitiated air. Before calculating the jet flow, anonreacting two-dimensional mixing layer is calculated to val-idate the compressibility correction to the turbulent viscosity.

Numerical calculations using the empirical correlation,Eq. (4), that relates the turbulent viscosity to the convectiveMach number are compared with experimental data13 for atwo-dimensional mixing layer. The shear-layer growth ratenormalized by the spreading rate at zero convective Machnumber is shown in Fig. 1. In the experiment, the mixing-layerspreading rates are obtained from pitot-pressure surveys for avariety of Mach number-gas combinations. Comparisons be-tween these data and numerical results reveal that the empiri-cal correlation describes reasonably well the spreading rate ofnonreactive mixing layers. One can observe from Fig. 1 thatthe convective Mach number does not have to be very large forcompressibility effects to be significant. A second noteworthyfeature is that for Mc>0.75 the growth rate appears to ap-proach asymptotically a value roughly 20% of its incompress-

Through proper combinations of reactive scalars, a singleglobal reaction step can be derived from a detailed reactionmechanism17'18 that contains the seven essential reaction steps.The progress of this global reaction step can be described by aprogress variable q, which has a value of unity when thereactions reach the equilibrium state and a value of zero whenthere is no reaction at all. With the introduction of an addi-tional reactive scalar, the thermochemical properties are nowfunctions of three scalars: /, h, and q. For turbulent reactingflows, the statistical information about the distribution of thethermodynamic states is needed in order to determine meanvalues. This information is provided by assuming a certainshape for the joint PDF among the scalars with some parame-ters to be determined by the moments. For simplicity, thethree scalars are assumed to be independent, and a clippedGaussian distribution9 is assumed for the mixture fraction, athree-delta function4'11 for the progress variable, and a single-delta function4'11 for the mixture enthalpy. Determination ofthese PDF functions only requires the mean and variance ofscalars /, h, and q.

To calculate the mean values of q and its variance p, twomore transport equations are necessary. For axisymmetricflows, the parabolic modeled equations are represented as

0.8-

0.4-

o o

•D

O Papamoschou data• Compres. effectsO No correction

OJSS 0.50 0.75 135 1.50 1.75

Fig. 1 Comparisons between the numerical results of the presentmodel and experimental data by Papamoschou and Roshko.13 Theshear-layer growth rate normalized by the spreading rate at zeroconvective Mach number is for a nonreacting two-dimensional mixinglayer.

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398 VILLASENOR, CHEN, AND PITZ: EXPANDED SUPERSONIC TURBULENT JET

OUTER JETM=1.9. u=1570 m/sP=0.1 MPa, T=1495 K

d=0.009525 m D=0.0653 mInjector lip thickness=0.0015 m

INNER JETM=2.0, u=2700m/sP=0.1 MPa, T=314KY/ra=1.0,Y0,=0.0,

CONVECTIVE MACH NUMBER

Mi=ui/ai=3.245, Aw=u2/Ui=0.557

Mc:

Fig. 2 Inner jet and outer jet inflow conditions for the scale super-sonic burner used by Beach reported in Evans et al.1 The convectiveMach number is evaluated at the inflow conditions.

0.2

Fig. 3 Initial total pressure profile.

ible value. Also shown in Fig. 1 is the numerical solutioncorresponding to the normalized spreading rates for the sameflow conditions but without the compressibility correction.As expected, the ratio 6 V6Mc = o remains almost, constant andvery close to 1. This comparison illustrates that the standardk — c - g turbulence model is inappropriate for supersonicflows.

For reacting supersonic axisymmetric jets, the flow condi-tions in experiments of Beach reported in Evans et al.1 areused to evaluate the performance of the parabolic code inaxisymmetric reacting flows. A schematic of their coaxial jetflowfield examined in this work is shown in Fig. 2; the temper-ature and other exit conditions for the nozzle are also given inFig. 2. To be consistent with the experimental data, the out-side diameter (d = 0.009525 m) of the H2 injector is used tonormalize the radial profiles. The present calculations assumethat the gases expand ideally into atmospheric air so thatshock waves are not present. Furthermore, uniform condi-tions are used at the inflow boundaries for both the inner and

0.0-

0.0-

0.4-

0.2-

> ExperimentCalculated

0.2 0.4 0.6 0.0 1J2 1.4 1.6r/d

Fig. 4a Radial profile of the predicted and deduced mixture frac-tions for the nonequilibrium model at x/d = 8.26.

0.0-

0.0-

0.4-

0.2-

• ExperimentCalculated

1.2 1.4 1.6

Fig. 4b Radial profile of the predicted and deduced mixture frac-tions for the nonequilibrium model at x/d - 21.7.

0.0-

0.0-

0.4-

x/d=27.9

ExperimentCalculated

03

Fig. 4c Radial profile of the predicted and deduced mixture frac-tions for the nonequilibrium model at x/d = 27.9.

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VILLASENOR, CHEN, AND PITZ: EXPANDED SUPERSONIC TURBULENT JET 399

outer jets. The numerical computation for solving the trans-formed,20 coupled partial differential equations is carried outby using a block-tridiagonal solver.21 Fifty grids across half ofthe jet are used and typically 2000 marching steps are per-formed. Using a VAX 8650 machine, about 40 min of CPUtime are required to calculate the table and approximately 20min to solve the reacting flow.

The calculations are initiated at x/d = 0.33, where the flowmeasurements were made. The initial velocity profile is com-puted from the tabulated experimental pitot-pressure dataprovided by Evans et al.,1 assuming a constant static pressureof 1 atm. The procedure for calculating the k and e initialprofiles is described by Evans et al.1

Several test runs were conducted showing that results aresensitive to the initial k and e values. Because of the uncer-tainty in specifying the initial k and e values, the dissipationlength parameter is chosen to achieve good agreement with theexperimental data. The value of /e = 0.24 mm is used for theequilibrium and nonequilibrium models so that the calculatedprofiles predicted by both chemistry models can be compared.In Fig. 3, the measured total pressure is compared to the initialprofile for total pressure used in the present calculations forthe nonequilibrium and equilibrium cases. The dip in theprofile starting at the inside H2 injector radius (r/d = 0.33) tothe outside H2 injector radius (r/d = 0.5) is due to the effectof the thick H2 injector wall.

One of the most important features of the mixture fractionis that, under the assumption of equal diffusivity, it is inde-pendent of the progress of chemical reaction and indicateshow well the fuel and oxidizer streams are mixed. To assessthe model's capability to predict the mixing process, radialprofiles of mixture fraction are plotted at three locations:x/d = 8.26, 21.7, and 27.9. A comparison of the experimentaldata and the numerical predictions is presented in Figs. 4 forthe nonequilibrium code. The equilibrium results are similar.Experimental data for the mixture fraction are deduced fromthe measured mass fractions of the three major species usingH as the conserved element

/ = - (8)H20

where superscripts / and oo denote the fuel and outer streamsat the nozzle exit, respectively. As can be seen from Figs. 4band 4c, the predictions are in satisfactory agreement with theexperimental data, except in Fig. 4a, where the model under-predicts the degree of mixing in the inner part of the jet.

0.8-

0.6-

0.4-

0.2

• ExperimentCompress, effectsNo compress, effects

0.2 0.4 0.6 0.8r/d

1.4 1.6

Fig. 5 Predicted and deduced mixture fractions for the nonequi-librium model with and without the compressibility correction atx/d = 15.5.

The requirement for the compressibility correction is real-ized when the mixture fraction profiles with and without thecompressibility effects are compared at location x/d - 15.5(see Fig. 5). The disagreement between the two solutionsshows that the turbulent mixing rates are incorrectly predictedwhen the compressibility effects are not present in the turbu-lence transport model. The compressibility correction to thecomputed mixture fraction distribution at the far-downstreamregions (x/d>2l.l) is negligible. This trend is expected sincethe supersonic spreading rates approach the incompressibilitylimit as the convective Mach number diminishes downstream.

The calculated profiles of the three minor species at locationx/d - 15.5 are now examined for the finite rate chemistry andfast chemistry cases. In Fig. 6, the mass fraction distributionsof the H, O, and OH radicals are shown for the nonequi-librium model, whereas Fig. 7 shows for the same minorspecies the profiles with the equilibrium model. For the partialequilibrium model, the concentrations of H, O, and OH andtheir respective integrated areas are much larger. For kineti-cally controlled processes, the reaction zone occurs over amuch wider domain within the flowfield. In particular, theOH mass fraction distribution has a very tall and sharp peak.Also note that the H mass fraction penetrates much fartherinto the rich region of the jet than in the equilibrium case.

0.030

0.025-

0.020-

0.0/5-

0.0/0-

0.005-

0.000

A/ \

£...OH

1 12 1.4 1.60 OJS 0.4 0.6 0.8r/d

Fig. 6 Radial profiles of the predicted mass fractions of three minorspecies for the nonequilibrium model at x/d = 15.5.

0.030

0.025-

0.0*0-

0.0/5-

0.0/0-

0.005-

0.000

£-_

H_OH

A\

1.2 1.4 1.60.2 0.4 0.6 0.8r/d

Fig. 7 Radial profiles of the predicted mass fractions of three minorspecies for the equilibrium model at x/d = 15.5.

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400 VILLASENOR, CHEN, AND PITZ: EXPANDED SUPERSONIC TURBULENT JET

0.0-

i0.6-

0.4-

OJZ-

03 0.4 0.6 0.8 1 US 1.4 1.6

Fig. 8a Radial profiles of the predicted and measured mass fractionsof major species for the nonequilibrium model at x/d = 8.26.

Fig. 8b Radial profiles of the predicted and measured mass fractionsof major species for the nonequilibrium model at x/d = 15.5.

0 OJ2 0.4 0.6 0.8 1 1Jt 1.4 1.6

Fig. 8c Radial profiles of the predicted and measured mass fractionsof major species for the nonequilibrium model at x/d = 27.9.

Referring to Fig. 6, one can see that substantial levels of the Hatom mass fraction are produced along the entire radial do-main and this is not consistent with the equilibrium results.

The predicted centerline temperatures (not shown) for bothchemistry models at x/d = 8.26, for example, are about 340 K,and at such temperatures there should be no H atoms (Y$ =10~35). However the nonequilibrium model still predicts sub-stantial H atom levels at this position (F£oneq = 1.28 x 10~5).Evidently, the partial equilibrium assumption forces bimolee-ular reactions to proceed even at low temperatures, leading tothe unrealistic H atom concentration in the rich cold centralcone of the flame. This leads to the conclusion that the partialequilibrium assumption breaks down at low temperatures.Despite the low-temperature restriction, the partial equi-librium assumption works well after the flow is ignited and thereactions are proceeding at temperatures above 1200 K.

All of the comparisons made in this work are well beyondthe ignition point of the supersonic flame. This is supportedby experimental data previously obtained by Beach22 on super-sonic mixing and combustion in an identical hydrogen jet in acoaxial high-temperature test gas. The apparatus and instru-mentation are the same as the one described earlier (see Fig. 2),where supersonic hydrogen (Mach 2) is injected into an uncon-fined Mach 2 jet of air at 2200 K total temperature. The Machnumbers of the hydrogen jets in both experiments are thesame, but the Mach number and stagnation temperature of thevitiated air in our case are 5% smaller and 1.3% larger thanthose used by Beach.22 Photographs of the flowfield at testconditions show quite clearly that there is a measurable delaydistance before ignition from the hydrogen-air reaction can beseen. This distance is approximately 2.4 diameters (based onthe injector outside diameter). This suggests that for x-/d>2Athere is ignition, and in our flow, the simplified partial equi-librium model used here is a good approximation to describingthe combustion process in this H2-air supersonic flame.

In general, the concentrations of minor species predicted bythe nonequilibrium model are higher than those calculatedwith the equilibrium model. However, farther downstream(x/d = 27.9), both chemistry models predict very closely thesame amount of radicals, indicating that an equilibrium statehas been reached for the minor species. To evaluate properlythe validity of the results of the nonequilibrium model, theminor species need to be measured.

Figures 8a-c compare the predictions and experimental dataof the major species concentration distributions for the non-equilibrium model at x/d - 8.26, 15.5, and 27.9, respectively.The numerical results match the experimental data well inregions near the H2 jet, but a fairly substantial differencebetween the measured and predicted major species mass frac-tion values is evident at the flame front. The relative positionof the peak H2O concentration reveals that, indeed, O2 isconsumed more rapidly in the calculated results.

Several potential sources of error in the experimental proce-dure can be identified. Briefly, the mass fractions of H2, O2,and N2 were measured by gas chromatograph. Furthermore,water concentrations are deduced from the dry samples bydifference. It is not clear that the gas sampling probes quenchthe reaction and that the constituents reaching the samplebottle are those present ahead of the probe. As pointed out byEvans et al.,1 a quantitative indication of the validity of thisapproach is given by comparison of the measured freestreamlevels of nitrogen, oxygen, and water vapor as compared withthe inflow values for these species. From the reported mea-sured values at freestream and the inflow conditions, one candetermine that the error in the measurements ranges from 5 to15%. For this reason, nonintrusive methods are preferredtoday, but were not available at the time of the measurements.

The large grid spacing seen in the predicted major speciesprofiles for r/d>0.6 is caused by the rapid expansion of thegrid system in the computational domain at the outer jetstream where gradients of the flow variables are considered tobe negligible, and less accuracy is required.

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VILLASENOR, CHEN, AND PITZ: EXPANDED SUPERSONIC TURBULENT JET 401

0.8-

0.6-

0.4-

0.2-

H,0OH.

Evans et.alPresent model

0.2 0.4 0.6 0.8r/d

1.2 1.4

Fig. 9 Numerical predictions of the mass fraction profiles of twomajor species with the eddy-breakup model and the nonequilibriummodel at location x/d = 15.5.

Mass fraction profiles of measured and predicted majorspecies for the equilibrium model (not shown) are similar tothe nonequilibrium results. Slight differences between theequilibrium and nonequilibrium chemical models are identi-fied mainly in the fuel-rich region where the equilibrium codepredicts more H2 in the centerline. The general agreementis due to the fact that the predictions of stable componentsand temperature for the partial equilibrium assumption onlyshows a weak dependence on the progress variable.

It is very profitable to see the improvements of the newlydeveloped nonequilibrium code in contrast to previous pre-dictions of the same flow. Our solution is compared withnumerical predictions made by Evans et'al.1 for two reasons.First, their system of transport equations is of parabolic formand, hence, the equations are similar to the ones solved in thepresent model. Secondr their reaction kinetics, in the eddy-breakup model, include only one global reaction (O2 + 2H2^2H2O). The eddy-breakup model is based on the concept ofunmixedness, which plays a major role in turbulent flames.The rate constant for the global reaction (CEBu) is not univer-sal and is chosen to achieve the best fit to the data. Evanschooses a single value of CEBu that is valid for the wholeflowfield; however, CEBu varies for different flowfields by anorder of magnitude (Ref. 1). Figure 9 shows a comparison ofnumerical predictions of the eddy-breakup model and thenonequilibrium model with measurements for H2O and H2 atx/d - 15.5. The H2 distribution predicted by this PDF modelis far better than the Evans et al. solution. The Evans et al.solution does not include compressibility effects and overpre-dicts the rate of mixing.

The use of the eddy-breakup formulation predicts waterconcentrations that are lower than the experimental measure-ments. With the use of the PDF model, the agreement isreasonably good, except from r/d = 0.5 to 0.9 where greaterlevels of H2O are predicted. The partial equilibrium model insuperior because it is able to produce H2O faster in the centerof a flame where temperature and concentrations are high andproduce more slowly at the edges. A major disadvantage ofthe eddy-breakup model is that the model has to be adjustedempirically by choice of a constant (CEBU) in the productionterm of the single global reaction. Little guidance is availableas to how to pick this constant.

ConclusionsA numerical model for predicting ideally expanded super-

sonic turbulent jets with nonpremixed combustion of H2-airhas been developed. Two chemical reaction schemes, one with

equilibrium chemistry and the other with a simplified finiterate kinetics, have been incorporated into the present model.The two-equation model adopted for the turbulence model isa modified version of the k - e - g model tailored for super-sonic flows. This program is the first parabolic code to includethe turbulence chemistry interaction effects for supersonicflows. A joint PDF describes the interaction between theturbulence and the chemistry due to the turbulent fluctuationsof the mixing and reaction scalars. The reduction of the shear-layer growth rate at high speed is quantified by a compress-ibility correction that is a function of the convective Machnumber. The numerical computation is accelerated by theconstruction of lookup tables that provide all of the instanta-neous thermochemical values.

Comparisons of predictions and experimental data byBeach (see Ref. 1) for mixture fraction indicate that the nu-merical model predicts reasonably well the spreading rateswith a modified turbulent diffusion coefficient. Incorporationof the compressibility correction has a strong effect on theturbulence transport process. This correlation makes the para-bolic code more robust and versatile, since now it is able todeal with flows in the subsonic and supersonic regimes.

Although the partial equilibrium (PE) model is unable topredict ignition, in the experiment by Beach, ignition occursearly in the flame (2.5 diameters downstream). All of themodel/data comparisons of major species are made fartherdownstream (8 diameters), where the PE model is a goodapproximation. Only in the cold rich zones (temperaturesbelow 1200 K) does the PE model break down and overpredictthe H atom concentrations. Temperature and major speciesprofiles with equilibrium chemistry differ very little fromthose with the partial equilibrium assumption due to a weakdependence of these variables on the reaction progress vari-able. The nonequilibrium chemistry model predicts substan-tially greater concentrations of the minor species.

The use of the thefmochemical tables reduces CPU timesignificantly but increases computer storage. The real benefitof the thermodynamic tables becomes obvious if more com-plex supersonic flows such as those in a scramjet are studied.A severe limitation of this code is the constant pressure as-sumption. Thus -, it can only be used in those situations inwhich the inflow pressures at the inner and outer streams areidentical or differ by a small amount, say 5%.

AcknowledgmentsThis work was supported by the U.S. Department of En-

ergy, Office of Basic Energy Sciences, Division of ChemicalSciences, and by National Science Foundation PresidentialYoung Investigator Grant CTS/8657130. The first author isgrateful for the financial support provided by the NationalScience Foundation and Sandia National Laboratories and thecomputing facilities provided by Sandia Combustion ResearchFacility, Livermore, California.

Referencess, J. S., Schexnayder, C. J., and Beach, H. L., "Application

of a Two-Dimensional Parabolic Computer Program to Prediction ofTurbulent Reacting Flows," NASA TP-1169, March 1978.

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3Eklund, D. R., Drummond, J. P., and Hassan. H. A., "Calcula-tion of Supersonic Turbulent Reacting Coaxial Jets," AIAA Journal,Vol. 28, No. 9, 1990, pp. 1633-1641.

4Janicka, J., and Kollmann, W., "A Two-Variable Formalism forthe Treatment of Chemical Reactions in Turbulent H2-Air DiffusionFlames," Seventeenth Symposium (International) on Combustion,The Combustion Institute, Pittsburgh, PA, 1979, pp. 412-430.

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12Dash, S., Weilersteen, G., and Vaglio-Laurin, R., "Compressibil-ity Effects in Free Turbulent Shear Flows," Air Force Office ofScientific Research, TR-75-1436, Aug. 1975.

13Papamoschou, D., and Roshko, A., "The Compressible Turbu-lent Shear Layer: An Experimental Study," Journal of Fluid Mechan-ics, Vol. 197, 1988, pp. 453-477.

14Dimotakis, P. E., "Turbulent Free Shear Layer Mixing," AIAAPaper 89-0262, Jan. 1989.

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Layers," AIAA Journal, Vol. 21, 1983, pp. 926, 927.16Launder, B. E., and Spalding, D. B., "The Numerical Computa-

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17Dixon-Lewis, G., Golds worthy, F. A., and Greenburg, J. B.,"Flame Structure and Flame Reaction Kinetics; IX Calculation ofProperties of Multi-Radical Premixed Flames," Proceedings of theRoyal Society of London, Series A: Mathematical and Physical Sci-ences, Vol. 346, 1975, pp. 261-278.

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19Kee, R. J., Miller, J. A., and Jefferson, T. H., "CHEMKIN: AGeneral-Purpose Problem-Independent Transportable, FortranChemical Kinetics Code Package," Sandia National Lab. Rept.SAND 80-8003, March 1980.

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21Chen, J.-Y., Kollmann, W., and Dibble, R. W., "NumericalComputation of Turbulent Free-Shear Flows Using a Block-Tridiago-nal Solver for a Staggered Grid System," Proceedings of the 18thAnnual Pittsburgh Conference, Vol. 18, Instrument Society of Amer-ica, Pittsburgh, PA, 1987, pp. 1833-1838.

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