Modeling temperature and species fluctuations in turbulent, reacting flow

~

Computing Systems in Engineering Vol. 5, No. 2, pp. 117 133, 1994

Pergamon

Elsevier Science Ltd Printed in Great Britain.

MODELING TEMPERATURE AND SPECIES FLUCTUATIONS IN TURBULENT, REACTING FLOW R. L.

GAFFNEY,Jr,? J. A. WHIrE,t S. S. GIRIMAJI?a n d J. P. DRUMMOND~ tAnalytical Services and Materials, Inc., Hampton, VA 23666, U.S.A. ~NASA Langley Research Center, Hampton, VA 23665, U.S.A.

Abstract--Assumed Gaussian and fl probability density functions (PDFs) for temperature are used with series expansions of the reaction-rate coefficients to compute the mean reaction-rate coefficients in a turbulent, reacting flow. The series-expansion/assumed PDF approach does not require any numerical integration, which substantially reduces computational cost with little loss of accuracy. An assumed multivariate fl PDF for species is investigated for use in modeling the interaction of species fluctuations and chemical combustion. The multivariate fl PDF is initially evaluated through a parametric study. Results of the parametric study indicate that species fluctuations can increase or decrease the magnitude of the species production term, depending on the type of reaction. The models are then tested on a two-dimensional high-speed turbulent reacting hydrogen-air mixing layer. For the conditions tested the numerical simulations indicate that the net effect of species fluctuations is to reduce the mean species production rate.

NOMENCLATURE A B C, Fc Ft

Fp .~ g~ K,. k kb kt

M nr ns

P Pc Pt Pr

Pp Q R, S S, T Ta

u~ W, x, fl, p F 0 v) v',' /~

constant in Arrhenius expression constant in Arrhenius expression concentration of species i species concentration contribution to the species production term species mass-fraction contribution to the species production term density contribution to the species production term mass fraction of species i Gibbs free energy of species i equilibrium constant turbulent kinetic energy backward reaction-rate coefficient forward reaction-rate coefficient sum of stoichiometric coefficients number of chemical reactions number of chemical species probability density function (PDF) species concentration PDF species mass fraction PDF temperature PDF density PDF sum of species variances (turbulent scalar energy) universal gas constant sum of the square of the mean species mass fractions Schmidt number temperature activation temperature velocity in the x~ direction molecular weight of species i Cartesian coordinates coefficient in assumed multivariate fl PDF dissipation rate of turbulent kinetic energy density gamma function non-dimensional temperature stoichiometric coefficient of species i that acts as a reactant stoichiometric coefficient of species i that acts as a product coeffÉcient of viscosity

tb,

production of species i caused by chemical reaction

Other marks

"

denotes denotes denotes denotes

an averaged quantity a Favre-averaged quantity Reynolds fluctuation Favre fluctuation INTRODUCTION

Turbulence plays a n i m p o r t a n t role in the mixing a n d c o m b u s t i o n of n o n - p r e m i x e d flows. 1 A n u n d e r s t a n d ing of the interaction of turbulence a n d chemical reaction is i m p o r t a n t in the optimization of c o m b u s tor designs. This knowledge is particularly i m p o r t a n t in the design o f high-speed c o m b u s t o r s such as those p r o p o s e d for the N a t i o n a l Aero-Space Plane, where c o m b u s t o r efficiency is crucial to the success of the program. The goal of the current work is to understand the interaction o f turbulence a n d chemical reaction to guide c o m b u s t o r design a n d to derive models t h a t can predict this interaction. T u r b u l e n t c o m b u s t i o n can be broadly categorized into two groups: premixed a n d n o n - p r e m i x e d combustion. In the former, fuel a n d oxidizer are initially well mixed, a n d a flame propagates t h r o u g h the homogeneous, t u r b u l e n t mixture. In the latter, which is the focus of the present study, the fuel a n d oxidizer are initially segregated a n d react as they mix. The progress a n d challenges in the u n d e r s t a n d i n g a n d modeling o f t u r b u l e n t c o m b u s t i o n are chronicled in the reviews o f Bilger 2 (non-premixed) and Pope 3 (premixed). A n accurate evaluation of the m e a n species production rate in flows with t e m p e r a t u r e a n d species fluctuations is difficult due to the non-linear dependence of the p r o d u c t i o n rate on t e m p e r a t u r e a n d 117

R.L. GAFFNEYet al.

118

species concentrations. If the joint probability density function (PDF) of temperature and species concentrations is known at each point in space and time, then the mean production term can be determined. The PDF can either be calculated by solving its modeled evolution equation 4 or specified by assuming a functional form with parameters that depend on the mean temperature and species concentrations and their first few moments. Solving an evolution equation for the P D F is more general but is computationally intensive for even simple flow geometries. The assumed PDF approach offers greater computational viability at the expense of generality. The objective of our work is to understand how temperature and species fluctuations affect chemical reaction and to model the effects that they have on combustion in a turbulent, reacting flow. We would like our models to be applicable to reaction mechanisms with multiple reaction paths for use in finite-rate chemistry calculations. In addition, we would like our models to require a minimal of computational effort to evaluate. We will accomplish this by making use of assumed PDFs. We propose to model the mean species production rate by first decomposing it into parts and then modeling each part separately. If a function is separable as

F(T, C) = FT(T)Fc(C ),

(1)

then

F(T, C) = Fr(T)Fc(C) + Fr(T)Fc(C) - Fr(~Fc(C) + c.c.t.

(2)

where the c.c.t, are cross-correlation terms that involve products of temperature moments Mr and species moments Mc c.c.t.= MTM c.

(3)

The temperature and species moments are

(4) and

Mc=[~=~ 1 /"~

,

(5)

The first term on the right-hand side of Eq. (2) is only affected by temperature fluctuations and not by species fluctuations. Similarly, the second term on the right-hand side of Eq. (2) is only affected by species fluctuations and not by temperature fluctuations. We will call these two terms the temperature term and the species term, respectively. Because the terms that represent the production of chemical species due to the forward and backward reactions are separable,

we can decompose the forward and backward contributions into parts in the above manner. We then model the temperature terms, the species terms, and the cross-correlation terms in three steps. The first step (which has been completed 5) is to investigate and model the effects of temperature fluctuations on the mean species production rate. In this step, we neglect species fluctuations so that Eq. (2) reduces to

F(T, C) = Fr(T)Fc(C-').

(6)

The second step (the subject of the current paper) is to investigate and model the effects of species fluctuations. In this step, we include our previous work with temperature fluctuations, but assume that the temperature and species concentrations are statistically independent. This assumption allows us to model the cross-correlation terms in Eq. (2) as the product of temperature moments and species moments: c.c.t.= M r M c.

(7)

Equation (2) then reduces to

F(T, C) = Fr(T) Fc(C).

(8)

The final step (yet to be completed) is to investigate and model the cross-correlation terms without assuming that the temperature and species are statistically independent. We will accomplish this by modeling the most significant cross-correlation terms with traditional moment methods. Although we present numerical results with each step, the model for the mean production rate is not complete until all three steps have been completed. The first two steps help us to understand only some of the effects that temperature and species fluctuations have on the mean species production rate. In a previous paper, 5 we analyzed the mean reaction-rate coefficients [the temperature term in Eq. (2)] by modeling the effects of temperature fluctuations with assumed Gaussian and fl PDFs. We found that temperature fluctuations can make a significant difference in the size of the mean reaction-rate coefficient. Three factors contributed to the change in the magnitude of the mean reaction-rate coefficient: the mean temperature, the intensity of the temperature fluctuations, and the shape of the reaction-rate coefficient curve (how rapidly it increases and decreases as a function of temperature). In this study, the mean reaction-rate coefficients were computed by numerically integrating the product of the reactionrate coefficient and the assumed temperature PDF. This approach proved to be very expensive computationally. In the current work, we present an alternate way to compute the mean reaction-rate coefficients by utilizing a series expansion of the reaction-rate coefficient and an assumed PDF. This hybrid method is much

Modeling turbulent, reacting flow less expensive computationally. Next, we investigate the effects of species fluctuations on the combustion process by assuming that the species mass-fraction fluctuations can be statistically described by a multivariate fl PDF. We then conduct a parametric study (by varying the mean species distribution and the intensity of species fluctuations) to determine their effect on the mean species production rate. Finally, a numerical simulation of a turbulent, reacting hydrogen-air mixing layer is performed with the models. CHEMICAL

REACTION

In a high-speed chemically reacting flow, the timescales associated with the chemical reactions can be the same order of magnitude as the fluid time-scales. If this is the case, finite-rate chemistry must be used in any numerical simulation. The species production rate ~b~is dependent on the temperature T and species concentrations (7,.:

,o°=w.

(vD-v D) k~ c ? - ~ , i= 1

}

c~;). i=1

(9) In this expression, W, is the molecular weight of species n, v;/and v~ are the stoichiometric coefficients of the reactants and products for species i and reaction j, and k 6 and kb~ are the forward and backward reaction-rate coefficients for reactionj. The forward reaction-rate coefficient is determined from an Arrhenius expression , ko(T) = A DTs0exp ( T--TAD )

(lO)

where A~, B6 and Ta 6 are constants for reaction j. The backward reaction-rate coefficient is determined either from an Arrhenius expression (if the constants are known) or from the forward reaction-rate coefficient and the equilibrium constant K~, as

119

and

Ans= ~

(11)

The equilibrium constant is a function of the temperature and the Gibbs free energy g~ of each species that participates in the reaction:

(14)

i=1

In the current work, the Gibbs free energies are determined from curve fits.6 In a turbulent flow in which chemical species are mixing and reacting, the temperature and species concentrations fluctuate, which causes fluctuations in the species production rate. Modeling the effects that these fluctuations have on the mean species production rate is difficult because of the highly non-linear dependence of this term on the temperature and species concentrations.

PRODUCTION

RATE

MODELING

If the joint PDF of the temperature and the species concentrations P(T,C) is known, then the mean species production rate is found by integrating the product of the production rate and the PDF as

go, =

I; I;;;fo ••"

UJ,P dT dC I dC2. • •dC,,,.

(15)

The difficulty with this approach is the determination of the joint P D F that correctly accounts for all of the important physical processes. The most general way to accomplish this is to solve an evolution equation for the PDF. Pope 4 and others 7'8 have taken this approach. We have chosen to take a slightly different approach. If we let

Fr(T)j)=k6(T)

Fr(T)h=kbj(T),

Fc(C)j) = f i C~.i, Fc(C)bj = f i CV;',, i=l

kbj(T) = kj) (T) . K:,(T)

v ~ - ~ v~j.

i=1

\R.T)

(17)

t=l

then the production rate can be written as

~. = wo ~ (v'D- vD) j~l

x [Fr(T)DFc(C)h - Fr(T)~jFc(C)h,]. [' - AGf'I Kej = (R, T) - ~ j e x p t ~ / ,

(16)

(18)

(12) The mean species production rate is then

where R, is the universal gas constant,

go. = w. ~ (v;)- v'.A [~ I

AGj= ~ v:~gi- ~ v~gi. i=l

CSE 5,'2--B

i=1

(13)

x [Fr(T)gFc(C) k -- Fr(T)~Fc(C) 6].

(19)

R.L. GAFFNEYet al.

120

Using Eq. (2), the mean species production rate can be written as

go, = IV,, ~ (%j"-v.,){[Fr(' T)fjFc(C)hj~l

+ Fr(T)j)Fc(C) h -- Fr(T)jiFc(•)j) +

(c.c.t.)j)l

A number of assumed PDFs have been used in the past. Among those previously investigated are the double delta PDF, 9 a clipped Gaussian, 1° and the fl PDF31 Girimaff 2 has shown that the fl PDF can accurately describe the evolution of a PDF for the turbulent mixing of two scalars in non-reacting homogeneous turbulent flows (double delta through Gaussian).

-- [Fr(T)b, Fc(C-~)b, + F r ( T ) b f c(C)h, -

Fr(T)bfc(C)b , + (c.c.t.)h, 1},

TEMPERATURE FLUCTUATIONS

(20)

where (c.c.t.)j) -= (Mr)sj (Mc)j?

(21)

(c.c.t.)h, = (Mr)oj (Mc)~j.

(22)

If we assume that the temperature and species are statistically independent, then the cross-correlation terms become (c.c.t.)j) = (Mr)~ (Mc)fj

(23)

(c.c.t.)h~ = (Mr)hi (Mc)b~,

(24)

and the mean production rate reduces to

In a previous paper] we used an assumed Gaussian PDF and an assumed fl PDF to investigate the effects of temperature fluctuations on the reaction-rate coefficients in the species production term. The effects of temperature fluctuation were important for the conditions considered; however, the integration of the product of the PDF and the reaction-rate coefficients at each point in the computational grid was very expensive. An alternative approach to using a PDF to compute the mean of a function is to use a series-expansion method. In this approach, the independent variables are decomposed into mean and fluctuating parts. The function is then expanded in a series about the mean. u With this approach, the mean forward reaction-rate coefficient can be written as

= w. ~ (v~-G) j=l

x [Fr(T)~Fc(C)j) - Fr(T)bfc(C)b,].

(25)

The mean temperature term and the mean species term can be modeled with marginal PDFs for temperature Pr and species Pc, respectively, as

Fr(T)j; = kj; (T) = Fr(T)~j = ks, ( r ) =

k6 (T)P r dT,

k~, ( r ) P r dT,

(26)

(27)

The n th term in the series has the form

L 5o°°'LT) J'

where the coefficients am," are functions of Bj. This series is convergent if

--•<

i.

× Pc dCi dC2"" "dC .... (28)

fo f:I:

Fc(Ck, . . . .

G(Cg

x Pc dCi dC2" " " dC,s.

(29)

If the general shape of a PDF is known, then an assumed PDF can be used. In this approach, the PDF is specified as a function of n parameters. The parameters are determined by requiring that n moments of the PDF equal the moments computed from some other method, such as the solution of n moment transport equations.

(31)

(32)

For a given fluctuation intensity T'/T, the series is slowest to converge when the activation temperature Tar is larger than the mean temperature T. The term in brackets in Eq. (30) can be interpreted as a turbulent correction to the reaction-rate coefficient evaluated at the mean temperature. Unfortunately, this method introduces an infinite number of higher moments. A simple truncation of the series after the first two or three terms is tempting; however, Libby and Williamsj point out that by neglecting the third and higher moments, severe restrictions are placed on the applicability of the expression. Although applicability is increased if higher moments are retained, solving transport equations for large numbers of moments is very expensive computationally.

Modeling turbulent, reacting flow

121

Contour of IOg~o (kf turbulent / kf laminar) Gaussian P D F 0.300

/ /

•

"~ t''~

500

1000

1500 2000 Mean t e m p e m t u ~ T, K

2500

u

8.00

T

7.72

s

7.45

R

7.17

o

6.90

P 0 N M L K

H G F

6.62 6.34 6.07 5.79 5.52 5.24 4.97 4.69 4.41 4.14 3.86

E D C B A 9 8 7 6 5 4 3 2

3.59 3.31 3.03 2.76 2.48 2.21 1.93 1.66 1.38 1.10 0.83 0.55 0.28

1

ooo

Fig. l. Forward reaction-rate amplification-factor contour plot for reaction 1 computed by numerically integrating the Gaussian PDF.

By combining the series-expansion m e t h o d and the assumed P D F approach, the c o m p u t a t i o n a l costs can be reduced significantly with little reduction in accuracy, If the P D F o f a variable is known, then all o f the m o m e n t s can be determined from the PDF:

(T')" =

~0¸~=( T -

T)"Pr(T) dT.

(33)

F o r example, m o m e n t s o f the Gaussian P D F are

I0 ./2 (T')" = I(-T'T')'/2 =I~I( 2 i - 1 )

when n is odd, when n is even.

(34) The m o m e n t s that are determined from the assumed P D F can be substituted into the series expansion to c o m p u t e the m e a n reaction-rate coefficient. The number o f terms retained in the series can be as few or as m a n y as necessary and will depend on the reaction

Table 1. Reaction number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Reaction

A

b

T,

02 ~ 2OH H + O: ~ OH + O OH + H 2 ~- H20 + H O + H 2~- OH + H 2OH ~ H20 + O H + OH + M ~-~ H20 + M 2H + M ~ H 2 + M O2 + H + M ~ H O 2 + M OH q.- HO 2 ~ 02 -4- H20 H + HO 2~.~ H2 + Q2 H + HO 2~ 2OH O + HO 2 ~- 02 + OH 2HO2 ~ 02 + HzO2 H 2 + HO 2~ H + H202 OH + H20~ ~ H20 + HO2 H + H202 ~- 1-120 + HO., O + H~O 2~ OH + HO2 H 202 + M ~ 2OH + M

0.170e 14 0.120e18 0.220e14 0.506e05 0.630e13 0.221e23 0.730e18 0.230e19 0.200e14 0.130e14 0.150e15 0.200e14 0.200e13 0.301e12 0.700e13 0.100el4 0.280e14 0.121 e 18

0.0 --0.91 0.0 2.67 0.0 -2.0 - 1.0 - 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

24157.0 8310.5 2591.8 3165.6 548.6 0.0 0.0 0.0 0.0 0.0 503.3 0.0 0.0 9411.2 722.2 1801.7 3220.9 22899.0

H2+

122

R.L. GAFFNEYet al.

set. In Fig. 1, we present a contour plot of the log (base 10) of the amplification factor for the forward reaction-rate coefficient for reaction 1 of the 9species, 18-reaction model given in Table 1. The amplification factor is defined as

where (__ I)i-1 i 1

I-I (n - j

an,i

i l l = OFL 0(1a,-~- B ) amplification factor - ks(T)

+ 1),

(i -- 1)! j= l

1],

(37) (38)

(35)

kA~

and This figure was made by numerically integrating the product of the reaction-rate coefficient and the Gaussian P D F to compute the mean reaction-rate coefficient. In Figs 2(a)-(c), we see the same contour plot generated with the series-expansion/PDF method with 2, 6, and 10 terms, respectively, to compute the mean reaction-rate coefficient. By retaining 10 terms in the series, amplifications are produced that are very close to the amplifications predicted by the numerical integration. Reaction 1 has the largest activation temperature Ta of the reaction set so it has the slowest series convergence. Reactions that have smaller activation temperatures need fewer terms to reproduce the amplifications predicted by the numerical integration. The fl P D F can also be used to give the higher moments for the series-expansion method. The moments of the fl PDF (for the variable 0) are n+l.~(i_l)nT-[

(0')" = ~=,~ u..,

i

(ill -{- n ~ ! ~ j )

1=1o (fl, + f12 + n -- i - - j ) '

(36)

The variable 0 is limited to values between zero and one. The fl P D F can be used for temperature by transforming the dimensional temperature T to a non-dimensional temperature 0: Z-

(T') n = (Tma x -- Tmin)n(o') n.

4/

0.250

u

s oo

T

s

7.72 7.45

R

7.17

Q P

6.90 6.62

O N M

6.34 6.07 5.79

L K

5.52

0.200

5.24 4.97 4.69

m e0.150

4

0.100

G F E D

4,14 3.86 3.59 3.31

C H B

3.03 4.41 2.76 2.48 2.21 1.93 1.66

A 9 8 7

•

500

1000

(41)

The backward reaction-rate coefficients can be treated in the same manner as the forward reactionrate coefficients if an Arrhenius expression is available for the backward reaction-rate coefficient. In the current work, we compute the backward reactionrate coefficient from the forward reaction-rate coefficient and the equilibrium constant [Eq. (11)]. Note

Series expansion (2 terms) with Gaussian P D F

0 050

(40)

The (dimensional) temperature moments are

Contour of Iog16 (kf turbulent / kf laminar)

oo o / /

Tmirl

0 -- Tmax_ Tmin.

1500

2000

Mean temperature T, K

Fig. 2(a).

2500

6

1.38

5 4 3 2

1.10 0.83 0.55 0.28 o.oo

1

Modeling turbulent, reacting flow

123

Contour of Ioglo (kf turbulent / k~ laminar) Series expansion (6 terms) with Gau.~ian PDF

u

8.00

T

7.72

s

R

0.300

•

6.90

P O N M L K

1

6,62 6.34 6.07 5.79 5.52 5.24 4.97 4.69 4.41 4.14 3,86 3.s9 3.31 3.03 2.76 2.48 2.21 1.93 1.66 1.38 1.10 0.83 0.55 0.28 o.oo

u

8.oo

T

s

7.72 7.45

R

7,17

N M

6.07 5.79

L K

5.52 5.24 4.97 4.69

H G F E

4.41 4.14 3,86 3.59

D C B

A

331 3.O3 2.76 2.48

9 8 7 6 5 4 3 2

2,21 1.93 1,66 1.38 1.10 0.83 0.Sb 0.28

H G F

E

E

1000

1500 2000 Mean t e m ~ m ~ r e T, K

D C B A 9 8 7 6 5 4 3 2

2500

Contour of Ioglo (kf turbulent / kf laminar) Series expansion (10 terms) with Gaussian PDF

/

"

0.250

0.200 "~'~ 0.150

0.100 l ~ / / / / / / , 5 " / /

~

I

0.050

500

1000

7.17

O

"~

500

7.45

1500 2000 Mean tempera~m T, K

2500

1

o.oo

Fig, 2(b, c). Fig. 2. (a) Forward reaction-rate amplification-factor contour plot for reaction l computed by the series-expansion/assumed-Gaussian-PDF method with two terms in the series. (b) Forward reaction-rate amplification-factor contour plot for reaction 1 computed by the 8eries-expansion/assumed-GaussianPDF method with six terms in the series. (c) Forward reaction-rate amplification-factor contour plot for reaction 1 computed by the series-expansion/assumed-Gaussian-PDF method with 10 terms in the series.

R.L. GAFFNEYet al.

124

that we can write the backward reaction-rate coefficient as an Arrhenius expression

{ - Tab~ kb=AhTShexp~----~-),

(42)

We now require a joint P D F for the density and the species mass fractions. To simplify the problem, we decouple the density from the species mass fraction so that the joint P D F for species concentrations becomes

Pc(C,. C2 . . . . . C,,) = Pp(p)Pf(f, ,f2 . . . . . f.s).

where

(50)

Ab = Ay(R.) a",

(43)

B b = BI + An,

(44)

We can further simplify the problem by assuming that the P D F for the density is a Dirac delta function ep = b(p --fi),

and

(51)

then

Tab = Tar

(F~)j; = ~

AG

R.

(F~),, = ~ , ,

va Mb,= A series-expansion of this term is complicated because of the dependence of the Gibbs free energy on the temperature. One way around this complication is to neglect the effects of temperature fluctuations on the Gibbs free energy and evaluate AG at the mean temperature. The series-expansion method can then be used to include the effect of temperature fluctuations on the remainder of the term. A n alternative is to curve fit the backward reactionrate coefficient with an Arrhenius expression. The series-expansion can then be applied to the backward reaction-rate coefficient to introduce the effects of temperature fluctuations. The mean backward reaction-rate coefficient has been computed using both of these methods with similar results.

i=l

At times, species mass fractions are more convenient to work with than species concentrations. The species mass fractions f. and species concentrations are related by the relationship

esE

,f2 . . . . .

As) =

C,= p f . w,

x 6(1 - f ~ --f2 . . . . f~s), (54)

with ns

ns

,

i=1

(47)

i=1

where

(Fp)b~ = f i p~'),

(48)

i=1

and

i=1

Q= Ef;f;.

(56)

i=1

The sum of the species variances Q is designated as the turbulent scalar energy because of its similarity to the turbulent kinetic energy. The mathematical constraint

(Fc)bj=

ns

r(~,)r(~2).., r(~.,)

(46)

We can then write

(Ff)~ = 1-[ f ~

r ( f l , + fl: + . . . + / L )

×f~, ,f~2-,...f~.~s-,

i=1

i=1

(53)

This reduces our problem to specifying a joint P D F for species mass fractions. GirimajP 2 suggests a multivariate fl P D F for modeling the species mass fractions during turbulent mixing as

S= ~,~

(Fp)j, = f i p~

v:; i=l

where

ASSUMED MULTIVARIATE ~ PDF

-

(52)

(45)

Q ~< 1 - s,

(57)

ensures that the values of fli are non-negative. This constraint has the physically correct limit when S = l, which occurs only when one of the mass fractions equals unity (all other mass fractions equal zero). When only one species exists, no fluctuation can occur in the species mass fraction, and Q must equal zero. This model gives expressions for the individual variances and covariances. They are

ns

(Ff)bj = I-[ f[~. i=1

(49)

(58)

Modeling turbulent, reacting flow

125 PARAMETRIC STUDY

and

f;fj = -~

(59)

(i # j ) .

We see that this model predicts all covariances to be negative. Generally, species that are premixed are positively correlated, and species that are unmixed are negatively correlated. Hence, this model is only appropriate for non-premixed flows. In the hydrogen-air reaction system investigated in this paper, only non-premixed species undergo reaction. Nitrogen, which is premixed with oxygen to form air, is considered inert and only participates in the reaction process as a third body. As a result, the nitrogen-oxygen covariance plays a minor role, and the model can be applied to this reaction mechanism. We can use this PDF to compute the mean species term ~-. Fortunately, these terms can be integrated analytically with the result that for non-third-body reactions 14

A2~ A + A.

1-I

i=1 k = l

~t~,~

(60)

1-[(B + M j i - k )

The species contribution to the production rate from the forward reaction is

F •~ U A 2 •

(67)

~- =~4:.

(68)

Because this term is linear in fA,, turbulent fluctuations do not affect its size. Now, consider the reverse of the above reaction

k=l

and

A + A ~Az. (rr)~ ~ =

(66)

The mean of this term is

vb

(I

(Fr)~ =

We will analyze the species contribution Fj to the species production rate oh, by using the multivariate fl PDF to examine the effects of species fluctuations on the reaction process. If we look at the multivariate fl PDF, we see that the mean species production rate is a function of the mean species mass fractions and the turbulent scalar energy, which makes a thorough investigation very difficult because of the large number of free parameters (ns - 1 mean species plus the turbulent scalar energy). As a result, we will limit ourselves to reactions that involve only two species. Consider the dissociation reaction

i=1 k = l

(61)

Mbj

I~ (B + Mh, -- k)

(69)

We have two molecules of the same species that react. The species term for the forward reaction is

k=l

Ff = f 2 ,

where

(70)

which is quadratic in fA- The mean species term is

B= ~ , .

(62)

i=1

For third-body reactions, nk k = l Wkk i -

n=lH(~i'~-]A~'k--n)

,

1

(63)

IJ (B + M~ + 1 -,,) n=l

~ . = y ~ + f;oc;4.

(71)

Because the variance of species A ( f ~ f ] ) is always positive, turbulent fluctuations always increase the size of this term. We can come to the same conclusion by examining the shape of Fs.5 Equation (70) is a concave upward parabola, so we expect fluctuations in f4 to produce an increase in the magnitude of FI. The ]~ PDF gives the variance as

and (72) -~ k=l

(/~ + ~ ffk - n ) i-

n=l

,

(64)

The mean species term can then be written as

l-I ( B + M h j + l - n ) tl=l

(73)

where

t~k=V~/+6,k

I~i'~k=Vi~+f~k,

and r/k is the third-body efficiency for species k.

(65)

The term in square brackets is the amplification factor. We can see that as fA goes to zero the amplification factor goes to infinity like 1/fA, which does not mean that the mean species term becomes

126

R. L. GAFF~Y et al.

Contour of Ioglo (Ff turbulent/Ff laminar)

F E D

A + A --> Product

1.9310 1,7931 1.6552 1.3793 1.2414

o o,oI 0.8276 i

0.5517 0.4138

O.2O

0.2759 0,1379 0.0000

O.OO~

.

0.25

0.50

i

.

0.75

1.00

f, Fig. 3. Amplification-factor contour plot for the reaction A + A--+Product computed with the multivariate fl PDF. infinite. The turbulence has simply changed the asymptotic behavior of FI as fA goes to zero. If no species fluctuations exist, then Fs goes to zero like f ] . If species fluctuations do exist, then ['/goes to zero like f~. Figure 3 shows contours of the log (base 10) of the amplification factor for this reaction. The mean species mass fraction of species A is plotted along the horizontal axis, and the turbulent scalar energy is plotted along the vertical axis. The mass fraction of species A 2 is implied by the requirement that the mass fractions sum to one. The curvature of the upper boundary is a result of the constraint given by Eq. (57). This figure shows that when the turbulence level is zero (Q = 0) the amplification factor is 1 (no amplification). As the intensities of species fluctuations increase, the reaction is amplified, and the largest amplifications take place at fA--0. We now consider the reaction of two molecules of different species: A + B--*Products.

(74)

The forward species term is Fj = f ~ f B .

(75)

The mean of this term is F I = J~A~e + f 'Af 's .

(76)

This term has a turbulent contribution, which is the covariance f ' a f ' s . This term will either increase or

decrease in size, depending on the sign of the covariance. The fl PDF gives the covariance as f'Af's = --JTAfn 1 Q -- S '

(77)

so that the mean species term is

We can easily see that as the fluctuation intensities increase the species term will decrease and even go to zero. The constraint given by Eq. (57) limits the magnitude of the turbulent scalar energy so that the mean species term never goes below zero. Figure 4 shows contours of the amplification factor for this reaction. We have assumed that only species A and B exist in the mixture. This figure shows that when no species fluctuations exist, the amplification factor is 1. As the intensities of the species fluctuations increase, the amplification factor decreases to its minimum of 0. This result is interesting because it shows that the mean species production rate decreases with increasing species fluctuations. This decrease is counterintuitive because we expect an increase in the production rate as a result of turbulent mixing. Although turbulence does increase mixing, the mixture is large scale rather than on the molecular level (where chemical reaction takes place). The physical process of large-scale mixing can be illustrated by considering the mixing of species within

Modeling turbulent, reacting flow an eddy (see Fig. 5). Fuel and air streams that move at different velocities create shear at their interface, which causes the fuel and air to wind up into a vortex. The eddy contains approximately 50 percent fuel and 50 percent air. Although the mean mass fractions (on the length scale of the eddy) are flue, = 0.5 and)fair = 0.5, the fuel and air are not well mixed. Most of the fuel and air are segregated in the windings of the vortex and are mixed only along the interface of the windings. The condition where species are mixed on the large scale, but not on the smaller, molecular scale is referred to as "unmixedness". '5 Unmixedness can be thought of as an indication of how well the fuel and air are mixed on the molecular level, whereas mean values give an indication of how well species are mixed on a larger scale. In turbulence modeling, where equations for the mean quantities are solved, small-scale structures, such as turbulent eddies, are not resolved, and the turbulence model must account for the effects of the small-scale structures on the mean flow. PDF models, such as those used in the current work, are phenomenological models that statistically describe the effects of the small-scale structures. Species fluctuations are an indication of unmixedness. Consider again Fig. 5. If we examine the species mass fractions along a line taken through the center of the eddy, we find that the species mass fractions increase and decrease (fluctuate) along this line. If the fuel and air had been well mixed within the eddy, the mass fractions would be constant (no fluctuations) throughout the eddy. Species fluctuations occur when the species are mixed on the large scale, but not on the

127

smaller, molecular scale (i.e. whenever unmixedness occurs). The larger the species fluctuations, the larger the degree of unmixedness. Reaction rates are lower when unmixedness occurs because the species are not mixed on the molecular level. While most of the reactions in our reaction set are two-body reactions, some three-body reactions occur. We now consider the reaction of three molecules of the same species: A + A + A --*Products.

(79)

Our species term is ~.=f3.

(80)

The mean of this is ~=f]

+ 3f'afAf'A +f'Af'af'A"

(81)

The first two terms in this expression are always positive, but the sign of the third term is unknown. We can determine the effect offA fluctuations on this term by examining the shape of the function. Because FI is cubic in fA, the function is concave upward. Thus, fluctuations infA will always increase the size o f f r. The fl PDF gives the mean species term as

× [(.1 - S) + Q(Z/fA --

Contour of (Ff turbulent / Ff laminar) A + B--> Product

0.40

1

Q

F

0.94

E

0.88

D

0.81

C

0.75

B

0.69

9 8

0.56 0.50

1)].

0.44 0.30

0.38 0.31 0.19 0.12

0.20

0.06 0,10

00o

-; : : i i 0.25

7

0.50

0.75

1,00

f, Fig. 4. Amplification-factor contour plot for the reaction A + B--,Product computed with the multivariate fl PDF. CSE 5 / 2 ~

(82)

R.L. GAFFNEYet

128

al.

Fy=f'2Afn[l + l - - f A ( Q "~I(1--S--Q'] -~-~ \ - i - ~ ] J \ I Z - S ~ - - Q f

(N

(86)

This expression is always positive and has a singular behavior a t ~ = 0. The asymptotic behavior of P / a s J'A goes to zero has been changed by the turbulence from a n f 2 type of behavior to that o f ~ . The contours of the amplification factor are shown in Fig. 7 for this reaction. NUMERICALSIMULATION

1

fN2 _/~-_ _ I I I

f021

:I

c--~

c~

I I I I

l I I I

I I I I

I t I I

I

I

I

I I I I

I

k

1f

~3

i-q

LJ

I I I I I

~.._

fH_2

J

~

J

Fig. 5, Eddy in a two-dimensional mixing layer with profiles of species distributions taken along a cut line made through the eddy. We notice two things about this expression. First, the term is always positive. Second, like the two-body reaction, Eq. (73), the amplification factor is singular at fA = 0. Fluctuations have changed the asymptotic behavior of this term from f 3A to ~A. Figure 6 shows contours of the log of the amplification factor for this reaction. The species term for the three-body reaction 2A + B--,Products

(83)

The parametric study is a useful tool for investigating and understanding the effects of species fluctuations on individual reaction rates. Unfortunately, non-linear coupling within the governing equations makes a determination of the effect that species fluctuations have on the complete reaction process difficult. In order to determine the overall effect, we need to numerically simulate a turbulent, reacting flow. To accomplish this simulation, we solve the Favreaveraged Navier-Stokes equations and ns - 1 Favreaveraged species continuity equations, where ns is the number of chemical species. The Reynolds stresses are modeled with a k - E turbulence model 16that includes the compressible dissipation model of Sarkar et al.: and the pressure dilatation model of Sarkar.lS In these calculations, we implicitly include the effect of density fluctuations through the use of Favre-averaged variables. The mean forward reaction-rate coefficient is computed with the series-expansion/fl PDF method. The mean backward reaction-rate coefficient is computed by neglecting the effect of temperature fluctuations on the Gibbs free energy and applying the series expansion to the resulting term. The Favre-averaged temperature is determined from the solution of the Navier-Stokes equations, and the Favre-averaged temperature variance is computed from the solution of a Favre-averaged enthalpy variance equation) The mean species contribution to the species production term is computed with the multivariate fl PDF for species. The Favre-averaged species mass fractions are determined from the solution of the species continuity equations. The Favre-averaged turbulent scalar energy is computed from an equation for the sum of the species variances.

Turbulent scalar energy equation

is

F: =f~fn.

(84)

Ff =S~a~B+ 2fAf Af'n + fsf'Af'A + f'Af'Af's.

(85)

The mean of this is

The first and third terms in this expression are always positive. The signs of the second and fourth terms are unknown. As a result, a determination of whether species fluctuations will cause an increase or a decrease in F: is difficult. The fl P D F gives this term as

An e.~ation for the Favre-averaged species variances ~'f~' can be derived in a manner similar to the derivation of equations for the Reynolds stresses. If the equations for the individual variances are summed, then an equation results for the turbulent scalar energy

O:

O(fiO)+ O(fi~,O.) + 2~ Ot

gx2

pu:'Q- :D

i= 1

ax:/I

Modeling turbulent, reacting flow

129

Schmidt numbers. The source term on the righthand side needs to be modeled. Note that we can write -

l

(87)

,,.

=

where u i is the velocity in the xi direction and D is the species diffusion coefficient (we have assumed Fick's law). This equation has several terms that must be modeled. The modeled equation is c~(PO) -~

~

~xj

+ 2 ~ -a'

+ 2

g

+ Pr']O01

(88)

i=1

i=t

j=l

X

sc /ox,j

= 2 ~ ~'¢be,

(90)

The term f,&~ is known, and the second term can be determined from the PDF. Girimaji ~2 has evaluated this term with this result:

i=l

i~, puTf; axJ

c3 r ( f i

f;' o~, = f,.~b, - ~dg,.

where /7 is the Favre-averaged turbulent kinetic energy, e is the dissipation rate of turbulent kinetic energy, and CT= 1. The turbulent transport term pu~f~' in this equation is modeled with a simple gradient diffusion assumption:

-

(91)

where

--

/'/~,+ yD.k + tbst ,, b,,-x-¢,)

(92)

and ~) pr

(~.)b,=(Fi)b, tB---~b~) PuTf;' -

Sc r &vj'

tbstb,j.

(93)

(89) F o r non-third-body reactions,

where # and/~r are the laminar and turbulent viscosities and Sc and Scr are the laminar and turbulent

tbst~, = 0

Contour of IOglo (Ff turbulent/Ff laminar) A + A + A --> Product

tbstb0 = 0.

F

3.86

E

3.59

D

3.31 2.76 2.48 2.21

0.40 Q

1.93 1.66

0.28

0.00

0.00 ~

i

0.25

0.50

0.75

1.00

t, Fig. 6. Amplification-factor contour plot for the reaction A + A + A ~ P r o d u c t computed with the multivariate fl PDF.

(94)

R. L. GAFFNEYet

130

al.

Contour of (Ff turbulent / Ff laminar) P 0 N

A + A + B --> Product

/

o.,o L--

\

t

O

.

\

0.30

16.65 15.96

15.27 13.88 13.18 12.49 11.80

11.1o 9.rl

9.02 7.63

6.94 6.24 5.55

0"1~ 0 0.00

i 0.25

0.69 01.39.00

0.50

0.75

1.00

f, Fig. 7. Amplification-factor contour plot for the reaction A +A + B-~Product computed with the multivariate fl PDF. RESULTS AND DISCUSSION

For third-body reactions,

[] I-I (fl.+ ~,:,,-k)

tbst~j = ~

n = l k=l M~+I

(B + M])) 1-I (B + M/j + l - k ) k=l

(95) and ns

P'~/i

U ]-I (fl,+#~,-k) tbstb~ =

n=lk=l Mb/ + 1

(B + Mbj) l-I (B + Mb, + l - k ) k=l

(96)

Differentiation and integration The convection terms in all of the equations are discretized with Roe's upwind flux difference splitring) 9 Higher order spatial accuracy is achieved by adding corrections to a first-order flux in a post-processing manner. 2° A third-order scheme with the minmod flux limiter was used for all the calculations in this paper. All spatial derivatives in the remaining terms are differenced with central differences. The equations were integrated in time with a three-stage Runge-Kutta scheme. The Runge-Kutta integrator and the upwind differencing are modifications that have been made to the SPARK 2~'22finite-rate chemistry code.

Our test case is a two-dimensional turbulent, reacting hydrogen-air mixing layer. The domain is 0.2 m in the streamwise direction and 0.05 m wide. The static pressure and temperature are 1 arm and 1000 K, respectively. The velocity of the hydrogen stream is 3800ms -l and the velocity of the airstream is 1200 m s- i. The infow velocity and species profiles are hyperbolic tangent profiles. The grid is 51 x 51. Several runs were made for these conditions. The first run neglected the effects of temperature and species fluctuations on chemical reaction by evaluating the species production term at the mean temperature and the mean species mass fractions. Although the flow is turbulent, we will refer to this case as the "laminar" chemistry case because we have neglected the effects of fluctuations in computing the mean species production rate. Next, a series of runs were made with the PDF and series-expansion/PDF methods to compute the mean forward and backward reaction-rate coefficients. These runs include the effects of temperature fluctu-

Table 2. Evaluation method

Gaussian PDF

Beta PDF

Numerical integration Series expansion (2 terms) Series expansion (4 terms) Series expansion (6 terms) Series expansion (8 terms) Series expansion (10 terms)

109.7 2.7 6.7 11.4 18.2 30.6

91.8 2.9 6.9 I 1.9 19.1 31.9

Modeling turbulent, reacting flow 0.01 H=O Mass Fraction Contour Hydrogen

Case2

Case3

Case

1

[

Fig. 8. Water mass-fraction contours for cases 1, 2, and 3.

ations, but do not include the effects of species fluctuations. The first two runs of this series were made by numerically integrating the product of the reaction-rate coefficients and the Gaussian and fl PDFs. The remaining runs in this series were made with 2, 4, 6, 8 and 10 terms in the series-expansion/PDF method. An examination of the solutions indicated that the series expansion with six terms adequately reproduces the numerically integrated results for these flow conditions. Table 2 shows the per cent increase in computer time required for each method to compute the mean reaction-rate coefficients rather than the reaction-rate coefficients evaluated at the mean temperature. The series expansion with six terms only increases the computational cost by about 11 or 12 per cent versus approximately 100 per cent for the numerical integration. Even with 10 terms in the series, the series-expansion/PDF method is much less expensive than numerical integration. The final run used the series-expansion/fl PDF method (with six terms) to compute the mean reaction-rate coefficients (the temperature terms) and the multivariate fl PDF for species to include the effects

Mass

131

of species fluctuations. Figure 8 shows contours of the water mass fraction (contour level 0.01) for the three cases. The production of water is an indication of chemical reaction with heat release. The figure clearly indicates that turbulent fluctuations cut reaction time (defined here as the time to produce 1 per cent of water) for these conditions in half. The reduction in reaction time between cases 1 and 2 is caused by the effects of temperature fluctuation and is discussed previously.6 Of interest in this paper is case 3, which shows the effects of species fluctuations. The figure indicates that one effect of species fluctuations is to increase the reaction time for these conditions, although not as much as temperature fluctuations decrease the reaction time. If we look at the reaction set listed in Table 1, we see that most of the reactions are two-body reactions that involve different species. Our parametric study indicated that species fluctuations would reduce the size of the mean production term for this type of reaction. This reduction in the mean production term is due to the phenomena of unmixedness (as discussed in the parametric study) where chemical species undergo largescale mixing but remain unmixed on the molecular level. Since chemical reactions take place on the molecular level, unmixedness results in reduced chemical reaction rates and increased reaction times. For these particular conditions, the maximum value of the turbulent scalar energy Q computed in the flow field was about 0.04. If we look at the amplificationfactor contour plot for this type of reaction (Fig. 4), we see that species fluctuations reduce the production term by about 10 per cent, This is a much smaller

fraction profiles x = 0.05 m

(a)

-

-

-

- -

o 50 I--

~

I

I

0,0200

/

0.0225

,(

0.0250 y, m

Fig. 9(a).

H20 --

---e--.

H2

......

02

--o---

.~

. . . . . . 02

0,0275

Case I

OH

0.0300

Case 2

Case 3

R . L . GAEFN£¥ et al.

132

M a s s fraction profiles (b)

x=0.1 m 1.00

,,. -

0.75

/

/¢

/

~/

I I

//,~' /

o.5o ! !

H2

02

- -

.2o

Case I

....... -- ~ ' - H2 -:. ::-: o2 H20

Case 2

--OH

/i

/ ,',;)' / ,; / ~. .

-

- -

.

.

.

/ ,;.

.

.....

o.

/?.,

.... /

0201 . . . . I . . . .

"",

/ ,',' / ~,. . . . . .

~ . ~ ,

--°--..2

o2 --;-.

i2o Case 3

--.--

o.

," \ / _ _ ~ " ,~" ~ . : " ~ - , .

I 0.001:':

': : ~ - ~ " " : - ~ ' ~ . - ' : 0.0200 0.0250 0.0300

-- ; ' : ' : ' 0.0350

y, m

M a s s fraction profiles x=0.2m

(c)

1.00 o o

0.75

O

H2

o2

H20

-

I wz "

o.50

,~/

~/ t'y

0.25

-m--.t--,a.~n.4 , "§~.

-

~

H2

--2221.20 c.se 2 ......

o2

OH

-2:221 o2 case3 H20

"~ /

0.00 :_ .: . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.010 0.020

-I---

Case 1

OH

- - " - - OH

0.030

,= : , : i 0.040

y, m

Fig. 9(b, c). Fig. 9. (a) Mass-fraction profiles of several species at the x = 0.05 m location. (b) Mass-fraction profiles of several species at the x = 0.1 m location. (c) Mass-fraction profiles of several species at the x = 0.2 m location.

c h a n g e in the m a g n i t u d e o f the p r o d u c t i o n t e r m t h a n t h a t p r o d u c e d by t e m p e r a t u r e fluctuations, w h i c h for these c o n d i t i o n s increased the size o f the p r o d u c t i o n t e r m by an o r d e r o f m a g n i t u d e . F i g u r e s 9(a)-(c) s h o w the profiles o f the m a j o r species at t h r e e s t r e a m w i s e

locations: x = 0.05, 0.1, a n d 0.2 m, respectively. W e c a n see t h a t at the x = 0.05 m location, the species d i s t r i b u t i o n s are the s a m e for the t h r e e cases. This result s h o w s t h a t the t u r b u l e n t c h e m i s t r y m o d e l s d o n o t affect the m i x i n g o f the species. A t the l o c a t i o n

Modeling turbulent, reacting flow where x = 0.1 m, cases 2 and 3 have significant water production; case 1 does not. The increase in water production is the result of temperature fluctuations that are modeled in cases 2 and 3, but not in case 1. The difference in the water production rate between case 2 and case 3 is the result of species fluctuations that reduce the production rate of case 2. By outflow (x = 0.2 m), all three cases show similar levels of water production. We conclude that the differences in the reaction time are caused by changes in the mean species production rate produced by the temperature and species fluctuations. CONCLUSIONS The following conclusions can be drawn from this work: • The series-expansion/assumed P D F method is a viable, alternative method for computing the mean reaction-rate coefficients for the hydrogen-air reaction mechanism of Table 1. The series-expansion/assumed P D F method requires considerably less computational effort to compute the mean reaction-rate coefficients than does the numerical integration of the product of the reaction-rate coefficients and the assumed P D F . • The magnitude of the species term (either forward or backward) in Eq. (20) caused by the reaction of two or three molecules of the same species is always increased by fluctuations in the mass fraction of that species. • The magnitude of the species term (either forward or backward) in Eq. (20) caused by the reaction of two molecules of different species is either increased or decreased by fluctuations in the mass fractions of species involved in the reaction, depending on the sign of the covariance of their mass fractions. The multivariate/~ P D F always predicts that species fluctuations will decrease this term. • Temperature and species fluctuations affect reaction time by changing the magnitude of the mean species production rate. The results of the parametric study should be qualitatively correct even if the true P D F for species is different from the assumed multivariate/~ P D F . These conclusions are based on assumptions that were made in the course of this study. Species fluctuations will affect the mean production rate [through the species terms in Eq. (20)] as presented in the parametric study. However, these are not the only effects of species fluctuations. Species fluctuations also affect the mean production rate through the cross-correlation terms. F o r chemical reactions in which temperature and species concentrations are strongly correlated, the effects of species fluctuations on the cross-correlation terms may be significantly larger than the effects of species fluctuations on the species terms in Eq. (22). In this case, the results of the numerical simulations may be different. Future work will address this concern by

133

modeling the most significant cross-correlation terms with traditional moment methods. REFERENCES

1. P.A. Libby and F. A. Williams, "'Fundamental aspects," Topics in Applied Physics--Turbulent Reacting Flow 44, 1-43 (1980). 2. R. W. Bilger, "Turbulent diffusion flames," Annual Review of Fluid Mechanics 21, 101-135 (1989). 3. S. B. Pope, "Computations of turbulent combustion: progress and challenges," 23rd Symposium (International) On Combustion (I 990). 4. S. B. Pope, "PDF methods for turbulent reactive flows," Prog. Energy Combustion Science 11, 119-192 (1985). 5. R. L. Gaffney, Jr, J. A. White, S. S. Girimaji and J. P. Drummond, "Modeling turbulence/chemistry interactions using assumed PDF methods," AIAA-92-3638, July 1992. 6. B. J. McBride, S. Heimel, J. G. Ehlers and S. Gordon, Thermodynamic Properties to 6000' K for 210 Substances Involving the First 18 Elements, NASA SP-3001 (1963). 7. C. Dopazo, "Probability density function approach for a turbulent axisymmetric heated jet. Centerline evolution," Physics of Fluids 18 (April 1975). 8. A. T. Hsi, Y.-L. P. Tsai and M. S. Raju, "A PDF approach for compressible turbulent reacting flows," AIAA-93-0087, January 1993. 9. E. E. Khalil, D. B. Spalding and J. H. Whitelaw, "'The calculation of local flow properties in two-dimensional furnaces," International Journal of Heat and Mass Transfer 18, 775-791 (1975). 10. F. C. Lockwood and A. S. Naguib, "The prediction of the fluctuations in the properties of free round-jet, turbulent diffusion flames," Combustion and Flame 24, 109-124 (1975). 11. P. Rhodes, "'A probability distribution function for turbulent flows." in Turbulent Mixing in Non-Reactive and Reactive Flows (edited by S. N. B. Murthy). Plenum Press, New York, 1975. 12. S. S, Girimaji, "Assumed/~-pdfmodel for turbulent mixing: validation and extension to multiple scalar mixing," Combustion Science and Technology 78, 177 196 (I 99 I). 13. R. Borghi, "Chemical reactions calculations in turbulent flows: application to a co-containing turbojet plume," Advances in Geophysics 18B, 349-365 (1974). 14. S. S. Girimaji, "A simple recipe for modeling reactionrates in flows with turbulent-combustion," AIAA 911792, June 1991. 15. J.S. Evans and C. J. Schexnayder, Jr. "Influence ofchemical kinetics and unmixedness on burning in supersonic hydrogen flames," AIAA Journal 18, 188-193 (1980). 16, C. G. Speziale and S. Sarkar, "Second-order closure models for supersonic turbulent flows," AIAA 91-0217, 1991. 17. S. Sarkar, G. Erlebacher, M. Y. Hussaini and H. O. Kreiss, "The analysis and modeling of dilatation terms in compressible turbulence," Journal of Fluid Mechanics 227, 473-493 (1991). 18. S. Sarkar, "'Modeling the pressure-dilatation correlation," ICASE Report No. 91-42, 1991. 19. R. L. Gaffney, Jr, "Derivation of a flux difference splitting scheme for mixtures of thermally perfect gases in turbulent flows," (in preparation). 20. J. A, White, J. J. Korte and R. L. Gaffney, Jr, "Fluxdifference split parabolized Navie~Stokes algorithm for nonequilibrium chemically reacting flows," AIAA 930534, January 1993. 21. J. P. Drummond, "'A two-dimensional numerical simulation of a supersonic, chemically reacting mixing layer," NASA TM-4055, 1988. 22. M. H. Carpenter, "Three-dimensional computations of cross-flow injection in a supersonic flow," AIAA 891870, June 1989.