NASA Contractor Report 175044 

Prediction of Heat Release 

Effects on a Mixing Layer 

Mohammed Farshchi 

Sverdrup Technology, Inc. 

Lewis Research Center 

Cleveland, Ohio 

February 1986 

Prepared for the 

Lewis Research Center 

Under Grant NAS 3-24105 

NI\SI\ 

National Aeronautics and 

Space Administration 

1111111111111 1111 11111 11111 11111 1111111111111 

NF01236 

NASA-CR-175044 

19860014386 

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PREDICTION OF HEAT RELEASE EFFECTS ON A MIXING LAYER 

Mohammed Farshchi 

Sverdrup Technology, Inc. 

Lewis Research Center 

Cleveland, Ohio 44135 

SUMMARY 

A fully second-order closure model for turbulent reacting flows is suggested based on Favre statistics. For diffusion flames the local thermodynamic 

state is related to a single conserved scalar. The properties of pressure 

fluctuations are analyzed for turbulent flows with fluctuating density. Closure models for pressure correlations are discussed and modeled transport 

equations for Reynolds stresses, turbulent kinetic energy dissipation, density- velocity correlations, scalar moments and dissipation are presented and solved, 

together with the mean equations for momentum and mixture fraction. Solutions 

of these equations are compared with the experimental data for high heat 

release free mixing layers of fluorine and hydrogen in a nitrogen diluent. 

INTRODUCTION 

A test case for the evaluation of a fully second order closure model of 

turbulent reacting flows with moderate to large heat release has been provided 

by the recent experiments of Hermanson et al. (ref. 1) and Mungal and Dimotakis 

(ref. 2). The dynamic field consists of a two-dimensional mixing layer with 

gas phase free streams carrying hydrogen in a nitrogen diluent in one stream 

and fluorine in a nitrogen diluent in the other. The reaction is 

and is highly exothermic. The experimental results with the highest heat 

release correspond to fluorine concentration of up to 6 percent and hydrogen 

concentration of up to 24 percent, with a maximum adiabatic flame temperature 

rise of 940 K. 

An attempt will be made to assess the possible improvement in the prediction of the above flow field by accounting for both velocity and scalar 

fluctuation time scales and by inclusion of the chemical heat release effects 

in the modeling of the Reynold stresses, scalar fluxes, and dissipation 

equations. 

Closure Model 

The closure model consists of three parts: 

(1) The thermo-chemical model based on the shifting equilibrium flame 

model (ref. 3). This model requires a fast rate of chemical reaction and 

relates the value of all thermodynanic variables to a conserved scalar. There 

are several scalar variables which are conserved in a chemical reaction and can 

be used as a basis for describing the mixing in a nonpremixed reacting flow. 

The mass fraction of a given element is such a variable. A normalized 
conserved scalar is referred to as the mixture fraction. Hence the density 

p, temperature T, and composition Yk are local functions of the mixture 

fraction f. 

(2) The turbulence or moment model consisting of a set of closed Favre 

averaged equations of order one and two, including the equations for all 

Reynolds stress components, scalar fluxes, dissipation rates of turbulence 

kinetic energy, and the mixture fraction variance. 

(3) The coupling model which relates the thermo-chemical model and the 

turbulence model through the probability density function, p.d.f., of the mixture fraction. In general, the form of p.d.f. will depend on the flow conditions and will be coupled with the chemical heat release. If constraints are 

imposed by solving the transport equations of the mean and the variance of the 

mixture fraction, the errors that can be made using an arbitrary form of the 

p.d.f. are limited. A Favre Beta function will be used here to represent the 

p.d.f. of the mixture fraction. The means and higher moments of the thermodynamic variables at any pOint may be obtained from their local relationship 

with the mixture fraction obtained from the thermo-chemical model and the 

p.d.f. of the mixture fraction. 

NOMENCLATURE 

o velocity divergence 

f mixture fraction f = 7+ f" 

fs stoichiometric value of the mixture fraction 

gi gravitational acceleration vector 

k turbulent kinetic energy 

p pressure, p = p + pi 

~ Favre probability density function 

R ratio of turbulence velocity to scalar time scales 

T absolute temperature 

Tadb adiabatic flame temperature 

Tamb ambient temperature 

U longitudinal velocity component 

U - " i velocity vector, Ui = Ui + u

i 

V transverse velocity component 

~ position vector 

mass fraction of the kth species in the mixture 

2 
yo.s 

r 

°t 

Ow 

°1 

f: 

f:f 

p 

T 

<I' 

-

( ) , ( 

( -) , ( 

( ) 1 ' ( 

) I 

) II 

)2 

transverse location of the point at wh1ch (U - U2)/{U, - U2) = 0.5 

d1ffus1v1ty (same for all spec1es) 

integral thickness 

vorticity or maximum slope thickness 

shear layer 1 percent thickness of the mean temperature profile 

turbulent k1net1c energy d1ss1pat1on rate 

scalar variance dissipation rate 

density of the mixture 

time scale 

equivalence ratio 

conventional averag1ng 

density weighted averag1ng 

high speed and low speed s1des 

TURBULENCE HODEL 

Second-order closure methods are believed to be the optimum level of 

closure and have been successfully applied to constant density flows, including buoyancy effects (refs. 4 to 6). The presence of the density fluctuations 

in the variable density or combusting flows results in far more complex equations for the second moments. Application of density weighted (Favre) averaging to variable density flows results in a set of equations which .are s1m1lar 

to those of constant density flows. This extends the use of the well developed 

constant dens1ty second-order closure models to the variable density case and 

makes available well tested model expressions for many terms in the moment 

equat10ns (ref. 7). This is, however, not sufficient for the complete closure 

of the density weighted moment equations, since new correlations involving 

density and divergence of velocity appear in the set. The exact equations show 

that the var1at1on of dens1ty 1s felt v1a three mechan1sms: (1) the var1at1on 

of the mean density and the correlation of density with velocity; (2) the nonzero divergence of velocity; and (3) the correlations involving pressure which 

are related to density via mass, energy, and state equations. These correlations appear in the transport equations of the Reynolds stresses and scalar 

fluxes and play-an important physical role. 

Pressure Equation 

The instantaneous pressure equation follows from mass and momentum balance 

as 

2 2 2 ~p = attp - aij (pU,Uj ) + g, a, P + a,jT1j ( 1 ) 

3 
where T1j denotes the Newton1an stress tensor. In contrast to the constant density case, the type of this equation for known velocity is hyperbolic. It describes the change of pressure due to hydromechanical mot10n and acoustic 

propagation of waves. The density term however contains not only acoustic 

variations of pressure but contributions which are due to mixing and chemical 

reaction. Expanding the first and second terms of above equation and ut1l1zing 

the 1nstantaneous mass and momentum equations, the above equation can be 

rewritten as, 

1 ~p = -paiujajUi + ; ajpa j P - pDtD (2) 

where the substantial derivative is 

and 

The last two terms in equation (2) represent the molecular viscous effects on 

pressure and are ignored in comparison to other terms in high Reynolds number 

flows. The second term in equation (2) results in an integral equation of the 

Fredholem type of second-order. It can be shown that the iterated kernals for 

this equation exist, thus a solution can be obtained. This solution is rather 

complicated, but it produces an integral form with a kernal modified by the 

density fluctuations. This term will have negligible affect at high Reynolds 

numbers due to the lack of correlation. The first term on the right-hand side 

of equation (2) is the well known hydrodynamic source of pressure change, see 

Chou (ref. 8). Th1s would be the only term appearing 1n the case of constant 

density flows. In the context of the thermochemical model for diffusion flames 

introduced above, density is a local function of the mixture fraction. It is 

however advantageous to consider p-l, specific volume, instead of p. 

Therefore 

The pressure equation can be recast in terms of mixture fraction, 

Ap ~ -p3\Uj 3j U\ - pOt (p dfp-10tf) (3) 

In pure mixing of two gases of different molecular weights the variation of the 

specific volume as a function of the mixture fract10n is close to l1near, with 

its second derivative equal to zero. However in diffusion flames this relation 

is strongly nonlinear near the stoichiometric value of the mixture fraction. 

To the left and right of this value p-l(f) is nearly linear with different 

slopes. If this local relation is simplified to partially linear with a jump 

in the slope at the stoichiometric value of the mixture fraction, then the 

terms involving the first derivative of the specific volume reflect the effect 

of mixing, with 

4 
and the terms involving the second derivative of the specific volume reflect 

the effect of chemical heat release, with 

(4 ) 

( 5) 

where H(f - fs) denotes the Heaviside function, ~(f - fs) the Dirac function 

as generalized derivative of H(f) and fs is the stoichiometric value of 

the mixture fraction. utilizing the s~ecific volume, the equation of conservation of mixture fraction and considering the limiting case of high Reynolds 

number and Ko1mogorov's hypothesis of local isotropy (refs. 9 and 10) the 

pressure equation for diffusion flames is then reduced to 

2 2 2 2 -1 6p ~ -paiUjajajUi + 2p E dff(ln p) dffP (6 ) 

where E = raifaif. Based on the above model and equation (5) the second 

term on the right hand side of equation (6) is recognized as the heat release 

contribution to the pressure equation. This term behaves like a Dirac delta 

function and its magnitude is determined by the location and movement of the 

instantaneous flame front, f(!,t) = fs' since only near the flame front it 

has a large contribution and is small everywhere else. This simplified pressure equation analysis indicates that the fluctuations of the pressure contain 

the contributions of mixing and reaction in addition to velocity fluctuations. 

Janicka and Lumley (ref. 11) have argued that the mixing part can be neglected 

at high Reynolds numbers in pure mixing. An order of magnitude analysis shows 

that in the case of chemical heat release the mixing contribution can also be 

neglected. 

Solution of the Poisson equation, (eq. (6», for pressure can be obtained 

by application of Green's theorem. Thus far from walls, where the surface 

integral may be neglected 

where 

and 

Second-Order Closure 

The complete set of equations constituting a closed system of Favreaveraged moments including all density variation effects will be discussed 

briefly. 

5 

(7) 

( 8) 
The mean veloclty, at hlgh Reynolds number, satlsfles 

aj(;u1 Uj) = -alP - al~u~) + ~gl (9) 

No closure assumption is requlred at this level of closure. 

The modeled Reynolds stress equation is given as 

( 

-k ~ (--u-H)~ '1--11 - --II -~ , 2 

+ a, CsP ; U,Umam\UiUj / +; p'uiajP + p'UjaiP~ + 0ij + 0ij ('0) 

The Kolmogorov assumption of local isotropy at high Reynolds numbers is used 

here, so that the dissipation tensor is isotropic with its trace equal to twice 

the turbulence kinetic energy dissipation, c. The diffusion model of Daly and 

Harlow (ref. '2) is chosen over the more complicated model suggested by 

Hanjalic and Launder (ref. '3) due to their equivalent performances in constant 

density flows. Consideration of the exact solution of the pressure equation 

g1ven by equat10ns (7) and (8), suggests the following decomposition of the 

veloclty-pressure gradlent correlation 

The first term on the right-hand slde of the above equation is denoted by Q~j and represents th~ correlatlon of the velocity fluctuations with the 

hyarodynamics part of the pressure fluctuation gradient. This would be the 

only term present for the incompressible flow case and contalns the pure 

turbulence interaction IIreturn to isotropy,1I and mean strain rate interaction 

IIfast response ll contributions. The closure model of Launder, Reece, and Rodi 

(ref. 4) is used for thls term. In density weighted form this term is given as 

(~ - ~ cS1jk) -

c

2 + 8 

(1I'ij - ~ cS 1j1l') 1 - C II II 

°1j = -c, p k uiUj 11 

8C 2 - 2 ( - ~ cS 1j1l') -

30C - 2 2 -

11 l1 ij 55 pk( a

j

u

1 + a

1

u

j ) 

t2 + 

4 --... 40C2 + '2 &1jk! II II 

;al Ul 11 u

1 uj - 55 (11 ) 

(~ ~) II II _ II II _ ~ II II 

where - 1I'ij = - u1u,al uj + uj ul a,u1 and 11' = .-u1 uj aj U1 

- (u~a}l + u~aliil) --II II 

t!ij and k = '12 ui ui · 

6 
,The term Q'j represents the heat release ~art of the pressurestra1n rate corre1at10n. Ins1ght 1ntomode11ng of th1s term can be ga1ned by . careful analys1s of the pressure equat10n's 'solut10n (eq. (8». Cons1der1ng 

just the heat release port10n of the pressure solut10n we have 

II 1 JJC ( , , 2 2 2 1) II 

-u, aj P2 = 2~ J ~ra,f a, f dff (In p) dffP- u, (!) 

x aj ( I! _1!, l)dX~ dX; dX; (12) 

Due to the D1rac delta funct10n behav10r of the second der1vat1ve of the 

spec1f1c volume near the st01ch10metr1c value of the m1xture fract10n, th1s 

correlat10n 1s determ1ned by the 10cat10n and movement of the 1nstantaneous 

flame front, f(!,t) = fs' 1.e., only near the flame front does the second 

der1vat1ve of the spec1f1c volume contr1bute to the pressure equat10n. There- . fore, the 1ntegral representat10n of the heat release contr1but10n 1nd1cates 

dependence on the probab111ty of the flame sheet be1ng present at a g1ven 

10cat10n. On the other hand, the probabi11ty of m1xture fract10n be1ng 1n an 

1nterval about the sto1ch10metr1c value, ~(fs;!)df, is zero for pure ox1d1zer, 

where f = 0, and grows qu1ckly w1th fuel concentrat10n. It reaches 1ts max1-

mum at the flame front reg10n and dec11nes to zero 1n the pure fuel reg10n, 

where f = 1. Th1s suggests that a1f(fs;!) 1s large and posit1ve on the 

entra1nment s1de of the flame and becomes small and negat1ve past the flame 

front. Cons1der1ng the d1mens10nal propert1es of equat10n (12)1t becomes 

clear th~tthere are two t1me scales affect1ng th1s term. One 1s the scalar 

fluctuat10n t1me scale def1ned by the scalar var1ance and its diSs1pat10n, the 

other one 1s the dynam1c or veloc1ty fluctuat10n t1me scale, wh1ch 1s related 

to the turbulence large eddy structures. If tensor1al propert1es and cons1stency w1th the 11m1ts of zero fluctuat10ns and pure m1x1ng w1thout react10n 

are requ1red, the follow1ng poss1ble closure model emerges: 

-112 where R = 2 cf/f k/c denotes the rat10 of turbulence t1me scale to scaler 

f1uctuat10ns t1me scale and Cf = raiflla,fll -- 1s the rate of d1ss1pat10n 

of the scalar var1ance. Begu1er, Dekeyser, and Launder (ref. 14) showed from 

a survey of several non1sothermal, nonreact1ng turbulent shear flows that the 

t1me scale rat10 was approx1mately equal to 2.0. The 1mp11ed constancy of R 

1s used 1n most work a1med at ca1culat1ng scalar var1ances. However based on 

the ev1dence to the contrary prov1ded by Warhaft and Lumley (ref. 15) an exact 

transport equat10n for the scalar d1ss1pat10n 1s der1ved and modeled, so that 

R can be determ1ned at each p01nt. 

The.constant CR1 can be est1mated as 

CRl ~ C* --' d

2 1n P 8~ ff 

where C* 1s a measure of correlat10n among the terms 1n the 1ntegral 

(eq. (12» and should have a value much less than one due to the degree of 

7 
stat1st1ca1 1ndependence 1mp11ed 1n the mode11ng of th1s term~ d~f 1n P 

1s a chem1ca11y dependent term that var1es from zero (for nonreact1ng cases) 

to the order of 100 for react10ns w1th large heat release. Therefore the value 

of CRl depends on the energy release 1n the flow f1eld. For the hydrogen - a1r flame of D1bble, Kollmann, and Schefer (ref. 16) a value of 1.25 1s 

recommended (ref. 10). 

The rate of turbulence k1net1c energy d1ss1pat10n determ1nes the dynam1c 

length and t1me scales and sat1sf1es the follow1ng modeled equation 

(14 ) 

where the last term conta1ns the effect of chemical heat release on d1ss1pat1on. 

This complex process 1s tentatively taken 1nto account by analogy with the 

correspond1ng term (eq. (13» for the normal stresses. 

Dens1ty - Ve10c1ty correlat1ons appear both d1rect1y and through the 

modeled terms 1n the Reynolds stress transport equat10n (10). Derivation of 

exact equations and the closure of the equat10ns are d1scussed 1n deta11 by 

Farshch1 (ref. 10). The modeled transport equations for these correlations 1s 

g1ven by 

- II _ 

+ C2RUPP' uj aj U1 ( 15) 

T 1s the turbulent t1me scale, wh1ch 1s trad1t10nally set equal to the 

turbulence time scale k/&. However 1t can also be the t1me scal~ scalar 

fluctuat10ns or a comb1nat10n of the two t1me scales (ref. 17). p'2 is the 

var'ance of density and 1s obta1ned d1rectly us1ng the thermo-chem1cal model 

and the appropr1ate we1ghted 1ntegrat1on of the m1xture fract10n p.d.f. 

The transport equat10n of the m1xture fract10n's mean value requ1res no 

mode11ng and 1s g1ven as 

B 
al <;;TIl f) = -al (pu~) 

where the molecular diffusion effects are neglected in comparison with the 

turbulence diffusion at high Reynolds numbers. 

--- Next a transport equation for the scalar flux vector. U1f". is derived 

(16 ) 

and modeled (ref. 6). The scalar pressure ~radient terms (pressure scrambling) 

are modeled using the same line of argument'used for velocity-pressure gradient 

correlations in the Reynolds stress equation. The modeled form of this equation is given as 

al (;rrjY-') = -! ;uf=a/il + ;u-gal! + aj (Cc T;u~alu~) 

~)1/2 2 e lfll 

+ CRFk f R _ ai(Pf(fs;~» p 

(17 ) 

plfll appearing in the above equation can be obtained from the p.d.f. of the 

mixture fraction. 

Mixture fraction variance appears in the Beta function representing the 

probability density of the mixture fraction. For high Reynolds number flows 

the modeled mixture fraction variance equation is expressed as 

( 18) 

The variance of the mixture fraction is dissipated by molecular action at small 

scales. This dissipation rate is also the key to determination of scalar 

fluctuation time scale. The exact form of the scalar dissipation equation and 

its modeling is discussed in details by Farshchi et al. (ref. 18). However. 

regrouping some of the exact terms and applying an order of magnitude analysis 

-1 2 and ignoring terms of order Ret and less. where Re = k /u& is the turbulence Reynolds number. Farshchi (ref. 10) shows thatfo~ high heat release cases 

there is a source along with a sink of the scalar dissipation. The chemical 

source term for diffusion flames is given by 

d2 p.( ra fila fll)2 ff i 1 

with cp3 = 1.0. The completely modeled scalar dissipation equation is then 

given by: 

9 
2 

COl; ~ - CO2;cf ~ - C03 ; (d f P)2 c~ f 

( 19) 

All constants used in above equations are summarized in table I. 

APPLICATION AND COMPARISON WITH MEASUREMENTS 

Hermanson's (ref. 19) experimental studies of the effect of heat release 

in a planar, gaseous reacting mixing layer formed between free streams containing hydrogen and fluorine in nitrogen diluents are used for comparison. The 

density ratio of the free streams is kept equal to one by using as diluent a 

mixture of nitrogen and a small amount of helium, on the fluorine side, and a 

mixture of nitrogen with small amount of argon on the hydrogen side. The 

nominal high speed, hydrogen containing side, flow velocity is 22 mls and a 

free-stream speed ratio of U2/Ul = 0.4 is maintained. The high speed 

free-stream turbulence level was measured to be about 2/3 percent. The measuring station was positioned 0.457 m downstream of the splitter plate trailing 

edge. The Reynolds number at the measuring station based on the high speed 

free-stream velocity, the downstream distance and the cold free-stream kinematic viscosity was Rex = 6xl05. The Reynolds number based on high speed 

boundary layer momentum thickness just upstream of the trailing edge was 

estimated to be about 240. 

Two cases of moderate and high heat release are considered, corresponding 

to two equivalence ratios (ref. 2): 

(1) ~ = 1, corresponding to fluorine concentration of 6 percent and 

hydrogen concentration of 6 percent, with a maximum adiabatic flame temperature 

of 860 K. 

(2) ~ = 1/4, corresponding to fluorine concentration of 6 percent and 

hydrogen concentration of 24 percent, with a maximum adiabatic flame temperature of 1240 K. Thermodynamic tables relating density and temperature to the 

mixture fraction are obtained using CEC83 (ref. 20) computer program .. 

Numerical Method 

The numerical method used for this calculation is a modified version of 

the Patankar and Spalding method (ref. 21) where all the first order moments 

are defined on node points, and all higher order moments are defined at 

midpoints. The given initial velocity profiles have Blasius form and agree 

with the boundary layer momentum thickness value repBrted by the experiment. 

The initial dissipation rate, c, and the shear stress, ullv", are determined 

10 
ut1l1z1ng the turbulence equ1l1br1um assumpt10n, a slmple grad1ent flux model 

w1th molecular k1nemat1c v1scos1ty, and the ass1gned veloc1ty prof1le. Next 

the well known turbulence v1scos1ty model, 

wlth lamlnar vlscos1ty 1s used to determine 1n1t1al turbulent k1net1c energy 

prof1le. The normal stresses are equi-partitioned. The in1tial mean m1xture 

fract10n has a step prof1le and all other turbulent correlations are set to 

zero. The 10ngitud1nal mean pressure grad1ent must be prescribed analytically 

1n terms of the grad1ent of the outer boundaries velocities and 1s zero here. 

The transverse pressure grad1ent cannot be prescribed and is approx1mated by a 

slmpl1fied vers10n of the mean transverse momentum equat10n as 

- --"2 

a p = -a (pv ) y y 

The upper bound of the marching step of the maln procedure ls determlned at 

each forward pos1t1on w1th respect to the value of a character1st1c th1ckness 

of the m1xfng layer. To solve a typlcal problem w1th thls 13 equation turbulence model over a d1stance of 0.9 m with 80 node po1nt in the cross d1rection, 

the t1me needed on an IBM 370 1s about 30 m1n CPU. The p.d.f. 1ntegrat10n for 

the calculat10n of mean dens1ty, temperature, and other m1xture fract10n 

moments at each node takes 2/3 of above time. 

D1scuss10n of the Results 

The 1n1t1al predict10n of the homogeneous cold flow m1x1ng layer resulted 

1n a m1x1ng layer th1ckness 35 percent below 1ts experimental value at the 

ax1al measur1ng stat10n. To val1date the model constants and the method used 

here and due to the lack of veloc1ty correlat10ns measurements by Hermanson 

(ref. 19) 1t was dec1ded to use Browand and Lat1go's (ref. 22) experimental 

study. Two measures of th1ckness of the turbulent m1x1ng reg10n are employed. 

Flrst is the 1ntegral thickness defined as 

Second 1s the vort1c1ty thickness defined as 

( AU) 

6W = (ay U)max· 

F1gure 1 shows that there is about 16 percent d1fference between the experi- mental and calculated values of the. normalized integral thickness. The ax1al 

rate of change of vort1c1ty th1ckness 1s exper1mentally estimated to be between 

0.15 and 0.17 ~, where ~ = (Ul - U2)/(Ul + U2)' The calculated value, dx6w = 

11 
0.148 X, is at most 13 percent less than the experimental value. Comparison of normalized mean velocity and turbulence shear stress are presented in figures 2, 3, and 4. They show good agreement with experiment except near the 

high speed side where the calcul'at1on shows a faster approach to the outer 

values. 

Phys1cally the spread of the m1x1ng layer downstream of the splitter plate 

is determined by two different mechanisms. The spreading rate of the mixing 

layer at dynamical equ1l1br1um stage is one mechanism. This is reasonably well 

predicted by the calculation. The other and more crucial mechanism, which is 

not predicted by the calculation, involves the transition from attached boundary layers on the splitter plate to free mixing layer developing downstream of 

it. If the boundary layers are laminar then there is also a transition from 

laminar to turbulent flow in the developing region of the mixing layer. Birch 

(ref. 23) has attempted to model the developing region of the mixing layer by 

a multi-length scale gradient flux turbulence model. However, present calculations indicate that a single-length scale second-order model along with substantially reduced value of the initial turbulent kinetic energy dissipation 

would result in an increase of the spreading in the initial developing region. 

Even though the final spreading rate in the similarity region is the same as 

before, the thickness of the mixing layer is larger by up to 25 percent. The 

major short com1ng of th1s treatment is that the amount of initial reduction 

of & is not the same for all flow conditions and is a function of velocity 

ratio and character of the boundary layers on the splitter plate. Since in the 

study of heat release effects only relative values of the spreading are of 

importance the basic model without any modification can be used. 

To examine the contr1but1on of heat release terms 1ntroduced by pressure 

fluctuation correlations and those appear1ng in the scalar dissipation equation 

two sets of model constants are examined: 

(I) Constants CR1, CR2, CRF, C03, and Cp3 are all set to zero 

(II) The values given in table I are used for above constants 

The rest of the constants are kept the same in both cases. 

The comparison of mean axial velocities (fig. 5), and turbulent shear 

stresses (fig. 6), obtained for the large heat release case, ~ = 1/4, utilizing models (I) and (II) indicate that there is a little difference between the 

two models and the improvement obtained by the application of model (II) is 

almost negligible. This means that the heat release portion of the pressure 

fluctuation correlations has little effect on the turbulent velocity f1eld. 

This result can also be shown by the fact that at each level of heat release, 

models (I) and (II) predict very close values for the spreading rate of the 

mixing layer. The spreading rate, however, decreases with the heat release. 

Spreading rate is defined by d(YO.l - YO.9)/dx, i.e., the rate of spread 

of the distance between the points at which U equals O.l(Ul - U2) + U2 

and 0.9(Ul - U2) + U2. Table II summarizes the spreading rates at different 

levels of heat release. 

The moderate heat release case causes about 10 percent reduction in the 

spreading rate, however the increase in the heat release has only produced 

about 3 percent further reduction. The spreading rate levels off as the heat 

release is increased. This leveling corresponds with the leveling of the mean 

12 
density reduction reported by Hermanson (ref. 19) and leads to the conclusion 

that probably the single most important cause of the spreading rate reduction 

with heat release is the reduction of density and the subsequent reduction of 

the turbulent shear stress, _pU~I. Turbulent shear stresses normalized by 

free-stream density and velocity difference are compared in figure 7. There 

is a considerable reduction of shear stress between the cold and moderate heat 

release cases. Further increase of the heat release, however, results in a 

small reduction in the value of shear stress. 

Hermanson (ref. 19) reported a monotonic decrease of the vorticity thickness, 6w, with heat release without any influence of the equivalence ratio 

on this trend. The present equilibrium flame formulation of the chemical 

reaction is however dependent on the stoichiometric value of the mixture fraction and therefore on the equivalence iatio,~. For 6 percent fluorine and ~ = 1, the stoichiometric value of the mixture fraction is about 0.5 and for ~ = 1/4 it is about 0.2. It is in the neighborhood of this value that the 

mixture density reaches its minimum value. Figures 8 and 9 show the differences in the density variance distributions and the mean density minimum value 

10cat10n for the two cases of ~ = 1 and 1/4. The movement of the mean density minimum value causes the shift in the profile of turbulent shear stress 

and the location of its peak value as displayed in figure 7. Since the velocity profile is dependent on the gradient of the turbulent shear stress and not 

its absolute magnitude, the vorticity thickness predicted for the case of moderate heat release, ~ = 1, is smaller than the value predicted for the high 

heat release case, ~ = 1/4. Hermanson's (ref. 14) experimental results for 

6 percent fluorene and ~ = 1 also show some scatter in vorticity thickness 

measurements with some values that are smaller than those measured for the 

higher heat release case, ~ = 1/4. These values are compared with calculated 

results 'n table III. 

Considering the general trend of the experimental vorticity thickness 

behavior and assuming that above scatter is within acceptable measurement 

accuracy bounds, one concludes that the above discrepancy between the predicted 

and measured results and the dependence on the stoich10metric value of the 

mixture fraction can be resolved by accounting for intermittency effects by 

such methods used by Kent and Bilger (ref. 24). Byggstoy1 and Kollmann 

(ref. 25) have presented a model for the prediction of intermittency and 

initial exploratory calculations with such models point to better predictions. 

Figure 10 compares the predictions of model (I) and (II) for the ratio of 

scalar to velocity field time scales, indic~ting that the influence of modifications included in model (II) is larger on the scalar field than the velocity 

field. Even though the difference in density predictions (fig. 11), is not 

large, the density variance and the mixture fraction variance predictions 

(figs. 12 and 13), show as much as 36 percent jump in their values with model 

(II) near the stoichiometric value of the mixture fraction. This change in the 

magnitude of the mixture fraction variance causes more than 10 percent reduct10n 1n the peak value of the pred1cted temperature which corresponds very well 

with the experimental measurement, as shown in figure 14. The transverse 

coordinate in this figure is normalized by shear layer 1 percent thickness, 

which is defined as the transverse width of the layer at which the mean temperature is 1 percent of the maximum mean temperature rise. The shift between 

the predicted and measured temperature profiles is symmetric on both sides and 

is expected to be a consequence of a fixed horizontal high speed upper side 

wall and diverging low speed lower side wall in the experiment. 

13 
Compar1son w1th the 11m1ted exper1menta1 results ava11ab1e 1nd1cate that 

the mode11ng approach used here 1s a step 1n the r1ght d1rect10n. However, 

comprehens1ve measurements of the scalar f1e1d quant1t1es, such as m1xture 

fract10n and dens1ty var1ances and sca1ar-ve10c1ty corre1at10ns, are needed to 

exam1ne and fully evaluate the contr1but10n of the model presented here. 

CONCLUSIONS 

A fully second-order closure model for react1ng turbulent flows have been 

extended to 1nclude the effects of heat release on he turbulence flow f1eld. 

Th1s was done by ana1yz1ng the exact pressure equat10n of a turbulent nonprem1xed flame. The scalar f1eld t1me scale has also been d1rectly calculated by 

develop1ng an equat10n for the d1ss1pat1on of scalar var1ance wh1ch also 

1nc1udes the heat release effects. Two models, (I) not 1nc1ud1ng heat release 

terms and (II) 1nc1ud1ng the heat release terms were compared and 1nd1cate 

that: (a) The ve10c1ty f1e1d pred1ct10ns are very s1m11ar 1n both models, 

correctly pred1ct1ng the reduct10n 1n m1x1ng layer spread rate w1th heat 

release. The scalar f1e1d results of model (II) are 1n better agreement w1th 

the exper1mental data y1eld1ng a 10 percent 1mprovement 1n temperature pred1ct10ns. (b) The t1me scale rat10 1s sens1t1ve to dens1ty f1uctuat10ns. Th1s 

p01nts to a correct 1nc1us10n of heat release effects1n the equat10n of scalar 

var1ance d1ss1pat10n. Nevertheless, 1t would be premature to draw any conc1us10n about un1versa11ty of such a'c10sure and the values of the constants used 

here. Further exper1ments are needed to support or to 1mprove present 

assumpt10ns and mode11ng. 

REFERENCES 

1. Hermanson, J.C., Munga1, M.G., and D1motak1s, P.E., "Heat Release Effects 

on Sheat Layer Growth and Entra1nment," AIAA Paper 85-0142, Jan. 1985. 

2. Munga1, M.G. and D1motak1s, P.E., "Mixing and Combustion With Low Heat 

Release 1n a Turbulent Shear Layer," Journal of Flu1d Mechan1cs, Vol. 148, 

Nov. 1984, pp. 349-382. 

3. B11ger, R.W., "Turbulent Flows W1th Nonprem1xed Reactants," Turbulent 

React1ng Flows, P.A. L1bby and F.A. W1111ams, eds., Spr1nger, pp. 65-113. 

4. Launder, B.E., Reece, G.J., and Rod1, W., "Progress 1n the Development of 

a Reynolds-Stress Turbulence D1sc10sure," Journal of F1u1d Mechan1cs, 

Vol. 68, Part 3, pp. 537-566. 

5. Lumley, J.L., Pred1ct10n Methods for Turbulent Flows: Introduct10n, Von 

Karman Inst1tute Lecture Ser1es 76, Rhode st. Genese, Belg1um, 1978. 

6. Launder, B.E., "Heat and Mass Transport," Turbulence, P. Bradshaw, ed., 

Spr1nger, Ber11n, 1976, pp. 232-287. 

14 
7. Jones. W.P. "Models for Turbulent Flows W1th Var1able Dens1ty and 

Combust10n." Pred1ct10n Methods for Turbulent Flows. W. Kollman. ed., 

Hem1sphere, Wash1ngton, D.C., 1980, pp. 379-421. 

8. Chou. P.Y., "On Veloc1ty Correlat10ns and the Solut10ns of the Equat10ns 

of Turbulent Fluctuat10n." Quarterly of App11ed Mathemat1cs. Vol. 3, 1945. 

pp. 38-54. 

9. Kollmann, W., "Mass Transport 1n Turbulent Flames," Paper Ms-7250. AIChE 

Meet1ng.Los Angeles. CA, 1982. 

10. Farshch1. M., "Second-Order Closure for Turbulent Flames," Ph.D. Thes1s. 

Un1vers1ty of Ca11forn1a, Dav1s, 1984. 

11. Jan1cka. J. and Lumley. J.L .• "Second-Order Mode11ng 1n Non-Constant 

Dens1ty Flows," Report FDA 81-01. Cornell Un1v. Jan. 1981. 

12. Daly. B.J. and Harlow. F.H .• "Transport Equat10ns 1n Turbulence." Phys1cs of F1u1ds, Vol. 13, No. 11, Nov. 1970, pp. 2634-2649. 

13. Hanja11c, K. and Launder, B.E., "A Reynolds Stress Model of Turbulence and 

1ts App11cat10n to Th1n Shear Flows." Journal of Flu1d Mechan1cs, Vol. 52. 

Part 4, 1972, pp. 609-638. 

14. Bequ1er, C., Dekeyser, I. and Launder. B.E., "Rat10 of Scalar and Velocity 

D1ss1pat10n T1me Scales 1n Shear Flow Turbulence," Phy~1cs of Flu1ds. 

Vol. 21, No.3, Mar. 1978, pp. 307-310. 

15. Warhaft. Z. and Lumley, J.L., "An Exper1mental Study of the Decay of 

Temperature Fluctuat10ns 1n Gr1d-Generated Turbulence," Journal of F1u1d 

Mechan1cs. Vol. 88. Part 4, 1978. pp. 659-684. 

16. D1bb1e, R.W., Kollmann, W. and Schefer, R.W .• "Conserved Scalar Fluxes 

Measured 1n a Turbulent Nonprem1xed Flame by Comb1ned Laser Doppler 

Ve1oc1metry and Laser Raman Scatter1ng," Combust10n and Flame, Vol. 55, 

No.3, Mar. pp. 307-321. 

17. E1ghobash1, S.E. and Launder, B.E. "Turbu1ent T1me Scales and the 

D1ss1pat10n Rate of Temperature Var1ance 1n the Thermal M1x1ng Layer," 

Phys1cs of F1u1ds, Vol. 26, No.9. Sept. 1983, pp. 2415-2419. 

18. Farshch1. M .• Kollmann, W., D1bb1e, R.W. and Schefer, R.W., "Second-Order 

Closure for Turbulent Nonprem1xed Flames," F1fth Sympos1um on Turbulent 

Shear Flows, Cornell Un1vers1ty. Ithaca. NY, 1985, pp. 10.19-10.25. 

19. Hermanson, J.C .• "Heat Release Effects 1n a Turbulent. React1ng Shear 

Layer," Ph.D. Thes1s. Ca11forn1a Inst1tute of Technology, 1985. 

20. Gordon. S. and McBr1de, B.J., "Computer Program for Ca1cu1at1on of Complex 

Chem1ca1 Equ111br1um Compos1t10ns, Rocket Performance, Inc1dent and 

Reflected Shocks, and Chapman-Jouquet Detonat1ons," NASA SP-273, 1971. 

21. Patankar, S.V. and Spald1ng, D.B .• Heat and mass transfer 1n boundary 

layers. Intertext, London, 1970. 

15 
22. Browand, F.K. and Lat1go, ,B.D., IIGrowth of the Two-D1mens10nal M1x1ng 

Layer From a Turbulent:and Nonturbulent Boundary Layer," Phys1cs of 

Flu1ds, Vol. 22, No.6, June 1979, pp. 1011-1019. 

23. B1rch, S.F., "Mult1-Length Scale Turbulence Models," AFOSR-84-0249TR, 

1983. (AD-A140527) 

24. Kent, J.H. and B'1ger, R.W., "The Pred1ct10n of Turbulent D1ffus10n Flame 

Fields and Nitric Ox1de. Formation," 16th Sympos1um (International) on 

Combustion, The Combustion Inst1tute" Pittsburgh, PA, 1977, pp. 1643-1656. 

25. Byggstoyl, S. and Kollmann, W., "A Closure Model for Coniditioned stress 

Equations and its App11cation to Turbulent Shear Flows," (subm1tted to 

Physics of Fluids) 1986. 

16 
TABLE I. - MODEL CONSTANTS 

C

i C

1 C2 CR1 Ce Ce1 Ce2 

0.2 1.6 0.43 1.25 0.15 1.45 1.90 

C,3 CR2 

1.0 1.25 

C1RU C2RU CRF CRUD 

4.0 0.5 1.25 1.0 

C

c 

C1C 

0.18 3.2 

C2C CC2 COl CO2 CD3 

0.51.182.20.81.0 

TABLE II. - SPREADING RATES OF THE MIXING 

LAYER AT DIFFERENT LEVELS OF 

HEAT RELEASE 

Heat release level d(YO.1 - YO.9)/dx 

Cold flow 0.0562 

Moderate heat release .0505 (~ = 1. Tadb = 860 K) High heat release .0488 (~ = 1.4. Tadb = 1240 K) 

TABLE 111.- COMPARISON OF NORMALIZED VORTICITY 

THICKNESS ~w/~wo WITH MEASUREMENTS 

[~wo is the cold flow vorticity thickness.] 

Experimental (ref. 14) Present 

measurements calculation 

~ = 1 moderate 0.7645 0.8814 

heat release .8066 .8814 

.9067 .8814 

~ = 1/4 high .8826 .9627 

heat release 
40 

30 

..... 

~ 20 10 

10 

o 

1.0 

,8 

N ,6 ;::) 

I 

...... 

.2 

N 

;::) 

.4 I 

I;::) 

.2 

0 

o 

o 

400 

o 

00 

o 

o 

o 

o 

o 

o 

o 

BROWAND & LATlGO, 1979 

PRES ENT CA LCU LA TlON 

800 1200 1600 2000 

x/6ti 

Figure 1. - Normalized local integral thickness as a function 

of downstream distance for X = 0,695 and li ti = 0.46 mm. 

0 

0 BROWAND & LATlGO, 1979 

PRESENT CALCULATION 

-1.2 -.8 -.4 0 .4 .8 1.2 

(Y-Y.5)IIY.I-Y.9 1 

Figure 2. - Longitudinal mean velocity. 
.012 

N 

o BROWAND & LATlGO, 1979 

PRESENT CALCULATION 

N 

::> .008 I 

.-f 

2 

-

!

> 

-::::l .004 

N 

::> 

I 

.-f 

::> 

-N 

::> 

I 

I::> 

O~--~~--~--~----L---~~~ 

-1. 2 -• 8 -.4 0 . 4 . 8 1. 2 

1.0 

.8 

.6 

.4 

.2 

(y-y. 5)/ IY.1-Y. 91 

Figure 3.:- Distribution of shear stress. 

o HERMANSON, 1985 

PRESENT CALCULATION 

o~ __ ~~ ____ ~ ____ ~ ____ ~ ____ ~ 

-1.2 -.4 0 .8 

(y-y .5)/ Iy.- Cy• 91. 

Figure 4. - Longitudi nal mean velocity, cold flow. 
1.0 

.8 

~ .6 I 

::2 -

N 

I~ .4 

N_ 

::::> 

<I 

-0 

~ 

~~ I a. 

I 

.2 

o HERMANSON, 1985 

CALCULATION, MODEL II 

----- CALCULATION, MODEL I 

OL-__ ~L-~~ __ ~ __ ~ ____ ~ __ ~ 

-1.2 -.8 -.4 0 .4 .8 1.2 

• 0040 

.0030 

.0020 

.0010 

(y-y. 5)/ I y. r Y• 91 

Figure 5. - Comparison of velocity prediction of models 

I and II with the experiment for the high heat release 

case, ({J = 1/4 . 

11.2 -.8 -.4 0 .4 .8 

(Y-Y. 5)/1 y. rY.9 1 

Figure 6. - Comparison of turbulent shear stress prediction of model I and II for high heat release case, ({J = 1/4. 
.012 

N......,. 

::J .008 

:::! 

cE 

-

10.. 

a 

N I.eN 

Ie.. 

.004 

. 250 

.200 

.150 

.100 

.050 

COLD FLOW 

MODRATE HEAT RELEASE, q; = 1 

HIGH HEAT RELEASE, q; = 1/4 

\ 

Figure 7. - Comparison of turbulent shear stress for 

different heat release values, using model I • 

____ HIGH HEAT RELEASE, q; = 1/4 

MODRATE HEAT RELEASE, q; = 1 

'" I \ 

I \ 

I \ 

I \ 

I \ 

I \ 

I \ 

I \ 

I \ 

I \ 

I \ 

I \ 

I \ I 

I 

I 

I 

I 

~1~.~2--~~----~----~----~----~ 

Figure 8. - Comparison of density variances for different 

equivalence ratios, using model I. 
1.6 

1.2 

10. .8 

.4 

---- HIGH HEAT RELEASE, q; = 1/4 

MODRATE HEAT RELEASE, q; = 1 

\ 

\ / 

\ / 

\ / 

\ / 

\ ",/' 

, " " ,------....-.,,'" 

~L.2-----L-----L~---L-----L~--~ 

.8 

.6 

e: .4 ..... 

.2 

Figure 9. - Comparison of mean densities for different 

equivalence ratios, using model I. 

".---...... 

/" " 

/ / " '\ 

/ \ / \ / \ 

/ MODEL II ~ 

I ---- MODEL I 

Figure 10. - Ratio of scalar to velocity time scales for high 

heat release case, q; = 114. 
10. 

1.0 

.8 

.6 

.4 

MODEL II 

.2 ---- MODEL I 

o~ ____ ~ ____ ~ ____ ~ ____ ~ ____ ~ 

-1. 2 -.8 -.4 0 .4 

• 10 

.08 

.06 

.04 

.02 

(y-y. 5)/ Iy. rY.9 1 

Figure 11. - Comparison of density predictions for high 

heat release case, q; = 114 • 

MODEL II 

--- MODELl 

.8 

O.L-~~~ ____ ~~ __ ~ ____ ~ ____ ~ 

-1. 2 

Figure 12. - Comparison of density variance predictions 

for high heat release case, q; = 114. 
.c 

E 

l-'" I 

.c 

"0 

1-'" 

.c 

E 

l-'" I 

II-

.050 

.040 

.030 

.020 

.010 MODEL II 

MODEL I 

OL---~~-----L----~----~----~ 

-1.2 -.4 0 .8 

.8 

.6 

.4 

.2 

(y-y. 5)/ I Y.1-Y• 91 

Figure 13. - Mixture fraction variance for high heat 

release case, lp = 114. 

o HERMANSON, 1985 

CALCULATION, MODEL II 

----- CALCULATION, MODEL I 

~.~8~~~~--~~--~~--~ o .4 .8 

(Y-Y.5)161 

Figure 14. - Temperature comparison for high 

heat release case, rp = 114. 
1. Report No. 2. Government Accession No. 

NASA CR-175044 

4. Title and Subtitle 

Prediction of Heat Release Effects on a 

Mixing Layer 

7. Author(s) 

Mohammed Farshchi 

9. Performing Organization Name and Address 

Sverdrup Technology, Inc. 

Lewis Research Center 

Cleveland, Ohio 44135 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546 

15. Supplementary Notes 

3. Recipient's Catalog No. 

5. Report Date 

February 1986 

6. Performing Organization Code 

8. Performing Organization Report No. 

None 

10. Work Unit No. 

11. Contract or Grant No. 

NAS 3-24105 

13. Type of Report and Period Covered 

Contractor Report 

14. Sponsoring Agency Code 

505-62-21 (E-2845) 

Final report. Project Manager, Raymond E. Gaugler, Internal Fluid Mechanics 

Division, NASA Lewis Research Center, Cleveland, Ohio 44135. Prepared for the 

24th Aerospace Science Meeting, sponsored by the American Institute of 

Aeronautics and Astronautics, Reno, Nevada, January 6-9, 1986. 

16. Abstract 

A fully second-order closure model for turbulent reacting flows is suggested 

based on Favre statistics. For diffusion flames the local thermodynamic state 

is related to a single conserved scalar. The properties of pressure fluctuations 

are analyzed for turbulent flows with fluctuating density. Closure models for 

pressure correlations are discussed and modeled transport equations for Reynolds 

stresses, turbulent kinetic energy dissipation, density-velocity correlations, 

scalar moments and dissipation are presented and solved, together with the mean 

equations for momentum and mixture fraction. Solutions of these equations are 

compared with the experimental data for high heat release free mixing layers of 

fluorine and hydrogen in a nitrogen diluent. 

17. Key Words (Suggested by Author(s» 

Turbulence modeling; Diffusion flame; 

Free mixing layer 

18. Distribution Statement 

Unclassified - unlimited 

STAR Category 34 

19. Security Classlf. (of this report) 

Unclassified 

20. Security Classlf. (of this page) 

Unclassified 

21. No. of pages 

• For sale by the National Technical Information Service, Springfield, Virginia 22161 

22. Price· 
End of Document