NASA Contractor Report 177556

Extra Compressibility Terms for FavreAveraged Two-Equation Models of

Inhomogeneous Turbulent Flows

Morris W. Rubesin

MCAT Institute

Ames Research Center

Moffett Field, California

Prepared for

Ames Research Center

CONTRACT NCC2-585

June 1990

National Aeronautics and

Space Administration

Ames Research Center

Moffett Field, California 94035-1000

EXTRA COMPRESSIBILITY TERMS FOR FAVRE-AVERAGED

TWO-EQUATION MODELS OF INHOMOGENEOUS

TURBULENT FLOWS

Morris W. Rubesin

MCAT Institute

NASA Ames Research Center

_{ofl'elt Field, CA 94035

ABSTRACT

Forms of extra compressibility terms that result from use of Favre averaging of the

turbulence transporl equations for kinetic energy and dissipalion are derived. These forms

introduce three new modeling constants, a polytropic coefficient that defines the interelationships of the pressure, density, and enthalpy fluctuations and two constants in the

dissipation equation that account for the non-zero pressure-dilitation and mean pressure

gradients.

NOMENCLATURE

a = n_ean velocity of .sound

c_ -- modeling coefficient, Reynolds heat flux, Eq. (13) [0.351

C_ = modeling coefficient, eddy viscosity [0.09 i

h = enthalpy

H = total eT_thalpy, Eq. (5) or Eq. (9)

k = turbulence k'iT_etie eT_ergy

1,,"= (u'i' "i )/2

i,,= ? l/

M_ - free - st'rear_ 3tach number

n =polytropic coefficient, Eq. (10)

p = pressure

q) = n_olecular heat flux in the jth direction,-(fi/Pr)hj

Q, = Reynolds heat flux in ith direction

S,y = strain rate tensor, 1/2(vi,j + ua,i - (2/3)5,.juk,_.)

T_, = surface ter1_peratur_

To_ - free - strean_ stagnatioT_ temperature

ORIGINAL PAGE IS

OF POOR QUALITY
Tij = total stress tensor, viscous plus Reynolds stress

uj = velocity in jth direction

/3 = fluctuating density variance, Vf-_/_

_,j = Kronecker delta

= dissipation rate of kinetic energy

la = molecular viscosity

p = density

rij = viscous stress, 2ftS_j

A = ensemble mean variable

A' = fluctuating variable

.7t = mass weighted ensemble mean variable

A" = fluctuating variable in mass weighted expansion

A a = partial derivative with respect to time

A, = partial derivative with respect to the ith coordinate

INTRODUCTION

Favre, or mass-weighted, averaging, Ref. 1, has become the most popular approach

for establishing the field equations used in evaluating compressible turbulent flows. One

reason for this is that the equations for mass, momentum and energy conservation retain

the relative simplicity, term for term, of their incompressible counterparts, Ref 2. The effects of density fluctuations are submerged within the definitions of the Reynolds stresses

and heal or mass fluxes and the profusion of new terms that occurs when primitive variables are employed, e.g. Ref. 3, is avoided. The effects of density fluctuations, however,

become reintroduced into the problem when use is made of field equations representing

the turbulence moments and scale. These equations contain additional terms unique to

compressible flows that involve density fluctuations and the non-zero divergence of the

turbulent fluctuating velocities. The purpose of this paper is to reexamine the modeling

these compressibility terms, with emphasis being given to the terms that appear in twoequation models of turbulence. Of course, similar forms of these compressibility terms can

be applied to second-order closure models as well.

The possible importance of these compressibility terms was first suggested in the paper

by Wilcox and Alber, Ref. 4. These authors modeled the aggregate of these terms in the

turbulence kinetic energy equation with an analysis that employed the assumptions that

pressure and total temperature fluctuations can be neglected. They developed a model

for these terms, expressed in mean variables, that introduced a single additional modeling

coefficient. No corrections were applied to the scale equation. The turbulence model, with

a particular modeling coefficient, was applied to some selected supersonic flows, including
near wake boundary layers with Mach numbers up Io 8. The computed results agreed

resonableywell with the experimentaldata, but the absenceof computed results without

the additional compressibility term in this reference did not pernfit firm conclusions t.o be

drawn by a reader as to the value of including this term.

Oh, Ref. 5, reexamined these compressibility terms in the turbulence kinetic energy

equation with the view of improving computations of the spread rate of a free-shear layer

formed bv a single supersonic stream. As in Ref. 4, no modifications were made to the

scale of turbulence, which, in lhis case, was prescribed algebraically. Oh's analysis of

the compressibility effects was based on an assumed delailed st ruclure of l wo-dimensionM

shocklets developed by the turbulenl eddies in relative supersonic motion. In addition, a

form of Taylor's hyt_othesis for the turbulent motions was invoked. Alt.hotlgh, the process

introduced two new modeling coefficients, one could be neglected by Oh as it. was a factor

of the term involving the mean pressure gradients, which were set to zero in the free-shear

problem. The second modeling coefficient was shown to produce a diminished spread

rate with increased Mach number, as occurs within the experimental data. It was found,

however, that good agreement with the data over the range of Mach number from ] I.o

4 would have required altering the value of the coefficient by about 40% over the Mach

number range.

To avoid conjectures, unsupported by experimental data, regarding the detailed structure of shock waves created by colliding three-dimensional eddies, Rubesin, Ref. 6, derived

formulations for the extra compressibility terms in the turbulence kinetic energy equation by introducing the concept that the state conditions wit.hin an eddy could be related

polytropically. This was another way of accounting for non-isentropic processes, shocklets,

etc., that occur within the turbulence, but. without introducing structural assumptions. It

was necessary, however, to make assumptions regarding the fluctuation of the total temperature and the variance of the density fluctuations. In Ref. 6, it was assumed lhat the

total temperature did not fluctuate and thai the variance of the density fluctuations varied

very slowly along streamlines. The resulting fornmlations for the compressibility terms in

the kinetic energy equation contained a single modeling coefficient., namely, the polytropic

coefficient.

The work of Ref. 6, was extended by Vandromme, Ref. 7, to retain the effects of

large mean density variations across streamlines and to introduce extra compressibility

terms into the turbulence scale equation, i.e. the rate of dissipation of turbulence kinetic

energy, as well as in the kinetic energy equation. In addition to the polytropic coefficient,

Vandromme introduced a modeling coefficiem in the added term in the dissipation rate

equation. Vaildromme found that the spread rate of the free-shear layer in supersonic flow

could be represented very well over the Mach number range from 0 to 4 with single values

of the polytropic coefficient and the modeling coefficient in the dissipation equation.

There have been several papers addressed to the evalualion of compressibility effects

within internal combustion engines, e.g. Refs. 8-10. In Refs. 8 and 9, the burden of

maintaining reasonable length scale behavior with a two-equation model during volumetric

compressions was placed on the dissipation equation, with the result that an an additional

divergence term was added to the dissipation equalion in k-( turbulence modeling. The

turbulence kinetic energy equation was left without additional terms, which, as argued in

ORIGINAL PAGE IS

OF POOR _UAL._T_
Ref. 10, is consistent with the low Mach nmnbers characteristic of internal combustion

engines. The difference, then, within this group of papers is in the values chosen for the

modeling coefficient multiplying tile divergence term.

In a recent paper. Ref. 11. Zeman identified the " titillation dissipation" as the

extra compressibility term in tile kinetic energy equation. In a manner similar to Oh,

Zeman utilized a modeled shock wave structure within the turbulent eddies to obtain the

functional form of the compressibilil.v correction. Assumptions regarding the shock wave

thickness relative to tile eddie,, and the protmbilitv dislrihulion function of the turbulence

Mach number led to a form of the c(mlpressitdlity term that contained two new modeling

coefficient s. one of them being thc kurtosis of the turbulence. It was shown that appropriate

constant values of lhese modeling c(,eflicienls in computations of the behavior of a freeshear layer could be madelo represent the data of Ref. 12 quite well.

From this brief review, il is apparent that various authors chose considerably different

routes to introduce compressibility terms into the field equations for either the turbulent

kinetic energy equation or the dissipation equation, or both. Under high Mach number

conditions, the order of magnitude arguments in Fief. 10 would suggest that these terms

should be included in both equations.

Because of tile flexibility intrc,duced into these models by tile introduction of one

or more additional modeling coefficients, agreement with particular experimental data, for

example, high-speed free-shear experiments, could be achieved by several disparate models,

Refs. 5, 7, and 11. In the absence of the application of these same models to more complex

flow situations, such as flows where shock waves impinge on boundary layers to produce

separation or where they interact with shear layers away from surfaces, it is not known

if the various models will differentiate and show one that is superior. To perform such a

comparative study, it will be best lo utilize a single computer code that contains a series

of the alternative turbulence inodels cited here which can be turned on and off at will.

The present paper is devoted to the development of one such model. It is an extension

of the work of Refs. (6) and (7), but with particular attention being given to the nonadiabatic character of high-Mach number flow fields and their large variations in the fluid

properties. In addition, the model allows inhomogenieties in the turbulence.

ANALYSIS

When Favre or mass-weighted ensemble averaging is applied to the conservation equations describing the turbulent flow of a compressible fluid, the following set of equations

results. The only physical assumplions applied to these equations are that the fluctuations in the fluid viscosity and thermal conductivity can be ignored. Turbulence effects

on molecular properties are not expected t.o be important where the molecular properties

themselves dominate.

Continuity

p,, + (_)aa),j = 0 (1)

Momentum in the ith direction
Here,the total stressis composedof molecular and turbulence contributions.

(2

Total energy

' ,2 - PUi uj (3

- p_yh" + p_'_:", 4j + -_7 qj - a.T.j - --."-.% - .."T''l,jj

,3

(4

The quantity /_ represents the mean total enthalpy, which is the sum of the mean

static enthalpy, h, and the kinetic energies of the mean motion plus the turbulence.

In addition, the turbulence kinetic energy equation can be written as

= " u"u"'fL " '-_ u" ' u"r!'.

--7 i,j ij Ui k],3 ÷ Ui,i i P,i

(5)

(6)

The additional compressibility terms in these equations, absent, for incompressible

fluids, are those involving u'i ' Tij_" qj" and u",,i.

Before evaluating the quantities that contain these additional compressibility terms,

it. is worthwhile to eliminate those of litIle consequence. In regions where the molecular

transport properties are important, it is to be expected that the effects of density fluctuations should be insignificant. With this assumption, the terms r'_.7, qj't' u"-iwij, and _,"_=i3,3. "

can be omitted from Eqs. (3). (4), and (6}.

These equations then reduce to

Tij = _-,j - P_ "i uj" (7)

and

- [p_" + p_'_," + c_3- _,T,j - _"<,_] (s)
(9)

The first step required in the modeling of the remaining additional terms is to determine a relationship between the fluctuations of velocities and lh{" fluctuations of the state

properties of the fluid. This is initiated by following Ref. 6 and assuming thai the fluid

behaves in a polytropic manner. Thus, the polytropic coetficienl, n, becomes a turbulence

modeling parameter.

Expressed in mass weighted variables, this assumption becomes

p' p' n pT" n ph "

..... n - - (10)

The last relationship is only exact for constant values of the specific heal, but will be

used here as a first approximation even when the specific heat does vary.

Next, a relationship between T" or h" and u'/ has to be established. In Ref. 6, it was

assumed that the total enthalpy at a point in space remained constant with time despile

the turbulence. This is expressed as

ff " u }'u 'i' H"--h"+ +{ 5 (11)

If the higher order terms are neglected, then

h" : (J2)

The experimental results cited in Ref. 13, however, indicate thai within an hypersonic

boundary layer, e.g. at Al e - 6.4 over a cooled surface where T_/To¢ = 0.46, the local

r.m.s, total temperature fluclualions are not zero, but can have magnitudes ranging from

< H">

1.4% < < 4.8% /)

with the larger variances occurring closer tothe surface. Equation (12}, therefore, has to

be modified to account for fluctuations in the total temperalure.

One could expect that tile local temperature or enl})a]py fluctualion would be dependent on the intensity of the turbulence and, also, on lhe mean static temperature or

enthalpy gradients. One such form, analagous to a mixing length formulation, is

h" = {la)

where a is a local mean parameter, yet to be determined.

If Eq. (13) is multiplied by' pu'_ and averaged, there resulls

Qj = puj"_".= -apu_,'/h , (14)

6
which relatesthe local heat flux 1othe Reynolds stresses and the mean enthalpy gradients.

This form is consislenl with the modeled forms developed in Ref. 14, namely

Qj = -c_ =puj u,. h,, (15)

with

c(. = 0.35

Consistency between Eqs. (14) and (15) requires

(16)

so thai, Eq.(13) becomes

_" ii~

h" = -c__u i h,, (17)

(

A more tensorially complex form of Eq. (la) could have been adopted to be consistenl

with the more recent, and empirically justified forms of Ref. 15. However, since the current

work of introducing the effects of compressibility into the k and { equations will later involve

thin layer approximations which force the principal fluxes of Refs.14 and 15 to be identical,

the simpler form of Eq. (13) will be used here.

The experimental data of Ref. 13 can be used t.o lest the validity of Eq. (17) provided

the differences in the physical assumptions and dependent variables used in the data reduction and this analysis can be reconciled. The data are expressed in unweighted ensemble

averaged quantities whereas mass weighted variables are used here. The quantities measured in the experiment through the use of a hot wire with a large range of overheat values

were the fluctuations of tola] temperature and axial mass flow. These quantities were then

converted lo static temperature and velocity fluctuations through the assumplions in Ref.

13 that slatic pressure fluctuations and higher order correlations could be neglected. Wilh

the latter assumption, the distinction between mass weighled and unweighled ensemble

averaged quantities vanishes. Consequently, a direcl comparison between the resulls of

Ref. 13 and Eq. (17) is possible.

In a boundary layer, the principal terms on the righl side of Eq. (17) are

h" = -c_l"u_'h 2 (18)

{

Similarly, the principal shear slress in the boundary layer, given by two equation modeling,

is

= (19)

When Eq. (19) is used 1o eliminate g in Eq. (18), the variances of the static enthalpy

and the velocity can be re]ated as

< J," >/i, O(/,/t-,,,) C. 1,, <,_,,a > t 1

< ,,';> ,,'a, (20)
where the differentialion is performed at a fixed station.

In evaluating the right member of Eq. (20). Ihe following values are employed:

c., = 0.35

(' =009 p

7"12 ( - = 0.a

P

which is consistent with the above value of (',, and the universal log law, and

< u!} > /' < u'1' >= 0.64

which corresponds 1o tile simulated dala of Ref. 16 at a g+ of about 140. The mean

enthalpy and velocily relalionships are laken directly from the data of 1Ref. 13. The

results from Eq. (20), which is t,ased on Eq. (17), are COml)ared with the corresponding

experimental dala of Ref. 13 ii, t"i_. 1. In addilion. Fig. 1 shows the relationship of the

enthalpy, velocity variances thai result from the use of Eq. (12) as evaluated in Ref. 6.

The use of Eq. (17) results in a ratio of the enthalpy and velocity variances that

represems the hypersonic boundary layer experimental dala of Ref. (13) reasonably well

over most of the boundary layer. 1_ is surprising that the computed results differ from the

data as much as they do in the log region of the boundary layer, y/5 < 0.2, where the

assumptions of lhe anisotropies and shear stress relationships listed above whould have

been expected to apply the best. In the region near the boundary layer edge, however, it

is not surprising that the computed results are high because no allowance has been made

for the shear stress to approach a zero value there. Nonetheless, these results based on

Eq. (17) are a significanl improvenmnt over those that result from the use of Eq. (12),

the basis of the work in Ref. 6. Accordingly, Eq. (17) is used in subsequent steps of this

analysis. Note that Eq. (17) indicates that the sign of the effect on the enthalpy fluctuation

caused by a velocity fluctuation depends on the sign of lhe local mean enthalpy gradient.

Alternative directions of heat flux apt)ear to have opposite effecl s on the turbulence kinetic

energy.

Evaluation of _];

l

From Eqs. (10) and (17

1-1 t

P

1 ph" 1 P k! ]_,.l ,,

(. 1 c,,_-ph=( = _lj

which can be used with the definition of u" to vield

I

(21)

i-}_ t/,','. : 1 c_ -1""t_'apt,",,"= (22)

i p (n- 1) _ _ h J '
Recall that

_2

_,_

(-_-1)

(23)

so that Eq. (22) can also be expressed as

_ [ 0,.,/.,,7,7

_ (24)

The brackeled terms represent moments of turbulence Mach numt)ers, which vanish

in incompressible flow, i. e. a _ oc. Thus u'i' is a measure of l he degree of compressibility

of the turbulence. It is also interesting that u'i' has a value of zero in the absence of any

local heat transfer, where ]_,) = 0. The latter point can be seen directly when Eqs. (24)

and (15) are combined to yield

u,._ - (3 1) Qi (25)

' (. - 1) _a 2

In most. aerodynamic applications, the principal mean pressure gradient is P,1, which

H

identifies u I as the main quantity required in the last term on the right of the kinetic

energy equation, Eq. (9). If the flow also behaves as a thin shear layer where h,2 is much

greater than h,l or h,a' Eq. (24) reduces to

X, I"__-ii "1

(26)

Since the Reynolds shear stress and the static enthalpy gradient in Eq. (26) can

each possess differem signs. _l__can be either positive or negative. For example, within

a planar, hot jet, h,2 and pU'l'll _' change sign simuhaneously on the cenlerline of the jet..

Thus u "1 retains a negative sign on either side of the centerline. Alternatively,.. in a high

speed boundary layer over a cooled surface, h has a maxinmm value within the bound_ary

layer, while tile shear stress maintains the same sign everywhere. Here, ttle sign of u_' is

different, below or above the point where t_ is maximum and the _" equalion will be affected

differently in these different zones of the boundary layer. This may explain, in paN, why

all, ached boundary layers seem to exhibit less compressibilily effects than do free shear

layers at. sinfilar Mach numbers.

Evaluation of p'ui, i'' in the Kinetic Energy Equation

In terms of mass weighted variables, the continuity equation for lhe fluctuating quantities is

!

P,, + (p'ao + o._),_ = 0 (27)

9
After somealgebraicmanipulations and neglectof higher order terms, Eq. (27) becomes

(p,z),, _ _j(p,2),j _ 2p,2uj,j- + 2p,jp'." _- 2_,'u"..j,j = 0 (28)

or

_,._,,, ,_..j . _,,,,, _ (F_), _4_,{/,,2).j (29)

P J,i = -p p P J 2p

Finally, relating the pressure and density fluctuations as in Eq. (8), allows writing

7t_

p' u" .... '- p' u" (30) J,3 - J,3 P

or, with Eq. (28), the pressure rate of strain term becomes

(31)

Eq. (31) shows that. the variance of the density fluctuations, =: }/p,2/_, plays a role in

establishing the magnitude of the pressure, rate of strain lerm appropriate to the kinetic

energy equation. In fact, when Eq. {31) is expressed directly in terms of the variance

-T

3, and use is made of the mean continuity equation and the definition of u j, Eq. (22)

simplifies to give

P j,j 2 '

Locally, the density variance can be established from Eqs. (10) and (17) as

2

(33)

or with higher order terms d,'opped

,) '(" (34)

In a thin shear layer, where h_ predominates, Eq. {34) reduces to

(35)

In the computation of a typical k-_ model, each of t.he quantities appearing in Eq. (35)

are known local quantities and can be used to establish the local field value of the density

variance required in Eq. (32).

10
The remaining terms in Eq. (9) that require modeling are the transport or diffusion

of kinetic energy

Dk =- (pu"k" u ''-I - " "" j _ + jp -- lti Tij), j (36)

and the turbulence dissipation

= u" r!'. (37)

The reasons for identifying the whole of Eq. (36) as a transport term is its divergence form

and its appearance in the total enthalpy equation, Eq. (8), where the pressure fluctuation is

an implicit part. of the enthalpy fluctuation. The dissipation term, Eq. (37), however, does

not occur in the total enthalpy equation since it merely represents an exchange between

mechanical energy and heat, the sum of which contributes to the dependent variable of

that equation. In incompressible flows, and in Ref. (11), these terms are combined and

their difference is modeled. Here, however, these terms will be treated separately because

their molecular terms are important in different parts of the boundary layer and this can

be used to advantage in the turbulence modeling of the additional compressibility terms.

The turbulent, parts of Dk are modeled to be consistent with the level of modeling in

Eq. (15), namely

where c, = 0.25, Ref. 17.

For a thin shear layer, the principal direction of gradients in the kinetic energy require

i = j = 2, so that Eq. (38) reduces to

II [ II II I It II ,

pu2^" 4- u2p = --Cs _p'u2U2]_,2 (39)

(

With tt tl -- , -- c, = 0.25 and pu2_12/pk "_ 0.36, Eq. (29) can be rewritten as

pu_k"+ u_p' = -.09}=-k 2 (40)

(

which is consistent with the simple gradient difl'usion form of the standard k - _ model

formulation.

The viscous part of Eq. (36) is

('u i i j ),J / ll i [ i,j 4- 1 j,i

J

(41)

where the effects of the fluctuating part. of the viscosity have been neglected. Because

these viscous terms are only important in comparison with their turbulent counterparts

very close to lhe surface where the turbulence has been damped, the extra compressibility

effects resulting from mass weighting are not expected to be important where the viscous

11
lerms dominate. The turbulence, therefore, in theselerms can be treated as solenoidal

and u"u"/2 = k. Eq. (41) then t)ecomes

l z

/ UIt it 1

* t

,.i

(42)

In incompressible flows, when the diffusion and dissipation terms are combined, only the

first term on the right of Eq. (42) remains to define the viscous diffusion of the turbulence

kinetic energy. This turns out lo be a reasonable approximation even in inhomogeneous

compressible turbulent flows. In the near wall region, the principal gradients of the turbulence moments arc normal 1() the surface, where i = j = 2. At the surface, k --_ !/2 ,

whereas u,u,, " " _ 94 , so that the latter call be neglected. The second lerln becomes a larger

conl ributor wilh distance fron_ the surface, however, there the contributions of both terms

in Eq. (42) become less signiticanl in comparison with the lurbulent transport. ]t has

been estimated with silnple mixing length arguments that tile second term in Eq. (42)

increases the local transport of kinelic energy by a nmximum of 13 percent at. a value of

4

1/ = 12. Over the entire sublayer, 0 < _/ < (i0, this term increases the diffusion of _' only

by about. 4 percent, so thai Eq. (421 can be simplitied to

-- (at,,,),j (43)

which can be used in both Eqs. (8i and (9).

With Eqs. (38) and (43)

Da, I" u",,"i 4- fi_,j] (44) cs ;_p o , ^','

,J

or in a thin layer

Dk

J

(45)

The dissipation term, Eq. (3T), expands to

--7,-- 2---,,

- i? -iU- _ ,, ,, 2

_ t t ,,: i,.3 + fi(u" _2 (46) [llli,jtt3,i 3 k,/,, e

It is noted that the "dilitalion dissipation", defined in Ref. 11 and modeled there,

appears as the third term on the right of gq. (46). Here, to account for inhomogeneous

turbulence, the dissipation will be treated as consisting of all three terms in Eq. (46) and

its difference from tile first term. lhe _ of the standard/_' -e model, will require modification

of the form of the e equation.

12
Thosereadersfamiliar with the details of Ref. 11 will note that the "dilitation dissipation" term above has a different sign and coefficientfrom that of Ref. 11. The reason

for this is that the forln of the total dissipation of Ref. 11resulted frmn manipulation of

the secondtenn of Eq. (46) under the assumptionthat tile fluctuating turbulenl velocity

moments, not necessarilythe Reynolds stresseswhich include local density,are homogeneous. It is interesting that anidentical result occursif the difl'usionand dissipation terms

are first, combined,but then it. is required to assumethat bolh the velocity momentsand

the viscosity are homogeneous.The approachusedin this paper avoidstheserestrictions,

and allows for both inhomogeneousturbulence and physicalproperties that arevariable.

Ill summary, the k - e model has been modified fur the additional effects of compressibility in the following manner. The kinetic energy equation, Eq. (9), is rewritten

as

.... '- " -- ÷ D,. p' " - *_i,-v- P,i (47)

where Dk is given by Eq. (44), p' ui," i is given by Eqs. (32) and (34) or (35), and u" is

given by Eq. (24) or Eq. (26). The dissipation, g, is modeled with the field equation

e >, .... (48)

--_'e4 _tiP,i - peuj,j

where

rl Ir

D, = (c,_(pu i _j)_,i + f_d),J (49)

To most clearly illustrate the additional terms introduced to account for the effects

of compressibility, equations (47) and (48) are given ill their high Reynolds number forms,

absent any near wall corrections. Eq. (48) contains three terms in addition 1o those

contained in the e equation of tile standard k-e model. The third term on the right has

been added to account for the expected deI)endence of turbulence length scale on passing

through a shock wave following arguments similar to those of Refs. (9); it does not contain

the factor of 1/3 reconlmended in Ref. (10) because the latter is only appropriate to

isotropic turbulence. This term is necessary even when the effects of fluctuations in density

are ignored. The other two terms, those containing ('(a and ('e4, account for the effects of

the flucluating density on altering the total dissit)ation of kinetic energy. Ill addition to

the polytropic coefficient, n, the two new modeling constants introduced are the (,',3 and

C_4. All the other modeling constants and relationships between J,', g, the eddy viscosity,

etc. are the same as in the usual incompressible k-e approaches.

CONCLUDING REMARKS

The forgoing anlysis to define the extra compressibility terms that are inlroduced

by fluctuating density in the Favre averaging process has resulled in the imroduction of

three new modeling constants: the poytropic coefficient, n, and two coefficienls in the

modified kinetic energy dissipation equation; ('(a and ('_4- In the absence of validation

13
computations that include these terlns for experimental conditions where they can be

important, il isn't possible at this time to give recommendations for their appropriate

values. In Ref. 6, it was recmmnended that n=l.2. The basis for this recommendation

was no more profound than that n = 1.2 is consistenl with polytropic irreversable expansion

relationships in thermodynandc cycles often cited in standard thernlodynamic text books,.

e.g. Ref. 18. On the other hand, when the behavior accross a normal shock wave is

considered to be polytropic, il is found thal ,_ > ";.. the isenlrot)ic coeflqcient. In view of

these differences and in the absence of any empirical guidance, it is believed that starting

wilh n 1.4 is reasonable. 11 is the inlenl of the author lo establish the value of('_3 first,

through comparisons of compuialions with experimental data few flow fields involving zero

mean pressure gradients, thereby eliminating the term conlaining (',4. Candidate flows for

this are single or double stream free-shear layers and attached fiat plate boundary layers,

all in high speed flows. The effects of the added terms are expected to be less for the

aIlached flow. Once consist enl values of n and (',3 are established, test cases involving

strong pressure gradients will be pursued to establish ('_4.

REFERENCES

1. Favre, A.,"Equations des (;az Turbulents C,ompressibles," .1. Mecan., Vol. 4, No.

3, Sepl. 1965, pp. 361-390.

2. Rubesin, M. W. and Rose, W. C., "The Turbulent Mean-Flow, Reynolds-Stress,

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 /'kSA Na_onal Aerc_a_lcs and Report Documentation Page

Sp_c@ AdmiRistrat_on

1. Report No.

NASA CR-177556

4. Title and Subtitle

2. Government Accession No. 3. Recipienrs Catalog No.

5. Report Date

Extra Compressibility Terms for Favre-Averaged

Two-Equation Models of Inhomogeneous Turbulence

7. Author(s)

Morris W. Rubesin

9, Performing Organization Name and Address

MCAT Institute

Ames Research Center

Moffett Field, CA 94035-1000

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, DC 20546-0001

June 1990

6. Performing Organization Code

8. Performing Organization Report No.

A-90196

10. Work Unit No.

505-80-11

11. Contract or Grant No.

NCC2-585

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

15. Supplementary Notes

Point of Contact: John R. Viegas, Ames Research Center, M/S 229-1, Moffett Field, CA 94035-1000

(415) 604-5950 or FI'S 464-5950

16. Abstract

Forms of extra compressibility terms that result from use of Favre averaging of the turbulence

transport equations for kinetic energy and dissipation are derived. These forms introduce three new

modeling constants, a polytropic coefficient that defines the interelationships of the pressure, density,

and enthalpy fluctuations and two constants in the dissipation equation that account for the non-zero

pressure-dilitation and mean pressure gradients.

17. Key Words (Suggested by Author(s))

Turbulence modeling

Compressibility

Favre averaging

18. Distribution Statement

Unclassified-Unlimited

Subject Category-34

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

2O

22. Price

A02

qASA FORM 1626 OCTaa

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