Two equation and higher order closures for compressible turbulence fail to capture the compressible wall layers' log scaling. Accounting for the distinction between Favre and Reynolds averaged variables in the compressible moment equations indicate that turbulent transport expressions obtained using the 'variable density approximation' are in error. The error is related to the enstrophy, a Reynolds averaged variable appearing in the equation for the Favre averaged k; recognizing this fact an expression for the transport of dissipation consistent with simple mixing length arguments is obtained. Within the (limited) context of a gradient transport hypothesis a rational form for the turbulent transport of the dissipation is found. Modestly better agreement with the well established compressible Van Driest log scaling is found in k - epsilon calculation

Morrison, J. H.

Ristorcelli, J. R.

English

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NASA Contractor Report 198305
ICASE Report No. 96-19
f
ICA
THE FAVRE- REYNOLDS AVERAGE DISTINCTION AND
A CONSISTENT GRADIENT TRANSPORT EXPRESSION
FOR THE DISSIPATION
J. R. Ristorcelli
J. H. Morrison
NASA Contract No. NAS1-19480
March 1996
Institute for Computer Applications in Science and Engineering
NASA Langley Research Center
Hampton, VA 23681-0001
Operated by Universities Space Research Association
National Aeronautics and
Space Administration
Langley Research Center
Hampton, Virginia 23681-0001
https://ntrs.nasa.gov/search.jsp?R=19960021627 2020-06-16T05:07:27+00:00Z

The Favre - Reynolds average distinction and a
consistent gradient transport expression for the dissipation
J.R. Ristorcelli
Institute for Computer Applications in Science and Engineering
NASA Langley Research Center, Hampton, VA
J.H. Morrison
Analytical Services & Materials, Hampton, VA
Abstract 1
Two equation and higher order closures for compressible turbulence fail to capture the compressible
wall layers' log scahng. Accounting for the distinction between Favre and Reynolds averaged vari-
ables in the compressible moment equations indicate that turbulent transport expressions obtained
using the "variable density approximation" are in error. The error is related to the enstrophy, a
Reynolds averaged variable appearing in the equation for the Favre averaged k; recognizing this fact
an expression for the transport of dissipation consistent with simple mixing length arguments is
obtained. Within the (bruited) context of a gradient transport hypothesis a rational form for the
turbulent transport of the dissipation is found. Modestly better agreement with the well established
compressible Van Driest log scahng is found in a k - E calculation.
1This research was supported by the National Aeronautics and Space Administration under NASA Contract No.
NAS1-19480 while the author was in residence at the Institute for Computer Applications in Science and Engineering
(ICASE), NASA Langley Research Center, Hampton, VA 23681-0001.

1. Introduction
Huang et al. [1], Wilcox [2] have indicated that current k-z turbulence closures do not adequateley
capture the compressible log layer behavior. Part of the problem with these sorts of turbulence clo-
sures appears to result from not adequately recognizing the distinction between Favre and Reynolds
averaged quantities - in the Favre averaged equations both Favre and Reynolds averaged variables
naturally appear. When mean density gradients are important, not accounting for the distinction
contributes to poor results.
In this article a derivation for the modelled dissipation equation for a compressible turbulent flow
is described. While not accounting for many of the complex effects of compressibility, discussed
in a general sense in Lele [4] or in the context of the fluctuating dilatation eg. Ristorcelli [5], the
present development, which involves no modeling assumptions, shows better agreement with the
compressible Van Dreist log law scaling. The usual gradient transfer expression for the turbulent
transport of the dissipation is found to be written as t'tV(fi_s) and not as vtfiVcs as obtained using
the so-called "variable density extension". Huang [3] has communicated that such a form for the
turbulent transport will produce better agreement with the Van Driest scaling. Wilcox [2] also
comments on the fact that, if pc were the dependent variable, then k - _ models would have closer
agreement with the Van Driest scaling. While there has been speculation, Huang [l] about the
properly conserved flow variable, there does not appear to be a clear indication as to what the
proper formulation for the dissipation equation should be. This note indicates the appropriate
form using a simple, but careful mathematical development.
2. Derivation
The nomenclature is now described. Upper case letters will be used to denote mean quantities and
lower case letters fluctuating quantites. Exceptions to this rule are the mean density, fi, (p has no
convenient upper case form) or the mean viscosity ft. Quantities with an asterisk denote the total
field, mean and fluctuating: p* = fi + p' or u_ = Ui + ui. Favre velocities will be denoted using
the set [_, vi] while Reynolds variables are denoted using the set [Ui, ui]. The dependent variables
are decomposed according to u_ = Ui + ui = Vi + vi where < ui >= 0, {vi} = 0, and p* = _ + p'
where < p' >= 0. The averaging operation is indicated using the angle brackets for time means,
< vivj >, and the curly brackets for the density-weighted or Favre mean {vivj}; the two averages
are related by fi{vivj} =< p*vivj >. Without loss of clarity the prime on the fluctuating density is
dropped. The portion of the second-order moment equations,
D
p--_ {vivj} = Pij + IIij + Tijk,k - < uj,pa?p > - < ui,pa'_p >, (1)
of interest are associated with the viscous dissipation type terms, where a_ = ft[ui,j + uj,i -2/3uq.q 5_j].
ThequantitiesPij, Ilij, Tijk,k are, respectively, the production pressure strain and turbulent trans-
port of the Favre averaged Reynolds stress, {vjvj}. Note that in the Reynolds stress equations the
terms arising from the surface forces terms appear naturally in Reynolds, ui, variables while the
problem is posed in Favre, vi, variables. Repeated application of the product rule for differentiation
and the definition of vorticity wi = qjkUj,k produces in the equation for the kinetic energy of the
turbulence the usual dissipation quantities. Keeping only terms of interest produces
-Dk=
p-_- --p{vivj}l,'},j + Tk -- f_ < wkwk > -- 3 fi < dd > (2)
where k = {vjvj}. The solenoidal dissipation has been rewritten ill terms of the enstrophy assuming
4
small scale isotropy. The dilatational dissipation is denoted ec = 5 < dd >. Our interest is with the
portion of the dissipation associated with the vortical modes of the flow. It is conventional to define
the solenoidal dissipation in terms of the Reynolds averaged enstrophy: pe_ = # < wkwk >. In the
"variable density approximation" and high Reynolds log layer limit the dissipation is assumed to
be described by the high Reynolds number form of the Favre averaged equation
D
 -ffi , - [ utc,[ae,,q ],, = p[c lPk - c 2e,] E,/k. (3)
where Pk -{v_vj}Sij. Sij ½[Vi,j +l/),i 2= = -sD_ij] in which D = Up,p is the mean dilatation which
is small in the log layer portion of this flow. Note the location of the mean density in the turbulent
transport term. The enstrophy's replacement by what is treated as a Favre averaged variable,
g,, must be done carefully. The conservation equation for the enstrophy, the primitive Reynolds
averaged variable appearing in the Reynolds stress equations, is, to lowest order,
4 w2
< o; 2 >,t + Uq < w 2 >,q + < UqW2 >,q = --_ < > D + 2 < wisij_i > -# < wi,kwi,k > (4)
The last two terms in the above equations are modeled by the right hands side of equation 3. The
gradient transport hypothesis for the enstrophy transport produces < uqw 2 >= -vt < w 2 >,q,
where vt = c_,k2/E. Replacing the enstrophy with the dissipation using Des = # < o;kwk > produces
D
[l/tO'_ "1 (pS),q ] _peD + P[celPk c,2e,] e,/k. (5)p -Te - ,q= _ _
The substantial derivative is along the Favre mean streamline. Note the location of the mean
density in the turbulent transport term. In general the appearance of the mean density inside the
first derivative will only be of importance when the mean density gradients are large, v___,,_ v___,
For weak density gradients the variable density assumption and the present derivation will give
the same computational result. Calculations, to be shown shortly, were performed with a standard
k - ¢ model with the usual gradient transport models for transport. The following values for the
empirical constants were used: c, = 0.09, c_1 = 1.44, c,2 = 1.92, ak = 1.0, a, = 1.17.
The betterperformanceof the k - w type of turbulence models comes from, in part, the fact that
a_ = c/k is not a specific (per unit mass) variable and the Favre form of the convection operator is
the correct expression for its turbulent transport.
3. Conclusion
One can follow Huang et al.'s [1] mean density gradient contribution analysis to asses the impact
of these new forms of the transport terms in the /," and z equations. The coefficients multiplying
the density gradient terms in their equation (20) are notably smaller using the present formulation.
Wilcox [2] has conducted a similar analysis in which he arrives at a similar conclusion. The
implication of both these analyses is that the deviation from the law of the wall will be smaller
with these changes to the transport terms. This is substantiated in Figure 1 in which a Mach 8
boundary layer calculation using a k - _ turbulence model is shown. The top line corresponds to
"variable density approximation": the lower line(s) which are virtually indistinguishable correspond
to the empirical compressible log law and an incompressible calculation. The middle lines are
with the new form of the transport in the dissipation equation given above: it shows a modest
improvement. The upper middle line is without the bulk dilatation term. The point of this article
is to present a rational form for the dissipation equation without adding ad hoc corrections. The
compressible moment equations are, of course, complicated by several additional issues: the effects
of the fluctuating dilatation and correlations involving the fluid property fluctuations, for example.
These are the subject of current research and are expected to make additional contributions towards
a better agreement with the well established Van Driest scaling.
To summarize: by careful accounting for the distinction between Favre and Reynolds averaged
variables a different expression for the turbulent transport of the dissipation has been derived. The
expression is consistent with mixing length arguments adequate for wall layers. Calculations for the
Mach 8 bounday layer show that following a rational procedure produces a closer agreement with
the log law of the wall using the conventional models for the other terms in the modeled dissipation
equation.
References
[1] Huang, P.G., P. Bradshaw, T.J. Coakley (1994). Turbulence models for compressible boundary
layers. AIAA J. 32:735.
[2] Wilcox, D.C. (1993). Turbulence modeling for CFD. DCW Industries, La Canada CA.
[3] Huang, P.G. (1995). Personal communication.
[4] Lele,S.K. (1994).Compressibilityeffectson turbulence.Ann. Rev. Fluid Mech. 26:211.
[5] Ristorcelli, J.R. (1995). A pseudo-sound constitutive relationship and closure for the dilata-
tional covariances in compressible turbulence: An analytical theory. ICASE report 95-22,
submitted J. Fluid Mech.
35
+
3O
25
20
15
10
_z;/÷
Figure 1. The mean profile for the Mach 8 boundary layer: upper line - variable density aproxima-
tion; lower lines - the log law and the incompressible calculation; middle lines - the present theory
with and without (upper) bulk dilatation.
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4. TITLE AND SUBTITLE
THE FAVRE-REYNOLDS AVERAGE DISTINCTION AND A
CONSISTENT GRADIENT TRANSPORT EXPRESSION FOR
THE DISSIPATION
6. AUTHOR(S)
J. R. Ristorcelli
J. H. Morrison
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
Institute for Computer Apphcations in Science and Engineering
Marl Stop 132C, NASA Langley Research Center
Hampton, VA 23681-0001
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Langley Research Center
Hampton, VA 23681-0001
5. FUNDING NUMBERS
C NAS1-19480
WU 505-90-52-01
8. PERFORMING ORGANIZATION
REPORT NUMBER
ICASE Report No. 96-19
I0. SPONSORING/MONITORING
AGENCY REPORT NUMBER
NASA CR-198305
ICASE Report No. 96-19
11. SUPPLEMENTARY NOTES
Langley Technical Monitor: Dennis M. Bushnell
Final Report
Submitted to Physics of Fluids.
12a. DISTRIBUTION/AVAILABILITY STATEMENT
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12b. DISTRIBUTION CODE
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13. ABSTRACT (Maximum 200 words)
Two equation and higher order closures for compressible turbulence fail to capture the compressible wall layers' log
scaling. Accounting for the distinction between Favre and Reynolds averaged variables in the compressible moment
equations indicate that turbulent transport expressions obtained using the "variable density approximation" are in
error. The error is related to the enstrophy, a Reynolds averaged variable appearing in the equation for the Favre
averaged k; recognizing this fact an expression for the transport of dissipation consistent with simple mixing length
arguments is obtained. Within the (limited) context of a gradient transport hypothesis a rational form for the
turbulent transport of the dissipation is found. Modestly better agreement with the well established compressible
Van Driest log scaling is found in k - _ calculation.
14. SUBJECT TERMS
Compressible Turbulence; Favre Average; Van Driest Scaling
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The Favre-Reynolds Average Distinction and a Consistent Gradient Transport Expression for the Dissipation

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