Center for TurbulenceResearch

Proceedings of the Summer Program

A new approachto turbulence mo deling

By B Perot AND P Moin

A new approach to Reynolds averaged turbulence mo deling is prop osed whichhas

a computational cost comparable to two equation mo dels but a predictive capability

approaching that of Reynolds stress transp ort mo dels This approach isolates the

crucial information contained within the Reynolds stress tensor and solves trans

p ort equations only for a set of reduced variables In this work direct numerical

simulation DNS data is used to analyze the nature of these newly prop osed tur

bulence quantities and the source terms which app ear in their resp ective transp ort

equations The physical relevance of these quantities is discussed and some initial

mo deling results for turbulentchannel ow are presented

Intro duction

Background

Two equation turbulence mo dels such as the k mo del and its variants are

widely used for industrial computations of complex ows The inadequacies of these

mo dels are well known but they continue to retain favor b ecause they are robust

and inexp ensive to implement The primary weakness of standard two equation

mo dels is the Boussinesq eddy viscosityhyp othesis this constitutive relationship is

often questionable in complex ows Algebraic Reynolds stress mo dels or nonlinear

eddy viscosity mo dels assume a more complex nonlinear constitutive relation for

the Reynolds stresses These mo dels are derived from the equilibrium form of the

full Reynolds stress transp ort equations While they can signicantly improve the

mo del p erformance under some conditions they also tend to b e less robust and

usually require more iterations to converge Sp eziale The work of Lund

Novikov on LES subgrid closure suggests that even in their most general

form nonlinear eddy viscosity mo dels are fundamentally incapable of completely

representing the Reynolds stresses Industrial interest in using full second moment

closures the Reynolds stress transp ort equations is hamp ered by the fact that

these equations are much more exp ensive to compute converge slowly and are

susceptible to numerical instability

In this work a turbulence mo del is explored which do es not require an assumed

constitutive relation for the Reynolds stresses and may b e considerably cheap er to

compute than standard second moment closures This approach is made p ossible

by abandoning the Reynolds stresses as the primary turbulence quantityofinterest

Aquasions Inc Canaan NH

Center for Turbulence Research

B Perot P Moin

The averaged NavierStokes equations only require the divergence of the Reynolds

stress tensor hence the Reynolds stress tensor carries twice as much information as

required by the mean ow Moving to a minimal set of turbulence variables reduces

the overall work by roughly half but intro duces a set of new turbulence variables

which at this time are p o orly understo o d This pro ject attempts to use DNS data

to b etter understand these new turbulence variables and their exact and mo deled

transp ort equations

Formulation

The averaged NavierStokes equations take the following form for incompressible

constantprop erty isothermal ow

r u a

u

u ru rp r S rR b

t

where u is the mean velo city p is the mean pressure is the kinematic viscosity

T

S ru ru is twice the rateofstrain tensor and R is the Reynolds stress

tensor The evolution of the Reynolds stress tensor is given by

R

T

u rR r R P rT rq rq

t

where P is the pro duction term is the homogeneous dissipation rate tensor

is the pressurestrain tensor T is the velo city triplecorrelation and q is the

velo citypressure correlation The last four source terms on the righthand side

must b e mo deled in order to close the system The pro duction term P is exactly

represented in terms of the Reynolds stresses and the mean velo city gradients This

is the standard description of the source terms but it is by no means unique and

there are numerous other arrangements

Note that turbulence eects in the mean momentum equation can b e represented

by a b o dy force f rR One could construct transp ort equations for this b o dy

force which has b een suggested byWu et al but mean momentum would no

longer b e simply conserved Toguarantee momentum conservation the b o dy force

is decomp osed using Helmholtz decomp osition into its soleno dal and dilatational

parts f r r
A constraint or gauge must b e imp osed on
to make the

decomp osition unique In this work we take r
With this choice of gauge

the relationship b etween and
and the Reynolds stress tensor is given by

r r rR a

r
r rR b

Note that the choice of gauge inuences the value of
but do es not aect how

inuences the mean ow

A new approa ch to turbulencemodeling

Using these relationships transp ort equations for and
can b e derived from

the Reynolds stress transp ort equations

u r r r q r rr rT Pr S a

t

u r
r
rq r rr r T Pr S b

t

These equations contain extra pro ductionlike source terms S and S whichcontain

mean velo city gradients Note that the pro duction term is not an explicit function

of and
except under limited circumstances and in general must b e mo deled

The inverse Laplacian r that app ears in these equations can b e thoughtofasan

integral op erator

Theoretical analysis

Turbulent pressure

Taking the divergence of Eq b the mean momentum equation gives the

classic Poisson equation for pressure

r p r u ru rrR

Since this is a linear equation the pressure can b e split conceptually into two terms

one can think of the mean pressure as b eing a sum of a mean ow pressure due to

the rst term on the righthand side

r P r u ru a

mean

and a turbulent pressure due to the second term on the righthand side

r P r rR b

tur b

Given the denition of and assuming that is zero when there is no turbulence

then it is clear that P For this reason will b e referred to as the

tur b

turbulent pressure This quantity is added to the mean pressure in the averaged

momentum equation which results in P p b eing the eective pressure

mean

for the averaged equations The quantity P tends to vary more smo othly than

mean

p which aids the numerical solution of these equations

For turbulentows with a single inhomogeneous direction the turbulent pressure

can b e directly related to the Reynolds stresses In this limit Eq a b ecomes

R where x is the direction of inhomogeneity This indicates that

R for these typ es of ows Note that R is p ositive semidenite so is always

greater than or equal to zero in this situation Positive is consistent with the

picture of turbulence as a collection of random vortices with lower pressure cores

emb edded in the mean ow It is not clear what the conditions for a negative

turbulent pressure would b e if this condition is indeed p ossible

B Perot P Moin

Turbulent vorticity

To understand the role of
it is instructive to lo ok again at turbulentows that

have a single inhomogeneous direction Under this restriction Eq b b ecomes

R where x is the direction of inhomogeneityIf
go es to zero

i

i
k k

when there is no turbulence then R or R and

i i
k k


R These are the o diagonal or shear stress comp onents of the Reynolds



stress tensor

For twodimensional mean ows with two inhomogeneous ow directions only

the third comp onentof
is nonzero and Eq b b ecomes

R R R R

 

Since
is resp onsible for vorticity generation it is appropriate that it b e aligned

with the vorticityintwodimensionalows As a rst level of approximation it is

not unreasonable to think of
as representing the average vorticity of a collection

of random vortices making up the turbulence and therefore
will b e referred to

as the turbulentvorticity

For twodimensional ows with a single inhomogeneous direction R



Note how the comp onents of
reect the dimensionality of the problem while the

mathematical expressions for these comp onents reects the degree of inhomogeneity

Relationship with the eddy viscosity hypothesis

The linear eddy viscosityhyp othesis for incompressible ows takes the form

T

R ru ru k I

T

where is the eddy viscosity I is the identity matrix and k is one half the trace

T

of the Reynolds stress tensor

Taking the divergence of Eq and rearranging terms gives

f r R r k u r r r uu rr

T T T

If the eddy viscosityvaries relatively slowly as is usually the case then the very last

term involving the second derivative of the eddy viscosity will b e small and can b e

neglected Under these circumstances the linear eddy viscosity mo del is equivalent

to the following mo del

k u r a

T

ru b

T

So to a rst approximation the turbulentvorticity
should b e roughly equal to the

mean vorticity times a p ositive eddy viscosity and the turbulent pressure should

b e roughly equal to two thirds of the turbulent kinetic energy These results are

entirely consistent with the ndings of the previous subsections

A new approa ch to turbulencemodeling

PHI

PSI

Figure Contours of turbulent pressure and negative turbulentvorticity

for the separating b oundary layer of Na Moin

Computational results

Equations a and b relating the turbulent pressure and turbulentvorticity

to the Reynolds stresses were used to calculate and
from DNS data for two

relatively complex twodimensional turbulentows a separating b oundary layer

Na Moin and owover a backward facing step Le Moin The

purp ose was to assess the b ehavior of these turbulence quantities in practical tur

bulent situations and to provide a database of these quantities for later comparison

with turbulence mo dels

Separatedboundary layer

The values of and are shown in Fig As mentioned previously for two



dimensional ows only the third comp onentof
is nonzero The owmoves from

left to right separates just b efore the midp oint of the computational domain and

then reattaches b efore the exit The contours are the same for b oth quantities and

range from U to U whereU is the inlet freestream velo city

Both the turbulent pressure and turbulentvorticity magnitudes increase in the

separating shear layer and the reattachment zone In addition b oth quantities

b ecome slightly negative in the region just in front to the left of the separating

shear layer and show some elliptic long range decay eects at the top of the

separation bubble There is some sp eculation at this time that these eects could

be numerical but there is also some reason to b elieve that they are a legitimate

result of the elliptic op erators which dene these variables Changes in the far

eld b oundary condition from zero value to zero normal gradient had no visibly

p erceptible eect on the values of and



The visual observation that and are roughly prop ortional is analogous to



the observation that k R originally develop ed byTownsend and

successfully used in the turbulence mo del of Bradshaw Ferriss Atwell

B Perot P Moin

R22

−R12

Figure Contours of the normal Reynolds stress R and negative turbulent

shear stress R for the separating b oundary layer of Na Moin

It is also consistent with the rst order notion of turbulence as a collection of

emb edded vortices with representing the average vortex core pressure and

representing the average vortex strength

In the case of a single inhomogeneous direction R and R It is



instructive therefore to compare the results shown in Fig with the R and R

comp onents of the Reynolds stress tensor shown in Fig The magnitudes of

the contours in Fig are the same as Fig This comparison clearly shows the

additional eects that result from inhomogeneity in the streamwise direction The

leading and trailing b oundary layers whichhavevery little streamwise inhomogene

ity are almost identical However the magnitudes of the turbulent pressure and

turbulentvorticity are enhanced in the separated shear layer due to the streamwise

inhomogeneity

Backward facing step

Computations of and for the backward facing step are shown in Fig The



ow is from left to right and there is an initial unphysical transient at the inow

as the inow b oundary condition b ecomes NavierStokes turbulence The b oundary

layer leading up to the backstep has mo derate levels of the turbulent pressure and

turbulentvorticity which closely agree with the values of R and R in that

region As with the separating b oundary layer the turbulent pressure and turbu

lentvorticity increase signicantly in the separated shear layer and reattachment

zone There is an area of slight p ositive turbulent pressure and negative turbulent

vorticity in the far eld ab out one step height ab ove the backstep corner This

mayormay not b e a numerical artifact and is discussed in the next section

El lipticity

Identifying the exact nature of the ellipticity of these new turbulence quantities is

imp ortant to understanding their overall b ehavior and how they should b e mo deled

A new approa ch to turbulencemodeling

PHI

PSI

Figure Contours of turbulent pressure and negative turbulentvorticity

for the backward facing step of Le Moin

When rewritten Eqs a and b b ecome

r rrR a

r rrR b

These are elliptic but order one op erators on the Reynolds stress tensor As demon

strated in x when there is only a single inhomogeneous direction these op erators

simply lead to various Reynolds stress comp onents Under these conditions they do

not pro duce action at a distance or long range eects normally asso ciated with

elliptic Poisson or Helmholtz op erators

For two and three inhomogeneous directions it is still not clear whether these

op erators pro duce long range eects There are certainly some situations in which

they do not One example is when the Reynolds stress tensor can b e represented

in the following form somewhat reminiscent of the linear eddy viscosity relation

R s v v where s is some scalar and v is a vector If this is the case

ij ij ij ji

then s r v and
r v and there are no long range elliptic

eects

In fact the presence of long range eects in and
is somewhat unsettling It

would suggest that these turbulence quantities can exist in regions where there is

no Reynolds stress Since r R r r
this would imply that a precise

cancellation of these long range eects must o ccur in regions where the Reynolds

stresses are small or negligible While the results presented in Fig and Fig

seem to show that long range elliptic eects do indeed take place they could also

be a numerical artifact The numerical solution of Eqs a and b requires

double dierentiation of the DNS data this pro duces compact Poisson equation

source terms that are only marginally resolved by the mesh It is our current

conjecture that these op erators are actually lo cal in nature and only serve to mix

B Perot P Moin

0.0050

0.0025

0.0000 phi budget

-0.0025

-0.0050 0.0 0.5 1.0 1.5 2.0

y/h

Figure Budget of the transp ort equation at a station roughly half way

through the recirculation bubble of the backward facing step xh dis

sipation or diusion velo city pressuregradient triple correlation term

pro duction p ositive or convection

various comp onents of the Reynolds stress tensor It is also conjectured from these

computational results that the turbulent pressure is a p ositive semidenite quantity

Note that the ellipticity discussed here is not the same as an ellipticityinthe

governing evolution equations for these quantities An elliptic term in the evolution

equations is b oth physical and desirable see Durbin Such a term mimics

long range pressure eects known to o ccur in the exact source terms The exact

evolution equations for and
describ ed b elow have just this elliptic prop erty

Turbulent pressure evolution

Considerable insight can b e obtained ab out the evolution of the turbulentpres

sure by considering the case of a single inhomogeneous direction It has b een shown

that under these circumstances R so the evolution is identical with the

Reynolds stress transp ort equation for the normal Reynolds stress R For the

case of turbulentchannel ow Mansour et al the R evolution is domi

nated by a balance b etween dissipation and pressurestrain with somewhat smaller

contributions from turbulent transp ort and viscous diusion There is considerable

interest in determining if these same trends continue for evolution in more com

plex situations since the ultimate goal is to construct a mo deled evolution equation

for this quantity

Figure shows the terms in the exact evolution equation for owover a back

ward facing step at a station roughly in the middle of the recirculation bubble

These terms were calculated in the same manner as the turbulent pressure Both

A new approa ch to turbulencemodeling

0.015

0.010

0.005

0.000 psi budget -0.005

-0.010

-0.015 0.0 0.5 1.0 1.5 2.0

y/h

Figure Budget of the transp ort equation at a station roughly half way



through the recirculation bubble of the backward facing step see Fig for caption

the detached shear layer and the backward moving b oundary layer are visible in the

statistics In the shear layer the exp ected dominance of dissipation and pressure

terms presumably dominated by pressurestrain is evident In the recirculating

b oundary layer turbulent transp ort and pressureterms probably dominated by

pressure transp ort are dominant It is interesting to note that the pro duction term

dominates in the middle of the recirculation bubble The fact that some of these

source terms are not exactly zero at roughly two step heights away from the b ottom

wall is thoughttobeanumerical artifact similar to those found when calculating

and
Some of the curves have an erratic nature due to the lack of statisti

cal samples This phenomena is also present in the unsmo othed Reynolds stress

transp ort equation budgets presented in Le Moin

Turbulent vorticity evolution

As with the turbulent pressure it is useful to consider the case of a single inhomo

geneous direction when analyzing the evolution of the turbulentvorticity Under

these circumstances evolves identically to the Reynolds shear stress R In



turbulentchannel ow the R evolution is dominated by a balance b etween pro

duction and pressurestrain with somewhat smaller contributions from turbulent

and pressure transp ort This trend continues in the evolution equation which



is shown in Fig for the backward facing step at a cross section roughly halfway

through the recirculization bubble xh The small value of the dissipation

is consistent with the fact that isotropic source terms can b e shown not contribute

to the evolution of

B Perot P Moin

Mo deling

Formulation

An initial prop osal for mo deled transp ort equations for the turbulent pressure

and turbulentvorticity are

a u r r r C

T

t T y

T

u r
r r


 b

T

t T y

where C y is the normal distance to the wall the timescale is given by T

and the eddy viscosityisgiven by j
jj j Dissipation and some

T T

redistribution is mo deled as an exp onential decay pro cess roughly corresp onding to

Rottas low Reynolds numb er dissipation mo del Turbulent and pressure transp ort

are collectively mo deled as enhanced diusive transp ort Pro duction and energy

redistribution are prop ortional to the turbulence pressure times the mean vorticity

for the turbulentvorticity and are prop ortional to the square of the turbulent

vorticity magnitude for the turbulent pressure High Reynolds numb er constants are

k at high Reynolds numb ers The low Reynolds number determined so that



constants which app ear with a are set to obtain exact asymptotic b ehavior and

go o d agreement with the channel ow simulations of the next section

Note that b oth and
have the same units An extra turbulent scale is currently

dened by using the mean ow timescale j j to dene the eddy viscosity The

solution of an additional scale transp ort equation suchaswould remedy a number

of p otential problems with the current mo del It could eliminate the singularityin

the eddy viscosityatzerovorticity removeany explicit references to the wall normal

distance and allow b etter decay rates for homogeneous isotropic turbulence The

disadvantage of this approach which will b e tested in the future is the added

computational cost and additional empiricism

Channel ow simulations

The mo del equations a and b were solved in conjunction with mean ow

equations for fully develop ed channel owatRe of and Since there is only

one inhomogeneous direction the turbulent pressure is prop ortional to the normal

Reynolds stress and is prop ortional to the turbulent shear stress Comparisons



of the mo del predictions and the DNS data of Kim Moin Moser are

shown in Fig

When a turbulentchannel ow is suddenly p erturb ed by a spanwise pressure gra

dient the ow suddenly b ecomes three dimensional and the turbulence intensities

rst drop b efore increasing due to the increased total shear Moin et al

Durbin mo deled this eect by adding a term to the dissipation equation

which increases the dissipation in these threedimensional ows The same quali

tative eect can b e obtained by dening the eddy viscosity in the prop osed mo del



as Intwodimensional ows this is identical to the previous denition

T

 

A new approa ch to turbulencemodeling

1.5 Mean Velocity (10e-1)

1.0

0.5

Normal Reynolds Stress (R22)

0.0

-0.5 Turbulent Shear Stress (R12)

0.0 0.5 1.0 1.5 2.0

Figure Mo del results solid lines and DNS data circles for turbulentchannel

ow Re

However in threedimensional ows the orientation of
will lag  and the eddy

viscosity will drop initially A smaller eddy viscosity leads to a smaller timescale

and increased dissipation Unfortunately the magnitude of this eect is severely

underestimated in the present mo del and a scale equation and a correction like

Durbins may b e required to mo del this eect accurately

Conclusions

This work prop oses abandoning the Reynolds stresses as primary turbulence

quantities in favor of a reduced set of turbulence variables namely the turbulent

pressure and the turbulentvorticity
The advantage of moving to these al

ternativevariables is the abilitytosimulate turbulentows with the accuracy of a

Reynolds stress transp ort mo del ie with no assumed constitutive relations but

at a signicantly reduced cost and simplied mo del complexity As the names im

ply these quantities are not simply mathematical constructs formulated to replace

the Reynolds stress tensor They are physically relevant quantities

At rst glance the op erators which relate and
to the Reynolds stress tensor

suggest the p ossibility of ellipticity or action at a distance However wehave

shown that under a number of dierent circumstances this do es not happ en and

conjecture that it may never happ en The physical relevance of these quantities

would b e complicated if they were nite when there was no turbulence Reynolds

stresses A pro of to this eect mayalsoprove our second conjecture that is a

p ositive denite quantity

The budgets for the transp ort equations of these new variables indicated that

the extra pro duction terms were not signicant and that these equations could b e

B Perot P Moin

mo deled analogously to the Reynolds stress transp ort equations An initial mo del

was constructed for these equations using basic mo deling constructs whichshowed

go o d results for turbulentchannel ow It is likely that for this shearing ow the

turbulent timescale is well represented by the mean owvorticity However for

more complex situations it is likely that an additional scale equation suchasan

equation will b e required

Acknowledgments

The authors would like to thank Paul Durbin for his comments on this work and

particularly for discussions concerning the ellipticity of these variables

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