A dynamic subgridscale eddy viscosity model

7/21/2019 A dynamic subgridscale eddy viscosity model

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A dynamic subgridscale eddy viscosity model

Massimo Germano, Ugo Piomelli, Parviz Moin, and William H. Cabot

Citation: Physics of Fluids A: Fluid Dynamics (1989-1993) 3, 1760 (1991); doi: 10.1063/1.857955

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http://scitation.aip.org/search?value1=Massimo+Germano&option1=authorhttp://scitation.aip.org/search?value1=Ugo+Piomelli&option1=authorhttp://scitation.aip.org/search?value1=Parviz+Moin&option1=authorhttp://scitation.aip.org/search?value1=William+H.+Cabot&option1=authorhttp://scitation.aip.org/content/aip/journal/pofa?ver=pdfcovhttp://dx.doi.org/10.1063/1.857955http://scitation.aip.org/content/aip/journal/pofa/3/7?ver=pdfcovhttp://scitation.aip.org/content/aip?ver=pdfcovhttp://scitation.aip.org/content/aip?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pofa/3/7?ver=pdfcovhttp://dx.doi.org/10.1063/1.857955http://scitation.aip.org/content/aip/journal/pofa?ver=pdfcovhttp://scitation.aip.org/search?value1=William+H.+Cabot&option1=authorhttp://scitation.aip.org/search?value1=Parviz+Moin&option1=authorhttp://scitation.aip.org/search?value1=Ugo+Piomelli&option1=authorhttp://scitation.aip.org/search?value1=Massimo+Germano&option1=authorhttp://scitation.aip.org/content/aip/journal/pofa?ver=pdfcov


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A dynamic subgrid-scale eddy viscosity model

Massimo Germane,@ Ugo Piomelli,b) Park Moin, and William H. Cabot

Centerfor Turbulence Research,Stanford, Cah~ornia 4305

(Received 14 November 1990;accepted7 March 1991)

One major drawback of the eddy viscosity subgrid-scale tressmodels used n large-eddy

simulations s their inability to representcorrectly with a single universal constant different

turbulent fields in rotating or sheared lows, near solid walls, or in transitional regimes. n the

presentwork a new eddy viscosity model is presentedwhich alleviatesmany of these

drawbacks.The model coefficient s computed dynamically as the calculation progressesather

than input apriori. The model is basedon an algebraic dentity between he subgrid-scale

stresses t two different filtered levelsand the resolved urbulent stresses. he subgrid-scale

stresses btained using the proposedmodel vanish in l aminar flow and at a solid boundary, and

have he correct asymptotic behavior n the near-wall region of a turbulent boundary ayer.

The results of large-eddysimulations of transitional and turbulent channel low that use he

proposedmodel are n good agreementwith the direct simulation data.

1. NTRODUCTION

In large-eddysimulations (LES) the effect of the large

scales s directly computed,and only the small subgridscales

are modeled. Since small scales end to be more isotropic

than thelargeones, t should be possible o parametrize hem

using simpler and more universal models than standard

Reynolds stress models. Thus, most subgrid-scale (SCS)

stressmodelsare basedon an eddy viscosity assumption. n

the most commonly used model, developed by Smagor-

insky, the eddy viscosity Y* is obtained by assuming hat

the small scales re n equilibrium, so hat energyproduction

and dissipation are in balance.This yields an expressionof

the form

vy- = KsA)*I~ t,

(1)

whereA is the filter width (which is proportional to the grid

size), Cs s the Smagorinsky constant, 13 = (23,,3,, ) /* is

the magnitudeof large-scale train-rate tensor

(2)

and iii is the large-scale elocity.

Lilly* determined hat, for homogeneoussotropic tur-

bulencewith cutoff in the inertial subrangeand A equal to

the grid size, Cse-0.23. In the presence f meanshear,how-

ever, this

value

was found to cause excessivedamping of

large-scale luctuations, and in his simulation of turbulent

channel flow, Deardorff3 used C, = 0.1 (also with filter

width equal to grid size). A

priori

tests by McMillan

et a1.4

on homogeneous urbulence confirmed that Cs decreases

with increasing strain rate. Mason and Callen however,

found that the value Cs = 0.2 gave good results if the grid

resolution wassufficiently fine, and concluded hat valuesof

Cs lower than 0.2 are required f the numerical resolution s

insufficient. Their results, however, were not confirmed by

Piomelli et aL6 who found the optimum value of Cs to be

Permanent address: Dipartimento di Ingegneria Aerospaziale, Politec-

nice di Torino, Turin, Italy.

Permanent address:Department of Mechanical Engineering, University

of Maryland, College Parli, Maryland 20742.

around 0.1 (again assuming he filter width to be equal o the

grid size) even with meshesmuch finer than those usedby

Mason and Callen. It should be noted, however, hat Ma -

son and Callen did not resolve he wall layer, while Piomelli

et aLb did.

Additional modifications to the Smagorinsky model

were made n the near-wall region of p lane channels o force

the subgrid-scalestresses o vanish at the solid boundary.

Moin and Kim,7 for example, used damping functions to

account for near-wall effects. Piomelli et al. chose the

damping function to ensure he proper asymptotic behavior

for the SGS shearstresses ear he wall, but found little dif-

ference with the results obtained with the standard Van

Driest damping usedby Moin and Kim and others.

Yakhot et aLa9 seda subgrid-scalemodel basedon the

renormalization group theory of Yakhot and Orszag in the

large-eddy imulation of channel low. Although the stresses

predicted by the model n its original formulation go to zero

at the wall without requiring any damping function, Yakhot

et al. included an ad hoc factor to take into account the

anisotropy of the small scales n the near-wall region. The

asymptotic behavior of the stresses redicted by this model

dependson the grid distribution in the wall-normal direc-

tion; for the grids commonly used,an incorrect asymptotic

behavior s obtained.

Large-eddy simulations of transition to turbulence in

boundary ayers and planechannel show that during the

early stagesof transition the Smagorinsky model predicts

excessive amping of the resolvedstructures, leading o in-

correct growth rates of the initial perturbations. To over-

come this difficulty an additional empiricism was intro-

duced in the form of an intermittency function which

modified the Smagorinskyconstant by effectively setting it

to zero during the linear and early nonlinear stages f transi-

tion,

This briefsurvey of the existing iterature indicates hat,

although modificationsof the Smagorinskymodel havebeen

successfullyapplied o the LES of transitional and turbulent

flows, it is not possible o model effectively with a single,

universal constant the variety of phenomenapresent n the

1760 Phys. Fluids A 3 (7), July 1991

0899-8213/91/071760-06$02.00

@ 1991 American Institute of Physics

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flous examined. The ad hoc manner in which the SGSeddy

viscosity has beenextrapolated to the wall is far from desir-

able. In addition the Smagorinskymodel cannot account for

energy flow from small scales o large scales backscatter),

which can be significant.

In this work a new, dynamic SGS stress model is pro-

posed hat attempts to overcome hesedeficiencies y locally

calculating the eddy viscosity coefficient to reflect closely

the state of the flow. This is done by sampling the smallest

resolvedscalesand using this information to model the sub-

grid scales.The model presentedhere requires a single nput

parameter and exhibits the proper asymptotic behavior near

solid boundariesor in laminar flow without requiring damp-

ing or intermittency functions. The model is also capableof

accounting for backscatter.

In the next section, the model will be presentedand its

characteristicsdiscussed. he model was estedboth apriori,

taking advantage of existing direct numerical simulation

(DNS) databases, ndaposterioriusing the model n an LES

calculation. The results of these estswill be discussed n Sec.

III. Concluding remarks will be made n Sec. V.

II. MATHEMATICAL FORMULATION

In large-eddy simulation, the large-scalequantities are

defined by the convolution of the velocity and pressure ields

with a filter function. For the purposesof this work we define

two filtering operators: one s thegrid filterz, denoted by an

overbar:

y(x) =

I

f( x)c( x,x)dx

(3)

(where the integral is extended o the ent;re computational

domain) while the other, the tea filter G, is denoted by a

tilde:

j-(x) = j-(x)& x,x)dx;

s

(4)

the filter width of the test filter is assumed o be larger than

that of the grid filter (i.e., the test filter corresponds to a

coarser mesh than the grid filter). Finally, let G = Gi?.

By applying the grid filter to the dimensionless ontinu-

ity and Navier-Stokes equations one obtains the filtered

equations of motions

(5)

(6)

In thefollowing,xorx, is thestreamwisedirection,yorx, is

the direction normal to the walls (which are located at

~7 + l), and z or x3 is the spanwise direction; further-

more, the distance from the nearest wall is denoted by JJ,~,.

The effects of the small scalesappear in the subgrid-scale

stress erm

r,,= u,u, -ii,ii,,

which must be m_odeled.

(7)

Now apply G to the equations of motion: the filtered

Navier-Stokes equations become

JT.. 1 a2ci

--L+---,

CYX, Re ax, 6 xj

(8)

where the subgrid-scalestress s now

Tc = uyj - ;,Gj;

(9)

finally, consider he resolved urbulent stress 4p, defined as

y, = -gj - S,Ej.

(10)

The resolved urbulent stresses re representativeof the con-

tribution to the Reynolds stresses y the scaleswhose ength

is intermediate between the grid filter width and the test

filter width, i.e., the small resolved scales.The quantities

given n (7), (9), and ( 10) are related by the algebraic rela-

tionL4

ytj = TJ - ?gY

(11)

which relates he resolved urbulent stressYii, which can be

calculated explicitly, to the subgrid-scalestresses t the test

and grid levels,TO and rTil.

The identity ( 11) can be exploited to derive more accu-

rate SGSstressmodelsby determining, for example, he val-

ue of the Smagorinsky coefficient most appropriate to the

instantaneous state of the flow. Assuming that the same

functional form can be used o parametrize both T, and rii

(the Smagorinsky model, for example), let M,, and mi, be

the models for the anisotropic parts of T, and r,,:

rij - (S,j/3)r,, --mu = - 2CKIS [Sij,

TO - (S,/3)T,, EM,, = - 2x@ I&:,,

where

(12)

(13)

i$ =+-($+$), 151=a,

(14)

his the characteristic filter width associatedwith G, and x is

the filter width associatedwith z. Substitution of ( 12) and

( 13) into ( 11) and contracting with sij gives

2Yi,SlJ= - 2c(P@ lS,S;, - PI5 I5$,, j,

(15)

from which C(x,y,z,t) can be obtained in principle. The

quantity in parentheses,however, can become zero, which

would make Cindeterminate or ill-conditioned. Apriori tests

in turbulent channel flow have shown this to be indeed the

case.For the channel flow, therefore, it was assumed hat C

is only a function ofy and t. To this end, the averageof both

sidesof (15) is taken over a plane parallel to the wall (indi-

cated by ( )) to yield

C(y,t)= - 1.

L-c,%d)

2 x2(lsl~mnsmn>~2(I~I~p;,,sp,>

(16)

the new dynamic eddy viscosity subgrid-scale tressmodel s

then given by

w,,m

m, =

(z/h)2(l~li?m,s;,,,> - (Is15pqspq)

IS IS,. (17)

In the present calculation, the sharp cutoff filter has been

used as both test and grid filter. In finite difference calcula-

tions the test-filtered flow quantities can be computed by

spatial averaging the calculated large-scalevariables over a

1761 Phys. Fluids A, Vol. 3, No. 7, July 1991

German0 eta/. 1761

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few grid cells, or example. n more generalsituations, more-

over, the plane averageshould also be replaced with appro-

priate local spaceand time averages. he model ( 17) implic-

itly assumes imilarity between he SGS stresses t the grid

and test evels,which are modeledusing the same unctional

expression,namely, the Smagorinskymodel.

A few remarks are n order regarding he properties and

the character of the subgrid-scale stress model given by

( 17). First, the model giveszero SGSstresseverywhere i;pr,

vanishes as ong as he denominator remains inite). Such s

the case n laminar flow or at solid boundaries.Furthermore,

it is easy o show that in the near-wall region mij is propor-

tional to the cubeof t,hedistance rom the wally,, regardless

of the choiceof h or A. This is the correct asymptotic behav-

ior for the ( 1,2) component of the subgrid-scalestress en-

sor, which, in this region, is the most significant one. To the

authors knowledge, his is the only model that satisfies his

property without the useof ad hoc damping functions. Final-

ly, the use of ( 17) implies that the modeled subgrid-scale

dissipation,esGs= m,zQ, is proportional to the averagedis-

sipation of the resolved turbulent stresses, L?,j,?Y , which

can be either positive or negative.Thus, the model doesnot

rule out backscatter. In the present ormulation backscatter

is not localized and may (or may not) occur at every point in

a plane; the use of local averaging n ( 15), however, would

allow the model to provide localized backscatter as well.

I -The only adjustable parameter n the model is the ratio

A/A > 1. The resolved turbulent stressescalculated using

small valuesof this ratio can be contaminated by numerical

errors; on the other hand, large values of it imply that the

stresses ue to large energy-carrying structures are used o

determine hz contribution of the subgrid scales. f the opti-

mal valueof A/A varies greatly from one low to another, the

applicability of the mode1 s reduced. In the next section,

large-eddysimulations of transitional and turbulent channel

flow are used o address his issue.

Ill. RESULTS AND DISCUSSION

A priori tests of the dynamic subgrid-scalestressmodel

( 17) were carried out to determine he accuracy with which

the model predicts the SGS stressesand dissipation. The

testswere performed using the DNS database f Kim et~l.~

for turbulent channel flow, and that of Zang efal. l6 for tran-

sitional flow. Reynolds numbers are, respectively,

Re = 3300 and 7900 (based on the centerline velocity U,

and channel half-width S) for the turbulent case, and

Re = 8000 or the transitional case (basedon initial center-

line velocity and channel half-width).

The sharp cutoff filter was applied as both grid and test

filter in the streamwiseand spanwisedirections. No explicit

test filtering was applied in the normal direction. Two com-

monly used definitions of the filter width were used:

---

x=A A A2 3, Z3=E,K2&,

(18)

and

p=yq +x; .+q p=iT: +g +g,

(19)

where b, and & are the filter widths in each coordinate

direction associatedwith aand E, respectively;E, was aken

I .

I.

16 1 I 8

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

, .

*I

_,

6

,, .: , I 9

-1.0

-0.8 -0.6

-0.4 -0.2 0.0

Y/6

FIG.

I. Plane-averagedubgrid-scalehear

tress (T,* ) and dissipation

(eSclg); Re = 3300 turbulent channel flow. A: exact;-: a = 2; ---:

a = 4; *. . .:a = $. (a) Dissipation; (b) SGS shear stress.

to ke equal o twice the grid spacing,Ax,, and vzrious values

of A, were examined. n the following the ratio A,/& will be

taken equal n all directions, and denoted by a (notice, how-

ever, that no explicit filtering is applied in the wall-normal

direction).

The mean subgrid-scaleshear stress ( r,2 ) and dissipa-

tion (escs are comparedwith the modeledones n Fig. 1 or

various filter widths in the turbulent channel flow. Equation

(19) was used to define the filter width. The choice a = 2

was found to yield the best results. However, actual large-

eddy simulations with the dynamic model appear o be very

insensitive o a (seebelow). With this choice b corresponds

to a wave number in the decaying region of the one-dimen-

sional energyspectrum, while z representsa wave number n

the hat region. In Fig. 2, the product Cx is plotted as func-

tion of the wall coord inate y+ = u,y,/~ [where

u, = (r&p2 is the friction velocity, 7, is the wall shear,

and p is the fluid density] ; the expected + behavior s evi-

dent. At thechannel center CzO.023 when ( 18) is used; he

square root of this value is 0.15, about 50% larger than the

value of C, used by DeardorK3 When ( 19) is used, at the

channelcenter Cz 7 X 10 4,which givesa value of 0.026 or

10-s

1o-5

& 10-7

1 o-9

lo-

FIG. 2. Variation ofCx* [defined n Eq. ( 17) ] with distance rom the wall;

Re = 33M) turbulent channel Row, a = 2.-: CT obtained using ( 18);

---:C~20btainedusing(19);~~*~:C~*-y+.

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-0.121

I

I

I

I

-1.0

-0.8 -0.6

-0.4 -0.2 0.0

Y/6

FIG. 3. Plane-averagedubgrid-scale issipation (es&; transitional flow,

t = 176.A: exact:-: a = 2.

the Smagori nsky o nstant.The issueof the sensitivity of the

numerical esults o the choiceof filter width and of a will be

addressedater. The mod el was also tested n transitional

flow for a = 2 (Fig. 3). T he SGS dissipation predicted by

the Smagorinskymodel or this case s many ordersof mag-

nitude larger, and peak smuch closer to the wall than the

exact one.

To further determine he accuracyof the dynamic SGS

model ( 17 , it wasalso estedaposteriori in the LES of tran-

sitional and fully develo ped urbulent channel low. Initial

conditionsconsisted f the parabolicmean low, on which a

2-D Tollmien-Schlichting (TS) modeof 2 % amplitude and

a 3-D TS modeof 0.02 % amplitude weresuperimposed. he

initial conditions and Reyn olds numbe r matched those of

the direct simulation of Ref. 16. The governin g equations

(5) and (6) were ntegrated n time usinga pseudospectral

Fouri er-Che byshev collocation method. Both filter

widths ( 18) and ( 19) wereused; he final resultswere nsen-

sitive to this choice ,so only those obtained using Eq. (18)

will be presented. he ratio

a

= 2 waschosen.At the initial

stages8 X 49X 8 grid points were used; he mesh was then

progressive ly efined up to 48

X

65

X

64 points; the dimen -

sionsof the computational domain were2?rSn the stream-

wisedirection and 41rS/3 n the spa nwise irection. Periodic

bound ary conditions were applied in the streamwi seand

spanwis e irections; no-slip conditions were applied at the

walls.

The time developmentof the mea n wall shear stress

(r,,,) is compa red n Fig. 4 with the DNS resultsof Ref. 16

and with the results of the LES of Ref. 12, which used a

Smagorinskymodel including Van Driest damping and an

1.6

0

r; 1.2

A

b* 0.8

V

0.0

0

50 100 150

200

250 300

FIG. 4. Time developmentof the plane-averaged all shearstress T,) in

Re = 8ooO ransitional channelRow.A: DNS;-: present esults; . . . .:

LES.

x00.16

2 0.12

-A

N 0.08

-2

v 0.04

0.00

-1.0

-0.8 -0.6

-0.4

-0.2

0.0

_.-_

L

1

>0.16

\ 0.12

: 0.08

: 0.04

0.00)

I

I I

,

-1.0

-0.8 -0.6

-0.4 -0.2 0.0

0.20,

o.ooJ ,

I

1 I

-1.0

-0.8 -0.6

-0.4

-0.2

0.0

Y/6

FIG. 5. Plane-averagedms turbulence ntensities (u*) * in Re = 8 000

transitional channel flow. A: Filtered DNS;16 -: presentcalculation;

....: LES. (a) t= 176; (b) t = 200; (c) f = 220.

ad hoc intermittency function; the present esultscompare

very well with the finely resolvedDNS. A coa rse irect sim-

ulation which canadequately esolve he early stages f tran-

sition (up to tr 170) cannot predict the drag crisis (Fig. 4 )

and the breakdownprocess.r2With the presentmodel, on

the other hand, he predictedpeakwall stresss within 3% of

the DN S result. The root-mea n-squa re luctuation of

(u~) (where uj = Zi - Ui and Vi = (iii)) and the

Reynoldsshearstress uu), show n n Figs. 5 and 6, are n

fair agreementwith the DNS results.The DNS resultshave

been iltered using he same ilter empl oyed n the LES cal-

culation. Discrepanc ies etween he LES and DNS resultsat

late stages f transition may be due to the fact that, at these

times,slight differencesn the prediction of the onsetof tran-

sition ma y result in significant differences n the instanta-

neou s ields. The capability of the model to predict aver age

backscatter s evidenced y the fact that for t< 185 he eddy

viscositywasnegative or significant regionsof the channel.

Once ully develo ped urbulent flow was achieved, ta-

tistics were accumulated.The Reynoldsnumbe r of the tur-

bulent flow wasRe = 6 100based n centerlinevelocity and

channel half-width. The m ean velocity profile is s hown in

Fig. 7, normalizedby the friction velocity U, and by the bulk

velocity U,,

u, =1

2s

s

66 (~)d~.

(20)

An inadequate esolution of the wall layer results n a low

value of wall stress hat is reflected n a hig h value of the

intercept of the logarithmic layer in Fig. 7(b). The overall

agree ment f the L ES results with the DNS data is fairly

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~~~~~~1

10 20 30 40

50

-4.0'

I

I

I

f

I

-1.0 -0.8

-0.6 -0.4 -0.2

0.0

-4.0'

I

t I

I

-1.0 -0.8

-0.6 -0.4 -0.2

0.0

Y/b

FIG. 6. Plane-averagedReynolds stress (uu) in Re = 8000 ransitional

channel low. A: Filtered DNS;16-: present calculation; * * * *: LES.lL

(a) t= 176;(b) t=200, (c) t=220.

good. The turbulence ntensities (uj) 2 normalized by the

friction velocity U, are shown in Fig. 8. The DNS results

havebeen iltered using he same ilter employed n the LES

calculation, In general, he dynamic model givesmore accu-

rate results than the Smagorinskymodel used by Piomelli

and Zang.

2 The peak of the streamwise urbulent kinetic

energyoccursneary + = 12,a valuealsoobtainedby experi-

FIG. 8. Turbulenceintensities (u;)~ in fully developed urbulent channel

flow. A: R e = 7900 filtered DNS;5 0: Re = 3300 filtered DNS; -:

presentcalculation;

f.*.: LES. (a) u; (b) u; (c) w.

ments and numerical simulations; the mean streak spacing

was ound to be/i + = 140,somewhat arger than the estab-

lished valueof 100,which is also expectedof large-eddy im-

ulations The skewness nd flatness actors of the three ve-

locity componentsare shown n Fig. 9. They compare airly

well with the DNS results. Note that in contrast to turbu-

lence ntensities, higher-orderstatistics from LES are com-

pared with unfiltered DNS results.

To investigate he effect of the parameter cyon the nu-

merical results we performed a calculation in which the val-

20

p '*O

(0)

15

+

'

-5-

x

0.5 h A

10

0.0

5

-0.5

0

lo-' 100

10' 102

- 1 o

-0.8 -0.6 -0.4 -0.2

0.0

Y+

1.2

1 o

no.8

5 0.6

0.4

0.2

0.0

-1.0 -0.8

-0.6 -0.4

-0.2

0.0

FIG. 7. Mean velocity profile in fully developed urbulent channel low. A:

FIG. 9. Skewnessand flatness actors of U, in fully developed urbulent

Re = 7900 iltered DNS;s c7:Re = 3300 iltered DNS:15 : presentcal-

channel Row. -: t4; f .:

u;- --: w from the present calculation. c1:

culation; . . .: LES. (a) Wall coordinates; (b) global coordinates. u; A: v; 0:ru from Re = 3 300 DNS.15 (a) Skewness; b) flatness.

I

16

-1 .o

-0.8 -0.6 -0.4 -0.2

0.0

Y/6

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uea = 4 was used.This amounts to changing he coefficient

of the first term in the denominator of Eq. ( 17) by almost a

factor of 4. The results were found to be very insensitive to

this parameter: differences n the mean and rms velocities

were ess han 3%; the wall stresses iffered by less han 6%.

The maximum resolved shear stresswas larger by approxi-

mately 4% in the calculation with a = 2, and the subgrid

scalecontribution smaller by the sameamount. The insensi-

tivity of the large-eddysimulation results to the value of a is

contrary to onesexpectations rom the

apriori

tests (Figs. 1

and 3) and castssomedoubt on the utility of apriori tests n

providing quantitative data for LES.

IV. CONCLUDING REMARKS

A new eddy viscosity subgrid-scale stress model has

been presented n which the smallest resolvedscalesare dy-

namically tested o predict the behavior of the subgrid scales.

This model is basedon the algebraic dentity ( 11) between

the resolved urbulent stresses nd the subgrid-scale tresses

obtained using two filters, the grid filter and the test filter.

The model coefficient s obtained dynamically as he calcula-

tions progress.This procedureexploits the spectral nforma-

tion on the energy content of the smallest resolved scales

provided by LES calculations to dynamically adjust the

model. The only input to the model is the ratio of test filter

width to grid filter width,

a.

Among the useful properties of

the model is its proper asymptotic behavior near the wall

without the use of ad hoc damping functions.

Large-eddy simulations of transitional and fully devel-

oped turbulent channel flow were also carried out. The re:

sults were in good agreement with those of direct simula-

tions, and better than those of LES that used the

Smagorinsky model with ad hoc damping and intermittency

functions. Doubling the value of a did not affect the numeri-

cal results significantly.

Investigation of the properties of this model when the

box filter is employed s desirable.The sensitivity of the re-

sults to the choice of

a

should also be further investigated n

flow configurations much different from those studied here.

Finally, the use of local spaceand time averages nstead of

the plane averageused o obtain (17) should be attempted.

ACKNOWLEDGMENT

Partial support for the second author (UP) was pro-

vided by the Office of Naval Research under Grant No.

N00014-89-J-1531.

J. Smagorinsky,on.Weather ev. 1,99 ( 1963).

*D. K. Lilly, NCAR Manuscript 123, 1966.

3J. W. Deardot%, J. Fluid Mech. 41,453 ( 1970).

40. J. McMillan, J. H. Ferziger, and R. S. Rogallo, AIAA Paper No. 80-

1339,198O.

P. J. Mason and N. S. Callen, J. Fluid Mech. 162,439 ( 1986).

6U. Piomelli, P. Moin, and J. H. Ferziger, Phys. Fluids 31, 1884 (1988).

P. Moin and J. Kim, J. Fluid Mech. 118,341 (1982).

*E. R. Van Driest, J. Aeronaut. Sci. 23, 1007 (1956).

9A. Yakhot, S. A. Orszag, V. Yakhot, and M. Israeli, J. Sci. Comput. 4,139

(1989).

V. Yakhot and S. A. Orszag, J. Sci. Comput. 1, 3 (1986).

U. Piomelli, T. A. Zang, C. G. Speziale,and M. Y. Hussaini, Phys. Fluids

A 2,257 (1990).

U. Piomelli and T. A. Zang, Comput. Phys. Commun. 65,224 ( 1991).

13U. Piomelli, W. H. Cabot, P. Moin, and S. Lee, Phys. Fluids A 3, 1766

(1991).

I4M. Germano, CTR Manuscript 116, 1990.

15J.Kim, P. Moin, and R. D. Moser, J. Fluid Mech. 177, 133 ( 1987). The

high Reynolds number DNS was performed after the publication of Ref.

11; he samenumerical method used or the lower Reynolds number case

was employed, and 256 X 193X 192grid points were used o give the same

resolution, in wall units, of the lower Reynolds number case.

16T A Zang, N. Gilbert, and L. Kleiser, in

Instability and Transition,

edit-.

ed by M. Y. Hussaini and R. G. Voigt (Springer-Verlag, New York,

1990), pp. 283-299.

T. A. Zang and M. Y. Hussaini, n

Nonlinear Wave Interactions in Fluids,

edited by R. W. Miksad, T. R. Akylas, andT. Herbert (ASME, New York,

1987), pp. 131-145.

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