Statistical ensemble of large eddy simulations - CiteSeerX

Under

consideration

for puSl_cation

Statistical By

D

i_ J. Fluid

ensemble A

N

IELE

C

A

Mech.

of large

RAT

AND

It,

Libre

2NASA

Ames

CP

Center,

stat_

_sem

_w. The

hfDzra atJon

b]e of laxge eddy provided

ensem b]e-averaged

vein

that

on

on_

depend

agog

and

can

thus be used

stat/st_

of the

fDr the subgrJd_aca]e inp_m

ent_d

with

aln ost hdepesdent enssu b]e cnntahs

i.

CA

:isrun

O

ER

S2

Belgium.

94035,

T his pl_duces of the

ogeneous

ow.

[IS.A.

not

_s:

odelpaxam

in portDnt

require any

eners

plImperty

of

spatial aver-

fDr buik_hg

dynam

dec_yilg

plane wake.

S's _

sam e in an

ows. A ]so, d_e 6ns_n b]e of LE S's

ensem b]e-ave_aged

odels fDr three

fDr the

e]ds Js used

bcalm

A n

that can be used

the tin e_devebp_g ber ofLE

G

sin ukzaneous_

Js that Jt does

ful)y iqhGm

of the num

Bruxelles,

Field,

Jc modeL

properties

stress tensor. The

and

. R

)

_xge-sza]e vebcJLy

varbusm

/sotr_pJc tu_bu]enae,

1050

:

i: proceduz_ h

M 2

Beige,

231,

s_ ul_tJons

of the dynam

dynam

EL

simulations

dqe di erent ]arge-sca]e vebc/ty

the stati_

the ensem b]e-ave_aged

provides

by

A RAY

Moffett

(1Received A

H

Euratom-Etat

de Bruxelles,

Research

IC

A.W

2Association Universit_

M

ALAN

eddy

Js3tr_p_ I t/s fDund

the stati_l

new

_ pzooedure

models has been

D/zbul_nce,

_9l_

that the resl]_s are

ens_n b]e p_mvJded

that the

at ]east 16 ree_t_ns.

Introduction Thenumberofdegreesoffreedom

spondstD

a tu_bul_It

dlree din ens_nal Stokes equat_ns sn a/l Reynolfls on_

needed

Gw Jsknc_n

tunbu]ent gove_nhg numbers.

systmm s. D Jrect num

tedlnrlues,

_RANS)

Large

s_ u]atJon

numberofd_of

have

There

Js thus an

Eddy

attzact_d

mud%

freedcma Jsreduced

usedtDde

isthe

]£erkemnelandH,

netheRANS

em-Jcals_ u%_tJons

e]dUi:

ht_mest

old--of

S1nu]atJon

0_) = whemeC

e]d u, thatcorre_2

_3 N S ) of the N av_r{

the evol/tJon of such syst_n s are thus ]in ited to m oderate_

a fraction of the tDt_inumber

these

to chazact_r/zeavebc_y

to hcL_ease as Re _/4 (R_! is theReyno]dsnumber)

_

deve_hg

fre_dcm _ES)

and

interest _ byus_g

techniques

Js actual_

Reyno]ds the past

a spat/al

whJd%

sin ulat_d. Am

Averaged

ong

Navi_x{Stokes

f_w decades.

InLES,

the

_g:

/dyG(x-y)u_),

istheLES

h

e]d.lnRANS,anensemb]eave_g_ngJs

(ii)

2 D.Ca_mt_M .M .R_,andA.A.Wray Inbothcases, theequations fDr_i or fDrU_ cont_ an unknown _ m odellhg.

T he purpose

two m ethods

of the appzoad_

to produce

T he p_sent

a statJstJcmlver_d3n

approad%

is m ot/vatsd by

fDr the degrees of freedcm oftufou]6nce. both

r_lg_tJons between duces the

as projgctg_e

w Jth a shg]e

the number ucD_atbns

otherhand,

.Hence,

unresoIIed Thede

scrn e conditigns

ofLE

and

equ_t.

h

in Sectbn

two

InSect/Dn

fram e_ork

3, wew

sca]eszz' :

usefnlinfDnn

_7_ mustbe

quantities m ade avaJ]ab]e dlrough

StatJstJcalensemb]e

the

cbs_ly

e]dsui

of studying

should

a statJstiml ensem

-

m ode/s fDr the

eJds isnotn_

tu_bu]6_t

obvJousand in Secti]n

ow w illbe supposed

that the know

in SectJon

last case, it Js _nown to devebp

On

]edge ofan

of the dynam

new

models

4.App]Jcati]n

ensem b]e

Jc procedure of

to the wake

that the know]edge that explhit/y

2

tD be

statistical quant/tJ6_. T he applbatJon

Js pz_s_nt_d

5. InthJs

atJon on

of[/{ abne.

u_

ze-

there is no clear sc_Je separation

a bcalversbn

ushg

di erenoe

operation

the sin u]atign. W e propose

ill/now

fDr devebpmg

et_rs are cam put_d

Se_ti3n

_r

can

3.

sin u]atJans ofa

averaged

ofan

inoorpoma_e

the enssm b]e.

ofLES's

fDr ]azge eddy

theNavJ_r{Stokesequat_bns.

sin u]atbn TheLES

ve]oc/L-y

eli _,

which

Stre_STij

:

-- t_ _1) :

f_{tlj

averaging

averaging

the knowledge

indep6_ldentLES

of realisations can be used

T he equatbn

ensemb]e

infDnn atJen fDr building

ensemb]e

2.

from

unresolved

expbred

to JsotropJc turbu]anoe

Js presented

ensemb]e

that aln ostno

scales {, shoe

on dqe m otg/ati)n

underwhJch

in w hiln m odelparam

ow

willbe

depend

S's yields a good

this approach

The

to extract sta_

nitJon ofequJu-al_tand probably

independent

ker_g.

LE S m odels. T he advantage

scales. This

S, models

bet of di erent ve_bcJty

know]edgeofthestatistizalpropertJesoftheLES

b]e of LE S's Js the abil_y

the

fr_n sta_theor_

]feting and

by so much

the statistizs ofthe

be heJpful in devebping

_nou]d

spat/al

of freedam

RAN

e]d. T here is,how ever, an in portant

the statistizs of the reso]ued them

LE S and

that assocJat_ a hum

6"ui :--u_ - (}i can be deduced

InLES

related _ between

ofdegIEes

and

frnm

ofLES.

assure ed that the

S

that requires

cam bine conc_0ts

elin inatBd areinspJrsd

operati]ns

LE S or RAN

ensstt b]e averaging

here is _

the fact that, in both

thathavebeen

I tJs thus in plb_]y

be regarded

devebped

t_nn

equat_n

explhgz]y depends

_)_E_+ _./fij_

(LE S ) is obt2mhed

:

by applying

thusdescrgDes

thee-vo]ut=bn

on the sm a/l sc2Jes through

-i_,_ + _4,Vzun_-

#3_u.

a spatial of a

iter tm ltered

the subgrg]

(21)

Stads txalensem b]eof_ eddy sinu]atbns 3 Forsire plbJty,weon_oons]der 91cc,n pr_]e ows,h whi:hp,thepressure divided by thedens_y, isdetenn bedby d_ehccm pr_il_-y condg_on. Theunknown _ensor Tu appears

h

ve]oc/by

the equation e]d. The

fDr d_e ]azge-sca]e vebcJty

purpose

nnlning

several sta_

pract_,

we

of this study

and

the advantages

i%dependent

LES's

of sgn ukaneously

fDr the

sam e

ow . In

fDr R

large-

=

-0z_ r +

- r:%j,

(22)

= I,...,R.

Itis worth

ment]oning

expens_e

than

LES

denote

and

sin u]atbn

and

that the use of an 6_ssm b]e of LE S's is not per sem

the use ofa by

shg]e

rea/_t/3n.

the

tin e at which

for obtaining

the

converged

tin e needed

shg]e

ulat_d

is prEsum

and

if some

all these

meaningful

ne what

though

Yet

is not yet avai]ab]e, _ equat/]ns

orby

2. The

condkSons

functions b_ (_) are

gJu_n.

and

tim e

an enssa b]e

tgn e. T bus,

fmr

in tgn e by the am ount t._/R.

the

transi_t

an overhead

phase

and

the

cost w illbe m odera_e.

negl_0]e.

and

Moreover,

if the

the boundary

e]ds{[

to the sam e experim

_

the ensemb]e.

equkra36nt

but

LE S

of sol/tJons fDr the N av_er{Stokes a

ow descrgoed

area/l

entalsituation

W e therE_e

independent

Js assure ed to be ful_ de ned ow

LES's

enlybeusEfmliftheLES

un_ueness

equation

_de6_ndent

to correspond

sta_

in w hJ_h the on

us also

T hen, the C P U

the additional

and

apzact_poi%tofv]_w

an LES

i. T he dora a_] _

of the

statiC.

statistics are to be e_tracted

of ex_ce

Let

istl + _ .With

cost/s total_

to obtain

equ_t

_]ds have

w illbe cons_de_

a proof

m ore

if there is no dJrenTJon ofhcln ogene/ty, the enssm b]e-averaged

ably the only way

]edge of an enssab]eofLES'scan

hdependent.

deve_ed.

the ensemb]e

ratio between

this add]tJonal

2 i. Statis t_lly T he know

_lly

the sta_.

to be advanced

uch

a stationary

the beginning

is thus R (it+ t_ /R ),w hx_n am ounts _

to conv_z_le statistizs is sm a]l then

LE S is not stationary

betwe6n

becomes

over both

r_a]/s_ti3n. Ifthe

In the exam pies trsatsd bek_,

applT]ach

period

to converge

sam pie, the ensem b]e only needs

l)_,tovera

this, _t us eonsdHer

statJstScsw 991 a shg]e LES

T he totalC P U cost fDr the ensemb]e of {R -

show

tumbula_oe

_t) requk_d

of reel_satJons, statistics ar_ accum equkra]6nt

To

t,_the tin e of the transit

by t_ the tim e 00eyond

required

de

on the s_ a/l-smm]e

(2 A) by the _Dlk_ i%g set ofequatJons

OtiS[ + Bju/u,

denote

]t depends

e]ds 3[ :

scale vebc_y

wherer

is to expbrE

equ_t

thus replace the equation

ui but

must

e_]ds.A lequations

by theNavJ6_r{Stokes

by the knowledge

of

is cr_nsi_er_]. _

of this dcrn a_

u_ (@T),t.):

b_ (t) where

the

4

D.CazatiM.M.Rcgers,and A.A .Wray 3.The]nJtJaleondkJons ui (x,0) : u(/0

_c model

spectra. The

fmr

to be cl_pped

of _h_s clJpphg

0 _i _r ]9 :

_he dynam

_now s good

agreement

_qsemb]e-everaged values and voJ/m

15.

only

results

the EAD

e-everaged

P

LES'swas

deviatJmns.

tuzba]snme

b]e of 16 32 :IfDrosd _/_bu]_nce

s that the mean

drops

(7 has

sn al]_r m agn_des

dJss_ati)n

the m sans and

2). Some

fDrwhi]h

dne vollm e-averaged

ensemble

LES's,

of the enssn b]e size are given in

befDre

have

reaches

fDr the

4 2. Fozr_d W e have run an _nss,

{see

the averaged

the energy

a 512 a D N S and

16 are indJsti_gui/nab]e

results arepbtte_

of C

C

values of C

: 1 while/£

16 sin ultaneous

increases. T his Js a]so re ectmd

16. H ow6-vEr, the oonsequ_oes

on physimlquantitg_ between

R

ofpo_ts

the ratio between

_3r dne total resolzed

fDr R

m_n

=

when

as a function

d_e _actJon

die cl_ped

0 2.9

{PDF)

fraction of negat_e

For _.st_nc_,

T he corn pazimn both

_rR

on_

s as well as the vol/m e-averaged

dzast_

function

= 16 .Hence,

st/llsk]ni cant even ]e_s s_ni

the

i: model

dlat with

Jc m odelperfmnn

m easurer_ ents of the spatial variabil_y Tab]e

dynam

eriml sin u]ati]ns.

encouraging.

the 6_sem b]e-averaged The

e-averaged

r_so]ued energy

and

LE S 'sw _th zero mo]ecular hhe standard

devi_t_bn

viscosLvy.

evo]ue sin i-

12

D .Camat_

TABLE

1. Average

and

M

.M

Rl

Before

1 2 4 8 16 32

0.018

standard

.Rcgers,and

A .A .Wray

clipping

After

clipping

0.29

0.089

0.19

0.031

01040

0.024 0.020 0.019

0.024 0.017 0.012

o

0.020

o27 00480.081

0.019 0.018 0,018

0,031 0,020 0.013

deviation versus

of tile inodel coefficient tile ensemble size,

(before

and

after

clipping)

0.5

0.0 0.0

5.0 Time

FI(;t!RE 3. Conq)arison of tile energy decay [/etween tile truncated DNS (solid line) and tile averaged energy predicted by tile set of LES's using EADP (dmshed line). The dotted lines correspond the averaged energy,/ :k one standard deviation _u_ predicted by the set of LES's using EA D P.

larlF fDr both hduoed

by

the vo]um

not htmoduce standard

spurbus

devJatJons

in the EADP vollm

rate.

_ertJal

range

exam

corre]atbns

inatbn

between

rein a_h sin ibm h

or

to

of the

with

obtmh

a very

results

m

gure

good

_

hdJcat_g

that dne coupling P

approach

does

do

not

at_

4 iadkates

enercjy

that

the

te

K okn observed

and(

observe ogorov

e]ds

of hdependent

the corn pensetvaed

spectrum

exp_-T

of the

dnat the LE S

ensemble

tm ccrn pare

") is the we

_nows the EAD

as those in the

intEm_st_g

32 :{ LES,

This

d_e di erent m 6m bets of the enssa b]e. T he

as _dependent

E(k)kr'/ae-2/:_,whereE(I, 0 f oourse,

mode]s.

eter thi_ugh

d_e two approaches,

sin ulatJons. Itis_

E(k):

sipatbn

ensem b]e-everaged of the m odelparam

r_n ain near]y

e-averaged

spe_:mam

e-and

the cnm putatJon

a we/1 oonstmnt.

\K oka ogorov

energy

is thedJsdevebped How

ever,

eonstant"

is reasonable.

5.

TestsJn T he

ow

dJr_ct num

wake consJdered

ows here

erJcalsimu]atJons

Js a tin e-_vo]uglg _4 oser&

Rogers

plane 1994;M

wake

fmr which

data

oser et ai 1997) and

frcm

both

large eddy

Stat/s

tim/ens_n

b]eof

_

eddy

7.0

13

s in u]mtbns

4.0

6.0 5.0 4.0 3.0

3o

o_

o

2,0

1.0

1,0

°'°o'.o

1_.o

2_.o

3_.o

0'00_.(

5.0

10.0

Time FI(;I*RE

4.

Resolved

turbulence: Dotted

lines

_tnd

in

the

and

hom

Rogers

ow

deta/iby

512

×

195

num

bet

128

based

a m

on

class

odes

of

the

are

eri_l

Re

=

p/u

=

2000.

Ina

s_

ti_

in

boxes)

forced

isotropic

dynamic

standard

procedure.

deviation

in

This

ow

is both

sta_

a m ore

d_n

anding

test

the

EADP

of the

ills

-

de

+ .v_

e-evotrhg

EA

and

on

D

a

EwJng

czo_

mapped

in

the

here

P

has

(1997).

i% the peri]di_

the

resokze

considered

Rogers&

is representmd

accu_t_y ux

wake

oser,

functbns

ass

plane

andM

varinb]es

tD

nonof the

section.

(1994)

Jacobipo_nmm

m

be

u]atJon

Fo_bas_

iqtmgzatmd

4- one

prevbus

Rogems

It =

Js

mean

thus

i_ the

requked

(right)

(open

avaibb]e.

_dependent by

spectrum

respectively.

are

oser&

of the

by

×

M

the

_ou]J

s studied

ise dir_cti3n_s

is represented

1997) and

num

spadald_pendence spanw

to model,

ogeneous

energy

volume-averaged

correspond

pseudospectmaldJr_ct

Jse and

compensated versus

dymtmic

ogeneous

in

and

boxes)

lines

inhmm

described

T he w

and

the

been

dm_hed

(Ghosg/&

stationary

T he

(left)

(solid

vohlme-averaged

sin ul_tJons

than

energy

ensemble-

15.0

k

stremm

-

dependence nite

dram

_bul_ce.

ain.

The

Up

tD

R eynolJs

cJt,

(U(_)-

ptane

U_)(ty

wake,

(52)

the

int_r_atEd

mass

ux

]tered

DNS

de

cit

Js

eonstant. LE an

S 's

of

hiti_

were

sam

e

conditbn,

ow have

pseudospectral

inhc_n dcrn

ain.

The

and

the Rogers

times

approprJat_

f_wer

enez

e hum

1998)

reported DNS,

fi3r the

ofm

but

and

procedure,

the

&

spatial

veJocJty

sam

EADP

LES'sexam

pazed

to theDNS.

wJth

Ghosal

ofm e num ined

Rogers

a (1997).

dependence

iszepresentsJ

number

odes

i= by

dJr_:bn

(1995).The

c_m

dynam

non-perigdJc

ber

m odes

the

the

cross-str_lm

Jin sam

ushg been

]3ke

ogeneous

ofCorral&

&

the

_ eJd

odes

of

t_zm

sofFouri_m

is then

ca]cu]ated

used

er]calm

_

her_.Thus

ethod

The the

theLES'swas have each

been LES

e]das

sinulatbns

vortJcJty

M

odeson using

a

the 64

×

adopted requires

the nitro

method 48

×

16

(C arati up

to

260

14

D .C azat_ M

.M

5 i. The I n the pz_Ent v_cosity

study, w e have

conc_ot.

T he

section (41). Inthism is expr_ed

in _

.Rogers

su/_jzJd-s(m]e mode]s

91vest,arid

rst one

three di er_nt m ode]s, all based

is the Sm agor91sky

ode/, the 91ertimlrange

made _ _ L,tSkl 5 kl -T his approxgn

a separade

equatbn

dk_ctly.

This

m odelbased

has

Jc prooedul_, m oti_Ded

M

sdon

91 the previous

viscosgcy yt _ --41:_ I/3

using d%e apprmxim

atJon is requked

(W ong

considered

&

h

usual_

at]on fmr the

ccrnpuDed.

Lilly 1994; Caratiet

scalhg

:

a thkd

model

ai

S because

However,

1995b)

h

LES

be pred_3_d an _at/ue

sud_ as

r Ti) _

odelA

trad/_onalLE

C, : C _-113 can

the din ansionalpzoduct

on the 91ertialrange

F inal_, w e have

fDr the eddy

Densorby

fDr the dissipation made is not

on the dynam

91troduaed

on the eddy

-_,r

dg_patJon

bas_

model

scalhg

s of the resolzed stra_nrat_ --r

,and A .A .Wr ay

_r

-2C'--4/3

' u"

(52)

_Dr w hJch the t_nsor T_ Js gJuen by the expres-

(2]3) M

odelB

:

Tu _

w here the brackets

91dJcade ensan b_zag91g

tages of this ]astm

odelhave

Inall_m

odel%

scale diss_atbn, avoid

been

the s_n

shce

over all ree_tJons.

disa/ssed

of O

a negatJue

O

v_os_y

(cl%pp91g procedure,

(eddy

,

(53)

T he posmb]e

advan-

91 section 23.

(or ofO, ) w illalso det_nn corresponds

nurn erScal 91stabi]__%_, the m odelparam

positive val/e

,5u Cn, r - (SU) --r )

_2C,,--4t:,

to a negative eter must

see Ghosalet

plus m o]ecular) is negatgm_.

he

the sk3n ofthe

eddy

v/szos_y. Inorder

to

then be set equal to a m 919n al

a]_ (1995)) at points where

For

subgr]d-

the Sm agor91sky

the total

m ode], the stability

cond/tJon /

C--2

--r--r \W2

[2,S%iSkl)+ _.> 0 depends

on the r_tzbn.

unJuersal

ow

in whkh

T his is an unde_]e

charact_

(_'is _Ideed

(5.4)

property

shoe

fDr allm erabers of the ens_n b]e. An the

same

:[Drall reactions

results _

(7 is supposed _atz_e the

to be a

fmnn u]atJon

_Ibw91g

stmbil_y

condition

(7-2 Inthe

i_ it ofan

in n_De number

t_nser am p]/tude wou]d reasonable stabil_y

2,Skl,Skl

ofr_a]i_tJons,

be aln ost unbounded.

to s_ piz in pose C

condition

mrax

is nabara/ly

>

0. Form

odelA,

+ I/.> them

ax_

0.

um

(55) ofthe r_so]ued

H enoe, fDr the Sm agol-hsky however,

stra91-raDe m ode], it is

the s/tuatJon is di erent. T he

the sam e in each r_a/isatJon , --4/:_ ,

+ Vll > 0,

(5.6)

stadstial_nsem b]sof]_tjeeddy sinu_tbns Forsinplbity,thesam eeonditbn hasbeenusedfmrmodelB.

15

52.TheJn_oondi_ns Inpracti_,h_condJti]ns fDrLEScanbebuiltekherby _g avai]ab]e) orby generat/qg a random ve]oc_y ell smds_iqg smm cussed

in Section 2 i). For the EA D P, w e have

e]ds. Inthe

case of Jsotropiz _/_bul_noe,

the i_it/al ell was to build

R

i_/tJal conditions

tin e-evo]ui%g

plane

sam e approach 19_rgenumber oonstrahts turbulent

.W e thus have

hhe smm e spectzum

random

hit_

fDr ilJtJ_g

ofquantiti_s

arEm

easurEd

that need

to be m a_ta_ed

_ather

than

under

and

ev-a]uatJng an

phases.

be generated

For

the

fDlk_hg

the

m ijht be consilered

(eg. pro

]_s ofm

ean

of the present

procedurE.

For

ilentimli_it/al

quant_

(1981)

ever, fi3rthe plane wake,

of them

ilitJ_tbn

to be smtis ed by

independent

m ai% puzpose

sta_

independent

approadn

eoul_

any number

but

(as d/s-

R ogalb's

the D N S. How

e_.). The

on the fact that the observed

are thus Jn_t

used

by all re_tJons

used a sin pie trick to generater

equiral_t that had

and

ccnd_-hns

used

kiqetiz energy, enstrophy,

is based and

wake,

spectrum

with

R

the on]F oonstr_t

as the one

test the EADP have

the energy

to generate

a DNS_henit Js e const_edqts

ate cc_n put_d

by using

R

vaJues

of

s_]dy

is tm

that reasgn

we

ells. 0 urprocedurE

through

plane

_ges

the change

(6_,6_ ), R

satisfy the requJrEml ent that this prooedure random

does

choizes

space. W

not

the LES

produce

the

subgril

model

all tin es. H ow ever, the model di er_nt mem

vebcity

r_ti]ns

be

sta_

two

e]ds az_ ilentJcal and

s w illhave

c]earihz

However,

:hJtig_loonditJons,

tm_n s, all the statJstJns would

_

that

equkra_t.

91dependent the

(5.7)

ekls are pz_]duoed

even

w id%

sin ply shi_

in

r_n ah

ilent_kml fDr

the desJrab]e e ect of de