Center for TurbulenceResearch

Annual Research Briefs

Analysis of discretization errors in LES

By Sandip Ghosal

Motivation and ob jectives

All numerical simulations of turbulence DNS or LES involve some discretization

errors The integrityofsuch simulations therefore dep end on our ability to quantify

and control such errors In the classical literature see eg Chu on analysis of

errors in partial dierential equations one typically studies simple linear equations

such as the wave equation or Laplaces equation The qualitative insight gained

from studying such simple situations is then used to design numerical metho ds

for more complex problems such as the NavierStokes equations Though suchan

approachmay seem reasonable as a rst approximation it should b e recognized

that strongly nonlinear problems such as turbulence have a feature that is absent

in linear problems This feature is the simultaneous presence of a continuum of

space and time scales Thus in an analysis of errors in the one dimensional wave

equation one may without loss of generality rescale the equations so that the

dep endentvariable is always of order unity This is not p ossible in the turbulence

problem since the amplitudes of the Fourier mo des of the velo city eld havea

continuous distribution The ob jective of the present researchistoprovide some

quantitative measures of numerical errors in such situations Though the fo cus of

this work is LES the metho ds intro duced here can b e just as easily applied to DNS

Errors due to discretization of the timevariable are neglected for the purp ose of

this analysis

Accomplishments

In this rep ort analytical expressions for the p ower sp ectra of errors due to the

spatial discretization of the NavierStokes equations are derived In x an ex

pression for the numerical error is presented as the sum of nitedierencing

aliasing and mo deling errors that have dierent origins In x expressions

for the p ower sp ectra of the rst two kinds of errors are derived as well as the corre

sp onding expressions for the subgrid and total nonlinear terms The essential to ol

that makes the derivation of such an analytical expression p ossible is the joint

normal hyp othesis for turbulentvelo cities The essential technique is identical to

that used by Batchelor in his derivation of the pressure sp ectrum of turbulence

from the energy sp ectrum Batchelor These results are applied to the

LES of turbulence in x to obtain some measure of numerical errors in nite

dierence schemes which are increasingly b eing used in turbulence computations

on ows with complex b oundaries This rep ort summarizes the essential results

the details of the mathematical development will b e presented elsewhere Ghosal

henceforth referred to as pap er

Present address CNLS MSB LANL Los Alamos NM

S Ghosal

Calculation of discretization errors

Any representation of the true velo city eld in a turbulentow on a nite grid

is necessarily approximate One must b e careful to distinguish b etween errors due

to the niteness of the representation and the discretization error of a numeri

cal scheme In a numerical simulation the velo city eld at any timestep can b e

regarded as an elementofavector space with a nite numb er of dimensions N

where N is the number of variables retained in the computation This is an ap

proximate representation in a subspace of the larger vector space that contains the

true solution The b est p ossible approximation to the true solution in the subspace

is the pro jection onto the subspace in fact that is the denition of a pro jection

op erator see eg Helmb erg The ideal or b est approximation to the

NavierStokes op erator in the nite subspace is that op erator that ensures that the

numerical solution remains lo cked to the pro jection of the true solution at all

times as b oth vectors move around in their resp ectivevector spaces It maybe

shown see pap er that this condition is achieved by sp ectral metho ds or prop

erly dealiased pseudosp ectral metho ds in the absence of subgrid mo deling errors

By discretization error Eofa numerical metho d we mean the deviation of the

right hand side evaluated with the metho d from what would have b een obtained

if the righthand side of the full NavierStokes equation were pro jected into the

computational subspace Thus for a sp ectral metho d used in conjunction with an

exact subgrid mo del E

In order to evaluate the formal expression for the error E one needs to intro

duce a basis The most advantageous choice is the D Fourierbasis since in Fourier

space dierentiations reduce to multiplication bywavevectors and numerical dier

entiation reduces to multiplication by mo died wavevectors see eg Vichnevetsky

Wenow restrict our attention to ows in a p erio dic cubical b ox Further

while considering nitedierence metho ds the grid will b e assumed uniform in ev

ery direction Let E k denote the comp onents of E in the Fourierbasis with i

i

to corresp onding to the x y and z directions resp ectively Then E kcanbe

i

written as

FD alias mo del

E kE kE kE k

i

i i i

The rst term arises b ecause of the inability of the nitedierencing op erator

x to accurately compute the gradient of shortwavelength waves We call this

k

the nitedierencing error It vanishes for a sp ectral metho d that can dierenti

ate waves of all wavelengths exactly The second term arises due to the metho d of

computation of the nonlinear term by taking pro ducts in physical space on a dis

crete lattice This is called the aliasing error and is well known in the literature

on pseudosp ectral metho ds Canuto et al Rogallo The last term is

the dierence b etween the true subgrid force and that computed using a subgrid

mo del We call this the mo deling error

In the following analysis the magnitude of the error E will b e characterized by

statistical prop erties such as its p ower sp ectral density Such statistical measures

can b e precisely dened only in the limit where the wavevector can assume a con

tinuum rather than a discrete set of values In physical space this implies that we

Discretization errors in LES

are considering the grid size and some characteristic scale of turbulence xed

and taking the limit as the size of the b ox L In actual simulations of course

the b ox size L is nite However L is taken much larger that or so that smo oth

power sp ectra can b e dened and computed statistical quantities are not changed

when the b ox size is increased further This ensures that the computed quantities

are indistinguishable from the ideal limit L For the purp ose of theoretical

analysis it is advantageous to take the limit L rst rather than at the end of

the computation Thus in the Fourierbasis the exact solution will b e characterized

by a continuum of wavevectors k R and the numerical solution will b e character

max max max max max max

ized by the subset k B where Bk k k k k k

x x y y z z

We will assume for simplicity that the grid length is the same in all three di

max max max

rections so that k k k k Further we will consider

m

x y z

the LES lterwidth and the grid length to b e identical This condition will b e

relaxed in x In the limit of innite b ox size the discrete Fourier transform

and its inverse take the form a factor of L is absorb ed in the denition of

the transform

Z

X

k x exp ik x x dkk exp ik x

B

x

where the summation is over all lattice p oints over the innite cubic lattice of

spacing and the integration over wave space ranges over all vectors k BThe

following useful identity is readily derived by taking the limit of innite b oxsize

X X

expiK x K a

x

a

where is the Dirac delta function is the set of wavevectors of the form

pk qk rk where p q and r are integers p ositive negative or zero and K

m m m

is anyvector not necessarily restricted to B This relation is familiar in solid state

physics see eg chapter pg of Jones March where the set go es by

the name Recipro cal Lattice When the lattice spacing the summation

over lattice p oints in b ecomes an integral over space and the usual continuous

Fouriertransformisrecovered In this limit the right hand side of relation b e

comes simply K and reduces to the familiar expansion of the deltafunction

in terms of exp onentials

Let us rst consider the eect of pro jecting the exact righthand side of the Navier

Stokes equation onto the Fourierbasis with wavevector k The ith comp onent is

given by

Z Z

dk dk k k ku k u k k k u k iP k

m n mn i imn

B B

where P has its usual meaning see eg Lesieur and is the exact

imn mn

subgrid stress in Fourierspace The Einstein summation convention for tensor

S Ghosal

indices is implied throughout this rep ort except where otherwise noted If the

exact derivative op erator x is replaced by the numerical dierentiation x

k k

multiplication bywavevectors k are replaced bymultiplication by the corresp onding

mo died wavevectors kThus we obtain

Z Z

h i

FD

E ki P k P k dk dk k k ku k u k

imn imn m n

i

B B

i h

M M

k k k u k k P k i P kT

i imn imn

mn mn

M

where T k is the mo died subgrid mo del obtained by replacing all multipli

mn

M

cation bywavevectors if any in the subgrid mo del kby the corresp onding

mn

multiplication by mo died wavevectors

To obtain the aliasing error we consider the eect of evaluating the nonlinear

term in physical space

M

iP k u xu xT k

imn m n

mn

On using the denition of the discrete Fourier transform wehave

X

u xu x exp ik x u xu x

m n m n

x

When u x and u x in are expanded in the Fourierbasis weget

m n

Z Z

X

dk dk u k u k exp ik k k x u xu x

m n m n

B B

x

The summation over lattice p oints can b e p erformed using

Z Z

X

u xu x dk dk u k u k k k k a

m n m n

B B

a

All the terms in the sum over a with the exception of a are clearly spurious

contributions and constitute the aliasing error Thus wehave

Z Z

X

alias

M

kiP k dk dk k k k au k u k T k E

imn m n

mn

i

B B

a

where consists of the vectors pk qk rk where p q and r can indep en

m m m

dently take on the values or but excluding the case p q r The reason

integer values of p q and r with mo dulus greater than are not included in is

that the relation a k k k cannot b e satised for suchvalues if k k k B

Discretization errors in LES

and hence the delta function ensures that they do not contribute to the sum The

M

last term T k is the contribution to the aliasing error from the subgrid mo del

mn

Obviously it dep ends on the nature of the mo del For a subgrid mo del that uses

a constant eddy viscosity this term is linear in the resolved elds and hence there

is no contribution to the aliasing error For more complicated mo dels such as the

Smagorinsky mo del it is dicult to evaluate this contribution analytically due to

the complicated nature of the nonlinearity

The expressions and for the nitedierencing and aliasing errors involve

M

the subgrid mo del Mo deling errors asso ciated with subgrid mo dels are dicult

ij

to estimate and further there is no obvious way to single out for this study any

one among the wide variety of subgrid mo dels in use It is therefore advantageous to

separate the issue of subgrid mo deling from the issue of discretization errors which

is the sub ject of this pap er In order to accomplish this weintro duce the concept

of the ideal subgrid mo del

M

xt

ij

ij

where xt is the exact subgrid stress One might think of the ideal subgrid

ij

mo del in the following way Imagine a DNS with an innitely greater resolution

running concurrently with the given LES Atevery timestep the exact subgrid stress

is computed from the DNS elds and supplied to the LES simulation as a function

of p osition The rest of the analysis in this pap er will b e presented for suchan

M

is already given as a function of p osition and time and idealized LES Since

ij

involves no computation it do es not contribute to aliasing errors Thus for such

M

an idealized LES T in The contribution from the subgrid mo del is not

mn

however zero for the nitedierencing errors even for the ideal mo del This is

b ecause the mo del is inaccurately dierentiated for computing the pressure and

the subgrid force The subgrid terms in for the ideal mo del are given by

M M

T Thus

mn

mn mn

ZZ

h i

FD

E ki P k P k dk dk k k ku k u k

imn imn m n

i

k k u k

i

The integration in now ranges over the entire wavespace Clearly for this

ideal mo del

h i

mo del

M

kiP k k k E

imn mn

mn

i

Power spectra

In this section analytical expressions for the p ower sp ectra of the nitedierencing

error aliasing error subgrid and total nonlinear term are derived

Finitedierencing error

FD

The p ower sp ectrum of the nitedierencing error is dened by E k where

n o

FD

E k

FD FD

hE kE lim k i

i i

V

k V

S Ghosal

f g denotes angular average in wavenumber space over the surface of the sphere

jkj k and V is the volume of the physical b ox containing the uid

From wehave

FD FD

k i kE hE

i i

Z

k k k k dk dk hu k u k k u k u k k i

imn m n

ipq p q

Z

i k k k k dk hu k u k k u ki

i

imn m n

jk k j hu ku ki

i

i

where h i denotes ensemble average denotes complex conjugate denotes the

imaginary part and k k P k P k The following two prop erties

imn imn imn

of the tensors follow immediately from the corresp onding prop erties of P

imn imn

imm imn inm

In order to make further analytical work p ossible with wenowintro duce

the Millionshchikovhyp othesis see eg Monin and Yaglom that in fully

develop ed turbulence the joint probability density function of any set of velo city

comp onents at arbitrary spacetime p oints can b e assumed to b e jointnormal The

jointnormal hyp othesis was originally evoked in turbulence in an attempt to close

the hierarchy of equations for moments see eg Lesieur Though this did

not succeed the jointnormal hyp othesis has b een successfully used in other con

texts Thus Batchelor Batchelor used it with success to predict the pressure

sp ectrum of isotropic turbulence The jointnormal hyp othesis implies in particular

hu x u x u x u x i hu x u x ihu x u x i

i j k l i j k l

hu x u x ihu x u x i

i k j l

hu x u x ihu x u x i

i l j k

and that all third order moments are zero Here uxt is the true velo city eld

dened at all space time p oints On taking the continuous Fourier transform of

and assuming the turbulence to b e homogeneous wehave

hu k u k u k u k i k k k k k k

i j k l ij kl

k k k k k k

ik jl

k k k k k k

il jk

where is the Fourier transform of the correlation tensor R x x hu x u x i

ij ij i j

On substituting into we get after some algebra see pap er

Z

FD

E k k k k k k k k k d k

imn ipq

mp nq

o

jk k j k

ii

Discretization errors in LES

Eq is the general result for homogeneous turbulence If in addition the

turbulence is isotropic simplies Batchelor to

ij

E k

k k k k

ij ij i j

k

where E k is the three dimensional energy sp ectrum and is the Kroneckerdelta

ij

symb ol The integral in the rst term of may b e written after substitution of

as

Z

J k k k k k d k

mpnq

mp nq

Z

k E P E Q

P Q P P Q Q Q P P P Q Q

mp nq m p nq n q mp m p n q

P Q

P Q k d Pd Q

This integral can b e simplied see pap er The result is

J kF k F k

mpnq mp nq mn pq pn mq

k k k k k k k k

n q m p n q m p

F k F k

nq mp

k k k

where

F k I I I I

I I I I F k

F k I I I I

I I I I F k

The terms I are dened as

m

Z Z

d d E kE kW I k

m m

j j

where the weights W are dened as follows

m

W

W

W

W

S Ghosal

Therefore after substituting in and using the prop erties and

imm

the following expression is obtained for the p ower sp ectrum of the

imn inm

nitedierencing error no summation over rep eated indices

FD

E k

X X

k k

m p

j k k j F k F k F k k k k k

imn imn

ipn

k

imn imnp

o n

X

k k k k

m p n q

k k E k jk k j k k F k

imn

ipq

k

imnpq

In the functions F k F k F k and F k are known once the energy

sp ectrum is sp ecied They are not aected by the choice of numerical schemes

On the other hand the co ecients of these functions in dep end only on the

numerical metho d through the dep endence of k on k and are quite indep endent

of the physical sp ectrum Thus given a sp ecic numerical scheme and energy

sp ectrum can b e used to compute the p ower sp ectrum of the nitedierencing

error This is done in x for various representativenumerical schemes

Aliasing errors

The p ower sp ectrum of the aliasing error is dened by

n o

alias

E k

alias alias

lim hE kE k i

i i

V

k V

From one obtains

Z Z Z Z

X

alias alias

hE kE k i P k P k dk dk dk dk

imn

i i ipq

B B B B

aa

hu k u k u k u k i k a k k k a k k

m n

p q

On applying the jointnormal hyp othesis one gets after some algebra see

pap er

alias

E k

Z Z

X

k P k P k dk dk k k k a k k

imn

ipq mp nq

B B

a

The integral in is dicult to handle analytically b ecause integration over the

cubical region B destroys the spherical symmetry of the problem exploited in the

FD

computation of E k in the last section In order to make analytical progress the

following approximation is intro duced The region B which is a cub e in kspace

Discretization errors in LES

is replaced by the largest sphere contained in it Clearly this pro cedure can b e

implemented simply by removing the sux B from the integral signs in and

replacing the energy sp ectrum E k by

n

E k if kk

min

m

E k

otherwise

The sup erscript min indicates that this pro cedure underestimates the true aliasing

error by failing to take account of the contribution of mo des close to the eight corners

of the cub e An alternative metho d that overestimates the error can b e provided by

replacing the cub e by the smallest sphere that contains it To obtain this estimate

min

one needs to use in place of E the following sp ectrum

p

max

E k if k k

m

E k

otherwise

The true aliasing error is then exp ected to lie b etween these twobounds With

the approximation so describ ed and with the energy sp ectrum dened as in

or the integral in may b e extended to the entire wave space Thus one

obtains

o n

X

alias

kJ k a P k P E k

mpnq imn

ipq

a

Substitution of the expression for J gives no summation over rep eated in

mpnq

dices

X X

alias

F K F K jP kj E k

imn

imn

a

X X

K K K K K K

m p m p n q

F K P kP kF K P k P k

imn imn

ipn ipq

K K

imnp imnpq

where K k a Note that in this case the F K do es dep end on the direction

i

of k so that the F K cannot b e extracted from the f g op eration Though the

i

summation over the set consists of terms for a cubical b ox one only

needs to evaluate terms due to symmetry Indeed the full set of aliasing mo des

a fall into three classes Rogallo

k k k

m m m

k k D k D fk k k D

m m m m m m

k k k

m m m

By symmetry all the contributions within each class are equal Therefore

alias alias alias

alias

E k E k E k E k

D D D

S Ghosal

alias alias

k is k is the contribution from any one of the D mo des E where E

D D

alias

the contribution from any one of the D mo des and E k is the contribution

D

from any one of the D mo des resp ectively If the mo died wavenumber kof

anumerical metho d and the energy sp ectrum of the turbulence E k are known

min max

maybeevaluated numerically using either E k orE k to get the lower

and upp er estimates for the aliasing error resp ectively

Subgrid and total contributions

The total nonlinear term N and the exact subgrid force S can b e readily written

down in terms of the Fourierbasis

Z

N kiP k dk dk k k ku k u k

i imn m n

and

ZZ Z Z

dk dk k k ku k u k S kiP k

m n i imn

B B

The p owersp ectra are dened as

S k

fhS kS k ig lim

i i

V

k V

N k

fhN kN k ig lim

i i

V

k V

where f g as usual denotes angular average over the sphere jkj k

FD

The evaluation of is similar to the calculation of E k inx One only

needs to replace in byP and drop the last term involving the

imn imn

viscosity The resulting expressions can b e further simplied using the prop erties

of the P tensors see pap er

imn

N k k F k F k F k

where F F and F are as dened in

The computation of S k once again requires us to restrict the k space integration

to a cubical domain which makes it dicult to handle the integrals analytically This

diculty is dealt with in precisely the same manner as was done in the computation

of the aliasing error The cubical domain in k space is replaced by a spherical region

of appropriate size This is completely equivalent to replacing the energy sp ectrum

min max

E k by a pseudosp ectrum E k orE k dened as

p

min

E k if k k

m

E k

otherwise

and

n

E k if kk

max

m

E k

otherwise

With this mo dication the calculation is exactly identical to that just presented

for the nonlinear term Thus one obtains

S k k F k F k F k

min max

where in the evaluation of the functions F the pseudosp ectrum E k orE k

i

should b e used in place of E k to obtain the lower and upp er b ounds resp ectively

Discretization errors in LES

10 1

10 0 k -1 E 10 y

10 -2

10 -3

-4

Energy densit 10

10 -5 -2 -1 0 1 2 3

10 10 10 10 10 10

Wavenumber k

Figure The VonKarman sp ectrum normalized so that the maximum energy

densityisat k and E

Application to LES

The results established in the previous sections will now b e applied to establish

quantitative measures of errors in LES In LES the grid spacing is typically much

larger than the Kolmogorov length so that molecular viscosityplays a negligible role

Therefore is set to zero throughout this section For the energy sp ectrum we

assume the VonKarman form

ak

E k

b k

where the constants a and b are chosen so that the maximum

of E k o ccurs at k and the maximum value E This can always b e

ensured by a prop er choice of length and timescales The VonKarman sp ectrum

has the prop erty E k k as k and E k k as k and is a fair

representation of inertial range turbulence A plot of this sp ectrum is shown in

Fig

Spectra

The p ower sp ectra N k and S k areevaluated numerically from and

resp ectively using the VonKarman sp ectrum We assume the LES lter to b e

equal to the grid spacing The results are shown in Fig for k and

m

where k It is seen that the p ower sp ectrum of the total nonlinear

m

term is reasonably at at high wavenumb ers while the subgrid contribution rises

monotonically to a maximum which app ears as a cusp when plotted on a linear

S Ghosal

10 2

10 1

0 y 10

10 -1

10 -2

10 -3

10 -4

10 -5 er sp ectral densit

w 10 -6 o P 10 -7

10 -8

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Wavenumber k

Figure The total nonlinear term compared to the lower N and upp er

H b ound of the true subgrid force for k and

m

scale at the cuto wavenumber k The subgrid contribution is seen to b e a

m

relatively small part of the total contribution from the nonlinear term

Subgrid mo deling is a very imp ortant part of largeeddy simulation A parametriza

tion of the interaction of the unresolved eddies with the resolved ones is expressed

as a subgrid mo del It is therefore desirable that the errors inherentinthenu

merical metho d b e much smaller than the physically motivated subgrid mo del We

now examine to what extentsuch an exp ectation is realized for a second order

centraldierence metho d implemented with the nonlinear term in divergence form

A second order centraldierence scheme is characterized by the mo died wavenum

ber k sink i or Eq is used to compute the p ower sp ectra

i i

FD

of the nitedierencing error E k for k and These results are com

m

pared to the p ower sp ectra of the resp ective subgrid terms in Fig Only two

values of k are shown for clarity The gures have the same qualitative app ear

m

ance for all values of k Thepower sp ectrum of the nitedierencing error rises

m

to a maximum at k k in the same manner as the subgrid contribution How

m

ever for all values of k the nitedierencing error is substantially larger than the

m

subgrid contribution over the entire wavenumb er range

Figure indicates that the error in a second order scheme cannot b e reduced to a

level b elow the subgrid contribution by suciently rening the grid As the grid is

rened k is increased b oth the error as well as the subgrid force decrease for all

m

wavenumb ers However the error continues to dominate the subgrid force through

out the wavenumb er range irresp ective of the resolution Let us now examine if this

situation can b e improved by using higher order centraldierence schemes Fig

ure sho ws the nitedierencing error evaluated using for a second fourth

Discretization errors in LES

10 2

10 1

0

y 10

10 -1

10 -2

10 -3

10 -4

-5 er sp ectral densit 10 w o 10 -6 P

10 -7

10 -8

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Wavenumber k

Figure Finitedierencing error for the second order centraldierence scheme

compared to the lower N and upp er H b ounds of the true subgrid force

for k and m

10 2

10 1

10 0 y

10 -1

10 -2

10 -3

10 -4

10 -5 er sp ectral densit

w -6 o 10 P 10 -7

10 -8

0 2 4 6 8

Wavenumber k

Figure Finitedierencing errors compared to the lower N and upp er

H b ounds of the subgrid force for k The numerical schemes considered

m

are second highest curve fourth sixth and eighth lowest curve order central dierences

S Ghosal

10 2

10 1

10 0 y

10 -1

10 -2

10 -3

10 -4

10 -5 er sp ectral densit

w -6 o 10 P 10 -7

10 -8

0 2 4 6 8

Wavenumber k

Figure The aliasing error for a secondorder centraldierence metho d

a fourthorder centraldierence metho d and an undealiased pseudo

sp ectral metho d compared to the lower N and upp er H b ounds of the

subgrid force Each metho d is represented by a pair of curves corresp onding to the

lower and upp er b ounds of the error

sixth and eighth order centraldierence scheme together with the subgrid term

computed using for a xed resolution k It is seen that higher order

m

schemes do lead to reduced levels of error However even with an eighth order

scheme the subgrid contribution is dominated bynumerical errors in ab out half of

the wavenumb er range

Figure shows the corresp onding comparison for the aliasing error computed

using In general increasing the order of a scheme has a relatively weak eect

on the aliasing error and the eect is primarily in the high wavenumb er region This

eect is in fact in the reverse direction compared to the nitedierencing error

That is the lowest pair of curves which corresp ond to a secondorder scheme have

the smallest aliasing error and the highest pair corresp onding to an undealiased

pseudosp ectral metho d have the largest The aliasing errors for sixth and eighth

order schemes are intermediate b etween the fourth and the pseudosp ectral they

have b een omitted from Fig for clarity The eect is of course quite easy to

understand In the one dimensional problem the aliasing part of the nonlinear

termismultiplied by the mo died wavenumber which approaches zero at the cut

o so that the aliasing error is also reduced to zero at k In the three dimensional

m

problem a similar situation applies except that the p ower sp ectrum do es not actually

go to zero on account of the averaging over wavenumber shells However the aliasing

error is reduced at high wavenumb ers for centraldierence schemes

Discretization errors in LES

Scaling laws

In this section the dep endence of some measure of global error on resolution

k isinvestigated An appropriate measure of the kind is

m

Z

k

m

E k dk

where stands for FD alias nl or sg corresp onding to the global nite

dierencing error aliasing error total nonlinear term or subgrid term resp ec

tively is closely related but not exactly equal to the the rms value whichis

given by the integral of the p ower sp ectrum over the entire wavenumb er range

The corresp ondence is not exact b ecause the mo des at the corners of the cub e

k k k k k k outside of the inscrib ed sphere of radius k

m m m m m m m

have not b een included in the denition Thus is a lower b ound of the true

rms value The can b e evaluated as a function of k bynumerically integrating

m

the p ower sp ectra E k presented earlier

Figure shows the lower and upp er b ounds measured by the corresp onding

for the subgrid force as a function of k The corresp onding quantity for the

sg m

total nonlinear term is also shown for comparison The subgrid contributions

nl

are seen to ob ey a p ower law A least square t gives

k Lower b ound

m

sg

Upp er b ound k

m

The total nonlinear term also app ears to followa power law A least square t in

this case gives

k

nl

m

The tted curves and are shown in Fig as dashed and solid lines resp ec

tively Thus the relative subgrid contribution is roughly k that

sg nl

m

is the role of the subgrid mo del decreases at higher resolution As an illustration

for an LES that resolves ab out a decade of scales b eyond the energy p eak the rms

value of the subgrid force according to this formula should b e in the approximate

range of the rms value of the total force

The following heuristic derivation Tennekes Lumley is sometimes

given for the scaling of the subgrid term The traceless subgrid stress is S

ij t ij

where is the eddyviscosityand S is the rate of strain The rate of dissipation

t ij

S jS j is a constant according to the classical Kolmogorov argument

ij ij t

p

Therefore j j jS j Now it seems reasonable to p ostulate that is the

ij t t t

q

pro duct of the gridspacing and the rms velo city of the subgrid eddies hu i

The latter can b e estimated from the Kolmogorovspectrum

Z Z

q

k k dk E k dk hu i

m

k k

m m

S Ghosal

10 2

10 1 sg or

nl 0 10

10 -1 0 1 2

10 10 10

k

m

Figure Global measure of the total nonlinear term and subgrid force

nl

lower b ound N upp er b ound H plotted as a function of the maximum

sg

resolved wavenumb er k The lines representpower law ts obtained by the least

m

square metho d

p

Thus so that j j The subgrid force whichis

t ij t

the derivativeof should then scale as j j k The scaling

ij ij m

exp onent in is reasonably close to what this rough argument predicts

It should b e noted that even though the subgrid stress decreases with increasing

resolution its derivative the subgrid force actually increases

Figure shows the integrated nitedierencing error plotted against k

FD m

There app ears to b e an asymptotic approachtoa power lawbehavior for large k

m

A least square p ower law t to the last three data p oints gives

Order

Order

Order k

FD

m

Order

Sp ectral

which are shown as solid lines in Fig The subgrid terms are also shown for

sg

comparison It is signicant that the exp onent in the dep endence of the integrated

error on resolution in turns out to b e indep endent of the order of the scheme A

higher order scheme reduces the error only through a reduced prefactor multiplying

the k term

m

Figure shows the integrated value of the aliasing error plotted against

alias

k The lines are p ower law ts to the data Only the second order scheme and

m

the pseudosp ectral scheme without dealiasing is shown The curves for the fourth

Discretization errors in LES

10 2

10 1 sg or

FD 0

10

10 -1 0 1 2

10 10 10

k

m

Figure Finitedierencing errors plotted as a function of the maximum

FD

resolved wavenumber k for central dierencing schemes of order topmost

m

and lowermost The solid lines are leastsquare p ower law ts Lower N

and upp er H b ounds of the subgrid force arealsoshown for comparison sg

10 2

10 1 sg or

alias 10 0

10 -1 0 1 2

10 10 10

k

m

Figure Upp er and lower b ounds of the aliasing error for a secondorder

alias

and undealiased pseudosp ectral metho d Solid and dashed lines are least

square p ower law ts Upp er H and lower N b ounds for the subgrid term

sg

are also shown for comparison

S Ghosal

sixth and eighth order schemes haveintermediate p ositions and have b een omitted

for clarity These ts are given by the following analytical expressions

k Lower b ound Pseudosp ectral

m

k Upp er b ound Pseudosp ectral

m

alias

k Lower b ound Secondorder

m

k Upp er b ound Secondorder

m

The imp ortant distinction from Fig is that here the curves are reversed Thus

the lowest curve corresp onds to the second order scheme and the highest corresp onds

to an undealiased pseudosp ectral scheme The subgrid term is also shown for

sg

comparison Of course for a sp ectral scheme prop erly dealiased with the rule

b oth the aliasing as well as the nitedierencing errors are identically zero

Discussions

The results of the ab ove analysis may b e summarized as follows In large eddy

simulation the net eect of the unresolved eddies on the resolved ones is repre

sented by a subgrid mo del The resulting equations which are the NavierStokes

equations augmented by an additional term the subgrid force is then solved nu

merically In such a pro cedure the presumption is that the asso ciated numerical

errors are small compared to the subgrid mo del b eing used Tokeep the analy

sis as simple as p ossible isotropic turbulence in a b ox with p erio dic b oundary

conditions was considered together with a simple numerical metho d an order n

n to centraldierence scheme with the nonlinear term in the divergence

form It was found that the p ower sp ectrum of the aliasing error is signicantly

larger than the subgrid term over most of the resolved wavenumb er range Higher

order schemes have the eect of increasing the aliasing error The nitedierencing

error for a secondorder scheme also remains signicantly larger than the subgrid

term over most of the resolved wavenumb er range The situation is improved by

going to higherorder schemes However even for an eighthorder scheme the error

dominates the subgrid term for almost half of the resolved wavenumb er range An

increase in grid resolution makes the errors increase faster than the subgrid force

so that the situation cannot b e improved by grid renement as long as the cuto

is in the inertial range

Wenow consider a p ossible remedy for this diculty In LES the NavierStokes

equations are rst ltered to remove all scales b elow some lterwidth The

f

resulting equations are then discretized on a grid of spacing In order that the

g

smallest resolved scales b e representable on the grid it is required that

g f

In practice one most often assumes to minimize computational cost and

g f

accepts the consequence that the marginal eddies may not b e well resolved As a

matter of fact this distinction b etween and is often ignored and one sp eaks

g f

of lterwidth and gridspacing interchangeablyHowever if one exp ects to ade

quately resolve all scales up to it is natural to require that beseveral times

f g

smaller than Rogallo Moin Thus we are led to consider an LES with

f

a lterwidth p erformed on a numerical grid of spacing Clearlyinany

f g f

Discretization errors in LES

10 2

10 1

0 y 10

10 -1

10 -2

10 -3

10 -4

10 -5 er sp ectral densit

w 10 -6 o P 10 -7

10 -8

0 2 4 6 8

Wavenumber k

Figure The nitedierencing error for a second order centraldierence

scheme with N for N upp ermost curve and lo wermost curve

f g

The lower N and upp er H b ounds of the subgrid force are shown for comparison

f

k is held xed m

10 2

10 1

0 y 10

10 -1

10 -2

10 -3

10 -4

10 -5 er sp ectral densit

w 10 -6 o P 10 -7

10 -8

0 2 4 6 8

Wavenumber k

Figure The nitedierencing error for for a second

f g

upp ermost fourth and eighth lowermost order centraldierence scheme The

lower N and upp er H b ounds of the subgrid force are shown for comparison

f

is held xed k m

S Ghosal

f g

such computation all Fouriermo des b etween k and k must b e

f g

m m

held at very low amplitudes for otherwise these contaminated mo des would so on

f

destroy the accuracy of computation of the mo des k through nonlinear inter

m

actions This mightbeachieved naturally by the eective dissipation range of

the eddyviscosityThismay also b e achieved by replacing the usual discretization

of the NavierStokes equations by the following alternative Lund

u F P F

i ij

F u u u

i j i

t x x x x x

j i j k k

where F represents a suitably designed ltering op eration that reduces the am

f g

plitudes of all mo des in the range k k to zero or very small values Compact

m m

lters for nitedierence schemes that are close to a sharp low pass lter in Fourier

space were rst considered by Lele They have b een used in the present

context by Lund Lund The nitedierencing op erator x is on the

j

ner grid The eect of this mo dication is easy to investigate in the present

g

formalism Thus for a second order metho d the in the expression for the mo d

ied wavenumb er need simply b e replaced with Figure shows the result of

g

such a computation for a secondorder centraldierence metho d with N

g f

f

where N and for a xed k It is seen that with N the

m

nitedierencing error is ab out one or two orders of magnitude b elow the subgrid

f

term throughout the wavenumb er range from k to k However taking

m

increases the numb er of grid p oints by a factor of and the

g f

total computational cost if the timestep t is limited by the CFL condition so

that t by a factor of It may therefore b e advisable to use instead

a higher order scheme in conjunction with a grid that is ner than the lterwidth

In Fig has b een xed at and the sp ectra of nitedierencing errors is

g f

plotted for a second fourth and eighth order scheme It is seen that for an eighth

order scheme the nitedierencing error is several orders of magnitude b elow the

subgrid term The increase in computational cost due to the rened grid is a factor

of Implementation of an eighth order scheme would also carry a p enalty

in terms of added cost However in view of the vastly increased accuracy the addi

tional cost may b e justied In addition to reducing the nitedierencing error the

ltering scheme completely removes the aliasing error This is b ecause mo des

k and k that alias to a mo de k must satisfy the relation k k k a where

a is a memb er of the recipro cal lattice Any comp onentofthevector on the

f

left of this equation can b e at most k so that the lefthand side cannot exceed

m

g f

or larger the Since at least one comp onent on the righthand side is k k

m m

f g

equation cannot b e satised if k k that is if there cannot

f g

m m

be any aliasing errors This is of course the well known dealiasing rule see

eg Canuto et al

Future plans

The analysis presented in this rep ort is kinematic in nature in the sense that

the departure of the righthand side of the NavierStokes op erator from its ideal

Discretization errors in LES

value is investigated The eect of this error on the dynamics of the solution and

ultimately on the prediction of averaged quantities is unknown However in the

light of the present ndings that these errors are comparable in size to the subgrid

term a careful and systematic study is required b efore nitedierence metho ds

can b e considered reliable Such a program of study should cho ose a sp ecic ow

for which reliable exp erimental data are available and for which issues suchas

sensitivity to initial and b oundary conditions are reliably known to b e unimp ortant

Numerical simulations should then b e p erformed using b oth sp ectral and various

nitedierence metho ds and the results compared to exp eriments and to each other

The eect of reducing errors using metho ds describ ed in x on relevant statistical

averages should b e studied

A study of this nature has recently b een undertaken byKravchenko and Moin

Kravchenko and Moin They used a channel ow sp ectral co de that uses

Bsplines in the wall normal direction and trigonometric basis functions in the

homogeneous directions By replacing the wavenumbers by the mo died wavenum

b ers in the homogeneous directions they were able to mimic various nite dierence

schemes Numerical exp eriments were run with various forms divergence rota

tional skewsymmetric of the nonlinear terms with staggered as well as nonstag

gered grids Aliasing errors in general were found to havea very serious eect on

the simulation causing the ow to laminarize in some cases as might b e exp ected

in the light of the present analysis The eect of aliasing errors on the simulation

as well as their size was found to dep end strongly on b oth the form of the nonlinear

term as well as the order of the scheme Aliasing errors had the most serious eect

for undealiased pseudosp ectral metho ds a result also consistent with the present

study The eect of aliasing errors on numerical simulations have also b een studied

by Blaisdell et al Zang Kim et al and Moser et al

among others using numerical simulations

Acknowledgments

Iwould like to thank Prof Parviz Moin for encouraging me to work on this

problem and for his supp ort and constructive suggestions in particular for sug

gesting the use of the VonKarman sp ectrum in x Prof Rob ert Moser Dr

Karim Shari Dr Alan Wray and Dr Michael Rogers read the manuscript and

made useful suggestions for which I am grateful The discussion in x is partly

motivated by a suggestion from Dr Thomas Lund

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