Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

AIAA 98-2803

MATHEMATICAL AND PHYSICAL CONSTRAINTS ON LES

Sandip Ghosal*

Los Alamos National LaboratoryLos Alamos, NM 87545, USA

This review attempts to summarize recent progress in thetheoretical foundations of LES. Most of the work reportedhere is motivated by conceptual difficulties encountered in ap-plying the LES method to inhomogeneous complex geometryflows. Among the topics covered are the problem of the lackof commutation between filtering and derivative operators forinhomogeneous flows, the issues of enforcing symmetry andreadability conditions in subgrid modeling and the problemof unacceptably high numerical errors in LES implementa-tions with finite difference methods.

I. INTRODUCTION

In direct numerical simulation (DNS) of Navier-Stokes(NS) turbulence, the numerical resolution is sufficientlyfine so as to resolve all scales of motion that carry sig-nificant energy. It is well known that such resolution re-quirements make DNS prohibitively expensive for manyaerospace applications. The Reynolds averaged Navier-Stokes (RANS) approach is much cheaper computation-ally but require nonuniversal closure models which are of-ten difficult to construct, especially in problems involvingcomplicated geometry and flow separation. An interme-diate approach is large eddy simulation (LES), where oneonly seeks to resolve those eddies that are large enoughto contain information about the geometry and dynamicsof the specific problem under investigation, and, regardall structures on a smaller scale as 'universal' followingthe viewpoint of Kolmogorov. The difficulty of this ap-proach is, there is no real separation between the largeand small scales, the division is merely a convention. Thisprevents a systematic approximate solution of the closureproblem along the lines e.g. of the Chapman-Enskog de-velopment for the Boltzmann's equation. In the latterexample, there exists a regime of interesting problemswhere the scale of variation of hydrodynamic variables ismuch larger than the molecular mean free path. Such ascale separation is not possible in the case of LES.

"Research Fellow. Center for Nonlinear Studies. Copyright©1998 by the American Institute of Aeronautics and Astro-nautics. Inc. All rights reserved.

There are two possible approaches to thinking aboutLES. In the first, for every dynamical variable / (e.g.pressure)_one explicitly defines a smoothed or 'filtered'variable / defined as an appropriate local average of /. sothat / is a smoother function that only follows the largescale structure of / and lacks the small scale fluctuations.One then derives from the Navier-Stokes equations a setof equations satisfied by the corresponding filtered vari-ables. These equations are exact as no approximation hasbeen made so far. However, the LES equations are un-closed. At this stage an approximate model is chosen forthe unclosed terms to give a set of model LES equations.These equations are then solved numerically.

A second possible approach is to simply discretize theNavier-Stokes equations using some appropriately chosennumerical scheme, on a grid that is too coarse to allow forany sensible resolution of the fine structure generated bythe exact Navier-Stokes equations. One then tries to ac-count for any problems created by this lack of resolutionby adding additional terms or modifying the numericalscheme itself to 'fix' these problems (such as insufficientenergy dissipation). Examples of this approach are sim-ulations that do not use any subgrid models at all butrely on the inherent dissipation of numerical schemes tomimic transfer of energy to smaller scales through triadinteractions.

The main advantage of the first approach is, it enablesone to address the issues of subgrid modeling and nu-merical methods separately thereby enabling one to dealwith two very difficult problems one at a timel Thishas a distinct advantage for any theoretical analysis ofeither subgrid modeling or numerical errors. The sec-ond approach may be cheaper (no overheads for subgridmodels!) when it works. However, when it doesn't itis more difficult to figure out exactly what went wrong(e.g. trying to improve grid resolution to resolve meanvariables may trigger an instability because of decreasednumerical dissipation!) In this paper we will assume theformer approach to LES simply because it is more struc-tured and therefore is a lot more amenable to theoreticalanalysis. There exists a not too precisely stated "hope",that, when one solves the Navier Stokes equations on adiscrete grid with a dissipative scheme, perhaps one ob-tains exactly the solution of a set of LES equations withsome combination of 'filtering' and 'subgrid model' that

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is "implicit''. That is. we do not know exactly what thefiltering scheme and subgrid models are, but there ex-ists a scheme and a model such that the correspondingLES equations when solved would give the numerical so-lution exactly. This belief however is not true, as hasbeen pointed out in the literature.1-19.

In the next section we discuss the basic equations forthe filtered variables. In the case of inhomogeneous flows,there is a "closure problem" that arises not due to thenonlinearity of the Navier-Stokes equations but becauseof the failure of the derivative operator to commute withthe filtering operation and/or because of the presenceof finite boundaries. The various approaches for dealingwith this problem is discussed. In § 3 we discuss theissue of subgrid modeling and the constraints imposedon the subgrid model by the mathematical structure ofthe Navier-Stokes equations. In § 4 we discuss the is-sue of discretization errors in LES. On account of thepresence of a continuum of spatial scales a new approachto error analysis is required with errors being character-ized by their power spectra rather than single numericalvalues. The power spectra depends on both the numer-ical method as well as the energy spectrum but may beexpressed in separable form. The principal results aresummarized in § 5.

For simplicity, and due to limitations of space and theauthor's expertise, discussion will be limited to incom-pressible turbulence. However, many of the results areeasily extendible to the general case of compressible flows.

II. THE BASIC EQUATIONS OF LES

We will follow the index notation for tensors with thesummation convention. The Navier-Stokes equations aretherefore written as follows:

UjUj (6)

+ Ui = --dtp + vdkk^iP (1)

(2)

where v and p are the (constant) kinematic viscosity anddensity respectively, u, is the velocity and p is the pres-sure. For any dynamical field /(x. t) the correspondingLES field is defined as

(3)

where G is some function ("the filter") with an effectivewidth of the order of unity and A is the "filter-width".On applying the operation (3) to (2) we derive the basicequation of LES:

i + M - -~dip +P

M = Q

(4)

(5)

where

is the (unclosed) subgrid stress. Note that the first termon the right hand side of (4) is a consequence of theconstancy of p.

A. The trouble with ^.homogeneous flows

The derivation of (4) and (5) are valid provided allthe differentiation operators commute with the filteringoperator, that is, if the following is true for any function/(*):

(7)

It is easily verified that (7) is true if the the filter width Ain (3) is a constant. However, for a wall bounded flow, ifthe distance from the wall is less than A/2 one will needto "truncate" the filter function 'G', or do somethingsimilar in order for the filtering operation (3) to be welldefined. Thus, either A must be variable or the func-tional form of G itself must change as one approacheswalls. This is entirely consistent with the definition ofLES, since the length scale below which eddies can be re-garded as "universal" decreases as one approaches walls.For the same reason a variable filter width is also essen-tial for the proper formulation of LES in inhomogeneousflows (of which wall bounded flows represent a specialcase). The fact that variable filter widths result in a vio-lation of (7) invalidating the basic equations (4) and (5)was recognized early2. The problem however has beenlargely ignored until recently, perhaps as a result of be-ing overshadowed by more serious problems related tonumerical methods. It was hoped that any errors thatresult in ignoring this problem would perhaps be com-parable to other sources of errors, viz. errors due tothe subgrid closure, truncation errors, aliasing errors andtime-stepping errors. While such an approach might havebeen not entirely unreasonable at the time, the rapid ad-vancement17'16 of LES in recent years has necessiated amore careful examination of this issue.

B. Navier-Stokes acquires extra terms

A systematic analysis of this "commutation error" maybe undertaken along the following lines3.

We consider the one dimensional case of a dynamicalvariable u(x), the generalization to three dimensions isobvious. A one to one transformation from an infinitedomain — oo < £ < +00 to a finite domain a < x < bmay be affected through a monotonic function

£ = /(*) (8)such that /(a) = —oo and f(b) = +00. In the £ spacethere are no boundaries so that filtering with a uniform

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filter width A is easily defined as in (3). Filtering inphysical space x is then defined as follows:

1 Transform variables x —»• £ in u(x).2 Filter u(f~l(£)) the usual way.3 Transform back £ —» x to get filtered field.

Mathematically, this operation amounts to the followingdefinition:

+oc(9)

(10)

It is easy to show that away from boundaries, (10) isapproximately equal to filtering with the kernel G butwith a variable filter- width 5(x) — A//'(x), that is,

It should be noted that in the definition (10), due to thefunction / approaching ±00 for x approaching bound-aries, the integrand never becomes undefined. However,if the right-hand side of (12) were used to define the fil-tering operation, the integrand would become undefinednear boundaries and one would have to restrict the in-tegration to a finite domain thereby creating additionalboundary terms. This difficulty is avoided in the defini-tion (10) by "transforming away" all boundaries to in-finity (an alternate approach that explicitly retains theboundary terms has been proposed by Fureby and Ta-bor28). The filter (10) actually differs markedly from theapproximate form (12) near boundaries, as the kernel be-comes visibly asymmetric3. It should be noted that thereasonable requirement that the filter-width, 8, shouldapproach zero near boundaries does not mean that thegrid spacing h should also approach zero (which is impos-sible to meet with finite computing resources!) In fullydeveloped turbulence, the smallest scales that need to beresolved is set by '<?'. However, near boundaries, 5 ->• 0,so that, the filtered field becomes indistinguishable fromthe true Navier-Stokes solution and the structures oneneeds to resolve are the smallest physical structures setby the viscous cut-off, '<?' (not SI). In general, the reso-lution requirement may be written as h < m.in[6, i}. Forhigh Reynolds number turbulence and far from bound-aries, this is equivalent to h < 6 while near walls theresolution requirement reduces to h < i.

An expression for the commutation error can easily bewritten down using the definition (10):

-dx dx A A

«W(y) 1 -

If G is symmetric (as with most commonly used filters)then it is easily shown using Taylor series analysis thatC[u] = C2(x)A2 + • • • where cz(x) is a coefficient indepen-dent of A and proportional to the second moment of G.Since the filter width and the grid spacing are usually ofcomparable magnitude, the error made in assuming thatthe derivative and filtering operations commute is of thesame order as the truncation error of a second order finitedifference scheme. In turbulence computations, high or-der methods such as spectral methods4 and more recentlycompact schemes6 are usually preferred over second or-der schemes. Therefore, the commutation error can bea serious threat to the accuracy of turbulence computa-tions, degrading the resolution of a high order method tono more than a second order scheme.

One method of correcting for the commutation erroris to approximate C[u] by a linear combination of deriva-tives u"(x), u""(x),--- and choose the coefficients suchthat the residual error is less than any desired order (inpractice the same order as the truncation error). Thefollowing approximation for example has fourth order ac-curacy:

where 8(x) = A//' (re) is the local filter- width introducedearlier and a = /C2G(C)dC- The expansion (14) canbe applied to derivatives in each direction in the Navier-Stokes equation and the equations for the filtered field ucan be written down. They are the "usual" LES equa-tions augmented by the higher order derivative terms(they will not be written down here but the interestedreader may find them in the original paper3).

The physical meaning of the additional terms may beeasily understood. For this, consider a Gaussian wavepacket u(x) traveling from left to right according to theevolution equation

du du(15)

Suppose that the nhysical space filter width 6(x) is in-creasing monotonically from left to right. Then as thewave travels it would encounter ever increasing filterwidths so that u would have a decreasing peak and broad-ening width as it propagates to the right even thoughu(x) travels unchanged in form. On applying (14) to(15) we find

du du (16)

(13)

where v = a52(6'/6) is positive. The term on the righthand side of (16) is a diffusion term which causes the nec-essary spreading of the wave packet u. The interpretationcan be confirmed by exactly evaluating u and comparingwith the prediction of (16) for a simple choice of <5(x).The opposite effect is observed for a wave packet moving

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from right to left. In this case the diffusion term is re-placed by an antidiffusion term, so that, the wave-packetincreases in amplitude and becomes more and more con-fined. However, for such a wave, if the domain of x is(—00,+00) the wave packet would diverge like a deltafunction due to the anti-diffusion term. This is clearly anincorrect prediction since u(x) should clearly converge tou(x) as the wave moves to the left and the filter-widthapproaches zero. This difficulty arises because as theprofile of u becomes sharper its higher order derivativesbecome larger and ultimately the "correction" term onthe right of (16) becomes of the same order as the otherterms invalidating the approximation unless further cor-rection terms are brought to play on the right hand sideof (16). For a semibounded domain (a, oo) (or boundeddomain), which is a more realistic situation, the problemdoes not arise, since, a wave moving to the left, thoughinitially amplifying, is ultimately reflected on reachingthe boundary and subsequently travels to the right withdecaying amplitude.

Though the effect of the correction term is easy to un-derstand in this simple situation, its effect in the contextof the full Navier-Stokes equations is unknown since LESwith the correction terms has never been done. An in-teresting model problem to study may be LES of thecollision of a vortex with a wall where one should beable to observe the creation (destruction) of fine struc-ture brought about by the additional terms as the vortexmoves towards (away) from the wall (the zone of finestresolution).

A question that immediately comes to mind is whetherboundary conditions in addition to the usual 'no slip'ones are needed for the LES equations near rigid walls,since by adding the extra terms we have increased theorder of the basic equations. To examine this questionlet us consider the simplest of boundary value problems

^u — n

with the boundary condition

u(0) = 1-

The solution is clearly

u(x) = 1

which implies

u(x) = 1.

(17)

(18)

(19)

(20)

Now let us see what is needed to recover this solutionfrom the equation for u obtained by applying the filteringoperation to (17) and using (14):

du~dx

dru(21)

Let us choose as an example the mapping f(x) = logx,so that v = aS2(6'/8) = aA2x. Thus, the equation for uin this elementary example is:

du—ax

d2u

which has the general solution

u(x) = 1 + Cx I+Q"

(22)

(23)

given the boundary condition u(0) = 1. It is clear that(20) cannot be derived as the unique solution to (22) nomatter what boundary condition is spanned for du/dx.This is because the equation (22) is sinj ar at x = 0 dueto vanishing of the coefficient of the second derivativeand the uniqueness theorem for solutions of second orderdifferential equations do not hold for solutions throughx = 0.

The correct condition needed to recover (20) is theboundedness condition

lim u(x. A2

A-+O ' (24)

should be finite if the limit is taken with fixed x. Apply-ing (24) to (23) readily implies C - 0 and (20) follows.A more practical way of enforcing (24) is to look for anasymptotic solution in the form

U = UQ (25)

which from the start rules out solutions that are singularas A -> 0. On substituting (25) in (22) we have thefollowing chain of equations:

dup~dxdun

dx

= 0

= OiX- dx2 (n=1.2, '

(26)

(27)

(28)

Since u —)• u as z —» 0. we have the boundary condi-tions un(0) = 1 if n = 0 and 0 otherwise. Thus, at eachstep we have a differential equation of the same orderas the original one and no additional boundary condi-tions are required. The procedure has been worked outin detail in the paper by Ghosai and Moin3 for the incom-pressible Navier-Stokes equations. However, the authorspresent the asymptotic method as an alternative to in-troducing additional boundary condition. This howeveris incorrect. As noted above, additional boundary con-ditions do not determine the solution uniquely and arenot needed. The correct condition is the requirement(24) which is enforced through the asymptotic expansion(25). The procedure does involve added computationalcost since each time-step would require one or more ad-ditional evaluations of terms on the righthand side andadditional storage may also be required. Therefore, it isattractive to examine, whether, it is possible to choosethe filter function G cleverly, so that, the filtering and dif-ferentiation do commute, with an error not greater thanthat the truncation error, so that, the additional termsare ignorable. This is discussed in the following section.

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C. Making the extra terms go away!

Harmen van der Ven' proposed a form of the filterfunction 'G' such that the commutation error is made assmall as desired, van der Ven worked with the filteringdefinition

u(x] = W}f-IG(wi)u(y]dy (29)

where S(x) is the space dependent filter-width. Aspointed out earlier, this definition is approximately thesame as the definition (10) away from boundaries, but,near boundaries the integrand becomes undefined, or,if the integration is stopped at the domain boundaries,"boundary terms" arise as a consequence of integrationby parts. Van der Ven noted this difficulty, but left theproblem open for future investigation. If we ignore thisboundary problem for the time-being, the commutationerror may be written as follows:

s'(x] r+ccC[u] = -4 / [G(0 + CG'(C)] f(x - #(*)).

0(X) ./-oc(30)

Van der Ven observed that if G is chosen as the solutionof the following equation:

where e is arbitrary, then the right-hand side of (30) maybe written after n integration by parts as follows

(32)

Thus, the commutation error can be made arbitrarilysmall. The equation (31) is easily solved by takingFourier-transforms. In particular, if n = 2m (a posi-tive natural number), and if e = (— l ) m a (a > 0), oneobtains the two parameter family of filters Gm(a, C) de-fined through the Fourier-Transform:

(33)

In particular m = 1 corresponds to a Gaussian fil-ter. For all of these filters, the commutation error~6'(x)S2m-l(x).

Van der Ven's analysis has recently been generalized byVasilyev et al.1 so as to contain the earlier work of Ghosaland Moin and van der Ven as special cases. Vasilyev et al.use the filtering definition (10), with /(a) = a and /(&) =/? but G is allowed to be a function of (T? — £)/A as wellas £ separately. They are able to write the commutationerror in the following form:

+ 00 + 00

(34)fc=l fc=0

where the moment M*. is denned as

^(35)

If the filter is chosen so that M0(0 = 1 and A4(£) = 0for k = 1,2. ---,n - I then from (34) it follows thatC[u] ~ An. There exists a wide class of such filters withn—l vanishing moments, as pointed out by Vasilyev et al.One example is the one parameter family (33) introducedby van der Ven. Another is the correlation function ofthe Daubechies scaling function used for constructing or-thonormal wavelet bases. Vasilyev et al. also show howto construct discrete versions of such filters on numericalgrids.

D. Grid &: test-filters, self-similarity and all that

In the classical Smagorinsky model

ij = -2CA2|S|Si;- (36)

with a constant C, though the concept of a filter is re-quired, one does not need to know how exactly it is de-fined because one does not need to use the filtering op-eration explicitly in solving the basic equations (4), (5)and (36).

The situation is different in various types of "mixedmodels"8-9. Here one decomposes rtj as follows

= Li + + (37)

where Lij = u^Uj — Uitij, Cij = Uiu'j + UjU\ and Rij =u'jU'j. One then argues that it is only the parts dj andRij that need to be modeled since Lij is known in termsof the LES field On. The Smagorinsky model is thereforeapplied only to the traceless part of dj + Rij , and. Lijis computed explicitly. In this formulation one does needto know explicitly what the 'filter' is since it is needed inthe formulation.

Another class of models where knowledge of the filteris required, at least indirectly, are the "dynamic mod-els"11'13-10'12. In the dynamic model one formally appliesa second filter "~" with filter- width A > A to (4) and(5) to generate a set of equations identical to (4) and(5) but with the operation " ~ " replacing the " ~ " and

replacing T^. One then argues thatthese equations are essentially identical to the LES equa-tions (4) and (5) except for the length-scale A. Therefore,one may use the same 'subgrid model' for T^ as for T^but simply use A as the relevant length scale instead ofA. There exists a relation between T^ and T^ as notedby Germane et al.ux

T — T. . <?•. (1K\J-nj — J-ij — TIJ. (38)

Taking the traceless part of this identity and substitutingthe Smagorinsky model in it, we obtain, upon using theself-similarity assumption for Tij,

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13J ij = -2CA2|S|Sy + 2A2C|S|5ij. (39)

This is an equation for C that can be used in variousways to determine C as is well known (see the referencesat the beginning of this paragraph). This fact was firstrecognized by Germane. Piomelli, Moin and Cabot whoshowed great ingenuity in figuring out a way to exploitthe identity to extract the 'unknown' coefficient 'C", atask that at first sight would seem impossible!

One of the assumptions in the above argument is thatthe filters "~" and "~" are self-similar, that is, the ker-nels are identical except for a scale factor. While thisis certainly true for the Gaussian filter as well as theFourier-cutoff filter, it is not true in general. In practi-cal implementations of the dynamic model, the so called"top-hat" filter (G(x) = 1 if \x\ < A/2 and 0 otherwise)is often used. However, the convolution of two "top-hat"filters is not another "top-hat" filter as one can easilyverify. This apparent inconsistency was pointed out byCarati et al.14 who also proposed an ingenious solutionto the dilemma reminiscent of "renormalization group"ideas. They considered filters constructed by infinite it-eration of base filters or "generators" GA'-

* GA/I * GA/& * (40)

where '*' denotes the convolution operator, and GA de-notes the kernel of the generator _with the filter widthparameter equal to A. Thus, if A = 2A, the test fil-ter G^ = £/2A * GA is by construction self-similar to thegrid filter. In practical implementations of the dynamicmodel one only uses the filter "^'^ that is Q, explic-itly. According to (40), once the test filter Q is chosen,the grid-filter is determined uniquely from the require-ment of self-similarity. In particular, as mentioned be-fore, the Fourier-cutoff filter and gaussian filter fall withinthe class of filters defined by (40). The top-hat filterdoes not. However, the top-hat filter may be used as thegenerator to generate self-similar filters according to theprescription (40).

III. CONSTRAINTS ON SUBGRID MODELS

As discussed in the introduction, as of this date, thereexists no systematic method for deriving a closure forthe subgrid stress,x that is, there exists no satisfactory"theory" of turbulence. Under the circumstances, thebusiness of "subgrid modeling" is essentially that of mak-ing an educated guess. It is therefore a wise strategy toreduce the possible choices somewhat by restricting theguesses to those that satisfy properties of the subgridstress that can be proven to be true. These restrictionsbelong to two classes, the symmetry requirements andthe realizability requirements, the subject of the remain-der of this section.

A. Symmetry requirements

The concept of symmetry is fundamental in physics.It means e.g. that nature does not distinguish betweendifferent points and directions in space so that the equa-tions describing physical phenomenon should have thesame form in reference frames translated or rotated withrespect to each other. The Navier-Stokes equations them-selves show many symmetries such as invariance undertranslation, rotation, reflection, galilean transformationsand scale invariance. One must not however jump tothe conclusion that the LES equations therefore, do or"should" exhibit the same symmetries. The point is, inperforming the "coarse graining" by means of the filteringoperation to go from Navier-Stokes to the LES equations,one may break one or more of these symmetries. For ex-ample, if one chooses different filter-widths for differentpoints of space the symmetry under translational invari-ance of the NS is broken. This is not something that is"wrong" or must be avoided, but it is merely a conse-quence of our choosing not to treat all points of spaceas equivalent but deliberately choosing different resolu-tions in different regions for good reason. Of course, thesubgrid model must reflect the same symmetries as theexact LES equations, but, in deciding which symmetriesthe subgrid model needs to satisfy, one must be carefulin recognizing which symmetries are "inherited" by theLES equations from the NS, and which symmetries are"broken".

Translational symmetry: If the origin of the co-ordinate system is shifted by an amount a, a givenpoint has co-ordinates x' = x — a in the new refer-ence frame. Clearly, the velocities are unaffected, so thatu'(x') = u(x). Now the question is, does the filtered fieldu satisfy the same symmetry, that is, is u'(x') = u(x)?By elementary transformation of variables it follows that

(41)

(42)

and this equals u(x) if and only if G(x — a, y — a) =G(x, y), that is, if G is a function of x - y only.

In inhomogeneous flows one often uses filters such as(10) and (29) which do not have the above form, andtranslational symmetry is therefore broken in the LESequations. As an example let us consider the problemof a wall bounded flow. _ If one uses a subgrid modelTij — (!/3>)6ijTkk = —2vtSij where the eddy-viscosity isdefined as ft oc t/3 when y is less than some length 'Z/' andconstant otherwise, the subgrid model would not be in-variant under translations in the y-direction. This, how-ever is perfectly legitimate since invariance under trans-lations in the y direction is a "broken symmetry" for theLES equations derived with a y dependent filter. Anyattempt to "enforce" this nonexistent symmetry in the

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subgrid model would result in incorrect behavior of theequations.

Rotational symmetry: Arbitrary rotations of the co-ordinate system with the origin fixed are described byx\ = AijXj, where Aij is a unitary matrix. It is wellknown that u; transforms as a vector u'i = AijUj. Thequestion that we should ask is whether ut has the sametransformation rule as u,;, that is, whether the filter "~"is a scalar operator. It is easy to convince oneself thatthis is not in general true, for

(43)

= G (Ax, Ay) AijUj(y)\det A\ dy, (44)

(45)

and the right-hand side equals AijUj(x) if and only if(note that |deij4| = 1 for rotations)

and the right-hand side equals AijUj(x.) if and only if(note that \detA\ = 1 for reflections)

G(Ax,Ay) = G(x,y), (50)

G(Ax,Ay) = G(x,y), (46)

that is. if G depends only on |x - y| the invariant underarbitrary rotations.

A consequence of this is, for filters that are not spher-ically symmetric, the filtered fields do not necessarily in-herit the vector or tensor properties of the underlyingfield. In particular, the subgrid stress rjj is not a tensor,that is T-J 7^ AimAjnTmn unless the filter is sphericallysymmetric. However, if the Reynolds number is suffi-ciently high so that the effective filtering zone character-ized by the length '<T still contains a very large number ofsubgrid eddies the filtered field would be almost indepen-dent of the shape of the kernel G. Thus, the departureof TJJ from the tensor transformation rule would be neg-ligible for robust turbulence away from walls. Invarianceunder rotations is therefore always preserved for spheri-cally symmetric filters and also preserved to a very goodapproximation for other filters provided the filtering zonecontains a sufficiently large sample of subgrid scales.

All subgrid models known to the author are designedso that the modeled subgrid stress has the tensorial prop-erty in situations where the filtering can be regarded asa scalar operation.

Parity invariance: Reflections or parity transforma-tions (replacing a "right-handed" system by a "left-handed" one) is described by the co-ordinate transfor-mation rule x'j = A^XJ where detA = — 1. The NSequations (and all of classical physics) are known to beinvariant under such "mirror reflections". The transfor-mation rule for the velocity in this case is

(47)

= IG (Ax, Ay) AijUj(y)\det A\ dy, (48)

(49)

that is, if G remains unchanged when one passes fromright-handed to left-handed systems.

A consequence of this is, asymmetric filters such as (10)may cause the reflection invariance of the NS equationsto be broken in the LES equations.

The subgrid models known to the author satisfy thissymmetry requirements when the filter is chosen so thatthe LES equations inherit this symmetry. Exceptionswould be any model that attempts to represent TJJthrough combinations such as UiUj (where Wj is the vor-ticity vector). This is because Wj is a pseudo-vector andUiUj therefore would not transform the same way as r^.

Galilean Invariance: The NS equations and all the ba-sic laws of physics are Galilean invariant, that is, allreference frames translating uniformly with respect toeach other are equivalent. Since the filter kernel of LESG(x,y) is time independent, Galilean invariance is sat-isfied whenever translational invariance is an inheritedsymmetry. Thus, for filters of the type G(x - y) theLES equations preserve the symmetry under arbitraryGalilean transformations. In situations where one direc-tion, say the y direction is inhomogeneous (requiring ydependent filter-widths) Galilean invariance is true onlyfor the homogeneous directions.

All subgrid models known to the author satisfy therequirements of Galilean invariance with the exceptionof the so called "mixed models" discussed earlier. It iseasily seen that each of the component terms Lij. djand Rij change on transforming to a frame uniformlytranslating with respect to the given one. However, theextra terms cancel on adding the components together sothat Ty is invariant. This precise cancellation no longerhappens when dj + Rij are replaced by a model thatdoes not necessarily produce the additional term neededfor the cancellation on transforming to the moving frame.The problem has been known for some time and has beenpointed out in the literature.

Scale Invariance: Scale invariance is simply a state-ment of the fact that our basic units of measuring spaceand time are arbitrarily chosen so that the equations de-scribing physical phenomenon should have the same formno matter what units are adopted. 'Dimensional analysis'used extensively in fluid dynamics is simply a statementof this scale invariance symmetry. Since the filter G isalways written in terms of dimensionless variables

(51)

(Go is dimensionless) in order for the filtered fields tohave the correct dimensions, the scale invariance symme-try is always inherited by the LES equations.

All subgrid models (unless they are dimensionally in-correct!) satisfy the scale invariance symmetry.

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B. Realizability

The requirement of readability of subgrid models wasfirst pointed out by Schumann23 in the context of RANS.The readability condition implies in particular that theturbulent kinetic energy k — (l/2)(u'iu'i) is non-negative,where { } denotes either ensemble average, space averageor time average. A more general statement of the realiz-ability condition is that all three eigenvalues, AI , A2 andAS of the stress tensor r^- must be non-negative. Thisfollows immediately on choosing a co-ordinate system inwhich Tij has the diagonal form (this is always possiblesince Tij is symmetric). Then we have

AI = ru = £ > 0

2 = r22 = £ ("2*4} > 0

S = T33 = ~(u'3u'3) > 0.

(52)

(53)

(54)

The equations (52) to (54) can be written in severalequivalent forms23. Some of these are

1. The quadratic form

is positive semi-definite (Q > 0).

2. The three principal invariants ofnegative:

0

(55)

are non-

(56)

(57)

(58)

L to(no summation over repeated indices, i,j3.)

3. The following chain of inequalities are true (theyare not independent):

T^ > 0 if i = j (59)T?j<TiiTjj lii^j (60)

det[Tij] > 0 (61)

(no summation over repeated indices).

4. The following conditions are true:

TU > 0 (62)TllT22 - T2, > 0 (63)

det[Tij] > 0 (64)

LES differs from the RANS approach in that the ensem-ble average ( ) must now be replaced by the filteringoperation " ~ x" . The proof of the readability require-ments rest on the following properties of the averagingoperator:

1. (a) = a

2. (a/) = a</>

3- (/ + <?> = (/} + <<?>

4. (/2) > 0

5. (/} is a constant

where / and g are any two dynamical fields and a and bare constants. It is easy to see that the linearity proper-ties (1). (2) and (3) also hold for the filtering operation.However, the property (4) is true only for non-negativefilters. Property (5) is not valid, the filtered field is slowlyvarying compared to the unfiltered field, but it is not aconstant. However the realizability conditions can beshown to be true in the LES case for non-negative filtersprovided one adopts the standard LES definition of thesubgrid stress, ry = uiuj — Uiiij. In situations wherethe filtering can be considered a scalar operator (see ear-lier discussion) we can write the definition of the subgridstress in the co-ordinate frame in which the stress tensorhas the diagonal form. Thus,

A2 = 722 = U\ —

3AS = 733 = "3 -

(65)

(66)

(67)

The right hand sides of each of these expressions arenon-negative if the filter G is non-negative. This can beproved in the following way. Let us assume that G > 0,and, without loss of generality, we choose to work withfilters that are normalized to unity, / G(x) dx = 1. If werepresent continuous integrals in terms of discrete sumsthen

(68)

where Wi are a set of weights such that u^ > 0 and£^ ̂ = 1, and "lim" denotes the usual operation ofpassing to the limit of an infinite number of infinitesi-mal subdivisions. We now note the following inequal-ity obtained on substitution of the n-dimensional vectorsa-i = \/wl and bi = ^/wlfi (note Wi must be non-negative)in the Cauchy-Schwarz inequality |a-b|<||a| | |b| | withrespect to the Euclidean metric:

(69)

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On using this inequality on the right hand side of (68) itfollows that all three eigenvalues AI, A2 and AS in (67) arenon-negative. Thus the realizability conditions (52) to(54) and its various equivalent forms are rigorously truein LES provided only non-negative filters are employed.The restriction to non-negative filters may be undesirablesince the "commuting filters" discussed earlier and sev-eral commonly used filters (such as "Fourier-cutoff") donot have this property. However, if the Reynolds numberof the turbulence is sufficiently high so that the "effec-tive filtering volume" contains a statistically significantrange of eddies, it is expected that the result of the fil-tering would be independent of the precise form of thefiltering kernel. Thus, in practice, the realizability con-ditions (52) to (54) can be assumed to be valid for LESeven though strict validity can only be demonstrated inthe case of non-negative filters.

In the standard algebraic closures of the incompress-ible version of the LES equations, only the traceless partof the subgrid stress is modeled, the isotropic part is ab-sorbed into the pressure term to give the "effective pres-sure" p + (2/3)fc (k is the subgrid kinetic energy) whichis determined from the continuity equation. Since k isnever known separately there is no way to check in thesemodels whether or not the realizability conditions aresatisfied. In one equation versions of the model10 onesolves an additional transport equation for k so that k isknown. The usual subgrid closure in this case is

!• (70)

If (sQ,s/3,s7) denote the principal strains ordered suchthat sa > S0 > s7 (note that sa + sp + s7 = 0 forincompressible turbulence so that sa > 0 and s7 < 0),the realizability conditions may be written as

Vk'3A|s7 3AsQ

(71)

It is difficult to supply a proof that any given one equa-tion closure satisfies (71). One is usually restricted toa "numerical proof" where statistics is collected at eachgrid point and the inequality (71) is checked to see if real-izability is violated at 'only' a small fraction of the com-putational volume. Sometimes the C value is 'clipped'if it falls outside the theoretically allowed range (71) butthis is not a satisfactory method for obvious reasons. Themore restricted realizability condition k > 0 can usuallybe theoretically guaranteed for one equation models10.

IV. ANALYSIS OF DISCRETIZATION ERRORS

The discussion so far was restricted to issues relatedto the derivation of the equations for the filtered fieldsfrom the Navier-Stokes equations. Once the basic equa-tions are written down, the next step is to discretize it

in time and space and solve it numerically to completethe LES procedure. For LES to have any credibility, thesources of errors in the numerical procedure must be un-derstood, quantified and controlled. In the author's view,rigorous error analysis of this kind needs to complementperformance tests where certain physical quantities arecomputed and compared with experiments in a full scalesimulation.

Careful error analysis in LES, and indeed, in DNS aswell, involves some special difficulties. The problem is,turbulence cannot be characterized by a single space andtime scale that can be normalized to unity. Consider forexample the one dimensional wave equation

du du (72)

If the space variable is discretized and the re-derivativeevaluated with a second order central difference scheme,it is well known that the resulting truncation error ~u'"A2/6. In simple problems, such as the propagationof a wave-packet, one can reasonably define unique char-acteristic length and time scales so that u'" is of orderunity if the equations are non-dimensionalized with thesecharacteristic scales. Thus, one may reasonably concludethat for the central difference scheme, the truncation er-ror ~ A2. This is no longer true in turbulence or inother nonlinear problems characterized by a broad spec-trum of scales. In LES or DNS one could write the aboveerror estimate in Fourier-space as fc3A2u(k)/6, however,the magnitude of this error depends on which Fourier-modes one looks at, and the magnitude of u(k) whichdepends on the turbulence spectrum. Therefore it is dif-ficult to judge, without a more precise analysis what thetrue magnitude of the errors are.

A method for more carefully quantifying numerical er-rors in such nonlinear "broad-band spectrum" type prob-lems may be developed along the following lines.15 To bespecific we consider a hypothetical numerical simulationusing a finite difference scheme of isotropic turbulence ina box with periodic boundary conditions. We consideronly the numerical error due to the spatial discretiza-tion at a given point in time. Time-discretization errorsand numerical stability is a rather large subject and isnot discussed here. The effect of time discretization onturbulence with a realistic energy spectrum has recentlybeen studied by Fabignon et al.22 using a generalizationof the Von Neumann analysis.

The essential idea in the analysis of space discretiza-tion errors in turbulence15 is to write the finite differenceimplementation of a numerical scheme in spectral spaceby using the "modified wave number" to represent thedifferencing scheme. The numerical error for a wave ofgiven wave-number k can be written as the sum of threeterms (A) The subgrid modeling error, (B) The trunca-tion error (C) The aliasing error. The origin of (A) isclear, it appears because the subgrid model does not ex-actly equal the true "subgrid stress," TJJ. Since we donot know what the true subgrid stress is, any theoretical

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treatment is difficult. The problem either needs to bestudied by "apriori testing" where the subgrid stressesare extracted from a well resolved DNS field and com-pared to the model25'5 or by "aposteriori testing" wherea full simulation is run (see the review article by Lesieurand Metais16 and references therein) and various statis-tics from the simulation are compared to DNS or exper-iments. Both of these approaches axe prevalent in theLES literature and will not be discussed here. Instead,we will assume that Tij is exactly known so that anynumerical errors incurred in computing the model willalso be neglected. The truncation error in differentiatingT^ on the right hand side of the momentum equation ishowever included.

The truncation error arises because numerical differ-entiation cannot find the exact derivative of a Fourier-mode. Thus, if we apply a finite-differencing operator '£>'to a wave A exp(ifcr) the result is ik'A exp(ikx) where k',is a function of k and the grid spacing '/i'. 'fc' is calledthe modified wave number of the finite difference opera-tor '£>'.

The aliasing error arises when nonlinear terms areevaluated by multiplying together variables defined ona discrete grid, generating higher harmonics. The wave-lengths of some of these waves are so short that theycannot be resolved on the grid. They get "misinter-preted" as a wave of much larger wavelength (they ac-quire an "alias" or duplicate identity!) This descriptioncan be presented in a more formal way15, in the literatureon spectral methods the origin of aliasing errors is wellknown4 and so is their potential to inflict serious damageon simulations1'. The aliasing error depends on how thenonlinear term is written. For example, d/dx (u2) is notthe same as 2u(du/dx) on discretization.

Once these error terms are written down, they wouldcontain quadratic terms in the velocities such as UiUjdue to the quadratic nonlinearity inherent in the Navier-Stokes equations. A useful thing to know about theseerrors is their power spectra. These too can be writtendown in a formal way but they now contain terms that arequartic in the velocities. One now uses the "quasinormalhypothesis" that works quite well at the kinematic level(though the long term effect of the deviations from quasi-normality has prevented its successful exploitation as aclosure scheme) to express fourth order velocity momentsby second order ones. Further, for isotropic turbulence,these second order moments can be re-expressed in termsof the energy spectrum. A similar strategy was used byBachelor27 to deduce the pressure spectrum of isotropicturbulence from the energy spectrum. The end result isa set of analytical formulas for the error spectra. Thesewill not be written down here but the reader may findthem in the original reference15.

0 2 4 6 8Wavenumber

FIG. 1. Power spectra of the truncation error for a (topto bottom) second, fourth, sixth and eighth order central dif-ference schemes (solid lines) compared to upper and lowerbounds of subgrid force (symbols).

Figure 1 shows the power spectrum of the truncationerror for central difference schemes of second, fourth,sixth and eighth order for a turbulence spectrum modeledwith the "Von-Karman spectrum" (E(k) ~ fc4 as k —» 0and E(k) ~ k~5/3 as k -»• oo) at a Reynolds number con-sidered essentially infinite. The normalization is chosenso that the energy spectrum has its maximum value ofunity at k = 1. The symbols represent upper and lowerbounds for the true subgrid force computed using similar"quasinormal analysis." It is clear that even for the highorder schemes, truncation errors are unacceptably large.The physical reason for this is, the modified wavenumberhas the largest deviation from the true wavenumber nearthe cut-off. However, in LES, the energy spectrum doesnot fall off sharply near the cut-off due to a dissipationrange as it does in DNS. In the standard implementa-tion of LES the "grid size" is usually considered equalto the "filter size." As a result, increasing the resolu-tion essentially changes the problem by bringing in newmodes. Both the subgrid term and the truncation errorchanges without changing the fact that the truncationerror dominates the subgrid force. This relation couldonly change when the LES converges to a DNS and allscales of motion are resolved. There is no concept of"grid independence" in the standard implementation ofLES!

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102

10'

10 20Wavenumber

FIG. 2. Power spectra of the truncation error for a secondorder central difference scheme for two different grid resolu-tions (solid lines) compared to upper and lower bounds ofsubgrid force (symbols).

Figure 2 shows how the truncation error spectrum forthe second order central difference scheme changes whenthe grid size is reduced by a factor of four. Also shownfor comparison are the power-spectra (upper and lowerbounds) for the subgrid force at the two different reso-lutions. The results support the statements made at theend of the last paragraph. Both the truncation error andthe subgrid force are altered by the increased resolutionbut the truncation error continues to dominate the sub-grid force.

102

0.0 2.0 4.0Wavenumber

FIG. 3. Power spectra of the aliasing error for an un-dealiased pseudo-spectral scheme (solid line), fourth ordercentral difference scheme (dashed line) and second order cen-tral difference scheme (dotted line) compared to upper andlower bounds of subgrid force (symbols).

Figure 3 shows the aliasing error spectra for centraldifference schemes of second and fourth order as well asthat of a (undealiased) pseudo-spectral scheme that hasno truncation error. The aliasing error is once again seento dominate the subgrid force. The schemes with thelarger truncation errors have somewhat reduced aliasingerror. This effect is well known17 and is due to the factthat the modified wavenumber of the approximate differ-encing schemes go to zero near the grid cut-off.

4.0Wavenumber

FIG. 4. Power spectra of the truncation error for a (topto bottom) second, fourth and eighth order central differencescheme (solid line) compared to upper and lower bounds ofsubgrid force (symbols) when the filter-width is taken as twicethe grid spacing.

One possible method of controlling these errors is il-lustrated by figure 4. Here the filter-width is taken astwice the grid spacing (the filter being assumed to be ofthe "Fourier-cutoff' type) and the truncation error spec-trum is plotted for the second, fourth and eigth ordercentral differencing scheme. In this case, the truncationerror is reduced several orders of magnitude below thesubgrid force for the eight order scheme. Further, for theFourier-cutoff filter, the aliasing error is reduced exactlyto zero (a manifestation of the familiar '3/2 dealiasingrule'). A filter to grid ratio of two however implies aneight fold increase of grid points and a further increasein computational time if the maximum time-step is lim-ited by the CFL condition.

The above analysis of errors is essentially kinematicin nature. It quantifies the magnitude of the error butgives no information about the long term dynamical ef-fect of these errors on the computation. Such informa-tion however is critical for the success of the numericalmethod. A systematic study of this issue in the contextof LES has recently been undertaken by Kravchenko andMoin18. The above authors used a channel flow codethat used a high order B-spline method in the wall nor-mal direction and a pseudo-spectral method in the re-

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maining two homogeneous directions. The effect of usingfinite-difference implementations in the homogeneous di-rections could be studied by replacing the wavenumbersby modified wavenumbers in the spectral evaluation ofderivatives in the code. The effect of aliasing errors couldbe simulated by simply omitting the dealiasing proce-dure. Further, by introducing appropriate phase shiftsin the Fourier modes, the code could be made to mimiccomputations on a staggered mesh.

As a result of a series of systematic numerical experi-ments the authors concluded that both aliasing and trun-cation errors of low order schemes do seriously degradeLES computations as conjectured earlier based on a the-oretical analysis15. (It should be noted that a similarproblem was observed and reported earlier24"26 in thecontext of mixing layer calculations.) Of these, the alias-ing error was found to be the source of the most seriousproblems since they interfered with the energy conservingnature of the scheme and could lead to unstable calcula-tions. The magnitude of the aliasing error depended onthe form in which the nonlinear term was written. The"skew-symmetric form" had the lowest aliasing error dueto some cancellations between the partial contributions.This dependance of the aliasing error on the form of thenonlinear term has also been studied earlier by Blaisdellet a/.21 These studies have motivated a move towardshigh order finite difference schemes for LES calculationsthat retain, at the discrete level, the energy, momentumand mass conserving properties of the basic LES equa-tions20

V. SUMMARY AND CONCLUSIONS

The numerical simulation of turbulence is an extremelycomputationally intensive enterprise and quickly satu-rates even the impressive gains in computer speeds thatmarks every new development in computer technology.This fact, coupled with the extraordinary theoretical dif-ficulties involved in developing any statistical closure the-ory of turbulence propels LES as the most versatile toolfor engineering calculations. The ootential of LES for vi-sualization of the essential structures in the flow as wellas for reliable quantitative predictions has been demon-strated, at least in relatively simple geometry. One ofthe challenges ahead would be to demonstrate unambigu-ously that LES is able to predict quantitatively statisticalmeasures of interest in the domain of truly complex en-gineering flows.

This transition naturally raises questions such as

1. Can the theoretical framework of LES originallyenvisaged for uniform grids be extended to includenonuniform grid distributions?

2. Are finite difference methods which are the mostconvenient to implement in complex geometry ad-equate for LES?

3. What is the best way to restrict the class of subgridmodels so that they have as many of the propertiesof the true subgrid stress build into them.

An attempt has been made in this review to summarizeas best as possible research on questions of this kind.The summary is necessarily partial due to the rapidlyevolving nature of the field and the limited ability of oneperson to keep up with the entire literature.

Acknowledgements: The author would like to thankThomas Lund for his helpful comments

1 O.V. Vasilyev, T.S. Lund and P. Moin "A General Class ofCommutative Filters for LES in Complex Geometries", J.Comp. Phys. (submitted).

2 P. Moin, W.C. Reynolds and J.H. Ferziger "Large Eddysimulation of incompressible turbulent channel flow," Dept.Mech. Eng. Rept. TF12, Stanford University (1978).

3 S. Ghosal and P. Moin, "The Basic Equations for the LargeEddy Simulation of Turbulent Flows in Complex Geome-try," J. Comp. Phys. 118, 24 (1995).

4 C. Canute, M.Y. Hussaini. A. Quarteroni, T.A. Zang"Spectral Methods in Fluid Dynamics," Berlin: Springer(1988).

5 R.M. Kerr, J.A. Domaradzki, and G. Barbier "Small-scaleproperties of nonlinear interactions and subgrid-scale en-ergy transfer in isotropic turbulence," Phys. Fluids 8, 197(1996).

6 S. Lele, "Compact Finite Difference Schemes with SpectralLike Resolution," J. Comp. Phys. 103, 16 (1992).

7 H. van der Ven, "A Family of Large Eddy Simulation (LES)filters with nonuniform filter widths," Phys. Fluids 7(5),1171 (1995).

8 Y. Zang, R.L. Street, J. Koseff, "A dynamic mixed subgrid-scale model and its application to turbulent recirculatingflows," Phys. Fluids A 5, 3186 (1993).

9 J. Bardina, J.H. Ferziger, W.C. Reynolds "Improved sub-grid scale models for large eddy simulation," AIAA Paper80-1357 (1980).

10 S. Ghosal, T.S. Lund, P. Moin and K. Akselvoll "A dy-namic localization model for the large eddy simulation ofturbulent flow," J. Fluid Mech. 286, 229 (1995).

11 M. Germano, U. Piomelli, P. Moin and W.H. Cabot "Adynamic subgrid-scale eddy viscosity model," Phys. FluidsA 3, 1760 (1991).

12 C. Meneveau, T.S. Lund and W.H. Cabot "A lagrangiandynamic subgrid-scale model of turbulence," J. FluidMech. 319, 353 (1996).

13 D.K. Lilly "A proposed modification of the Germanosubgrid-scale closure method," Phys. Fluids A 4, 633(1992).

14 D. Carati and E. Vandeneijnden "On the self-similarity as-sumption in dynamic-models for large-eddy simulations,"Phys. Fluids 9(7), 2165 (1997).

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15 S. Ghosal "An analysis of numerical errors in large eddysimulations of turbulence," J. Comp. Phys. 125, 187(1996).

16 M. Lesieur and O. Metais "New trends in large-eddy sim-ulations of turbulence" Annu. Rev. Fluid Mech. 28, 45(1996).

17 R.S. Rogallo and P. Moin "Numerical simulation of turbu-lent flows" Annu. Rev. Fluid Mech. 16, 99 (1984).

18 A.G. Kravchenko and P. Moin "On the effect of numericalerrors in large eddy simulation of turbulent flows" J. Comp.Phys. 131, 310 (1997).

19T.S. Lund "Discrete filters for LES" CTR Annu. Res.Briefs pg.83 (1997).

20 Y. Morinishi, T.S. Lund, O.V. Vasilyev and P. Moin "Fullyconservative high order finite difference schemes for incom-pressible flow" J. Comp. Phys. (to appear).

21 G.A. Blaisdell, E.T. Spyropoulos and J.H. Qin "The effectof the formulation of nonlinear terms on aliasing errors inspectral methods" Appl. Numer. Math. 20, 1 (1996).

22 Y. Fabignon, R.A. Beddini and Y. Lee "Analytic evalu-ation of finite difference methods for compressible directand large eddy simulations" Aerospace Science & Technol-ogy 1(6), 413 (1997).

23 U. Schumann "Realizability of Reynolds-stress turbulencemodels" Phys. Fluids 20(5), 721 (1977).

24 B. Vreman. B. Geurts and H. Kuerten "Discretization errordominance over subgrid terms in large eddy simulation ofcompressible shear layers in 2D" Comm. in Num. methodsEng. 10, 785 (1994).

25 B. Vreman. B. Geurts and H. Kuerten "A priori tests oflarge eddy simulation of the compressible plane mixinglayer" J. Eng. Math. 29, 299 (1995).

26 B. Vreman, B. Geurts and H. Kuerten "Comparison of nu-merical schemes in large eddy simulation of the tempo-ral mixing layer" Int. J. Num. methods in Fluids 29, 299(1996).

2' G. Batchelor "The theory of homogeneous turbulence"Cambridge Univ. Press, Cambridge, UK (1996).

28 C. Fureby and G. Tabor "Mathematical and physical con-straints on large-eddy simulations" Theor. & Comp. FluidDyn. 9. 85 (1997).

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