Large-eddy simulation: achievements and challengesugo/research/pas.pdfProgress in Aerospace Sciences 35 (1999) 335—362 Large-eddy simulation: achievements and challenges U. Piomelli*

Progress in Aerospace Sciences 35 (1999) 335—362

Large-eddy simulation: achievements and challenges

U. Piomelli*

Department of Mechanical Engineering, University of Maryland, College Park Campus, College Park, MD 20742, USA

Abstract

In this paper, the present state of the large-eddy simulation (LES) technique is discussed. Modeling and numericalissues that are under study will be described, and results of state-of-the-art calculations will be presented to highlight theresponse of the subgrid-scale models to important features of the flow field. Finally, some challenges for research andapplications of LES in the near future will be discussed. ( 1999. Published by Elsevier Science Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3362. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

2.1. The filtering operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3382.2. Filtered Navier—Stokes equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3382.3. The subgrid-scale stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

3. Subgrid-scale models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3393.1. Two-point closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3403.2. Scale-similar and mixed models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3413.3. One-equation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3413.4. Dynamic models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

4. Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3435. Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

5.1. Building-block flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3455.2. Transitional and relaminarizing flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3455.3. Three-dimensional flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3465.4. Separated flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3505.5. General remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

6. Challenges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3546.1. Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3546.2. Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3546.3. Compressible flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3556.4. Wall layer modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

* Tel.: 001 301 405 5254; fax: 001 301 314 9477; e-mail: ugo@ barbaresco.umd.edu.

0376-0421/99/$ — see front matter ( 1999. Published by Elsevier Science Ltd. All rights reserved.PII: S 0 3 7 6 - 0 4 2 1 ( 9 8 ) 0 0 0 1 4 - 1

Notation

a=

freestream speed of soundC, C

1, C

2,

Ck, Ce, Cl SGS stress model coefficients

E kinetic energy spectrum; also, total energye#u

iui/2

FM2

resolved-scale second-order velocity struc-ture function

G filter functionh enthalpy (,e#p/o)k wavenumber; also, thermal conductivitykm

cutoff wavenumberK turbulent kinetic energyK

4'4subgrid-scale energy (,q

kk/2)

Ko Kolmogorov constant¸ij

resolved turbulent stresses (,uNiuNj

Y !uKNiuNKj)

p pressureQ

jSGS heat flux (,coN (u

je'C!uJ

jeJ ))

Re Reynolds numberRo rotation numberSMij

resolved strain-rate tensor(,(uN

i/x

j#uN

j/x

i)/2)

¹ temperature¹

ij subtest stresses (,uiujY!uNK

iuNKj)

uq friction velocity (,(qw/o)1@2)

º"

average (bulk) velocityu, v, w velocity components in the x-, y- and z-

directionsui

velocity component in the xidirection

uNi

resolved velocity component in the xidirec-

tionu@i

SGS velocity component in the xi

direc-tion

x, y, z coordinatesxi

tensor coordinates, i"1, 2 or 3 for x, yor z

Greek symbolsdij

Kronecker’s tensorD grid sizeD1 principal (grid) filter widthDK test-filter widthe internal energy per unit massc specific heat ratiok molecular viscosityl kinematic viscosityl%

eddy viscosity for K—e modelslT

SGS eddy viscosityl Fourier transform of the SGS eddy viscosityl` normalized Fourier transform of the SGS

eddy viscosityo fluid densitypij

viscous stress tensorqij

subgrid-scale stress tensor

(,uiuj!uN

iuNj)

mij

resolved rotation-rate tensor(,(uN

i/x

j!uN

j/x

i)/2)

ui

component of the vorticity vector in thexidirection

u vorticity modulusX channel rotation rate

AbbreviationsDNS Direct numerical simulationLES Large-eddy simulationRANS Reynolds-averaged Navier—StokesSGS Subgrid-scale

Other operators)1 filtering) test-filtering; also Fourier transform)8 Favre-filteringS)T spatial, temporal or ensemble averaging

1. Introduction

Turbulence is a phenomenon that occurs frequentlyin nature; it has, therefore, been the subject of studyfor several centuries. In 1510, Leonardo da Vinci accom-panied a drawing of the vortices shed behind a bluntobstacle with the following observation [1]:

Observe the motion of the water surface, which re-sembles that of hair, that has two motions: one due tothe weight of the shaft, the other to the shape of thecurls; thus, water has eddying motions, one part ofwhich is due to the principal current, the other to therandom and reverse motion.

Although based entirely on speculation, and not ac-companied by a mathematical analysis, this observation

may be seen as a precursor to Reynolds’ decompositionof velocity, pressure and other variables into mean andfluctuating parts.

Over 100 years after Osborne Reynolds’ experiments,turbulence is still one of the outstanding problems inapplied mechanics. No useful analytical solutions of tur-bulent flows in geometries of engineering interest areavailable, although statistical theories of turbulence haveprovided good understanding of the scaling laws in vari-ous flow regimes. Experimental studies have also giveninsight in the understanding of the structure of turbulentflows. Flow visualization has been particularly usefulin the identification of the coherent eddies that areresponsible for most of the energy production, especiallyin regions of high shear. Measurement techniques haveprogressed significantly: it is now possible to obtain

336 U. Piomelli / Progress in Aerospace Sciences 35 (1999) 335—362

single-point measurements of velocity and velocitygradient components using Laser-Doppler velocimetryor multiple wire anemometers, or velocity distributionsin a plane, through Particle-Image or Particle-TrackingVelocimetry.

Analytical or numerical solution of turbulent flowproblems can be accomplished using various levels ofapproximation, yielding more or less detailed descrip-tions of the state of the flow. The simplest approach is touse semi-empirical correlations. Moody’s diagram, whichgives the skin friction factor for cylindrical pipes as afunction of Reynolds number and relative roughness, isan example of this approach, which is especially useful forglobal, control-volume analyses, but yields no informa-tion on local quantities and relies heavily on the avail-ability of experimental data in configurations similar tothe one under study.

A more sophisticated method involves the use ofReynolds’ averaging: the long-time average of a quantityf is defined as

S f T"1

¹Pt`T

t

f (q) dq, (1)

where ¹ is a time interval much longer than all the timescales of the turbulent flow. The averaging operationdefined above permits one to decompose any quantityinto its mean part, S f T, and a fluctuating part, f!S f T.If the averaging operation (1) is applied to the equationsof motion, one obtains the well-known Reynolds-aver-aged Navier—Stokes equations (RANS), that describe theevolution of the mean quantities. The effect of turbulentfluctuations appears in a Reynolds stress term that mustbe modeled to close the system. A very wide range ofmodels for the Reynolds stresses is available, rangingfrom simple, algebraic models, to K—e models, to full oralgebraic Reynolds stress closures. The solution of theRANS equations is now used in engineering applicationsto predict the flow in fairly complex configurations. Thisapproach, however, suffers from one principal shortcom-ing, the fact that the model must represent a very widerange of scales. While the small scales tend to dependonly on viscosity, and may be somewhat universal, thelarge ones are affected very strongly by the boundaryconditions (see, for instance, the difference between thespanwise rollers present in mixing layers and wakes andthe elongated streamwise vortices that are found in thenear-wall region of a turbulent boundary layer). Thus, itdoes not seem possible to model the effect of the largescales of turbulence in the same way in flows that are verydifferent.

The direct numerical simulation (DNS) of turbulenceis the most straightforward approach to the solution ofturbulent flows. In DNS the governing equations arediscretized and solved numerically. If the mesh is fineenough to resolve even the smallest scales of motion, andthe scheme is designed to minimize the numerical disper-

sion and dissipation errors, one can obtain an accuratethree-dimensional, time-dependent solution of the gov-erning equations completely free of modeling assump-tions, and in which the only errors are those introducedby the numerical approximation. DNS makes it possibleto compute and visualize any quantity of interest, includ-ing some that are difficult or impossible to measureexperimentally, and to study the spatial relationshipsbetween quantities of interest (for instance, vorticity andenergy production), to obtain insight on the detailedkinematics and dynamics of turbulent eddies. DNS hasbeen a very useful tool, over the past ten years, for thestudy of transitional and turbulent flow physics, but italso has some limitations. First, the use of highly accu-rate, high-order schemes is desirable to limit dispersionand dissipation errors; these schemes (spectral methods,for example) tend to have little flexibility in handlingcomplex geometries and general boundary conditions.Secondly, to resolve all scales of motion, one requiresa number of grid points proportional to the 9/4 power ofthe Reynolds number, Re, and the cost of the computa-tion scales like Re3. For these reasons, DNS has largelybeen limited to simple geometries (flat plate boundarylayers, homogeneous flows) at low Reynolds numbers,and its application to engineering-type problems withinthe next decade appears unlikely.

Large-eddy simulation (LES) is a technique intermedi-ate between the direct simulation of turbulent flows andthe solution of the Reynolds-averaged equations. In LESthe contribution of the large, energy-carrying structuresto momentum and energy transfer is computed exactly,and only the effect of the smallest scales of turbulence ismodeled. Since the small scales tend to be more homo-geneous and universal, and less affected by the boundaryconditions than the large ones, there is hope that theirmodels can be simpler and require fewer adjustmentswhen applied to different flows than similar models forthe RANS equations. The early applications of LES wereconfined to ‘‘building-block’’ flows: homogeneous turbu-lence, free shear flows, plane channel flows. Following theintroduction of new models, which allowed more accu-rate computations to be performed with less empiricismthan before, LES is now being used not only to calculatestandard, well-documented test cases, but also to studythe physical phenomena that occur in more complex,engineering-like applications.

The purpose of this article is to put in perspective thepresent state of development of this technique, and tooutline some of the challenges that lie ahead, in terms ofapplications, numerical and modeling issues. This over-view is not meant to be comprehensive, but rather tohighlight some issues of importance in the author’s opin-ion. More complete reviews of the area can be found inrecent articles [2,3] and books [4].

In the following sections, a brief overview of the formu-lation of LES, of current SGS models and of numerical

U. Piomelli / Progress in Aerospace Sciences 35 (1999) 335—362 337

issues will be given. Some of the important achievementsof LES to date will then be presented, followed by a dis-cussion of issues that need to be addressed to transfer theuse of LES from the research to the design environmentin the near future. Some conclusions will then be drawn.

2. Formulation

2.1. The filtering operation

To separate the large from the small scales, LES isbased on the definition of a filtering operation: a filtered(or resolved, or large-scale) variable, denoted by anoverbar, is defined as [5]

fM (x)"PD

f (x@) G (x, x@;DM ) dx@, (2)

where D is the entire domain, G is the filter function, andDM is the filter width, i.e., the wavelength of the smallestscale retained by the filtering operation. The filterfunction determines the size and structure of the smallscales.

It is easy to show that, if G is a function of x!x@ only,differentiation and the filtering operation commute.Otherwise (for example, in cases in which the filter widthis non-uniform in space), commutation errors result.Ghosal and Moin [6] found that these errors are o (DM 2).When second-order-accurate schemes are used, therefore,they are of the order of the truncation error; when higher-order schemes are employed, however, they may lowerthe accuracy of the calculations. Ghosal and Moin [6]proposed a correction that reduces the commutationerror to fourth-order; this correction acts as a diffusiveterm for a wave moving in the direction in which the filterwidth is increasing, and an anti-diffusive term if the wavemoves in the direction of decreasing filter width. In prac-tical applications, in which the grid stretching is notmonotonic, this may result in exponentially growingsolutions that may lead to numerical instabilities. Thiscorrection has never, to date, been tested. Vasyliev andLund [7] have developed a method to construct continu-ous and discrete filters that reduce the commutationerror to any desired order.

The most commonly used filter functions are the sharpFourier cutoff filter, best defined in wave space1 as

G] (k)"PD

G(x@) e~*kx{dx@"G1 if k4n/DM ,

0 otherwise,(3)

1Here, a quantity denoted by a caret, ) , is the complex Fouriercoefficient of the original quantity; in most cases, however,a caret will denote a test-filtered variable, as defined in Sections3.2 and 3.4.

the Gaussian filter,

G(x)"S6

nDM 2expA!

6x2

DM 2 B, (4)

and the tophat filter in real space:

G(x)"G1/DM if DxD4DM /2,

0 otherwise.(5)

These three filters and their Fourier transforms G] (k)are shown in Fig. 1. In practice the Gaussian filter isalways used in conjunction with a sharp Fourier cutoff;the truncation of the Gaussian at a non-negligible valueis the cause of the ringing observed in the figure. Foruniform filter width DM these filters are mean-preservingand commute with differentiation.

To illustrate the difference between the filters definedabove they are applied to a test function, and the spectraof the filtered variables are shown in Fig. 2. The top hatand Gaussian filters give similar results; in particular,they both smooth the large-scale fluctuations as well asthe small-scale ones, unlike the Fourier cutoff, whichonly affects the scales below the cutoff wave number.

2.2. Filtered Navier—Stokes equations

If the filtering operation (2) is applied to the governingequations, one obtains the filtered equations of motion.For incompressible flow of a Newtonian fluid2 they are

uNi

xi

"0, (6)

uNi

t#

xj

(uNiuNj)"!

1

opNx

i

!

qij

xj

#l2uN

ix

jx

j

. (7)

These equations govern the evolution of the large,energy-carrying, scales of motion. The effect of the smallscales appears through a subgrid-scale (SGS) stress term,

qij"u

iuj!uN

iuNj, (8)

that must be modeled.

2.3. The subgrid-scale stresses

If the subgrid-scale velocity u@i"u

i!uN

iis defined, the

SGS stresses (8) can be decomposed into three parts [5]:

qij"u

iuj!uN

iuNj"¸

ij#C

ij#R

ij, (9)

where ¸ij"uN

iuNj!uN

iuNj

are the Leonard stresses,

Cij"uN

iu@j#u@

juNiare the cross terms, and R

ij"u@

iu@jare

2Extensions to compressible flow will be considered later inSection 6.3.


Fig. 1. Typical filter functions. —— sharp Fourier cutoff; — — — truncated Gaussian; — ) — top hat. (a) real space; (b) Fourier space.

Fig. 2. Filtering of a test function. —— Unfiltered; #sharp Fourier cutoff; e Gaussian; n top hat; — — — k~5@3.

the SGS Reynolds stresses. The Leonard stresses repres-ent interactions between resolved scales that result insubgrid-scale contributions; they can be computed ex-plicitly, and, when the sharp cutoff filter is used, they arethe aliasing errors. The cross terms represent interactionsbetween resolved and unresolved scales, whereas the SGSReynolds stresses represent interactions between small,unresolved, scales. While the SGS stresses are invariantwith respect to a Galilean transformation, neither ¸

ijnor

Cij

are [8]. For this and other reasons (see [9]), thedecomposition (9) has largely been abandoned.

3. Subgrid-scale models

In LES the dissipative scales of motion are resolvedpoorly, or not at all. The main role of the subgrid-scale

model is, therefore, to remove energy from the resolvedscales, mimicking the drain that is associated with theenergy cascade. Most subgrid-scale models are eddy-viscosity models of the form

qij!

dij3

qkk"!2l

TSMij, (10)

that relate the subgrid-scale stresses qij

to the large-scalestrain-rate tensor SM

ij"(uN

i/x

j#uN

j/x

i)/2. In most

cases the equilibrium assumption (namely, that the smallscales are in equilibrium, and dissipate entirely and in-stantaneously all the energy they receive from the re-solved ones) is made to simplify the problem further andobtain an algebraic model for the eddy viscosity [10]:

lT"CD2DSM DSM

ij, DSM D"(2SM

ijSMij)1@2, (11)


Fig. 3. Non-dimensional spectral eddy viscosity as parameterized by Chollet [15] and Metais and Lesieur [16] for turbulence obeyinga k~m spectrum. The lines correspond to m"8/3 (lowest) through m"2/3 (highest); a solid line denotes a Kolmogorov spectrum(m"5/3).

where D is the grid size (for anisotropic grids, the cuberoot of the cell volume is usually employed). This modelis known as the ‘‘Smagorinsky model’’. The value of thecoefficient C can be determined from isotropic turbulencedecay [11]; if the cutoff is in the middle of a long inertial

subrange, the Smagorinsky constant C4"JC takes

values between 0.18 and 0.23 (and CK0.032—0.053). Inthe presence of shear, near solid boundaries or intransitional follows, however, it has been found that itmust be decreased. This has been accomplished by vari-ous types of ad hoc corrections such as van Driest damp-ing [12] or intermittency functions [13].

3.1. Two-point closures

Two-point closures have been an alternative way toderive SGS models. Kraichnan [14] computed the en-ergy transfer from the resolved to the unresolved scales,¹

46"(kDk

m), given a cutoff wave-number magnitude k

mly-

ing in an infinite inertial subrange, using a two-pointclosure model for isotropic turbulence. He defined the neteddy viscosity from the calculated subgrid-scale transfer:

l%(kDk

m)"!

¹46"

(kDkm)

2k2E(k), (12)

where E(k) is the kinetic energy spectrum. For km

lying inthe inertial subrange, the net eddy viscosity approachesa k-independent eddy viscosity for k;k

m. Near k

m, how-

ever, both a negative and a positive contribution aresignificant; the negative part corresponds to energytransfer from the small to the large scales (backscatter, oreddy noise). The net eddy viscosity increases with in-creasing k/k

mto a finite cusp at k/k

m"1.

Chollet and Lesieur [17] used the eddy-damped quasi-normal Markovian (EDQNM) theory to develop a SGS

model with similar results. The Chollet—Lesieur modeluses [k

mE(k

m)]1@2 as velocity scale, and k~1

mas length

scale for the eddy viscosity, which, in wave space, is givenby

l(k)"l`(k/km)[E(k

m)/k

m]1@2; (13)

l`(k/km) can be approximated by [15]

l`(k/km)"Ko~3@2[0.441#15.2 exp(!3.03k

m/k)], (14)

where the value of the Kolmogorov constant was chosento be Ko"1.4. The Chollet parameterization of thedimensionless eddy viscosity is shown in Fig. 3. The eddyviscosity goes to a k-independent constant for k/k

m;1,

and rises to a finite cusp at k/km"1.

The Chollet—Lesieur model produces zero eddy viscos-ity as long as there is no energy near the cutoff; it is,however, defined in wave space, which hampers its exten-sion to finite-difference schemes and to complex geomet-ries. To overcome this shortcoming, Metais and Lesieur[16] derived the ‘‘structure function model’’. Assuminga cutoff wave number in the inertial region of a Kol-mogorov spectrum, they expressed the energy spectrumat the cutoff, E (k

m), in terms of the resolved-scale second-

order velocity structure function,

FM2(x;D)"S[uN

i(x#r)!uN

i(x)][uN

i(x#r)!uN

i(x)]T, (15)

where S)T is an ensemble average taken over all pointssuch that DrD"D, and obtained

lT(x)"0.063D[FM

2(x;D)]1@2. (16)

It must be remarked that, if an isotropic grid is used,the structure function can be seen as a finite-difference


approximation of the velocity gradient tensor:

FM2K2D2

uNi

xj

uNi

xj

"2D2(SMijSMij#m

ijmij)

"D2(DSM D2#uiu

i) (17)

(where mij

is the anti-symmetric part of the velocity gradi-ent tensor, and u

ithe vorticity), which gives

lT(x)"0.063D2(DSM D2#DuD2)1@2. (18)

Thus, the structure function model in its original form isactually a Smagorinsky-like model with the strain ratereplaced by the velocity gradient tensor. For isotropicflows, the model is less dissipative than the Smagorinskymodel, in which values such as C

4"0.18!0.23 are com-

monly used; this is reflected in a more accurate predictionof the inertial range (see, for instance, Fig. 2 in the articleby Lesieur and Metais [2]). For sheared flows, however,the structure function may be excessively dissipative.Improved results were obtained using a ‘‘four-point for-mulation’’, in which only the differences in planes parallelto the wall were used [18], or by applying a Laplacianfilter to the velocity before computing the structure func-tion [19].

Metais and Lesieur [16] also proposed a modificationto the eddy viscosity in Eq. (14) to account for deviationsof the spectrum from the 5/3 Kolmogorov law. Assumingthat E(k)Jk~m gives a correction to the plateau level,which is decreased for m'5/3 (see Fig. 3), while main-taining a cusp-like behavior for kKk

m. This ‘‘spectral-

dynamic model’’ resulted in improved results intransitional flows, in the near-wall region of turbulentflows, or in regions of intermittent flow, where the spec-trum is steeper than k~5@3 (see [20]).

3.2. Scale-similar and mixed models

While eddy-viscosity models may be able to representthe global, dissipative, effects of the small scales in a satis-factory way, they cannot reproduce the details of thestresses (and the energy exchange) accurately on a locallevel, and in particular the correlation that exists betweenlarge-scale, energy (and Reynolds stress) producingevents and energy transfer to and from the small scales[21, 22]. Scale-similar and mixed models try to reproducethis correlation more accurately. They are based on theassumption that the most active subgrid scales are thosecloser to the cutoff wave number, and that the scaleswith which they interact most are those right above thecutoff [23]. The ‘‘largest subgrid scales’’ can be obtainedby filtering the SGS velocity u@

i"u

i!uN

ito yield

uN @i"uN

i!uO i . The SGS stresses can then be written as

qij"!2l

TSMij#uN

iuNj!uP

iuPj. (19)

The last two terms in Eq. (19) represent the ‘‘scale-similarmodel’’; the eddy-viscosity contribution provides the

dissipation that is underestimated by the scale-similarpart alone.

Another possible form of a scale-similar model can beobtained by applying a second filter, G] , with character-istic length DK 'DM , to the velocity field [22]. The SGSstresses can be parameterized as

qij"!2l

TSMij#uN

iuNj

Y !uNKiuNKj. (20)

This model has been applied by Anderson and Meneveau[24] to the simulation of homogeneous isotropic turbu-lence decay. Scale-similar and mixed models have beenrecently revisited in the framework of dynamic modelingideas [25—28].

3.3. One-equation models

One-equation models solve a transport equation forthe subgrid-scale energy K

4'4"q

kk/2 to obtain the velo-

city scale. The terms that require modeling are the turbu-lent transport and pressure diffusion, which are usuallymodeled jointly as an added diffusion, and the viscousdissipation, which is usually taken to be proportional toK3@2

4'4/DM . Consequently, a one-equation model can be cast

in the form

lT"ClK1@2

4'4D, (21)

K4'4

t#

xj

(K4'4

uNj)"#

xjCACk

K1@24'4

DM #lBK

4'4x

j

#qijuNiD

!CeK3@2

4'4DM

!qijSMij. (22)

The constants can be evaluated based on turbulencetheory [11, 29] or adjusted dynamically [30].

One-equation models have been used by several re-searchers [30—32]. Among their advantages is the factthat the independent definition of the velocity scale re-sults in a more accurate prescription of the SGS timescale compared to algebraic eddy-viscosity models. Inthose, DSM D~1 is generally used as the time scale; the strain-rate tensor, however, is strongly affected by the meanflow and by the largest structures, and these results ina length scale that does not respond properly to per-turbations from equilibrium. Altogether, however, theexpense involved in solving an additional equation doesnot seem to be justified by improvements in the accuracy.

3.4. Dynamic models

The introduction of dynamic modeling ideas [33] hasspurred significant progress in the subgrid-scalemodeling of non-equilibrium flows. In dynamic modelsthe coefficient(s) of the model are determined as thecalculation progresses, based on the energy content ofthe smallest resolved scale, rather than input a priorias in the standard Smagorinsky [10] model. This is


accomplished by defining a test filter (denoted by a caret)whose width DK is larger than the grid filter-width D

(typically, DK "2D). Dynamic adjustment of the modelcoefficients is based on the identity [34]

Lij"uN

iuNj

Y !uNKiuNKj"¹

ij!q

ijY , (23)

which relates the ‘‘resolved turbulent stresses’’ Lij

(thecontribution from the region between test-filter and grid-filter scale), the subgrid-scale stresses q

ijand the subtest

stresses ¹ij"u

iujY !uNK

iuNKj, which are obtained by applying

the test filter G] , of characteristic width DK , to the filteredNavier—Stokes equations (6) and (7).

Consider now an eddy-viscosity model to para-meterize both subgrid and subtest stresses, of the form

qij"!2C

%7aij, ¹

ij"!2C

%7bij. (24)

Upon substituting Eq. (24) into Eq. (23), the identity (23)can be satisfied only approximately, since the stresses arereplaced by modeling assumptions, and the system isoverdetermined (five independent equations are availableto determine a single coefficient). Lilly [35] proposedthat the error incurred when a single coefficient is used beminimized in a least-squares sense.3 The error is

eij"L

ij!¹

ij#q

ijY "L

ij#2C

%7M

ij. (25)

with Mij"b

ij!aY

ij. The least-squares minimization pro-

cedure requires that

E2

C%7

"

SeijeijT

C%7

"2Teij

eij

C%7U"0, (26)

where the brackets indicate an appropriate ensembleaverage; since e

ij/C

%7"2M

ij, this implies that

S(Lij#2C

%7M

ij)M

ijT"0 (27)

which gives

C%7"!

1

2

PLM

PMM

, (28)

where PEF"SE

ijFijT.

This procedure can be applied to mixed models, ormodels with more than one coefficient as well. For a one-coefficient mixed model of the form

qij"A

ij!2C

%7aij, ¹

ij"B

ij!2C

%7bij, (29)

the least-squares minimization procedure gives

C%7"!

1

2

PLM

!PNM

PMM

, (30)

3 It is assumed here that the coefficients, C%7

and later C44

aresmooth on the DK scale, and can, therefore, be extracted from thefiltering operation.

with Nij"B

ij!A

ijY . In the case of two model coefficients

qij"C

44A

ij!2C

%7aij, ¹

ij"C

44Bij!2C

%7bij, (31)

it is necessary to require that E2/C44"E2/C

%7"0 to

yield

C44(x, t)"!

PMN

PLM

!PMM

PLN

PMN

PMN

!PMM

PNN

, (32)

C%7(x, t)"!

1

2

PMN

PLN

!PNN

PLM

PMN

PMN

!PMM

PNN

. (33)

The ensemble average has the purpose of removing verysharp fluctuations of the coefficient, which tend to de-stabilize numerical calculations, and make the modelinconsistent, since the model coefficients cannot be ex-tracted from the filtering operation (i.e., the differencesC

%7aijY !C

%7aijY and C

44A

ijY !C44A

ijY become significant).

Germano et al. [33] averaged the model coefficient overall homogeneous directions, thereby removing com-pletely the mathematical inconsistency. Ghosal et al. [30]used an integral formulation of the identity (23) thatrigorously removed the mathematical inconsistency atthe expense of having to solve an integral equationat each time step (an expense comparable to the solutionof a Poisson equation, therefore significant). Localizedfiltering can be performed over the scale DK (somewhatjustifiable by the consideration that, if the same coeffic-ient is used to model both q

ijand ¹

ij, it must be smooth

on the test-filter scale). Zang et al. [25] performed thistype of averaging; the inclusion of the scale-similar partinto their model decreased the contribution of the eddy-viscosity term, and no spuriously high values of thecoefficient were observed.

Meneveau et al. [36] proposed a Lagrangian ensembleaverage based on the consideration that the memoryeffects should be calculated in a Lagrangian framework,following the fluid particle, rather than at an Eulerianpoint, which sees different particles, with different his-tories, at each instant. This average is defined as

If"S f T"P

t

~=

f (t@) ¼(t!t@) dt@, (34)

where the integral is carried out following a fluid pathline. If ¼(t) is chosen to be an exponential function(to give more weight to recent times) the integrals at time-step n are governed by a passive-scalar-type transportequation, and can be conveniently evaluated using asimple relaxation technique; for instance

InLM

"Pt

~=

Lij(t@) M

ij(t@) ¼(t!t@) dt@

"HMeLijnMn

ij#(1!e)In~1

LM(x!unDM t)N, (35)

where H is the Heaviside function, and the evaluationof the integrals at the Lagrangian point x!unDM t is


performed by linear interpolation, and e"(Dt/¹)/(1#Dt/¹).

4. Numerical methods

In large-eddy simulations the governing equations (6)and (7) are discretized and solved numerically. Althoughonly the large scales of motion are resolved, the range ofscales present is still significant. The need to resolve highwave-number turbulent fluctuations accurately impliesthat either low-order schemes must be used on very finemeshes, or higher-order schemes must be employed oncoarser meshes. High-order schemes are more expensive,in terms of computational resources, but the increase inaccuracy they allow (for a given mesh) often justifies theiruse. Spectral schemes are very accurate and have desir-able modified wave-number characteristics, but tend tobe more expensive than finite-difference schemes, andalso give little flexibility in the application of the bound-ary conditions, and are not readily adapted to complexgeometries. As the application of LES shifts from basic,building-block, flows in simple geometries towards morerealistic applications, however, the flexibility that finitedifferences enjoy in the treatment of complex geometriesand boundary conditions is resulting in more widespreadapplication of these methods.

The numerical error associated with finite-difference(or finite-volume) approaches can, however, have signifi-cant consequences for the accuracy of a calculation.Ghosal [37] applied finite-differencing schemes of vari-ous orders to a superposition of Fourier modes witha realistic spectrum to perform an error analysis of theseschemes in turbulent flows. He found that, for low-orderfinite-difference schemes, the truncation error is largerthan the SGS contribution, unless the filter width issignificantly larger than the grid size. This result hadpreviously been obtained, by a priori analyses of DNSdata for a turbulent mixing layer, by Vreman et al. [38],and has been confirmed a posteriori by Kravchenko andMoin [39]. It should, however, be remarked that, unlikesecond-order schemes, the SGS stresses are dissipative.For this reason, the error introduced by low-ordercentered schemes may be less fatal to LES calculationsthan modeling errors. Upwind or upwind-biasedschemes, which are dissipative, on the other hand, havebeen shown to affect very significantly the spectrum ofthe resolved structures [40].

For applications to complex geometries, single-block,Cartesian meshes are inadequate, since they do not givethe required flexibility. One alternative is the use ofbody-fitted curvilinear grids. LES codes in generalizedcoordinates have been used, among others by Zang et al.[25, 41] (who applied it to a Cartesian geometry, thelid-driven cavity [25], and to the study of coastal up-welling [41, 42]), Beaudan and Moin [43] and Jordan

[44, 45]. Jordan [44] examined the issue of filtering incurvilinear coordinates, and concluded that filtering thetransformed (in the generalized coordinates) equationsdirectly in the computational space is better than per-forming the filtering either of the transformed equationsin real space, or of the untransformed equations in Car-tesian space.

Even if curvilinear grids are used, the application ofLES to complex geometries might be limited by resolu-tion requirements. In the presence of a solid boundary,for instance, a very fine mesh is required to resolve thewall layer. If a single structured mesh is used, this require-ment results in an excessively fine resolution of the outerflow. In the calculations by Piomelli [46] the velocityspectra in the near-wall region were nearly flat, indicatingmarginal resolution, while in the outer flow they decayedby three or more orders of magnitude, as the resolutionapproached that of a DNS there. These considerationshave spurred the application of block-structured or un-structured algorithms in LES.

Kravchenko et al. [47] used zonal embedded meshesand a numerical method based on B-splines to computethe flow in a two-dimensional channel, and around a cir-cular cylinder. The use of the B-splines allows use of anarbitrarily high order of accuracy for the differentiation,and accurate interpolation at the interface between thezones. A typical grid for the channel flow simulations isshown in Fig. 4, which evidences the different spanwiseresolution in the various layers, in addition to the tradi-tional stretching in the wall-normal direction. The use ofzonal grids allowed Kravchenko et al. (1996) to increasethe Reynolds number of the calculations substantially:they performed an LES of the flow at Re

#"109 410;

using nine embedded zones allowed them to resolve thewall layer (the grid spacing in the zone closest to the solidboundary was *x`K130, *z`K20) using a total of2 million points. A single-zone mesh with the same res-olution would have under-resolved the wall layer. Themean velocity profile was in excellent agreement with theexperimental data. The behavior of the eddy viscosity(they used the dynamic eddy viscosity model [33, 35]) isshown in Fig. 5. Since in coarser meshes more energyresides in the subgrid-scale motions, the eddy viscosityincreases smoothly near the boundary between a finerand a coarser zone. In each zone, moreover, the SGSstress is very close to the value obtained from a single-zone calculation in which the resolution of the embeddedzone is matched.

Very few applications of LES that use unstructuredmeshes have been reported to date. Jansen [48]simulated isotropic turbulence and plane channel. Forthe plane channel, the results were in fair agreement withDNS data (the peak streamwise turbulence intensity, forinstance, was 15% higher than that obtained in theDNS), but slightly better than the results of finite-differ-ence calculations on the same mesh. Simulations of the


Fig. 4. Zonal embedded grid with fine grid zones near the walls and coarse zones in the middle of the channel. Reproduced withpermission from Kravchenko et al. [47].

Fig. 5. Subgrid-scale shear stresses in the LES of fully developed channel flow at Re#"46 300. )))))))) Zonal boundaries. The uniform-grid

size associated with — — — denotes a simulation that has the same resolution as the second zone; similarly the — — — grid denotesa simulation that has the same resolution as the third zone. Reproduced with permission from Kravchenko et al. [47].

flow over a low-Reynolds number airfoil using thismethod [49] were in fair agreement with experimentaldata. Knight et al. [50] computed isotropic turbulencedecay using tetrahedral meshes, and compared theSmagorinsky model with results obtained relying on thenumerical dissipation to drain energy from the largescales. They found that the inclusion of an SGS modelgave improved results.

While high-order schemes can be applied fairly easilyin simple geometries, in complex configurations their use

is rather difficult. Present applications of LES to relative-ly complex flows, therefore, tend to use second-orderschemes; the increasing use of LES on body-fitted gridsfor applications to flows of engineering interest, indicatesthat, at least in the immediate future, second-order accu-rate schemes are going to increase their popularity, at theexpense of the spectral methods that have been usedfrequently in the past. Explicit filtering of the governingequations, with filter widths larger than the grid size maybe required in such circumstances.


Fig. 6. Resolved vertical shear uN /y contours. Subharmonic transition in a flat-plate boundary layer. (a) to; (b) t

o#¹/4; (c) t

o#¹/2;

(d) to#3¹/4. ¹"136 is the period of the fundamental wave, and t

o"756. Reproduced with permission from Huai et al. [52].

5. Achievements

5.1. Building-block flows

The initial applications of LES were simple, building-block flows: homogeneous turbulence, mixing layers,plane channel flows. Reviews of some of these applica-tions can be found in the literature [2, 3, 51]. These con-figurations are still frequently used for model validation,but are becoming less popular due to the increased em-phasis on non-equilibrium, more complex (physically, ifnot geometrically) flows.

5.2. Transitional and relaminarizing flows

Many flows include regions of transition or relaminar-ization. Subgrid-scale models, usually based on highReynolds number dynamics, often have difficulty in theseregions. The eddy viscosity predicted by the Smagorinskymodel or the structure-function model, for instance, isnon-zero in laminar flows; the dissipation introduced bythe model during transition is unphysical, and has theeffect of damping the growth of the small perturbations.To force the SGS stresses to zero in laminar flows, inter-mittency factors or low-Reynolds-number correctionshave been used [13, 19, 53]. In the dynamic eddy-viscos-ity model, however, the coefficient vanishes in laminarflow, where ¸

ijis identically zero; this results in better

prediction of transition without ad hoc adjustments.An additional difficulty in transitional flows is that,

during the nonlinear interaction stages of the breakdown,very small structures (thin shear layers, for instance) aregenerated, that must be resolved even in an LES. Fig. 6shows the resolved vertical shear uN /y in an xy-planeduring subharmonic transition in a flat-plate boundarylayer. One can observe the development of a shear layer

(at xK640 in Fig. 6a) that is lifted from the wall anddevelops the kinks characteristic of the multiple-spikestages. The eddy viscosity (Fig. 7) is essentially zero in thelaminar region, begins to rise at xK700, and becomessignificant where the resolution is marginal. Between theshear layers, the eddy viscosity is small, whereas sharppeaks can be observed where the shear layers are stron-ger and small scales are being generated. With muchcoarser resolution than in DNS the development of thetransitional structures was predicted as well as the statist-ical quantities.

Situations in which the perturbations decay leading toa laminar or quasi-laminar state, also occur in engineer-ing applications; in turbulent channel flow, for example,system rotation acts to stabilize the flow near one wall,de-stabilize it near the other. Piomelli and Liu [56]applied a localized dynamic model to the study of rotat-ing channel flow, and found that the use of a coefficientthat was allowed to vary in all space directions, as well asin time, gives better prediction of the turbulent fluctu-ations than the plane-averaged model, especially on thestable side of the channel where the turbulent activity isconcentrated in the down-wash region of the longitudinalroll cells that are formed in this flow.

Lamballais et al. [20] also performed DNS and LESof rotating channel flow using the spectral-dynamicmodel. They explored a range of rotation numbersRo

""2Xd/º

"(where º

"is the average velocity in the

channel, d the channel half-height and X the rotationrate) between 0 and 1.5 (compared with the range04Ro

"40.21 examined by Piomelli and Liu [56]).

Lamballais et al. [20] obtained results in fairly goodagreement with DNS data and with previous LES calcu-lations [56].

The friction velocity uq"(q8/o)1@2 (where q

8is the

wall stress and o the fluid density) is shown for various


Fig. 7. Eddy viscosity contours. Subharmonic transition in a flat-plate boundary layer. (a) to; (b) t

o#¹/4; (c) t

o#¹/2; (d) t

o#3¹/4.

¹"136 is the period of the fundamental wave, and to"756. Reproduced with permission from Huai et al. [52].

Fig. 8. Friction velocity in rotating channel flow. ), — — —, — )— Experiments (Johnston et al. [54]); #DNS (Kristoffersen and Andersen[55]); j DNS (Lamballais et al. [20]); h resolved LES (Lamballais et al. [20]); ]resolved LES (Piomelli and Liu [56]); n LES with wallmodels (Balaras et al. [57]).

rotating-channel cases in Fig. 8. The resolved LES ofPiomelli and Liu [56] are in good agreement with DNSdata [20, 55], and with the experimental data on theunstable side of the channel. On the stable side all thenumerical calculations predict significantly higher fric-tion velocity than the experiments, in which full re-laminarizationwas observed. This may be due to residualpressure gradients that were present in the experimentdue to the small dimensions of the apparatus, which mayhave increased the tendency of the flow to relaminarize.Resolved LES of Piomelli and Liu [56] are, however, ingood agreement with DNS data [20, 55] on that side aswell. The LES of Lamballais et al. [20] underpredicts thewall shear on both sides of the channel, but shows trendsin good agreement with the DNS results. At the highestrotation rate (Ro

""1.5, not included in the figure) even

the unstable side exhibits a significant decrease in uq,predicted by both LES and DNS. Lamballais et al. [20]

attributed this to a strong tendency of the vortical struc-tures to be re-oriented in the streamwise direction on theunstable side of the channel (Fig. 9). These elongatedvortices on the stable side cause rotational motions in theyz-plane that result in a decrease in the streamwise rmsfluctuations (Fig. 10), a corresponding increase in the othertwo components, and an inversion of the flow anisotropy.

5.3. Three-dimensional flows

It was mentioned before that large-eddy simulationsare based on the assumptions that the small scales aremore isotropic, and less affected by the boundary condi-tions, than the large scales. This assumption justifies theuse of simple, equilibrium-based models, even in flowsin which the resolved scales are not in equilibrium. Bythe same token, LES should be more suitable thanthe Reynolds-averaged approach to study highly three-


Fig

.9.

Isos

urfac

esofth

evo

rtic

ity

modu

lus

u"

3º"/d

.R

e ""14

000.

Rep

roduc

edw

ith

per

mission

from

Lam

bal

lais

etal

.[2

0].


Fig. 10. Rms of the fluctuating velocity components (normalized by º"). —— streamwise, )))))))) wall-normal, — — — spanwise. Left figures:

DNS, Re""5000; right figures: LES, Re

""14 000. From top to bottom: Ro

""0, 0.17, 0.5 and 1.5. h is the channel half-width, d.

Reproduced with permission from Lamballais et al. [20].

dimensional or separated flows, especially those in whichthe gradient transport hypothesis, and consequently one-and two-equation models of turbulence, fails.

Piomelli et al. [58] used the velocity fields from a DNSof the flow in a three-dimensional boundary layer ob-tained by imposing an impulsive spanwise motion, withmagnitude equal to 47% of the initial mean centerlinevelocity, to the lower wall of a fully developed planechannel flow [59] to study the physics of the SGS stressessubjected to a three-dimensional perturbation. Thea priori tests showed that the SGS stresses react to theimposition of the secondary shear ¼/y more rapidlythan the large-scale ones, and return to equilibrium be-fore the resolved stresses do. The simulations of Sarghiniand Piomelli [28] confirmed the a priori tests results:dynamic and scale-similar models give more accurateprediction of the transient, non-equilibrium phenomena.Fig. 11 shows the time development of the wall stressq8

and turbulent kinetic energy K integrated over the

entire computational domain; both quantities are nor-malized by their initial values. The dynamic eddy-viscos-ity model [33, 35] (LSQ in the figure) and the mixedmodels [23] (denoted by BFR) predict the initial decreasein the turbulence quantities fairly accurately; in particu-lar, the initial decay and successive recovery of the turbu-lent kinetic energy are predicted well, as is the time scaleof this phenomenon. Due to the low-Reynolds number ofthis flow the SGS stresses are rather small; however, theuse of the dynamic eddy viscosity or of the mixed modelgives much better agreement with the DNS data [59]than a calculation in which no model was used (CDS).The Smagorinsky model [10] (SMG) dissipates too muchenergy due to the presence of the secondary shear; theincreased dissipation tends to cause relaminarization ofthe flow.

Liu et al. [60] computed the flow in a turbulent bound-ary layer on which a pair of strong counter-rotatingvortices was superimposed, using a localized dynamic


Fig. 11. Time development of integral quantities in the three-dimensional boundary layer. (a) Wall stress qw; (b) turbulent kinetic energy

K integrated over the computational domain; all quantities are normalized by their initial values. n DNS; —— no model (coarse DNS);—— — Smagorinsky model; )))))))) mixed model; — )— dynamic eddy viscosity model. Adapted from Sarghini and Piomelli [28].

Fig. 12. Contours of streamwise vorticity at the four streamwise stations. Gray: negative contours, black: positive contours. Reproducedwith permission from Liu et al. [60].

model. The vortices generate the extra strain components»/y, »/z, ¼/y and ¼/z. The mean streamwisevorticity development is predicted much more accuratelythan when K—e models are used, due to the fact thattwo-equation models cannot predict the gradients of thenormal stress anisotropy, which play an important rolein the development of X

x. The magnitude of the eddy

viscosity that would be used in K—e models is shown inFig. 13. A dot product is used to produce an eddy

viscosity:

l%,!

SuAiu@jTSS

ijT

2SSijTSS

ijT

. (36)

(using this ‘‘least-squares’’ definition, l%

is the eddy vis-cosity that would give the correct production rate, trans-ferring the correct amount of kinetic energy from themean flow to the turbulence). The figure compares this


Fig. 13. Contours of the eddy-viscosity at the second streamwise station: (a) l%from least-squares fit to LES results; (b) l

%"CkK2/e.

Reproduced with permission from Liu et al. [60].

quantity and an eddy viscosity calculated from the K—eformula, l

%"CkK2/e, with Ck"0.09. The values of

l%

predicted by the K—e formulation are close to thoseobtained directly from the LES away from the vortex;inside the vortex, however, they are roughly double theLES values. This explains the rapid decay of the stream-wise vorticity that is observed in K—e solutions.

Wu and Squires [61] performed LES calculations ofthe flow over a 2D bump. A canonical flat-plate bound-ary layer is introduced upstream of a convex surfaceprotruding from the wall, with two short concave regionsto join it smoothly to the wall. The boundary layerexperiences a series of sign changes in the pressure gradi-ent, which is first favorable, then adverse, then favorableagain, and relaxes back to equilibrium. Corresponding tothe first abrupt change in pressure gradient, an innerlayer is formed within the boundary layer, in whichturbulent kinetic energy production and dissipation areroughly in balance, at a value twice as large as in theoriginal boundary layer. This layer becomes thicker overthe bump summit, and the peak production and dissipa-tion occur a significant distance away from the wall.Downstream of the trailing edge a second inner regiondevelops. The agreement with the experiments is very

good (Fig. 14) except at the trailing edge, where themeasurements show more significant separation thanobtained from the calculations. Wu and Squires [61] alsoextended their calculations to compute the flow overa 3D bump obtained by sweeping the bump by 45$%' withrespect to the incoming free-stream.

5.4. Separated flows

Beaudan and Moin [43] used the dynamic andSmagorinsky SGS models for the LES of the wake ofa circular cylinder at Re"3 900 (based on cylinder dia-meter and freestream velocity). The dynamic model gaveresults in better agreement with the experiments than theSmagorinsky model, especially in the recirculation regionbehind the cylinder. If the boundary layer before separ-ation is subcritical, for instance, along the separationstreamline a shear layer develops that is often inviscidlyunstable; the grid, in this region, must be sufficientlyfine to capture the shear-layer roll-up, and the SGSmodel must give vanishing eddy viscosity in this region.The dynamic model correctly predicted zero viscosityalong the shear layers following separation, unlike theSmagorinsky model.


Fig. 14. Mean velocity along vertical traverses in the 2D bump. L Experiments; —— LES. Arrows indicate the correspondingtraverses. Reproduced with permission from Wu and Squires [61].

This calculation highlighted an important numericalissue, that of the effect of numerical dissipation on theresolved scales and on the subgrid-scale model. Mittaland Moin [40] performed calculations of the same flow,using a second-order central scheme instead of the fifth-order upwind-biased method employed by Beaudan andMoin [43]. They found that the numerical dissipationdue to the upwind scheme affected very significantlythe spectra (Fig. 15). However, the mean velocity andReynolds stresses were in only slightly better agreementwith the experiments, presumably because they are most-ly due to large scales contained in a small band offrequencies that was well resolved even when the upwind

scheme was used. Further work on this configuration wasperformed by Kravchenko and Moin [62], who useda B-spline block-structured approach that allowed themto resolve the boundary layer on the cylinder and thewake better than the finite-difference calculations, whileusing a lower total number of grid points. Their resultswere in better agreement with experimental data [63]than either the central difference [40] or the upwind [43]calculations. The streamwise velocity spectra they ob-tained 7 diameters downstream of the cylinder (Fig. 15)show the inertial subrange observed experimentally, andhave a more significant small-scale content than either ofthe finite-difference calculations. In terms of mean and


Fig. 15. One-dimensional frequency spectrum at x/D"7.0. —— B-splines; f experiment [63]; )))))))) central differences; — )— upwinddifferences; —— — 5/3 slope. Reproduced with permission from Kravchenko and Moin [62].

rms profiles, however, very little difference betweenthe three simulations was observed in the near wake;only in the far wake errors due to the truncation andthe dissipative character of the upwind scheme becamesignificant. This simulation also highlighted the effectof marginal spanwise grid resolution on the numericalresults.

The backward-facing step has been the subject of sev-eral calculations: Akselvoll and Moin [64] and Delcayreand Lesieur [65], among others, performed calculationsof this flow. Akselvoll and Moin [64] used the integralformulation of the dynamic model [30] and obtainedgood agreement with experimental [66], and DNS [67]data. They found that a very fine grid was required toresolve the instability of the shear layer emanating fromthe step. Delcayre and Lesieur [65], who used the selec-tive structure-function model [53], could not predict thereattachment length X

Rcorrectly, perhaps due to the

inflow condition (they used a mean velocity profile onwhich they superimposed random noise, rather thana realistic boundary layer velocity field as in the othernumerical calculations cited). The quantitative agree-ment in renormalized coordinates, (x!X

R)/H (where

H is the step height) was however better. Delcayre andLesieur [65] studied the topology of the flow field bymeans of flow visualization and probability-density func-tions. Fig. 16, in which Q"(m

ijmij!SM

ijSMij)/2'0 [m

ijwas

defined after (17) as the antisymmetric part of the velo-city-gradient tensor] defines regions in which rotation isdominant, illustrates a helical vortex pairing that leads toincreased three dimensionality of the flow within threestep-heights of the separation point, as well as the genera-tion of K and horseshoe vortices.

Akselvoll and Moin [68] and Pierce and Moin [69]computed the flow in a coaxial jet, a configuration fre-quently used as a combustor. In the non-swirling case[68] the two jets mix slowly; a fairly long recirculationzone is formed, and the mixing between the fuel and theoxidizer is rather slow. When swirl is introduced [69],a very different picture is observed: the swirl causes thestreamlines to turn towards the outer wall, giving rise toa stagnation point on the centerline immediately down-stream of the jet exit, and causing reattachment to occurmuch sooner, with more complete mixing.

Another complex flow that has been computed by LESis the asymmetric plane diffuser [70, 71]. This flow isparticularly difficult because of the disparity of scales:due to the expansion, the mean velocity and the Reynoldsnumber in the outlet region are 4.7 times lower than inthe inlet, and the inertial time scale is 4.72K22 timeslarger. Furthermore, an adverse pressure gradient existsthat leads to an unsteady separation. This simulationhighlights the effect of the boundary conditions on theaccuracy of the results. Kaltenbach [70] used fullydeveloped channel flow to supply the inflow data,which gave rather poor agreement with the experiments(Fig. 17): the separation point was predicted incorrectly,a very small reversed-flow region was observed, and therecovery region was also not predicted very accurately.Fatica and Mittal [71] used an inlet velocity profile thatmore closely matched the measured one, and gave muchimproved results: the velocity profiles were in goodagreement with the experimental data (Fig. 17) bothin the separated and in the re-attached flow regions.The separated flow region is unsteady: separation isinitiated at some point in the diffuser; the recirculation


Fig. 16. Backward-facing step. Vorticity modulus isosurfaces; DuD"1.5ºo/H and Q'0. Reproduced with permission from Delcayre

and Lesieur [65].

Fig. 17. Mean velocity profiles in the diffuser. —— Fatica and Mittal [71]; — — — Kaltenbach [70]; s experimental data. (a) Front part ofthe diffuser; (b) rear part of the diffuser. Reproduced with permission from Fatica and Mittal [71].

zone is then swept downstream and the process repro-duces itself.

5.5. General remarks

The configurations described in this section are amongthe most advanced LES applications. They involve flowsthat are, geometrically, rather simple; however they con-tain additional physical complexities (secondary shear,

pressure gradients, streamline curvature) that are alsorelevant, sometimes dominant, in more complex, engin-eering, flows. Some include perturbations from equilib-rium followed by relaxation to a new equilibrium, aprocess that models for the RANS equations have diffi-culty predicting accurately. The physical knowledgegained from these studies can be very beneficial for thedevelopment of lower-level turbulence models for use inthe design environment.


Table 1CPU time required by various models. The CPU time is nor-malized by that of an LES that used the Smagorinsky model

Model CPU

No model 0.93Smagorinsky 1.00Dynamic eddy viscosity 1.07Dynamic mixed 1.11Lagrangian eddy viscosity 1.21Lagrangian mixed (one coefficient) 1.28Lagrangian mixed (two coefficients) 1.31

6. Challenges

6.1. Modeling

If LES is to be used successfully in complex engineer-ing configurations, several issues must be resolved. Firstamong them, the development of accurate SGS modelsfor non-equilibrium flows. In the author’s opinion, a suc-cessful model should satisfy the following requirements:

(1) Predict the overall dissipation correctly.(2) Vanish in laminar flow.(3) Depend strongly on the smallest resolved scales

(rather than on the entire turbulent spectrum).(4) Predict the local energy transfer between resolved

and subgrid scales.

A number of new models have been introduced inrecent years that give improved results with less empiri-cal input by satisfying, to some extent, the requirementslisted above. The dynamic model [33] and its derivativeshave already been mentioned. The structure-functionmodel [16] also has been applied in a variety of flowswith success. Several mixed models [25, 28] have beenused in non-equilibrium flows.

A disturbing trend, in the author’s opinion, is theincreasing complexity of SGS models. Some of the newermodels [19, 22, 26, 27], for instance, require repeated fil-tering operations, the evaluation of two dynamic coeffi-cients, or both. The repeated filtering increases the CPUtime, removes some of the local character of the model,and may be difficult in complex geometries. The evalu-ation of several coefficients also increases the CPU timerequired by a calculation. LES requires considerablecomputing efforts, especially when it is applied to three-dimensional, spatially developing flows, which requiremillions of grid points and long averaging times toachieve convergence of the statistics. It is, therefore, im-perative that the SGS models do not increase the cost ofthe calculation to such an extent as to make it impracti-cal. Table 1 shows the CPU time used by various modelsin a pseudo-spectral channel code, compared to thatrequired by the Smagorinsky model and by a simulationwith no model.4 The calculation of the SGS model, forthe dynamic eddy viscosity model requires a smallamount of the total CPU, only 7% more than theSmagorinsky model. As more complex models are used,however, the calculation of the SGS stresses may requirea substantial amount of CPU time. Two-coefficient mod-els require over 30% of the CPU time only for the modelevaluation. It is not clear whether the benefits gained bysuch models outweigh their additional cost.

4The code used a third-order Runge-Kutta time advancementfor the convective and SGS terms, and the coefficient was evalu-ated dynamically only once per step.

6.2. Acoustics

Significant interest has arisen recently in the soundemitted by turbulence. This source was considered sec-ondary in applications involving aircraft, where the jetnoise was usually dominant; however, the progress in jetengine construction over recent years has been such thatthe airframe-noise problem is becoming more and morerelevant. Since in large-eddy simulations the unsteadyvelocity and pressure fields are computed, LES has thepotential to be a very useful tool in aeroacoustics, beingable to yield more accurate and complete results (espe-cially in terms of wave number and frequency resolution),than RANS calculations, at a fraction of the cost of DNS.

Mankbadi and coworkers [72] computed the soundemitted by a jet using Lighthill’s acoustic analogy[73, 74], which decouples the aerodynamic and acousticfields. The near-field velocity and pressure are computedfirst; the far-field acoustic field is then calculated from anintegral involving the second derivatives (either in spaceor in time) of the stress term

¹ij"ou

iuj#d

ij(p!a2

=o)!kA

ui

xj

#

uj

xiB. (37)

In LES the small scales of motion are eliminated by thefiltering operation; although their contribution to themomentum balance is small, and can be modeled fairlyaccurately, the second derivatives of ¹

ijthat affect the

pressure disturbance will be more significantly affectedby the small scales. Mankbadi et al. [72] assumed thatthe SGS field contribution to the sound source and theeffect of the filtering operation were negligible. However,Piomelli et al. [75], in a priori studies of the effect of theunresolved scales on ¹

ijand its derivatives, found that

the subgrid scales give a very substantial contribution tothe rms of the source. The resolved scales, however, givethe correct spatial distribution of the source (Fig. 18),although at much reduced amplitude. Thus, one shouldexpect the wave-number distribution of the filteredsource to have the correct shape (i.e., peaks at the correct


Fig. 18. Contours of the near-field source at y`"12. (a) 2¹ij/x

ix

j; (b) 2¹M

ij/x

ix

j, exact SGS stresses. Reproduced with permission

from Piomelli et al. [75].

wave-numbers) up to the cutoff wave-number. The highwave-number components of the source, which are re-moved by the filtering procedure, cannot be recovered bymore accurate SGS modeling. These components signifi-cantly affect the higher derivatives of ¹

ij, both with

respect to time or space; however, since these small-scalefluctuations are largely uncorrelated, they are not ex-pected to affect the far-field sound significantly, at leastwhen coherent structures are present.

This a priori test raises some concerns, but does notprovide conclusive evidence on the accuracy of LES forsound calculations. More study and a posteriori tests, inwhich the actual far-field sound (rather than the sourceonly) is calculated, are required to settle this issue. Issuessuch as the use of periodic boundary conditions may alsoaffect the results significantly.

Another noise-related issue that has seen some prelimi-nary work by LES is the study of the wall pressurefluctuations [76]. The streamwise wave number fre-quency spectra of the pressure fluctuations are shown inFig. 19. The LES prediction for the resolved part of thespectrum is in good agreement with the DNS results.Since most acoustic arrays (sonar arrays, for instance)are designed to sense energy at low wave numbers, thelack of resolution of the high wave numbers in thistechnique should not limit its application to this class ofproblems.

Given the substantial amount of new information thatcan be obtained by applying LES to the study of aero-acoustics, it is surprising that very little research on noiseemission has been carried out using LES so far. Althoughthe challenges are significant (high order of accuracy,very careful application of the boundary conditions, andperhaps separate modeling of the unresolved pressurefluctuations may be required), LES has the potential tocontribute to the understanding of the physics, as well asto the design of improved devices.

6.3. Compressible flows

The applications of LES to compressible flows havebeen far fewer than for incompressible cases. This is alsoan area that will conceivably see enhanced effort in thenear future.

6.3.1. Favre filteringIn compressible flows, it is convenient to use Favre

filtering [77, 78] to avoid the introduction of subgrid-scale terms in the equation of conservation of mass.A Favre-filtered variable is defined as

fI"of/oN ; (38)


Fig. 19. Wavenumber-frequency spectra of the wall pressure in plane channel flow (Req"180): (a) DNS; (b) LES. Reproduced withpermission from Chang et al. [76].

the Favre-filtered equations of motion can be written inthe form

oNt

#

xj

(oN uJj)"0, (39)

oN uJi

t#

xj

(oN uJiuJj)#

pNx

i

!

pJji

xj

"!

qji

xj

#

xj

(pNji!pJ

ji), (40)

hij hgigj

I II

where a perfect-gas equation of state is assumed, and

pNij"2kS

ij#(k

2!2

3k)d

ijSkk

(41)

pJij"2kJ SI

ij#(kJ

2!2

3kJ )d

ijSIkk; (42)

here SIij

is the Favre-filtered strain-rate tensor, k is themolecular viscosity, and k

2the bulk viscosity; kJ "k(¹I )

and kJ2"k

2(¹I ) are their values at the filtered temper-

ature ¹I .Two unclosed terms appear in (40): term I, the diver-

gence of the SGS stresses qij"oN (u

iuj

'C!uJ

iuJj), and term II,

which is due to the nonlinearity of the viscous stresses.While the former is modeled, the latter is invariablyneglected (i.e., it is assumed that pN

ji!pJ

ji"0). The trace

of the SGS stresses in compressible flows cannot beincluded in the modified pressure, and requires separatemodeling (although it is frequently neglected [19]).

In addition to the momentum and mass conservationequations, an energy equation is required. Several op-tions are available: one can solve the internal energy, theenthalpy or the total energy equation. Each of thesechoices results in a different set of terms that must bemodeled. The equation for the internal energy per unit

mass e, for instance, is

(oN eJ )t

#

xj

(oN uJjeJ )#

qJj

xj

#pN SIkk!pJ

jiSIij

"!

xj

[oN ( uje'C!uJ

jeJ )]!

xj

[qNj!qJ

j]

hggiggj hgigjIII IV

![pSkk!pN SI

kk]#[p

jiSij!pJ

jiSIij]; (43)

hggiggj hggiggjV VI

here

qNj"!k

¹

xj

, qJj"!kI

¹Ix

j

, (44)

are heat fluxes, and kI "k (¹I ) is the value of the thermalconductivity obtained using the filtered temperature.

The under-braced terms must be modeled. Term III isthe divergence of the subgrid-scale heat flux:

oN ( uje'C!uJ

jeJ )"Q

j/c. (45)

Term IV is usually neglected, as was the analogous termin the momentum equation. Term V is the SGS pressure-dilatation, and term VI is the SGS contribution to theviscous dissipation. In past applications [80, 81] the sub-grid-scale heat-flux Q

iwas modeled, while terms V and

VI were neglected (as well as the diffusion nonlinearitiesIV). Vreman et al. [82] performed a priori tests usingDNS data obtained from the calculation of a mixinglayer at Mach numbers in the range 0.2—0.6, and con-cluded that neglecting the nonlinearities of the diffusionterms in the momentum and energy equations (terms IIand IV) is acceptable; they found, however, that the SGSpressure dilatation and SGS viscous dissipation are ofthe same order as the divergence of the SGS heat flux, Q

j,


and that modeling term VI improves the results, espe-cially at moderate or high Mach numbers.

Similar terms arise if the equation for the enthalpyh"e#p/o is used:

(oN h3 )t

#

xj

(oN uJjh3 )#

qJj

xj

!

pNt

!uJj

pNx

j

!pJjiSIij

"!

xj

[oN (ujh

'C!uJ

jh3 )]!

xj

[qNj!qJ

j]

hggiggj hgigjVII IV

#Cuj

p

xj

!uJj

pNx

jD#[p

jiSij!pJ

jiSIij]. (46)

hggiggj hgigjVIII VI

Terms IV and VI also appear in the internal energyequation; term VII is equal to the divergence of Q

j. The

velocity-pressure gradient term VIII can be decomposedto yield the pressure-dilatation term, and a pressure-diffusion part that can be related to Q

j:

uj

p

xj

!uJj

pNx

j

"

xj

[puj!pN uJ

j]![pS

jj!pN SI

jj]

"

c!1

cQ

jx

j

![pSjj!pN SI

jj], (47)

where the equation of state has been used to give

puj!pN uJ

j"oN R(¹u

j

'C!¹I uJ

j)"

c!1

cQ

j. (48)

The pressure-dilatation term has been neglected in thepast [83, 84].

The third option is to use an equation for the filteredtotal energy per unit mass EI "eJ#u

juj

'C/2:

t(oN EI )#

xjAoN uJ jEI B#

qJj

xj

#

xj

(pN uJj)!

xj

(pJjiuJj)

"!

xj

[oN (ujE

'C!oN uJ

jEI )]!

xj

[qNj!qJ

j]

hgggigggj hgigjIX IV

!

xj

[puj!pN uJ

j]#

xj

[pjiui!pJ

jiuJi]. (49)

hggiggj hggiggjX XI

In this case the convective term IX gives rise to twounclosed terms:

oN (ujE

'C!oN uJ

jEI )"oN ( u

je'C!uJ

jeJ )#oN A

ujukuk

'C

2!

uJjukuk

'C

2 B"

Qj

c#oN D

j, (50)

where

Dj"1

2(u

jukuk

'C!uJ

jukuk

'C )"12( u

jukuk

'C!uJ

juJkuJk!uJ

jqkk) (51)

is similar to the turbulence diffusion that appears in thesubgrid-scale kinetic energy equation. Knight et al. [50]proposed that D

jKq

jkuJk. The other two unclosed terms

in Eq. (50) do not require separate modeling, since theyinclude the already-modeled SGS stresses and heat flux.Term IV is usually neglected, term X can be expressed interms of the SGS heat flux Q

jusing Eq. (48); term XI is

analogous to the SGS viscous dissipation VI in the inter-nal energy or enthalpy equations, and is probably notnegligible.

Vreman et al. [82] derived an equation for a modifiedtotal energy, the total energy of the filtered field,oEY "oN (eJ#uJ

kuJk/2). In this transport equation the pres-

sure-dilatation term V, the SGS viscous dissipation VI,and a term of the form uJ

j(qf

ij/x

i) require modeling.

In summary, in compressible flows the following termsare unclosed:

(1) SGS stresses qij

(from I); various models have beenproposed, based on incompressible models (eddy-viscosity [85], mixed [83], dynamic [80, 81], andstructure function [18]). The need to predict thenormal stresses increases the model complexity (anadditional model coefficient may be required).

(2) SGS heat flux Qj

(from III, VII or IX); it can beobtained from the SGS stress term by using eithera constant or a dynamically-adjusted turbulentPrandtl number [80] or a mixed model [83].

(3) Pressure dilatation, V (the velocity—pressure gradientterm can be reduced to the pressure dilatation, andthe pressure diffusion can be modeled directly interms of Q

j). Vreman et al. [82] proposed a scale-

similar model for this term.(4) Viscous dissipation (term VI) or viscous work of

(term XI), depending on the equation chosen. Vremanet al. [82] proposed a scale-similar model for thisterm as well.

(5) Turbulent diffusion Djin Eq. (51). This appears only

if the total energy equation is used.(6) Terms arising from the nonlinearity of the diffusive

fluxes (II and V). It is probably safe to neglect them atlow or moderate temperatures.

If the diffusive nonlinearities are neglected, dependingon the set of equations chosen, four of the above termsmust be modeled: models for the SGS stresses and heatflux are always required, as well as a model for one of theviscous terms. The internal energy, enthalpy and modi-fied energy equations require modeling of the pressuredilatation, while the total energy requires modeling of thedivergence of the turbulent diffusion D

j. It is not known

whether there are advantages to either approach.


Fig. 20. Comparison of unclosed terms in the energy equations (P. Martın, personal communication).

While Vreman et al. [82] found most of these terms tobe important in a mixing layer, a comparison of theunclosed terms in the energy equation is shown in Fig. 20for a 1283 simulation of homogeneous isotropic decaywith turbulent Mach number M

t"0.52, Reynolds num-

ber (based on the Taylor micro-scale j) Rej"34.5, anda filter width DM /D"4 (P. Martın, personal communica-tion). The pressure-diffusion term, which does not requireseparate modeling but can be expressed in terms of Q

jis

significant, while the pressure-dilatation term is negli-gible. The viscous terms are also negligible, while theturbulent diffusion is significant.

Subgrid-scale stress modeling for compressible flows isat a much less advanced stage than for incompressibleones, partly due to the lack of suitable databases fora priori tests. Correspondingly, there are few baselinecalculations of building-block flows, and very few a pos-teriori evaluations of SGS models.

6.4. Wall layer modeling

Perhaps the most urgent challenge that needs to bemet, in order to apply LES to technologically relevantflows, is the modeling of the wall layer. The presence ofa solid boundary affects the physics of the subgrid scalesin several ways. First, the growth of the small scales isinhibited by the presence of the wall. Secondly, the ex-change mechanisms between the resolved and unresolvedscales are altered; in the near-wall region the subgridscales may contain some significant Reynolds-stress pro-ducing events, and the SGS model must account forthem. Finally, the length scale of the energy-carryinglarge structures is Reynolds-number dependent near thewall.

If the wall layer is resolved, the important energy-producing events must be captured. Since their dimen-sions scale with the Reynolds number, so will the cost of


Fig. 21. Sketch of the wall-layer model concept.

an LES calculation. Chapman [86] estimated that theresolution required for the outer layer of a boundarylayer is proportional to Re0.4, while for the wall layer(which, in aeronautical applications, only accounts forapproximately 1% of the boundary layer thickness) thenumber of points needed increases at least like Re1.8. Ifa single structured mesh is used, this requirement resultsin an excessively fine resolution of the outer flow, as seenearlier. The use of block-structured meshes may alleviatethis problem, but not remove it altogether.

Alternatively, approximate boundary conditions, orwall models, may be used. When the grid is not fineenough to resolve the near-wall eddies, the wall layermust be modeled by specifying a correlation between thevelocity in the outer flow and the stress at the wall. Thisapproach allows the first grid point to be located in thelogarithmic layer, and, since the energy-producing vor-tical structures in the viscous and buffer regions do nothave to be resolved, it permits the use of coarser meshesin the other directions as well: *x`K100—600,*z`K100—300. The modeling of the wall-layer physics,however, introduces further empiricism in the calcu-lations.

The basic assumption behind this approach is that theinteraction between the modeled, near-wall region, andthe resolved, outer region, is weak. Some support for thisassumption can be obtained, for instance, from a recentpaper of Brooke and Hanratty [87], in which the way thenear-wall vortices are born is investigated utilizing DNSdatabases from a turbulent channel flow. They foundthat the flow structures in the viscous wall region, whichare responsible for most of the shear stress production,regenerate themselves; no interaction with the outer layerstructures was detected.

Most wall models used in the past assume that thedynamics of the wall layer are universal, and that some

generalized law-of-the-wall can be imposed [31, 88].Balaras et al. [51] have proposed a new zonal approachbased on the consideration that, as the Reynolds numberincreases, a significant disparity of length and time-scalesdevelops between the near-wall region and the outerflow. If the grid-cell dimensions are large compared to thetypical eddy size in the near-wall layer, the cell closest tothe wall will contain a very large number of such eddies(Fig. 21), large enough to be considered a statisticallysignificant sample. Furthermore, since these eddies aresmall compared to the grid size, their life-cycle is shortcompared to the time step. Thus, the filtered (i.e., grid-averaged) velocity is not affected directly by each of theseeddies, but only by their combined effect, and it is notnecessary to resolve separately each Reynolds-stress-producing event that takes place in the wall-layer, butonly to account for them in a statistical sense, througha RANS-like model.

In the outer flow, then, the filtered Navier—Stokesequations are solved using as the wall-boundary condi-tion the wall stress supplied by the inner-layer calcu-lation. In the inner layer (i.e., from the first grid point, tothe wall), the boundary-layer equations for the mean (i.e.,averaged over the cell, in the plane parallel to the solidboundary) velocity are solved. The velocity profile ob-tained in this manner yields the cell-averaged viscousstress at the wall, required for the calculation of the outerflow at the next time-step.

The two-layer model was applied by Balaras et al.[57] to the flow in a plane channel, in a square ductand in a rotating channel. The model gave better resultsthan models based on the law-of-the-wall, most notablyfor the rotating channel flow, in which significant devi-ations from the logarithmic profile are observed, thatwere captured well by the two-layer model (see Fig. 8).The calculation of the flow in the square duct also gave


Fig. 22. Square duct flow; qw/Sq

wT profile along the lower wall. #Two-layer model; n shifted model; ——, — — — experimental data.

Reproduced with permission from Balaras et al. [57].

results in good agreement with the experimental data(Fig. 22).

Backward-facing step calculations were also per-formed by Balaras (personal communication) and byCabot [89, 90] using the Balaras et al. [57] formulation.The flow parameters matched those of the low-Reynoldsnumber calculations of Akselvoll and Moin [68]. Pre-liminary results indicate that the LES gives fairly goodprediction of the mean flow: the reattachment region ispredicted to within 5% of the value obtained from theresolved LES, at a fraction of the cost (the simulationthat uses the approximate boundary conditions requiresabout two hours on a Pentium II 300 MHz processor,whereas the resolved LES [68] used 20-30 Cray CPUhours). Mean velocity contours and turbulent kineticenergy contours and streamlines are shown in Fig. 23.Consistent with the results of Cabot [90] the reversedflow in the recirculation zone is too fast; the outer flow,however, which is mainly driven by the large spanwiserollers formed at the step, is very well predicted.

7. Conclusions

Large-eddy simulation has already demonstrated itscapabilities in calculations of relatively complex flows, atReynolds numbers that could not be reached by DNS. Atpresent, to maximize the returns, LES should be appliedto problems in which its cost is comparable to that of thesolution of the RANS equations, or to problems in whichlower-level turbulence models fail. Such problems in-clude unsteady or three-dimensional boundary layers,separated flows and flows involving geometries withsharp corners. These are problems in relatively simplegeometries, that, however, isolate one (or a few) of thefactors that are expected to be relevant in such configura-tions, albeit within a simplified geometry. Large-eddy

simulation of these flows can improve the understandingof the turbulence physics, and also be used to providedata for the development of more accurate lower-levelmodels (especially pressure statistics, which are difficultto measure experimentally).

Among its next targets are, first and foremost, flowsthat include additional geometric, as well as physical,complexities. Interest in compressible flows and inaeroacoustics is also increasing. The latter is one of theareas in fluid dynamics that receive the most substantialfinancial support from funding sources. The challengesthat need to be faced to achieve significant advancementsin these areas include the development of near-wallmodels, of accurate non-equilibrium and compressibleSGS models, and of high-order, energy-conservingmethods in curvilinear coordinates and on unstructuredmeshes.

Progress in computer technology has made it possibleto perform significant calculations on affordable desktopworkstations. The simulations that required hundreds ofCray XMP CPU hours when the author was working onhis dissertation, 12 years ago, are now routinely carriedout by his students on Pentium machines. Further tech-nological advances will benefit both the affordability ofrun-of-the-mill calculations, and the possibility of per-forming larger, leading-edge calculations in parallel envi-ronments.

Large-eddy simulation should not be construed asa fool-proof tool to obtain answers to turbulent flowproblems, nor, yet, as a design tool that can be used forreal-time optimization. With these caveats in mind, how-ever, the outlook for this technique is reasonably bright.The notable advancements in SGS modeling over the lastfew years, and the increasing number of researchers thatare applying their talents in this area are both a measureof past achievements, and a reason for optimism for thefuture.


Fig. 23. Flow over the backward-facing step; Re"5,100. Top: mean velocity contours; middle: contours of q2"Su@iu@iT; bottom:

streamlines. (E. Balaras, personal communication).

Acknowledgements

The support by the Office of Naval Research underGrant N-00014-91-J-1638 monitored by Dr. L. PatrickPurtell, and by the NASA under Grants NAG 1-1828,monitored by Dr. Craig L. Streett, and 1-1880,monitored by Dr. Michele G. Macaraeg, is gratefullyacknowledged. The author thanks Profs. Marcel Lesieurand Kyle Squires, Drs. M. Fatica, A. Kravchenko,F. Delcayre and E. Lamballais for permission to re-produce their results, Prof. Peter Bradshaw andDr. Andrea Pascarelli for much useful input, and sev-eral former and present graduate students for theircontributions.

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Documents

Large-eddy simulation: achievements and challengesugo/research/pas.pdfProgress in Aerospace Sciences 35 (1999) 335—362 Large-eddy simulation: achievements and challenges U. Piomelli*