Unformatted text preview: Introduction to Large Eddy Simulation of Turbulent Flows 1
J. Frohlich, W. Rodi Institute for Hydromechanics, University of Karlsruhe, Kaiserstra e 12, D76128 Karlsruhe, Germany Contents
1 Introduction
1.1 Resolution requirements of DNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The basic idea of LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 3.5 Schumann's approach Filtering . . . . . . . . Variable lter size . . Implicit versus explicit ..... ..... ..... ltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 3 4 5 6 2 Governing equations and Filtering 3 3 Subgrid{scale modelling Introduction . . . . . . . . . . . . . . . . . . Smagorinsky model . . . . . . . . . . . . . . Dynamic procedure . . . . . . . . . . . . . . Scale similarity models . . . . . . . . . . . . Further models and comparative discussion .6 .7 .8 . 10 . 10 6 4 Numerical methods 4.1 Discretization schemes in space and time . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Analysis of numerical schemes for LES . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.3 Further developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.1 5.2 5.3 5.4 5.5 Resolution of the near{wall region Wall functions . . . . . . . . . . . Other approaches . . . . . . . . . . In ow and out ow conditions . . . Sample computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Boundary conditions 14 15 15 16 17 17 6 Concluding remarks
1 19 to appear in B.E. Launder, N.D. Sandham (eds.): Closure Stratigies for Turbulent and Transitional Flows, Cambridge University Press 1 Introduction
This chapter is meant as an introduction to Large{Eddy Simulation (LES) for readers not familiar with it. It therefore presents some classical material in a concise way and supplements it with pointers to recent trends and literature. For the same reason we shall focus on issues of methodology rather than applications. The latter are covered elsewhere in this volume. Furthermore, LES is closely related to direct numerical simulation (DNS) which is also discussed in several chapters of this volume. Hence, we concentrate as much as possible on those features which are particular to LES and which distinguish it from other computational methods. For the present text we have assembled material from research papers, earlier introductions and reviews (Ferziger(1996), Hartel (1996), Piomelli (1998) ), and our own results. The selection and presentation is of course biased by the authors' own point of view. Supplementary material is available in the cited references. 1.1 Resolution requirements of DNS
The principal di culty of computing and modelling turbulent ows resides in the dominance of non{linear e ects and the continuous and wide spectrum of observed scales. Without going into details (the reader might consult classical text books such as Tennekes & Lumley (1972)) we just recall here that the ratio of the size of the largest turbulent eddies in a=4 ow, L, to that of the smallest ones determined by viscosity, , behaves like L= Re3 . Here, Reu = u L= with u being a characteristic velocity uctuation and u the kinematic viscosity. Let us consider as an example a plane channel, a prototype of an internal ow. Reynolds (1989) estimated Reu Re0:9 from u u c1=2, cf Re 0:2, where f Re is based on the center line velocity and the channel height. In a DNS no turbulence model is applied so that motions of all size have to be resolved numerically by a grid which is su ciently ne. Hence, the computational requirements increase rapidly with Re. According to this estimate a DNS of channel ow at Re = 106 for example would take around hundred years on a computer running at several GFLOPS. This is obviously not feasible. Moreover, in an expensive DNS a huge amount of information would be generated which is mostly not required by the practical user. He or she would mostly be content with knowing the average ow and some lower moments to a precision of a few percent. Hence, for many applications a DNS which is of great value for theoretical investigations and model testing is not only una ordable but would also result in computational overkill.
0 0 0 0 0 0 ; 1.2 The basic idea of LES
Suppose somebody wants to perform a DNS but the grid that would be required exceeds the capacity of the available computer so a coarser grid is used. This coarser grid is able to resolve the larger eddies in the ow but not the ones which are smaller than one or two cells. From a physical point of view, however, there is an interaction between the motions on all scales so that the result for the large scales would generally be wrong without taking into account the in uence of the ne scales on the large ones. This requires a so{called subgrid{scale model as discussed below. Hence, LES can be viewed as a `poor man's DNS'. The poor man, however, has to compensate by cleverness in that a model for the unresolved motion has to be devised and an intricate coupling between physical and numerical modelling is generated. On the other hand, the resolution of the large scales of the ow while modelling only the small ones { not the entire spectrum { is an advantage of the LES approach compared to methods based on the Reynolds averaged Navier{Stokes equations (RANS). The latter methods often have di culties when applied to complex ows with pronounced vortex shedding or special in uences of buoyancy, curvature, rotation or compression. Finally, LES gives access to the dominant unsteady motion so that it can, for example, be used to study aero{acoustics, uid{structure coupling or the control of turbulence by an appropriate unsteady forcing. 2 Governing equations and Filtering
The Navier{Stokes equations (NSE) constitute the starting point for any turbulence simulation. Here, we consider incompressible, constant{density uids for which these equations read @ui = 0 @xi @ui + @ (uiuj ) + @ = @ ( 2Sij ) @t @xj @ xi @xj where Sij = (@uj [email protected] + @[email protected] )=2 is the strain{rate tensor and
!1 (1) (2) = p= . For later reference we introduce Reynolds averaging which isR used in statistical turbulence mod1 elling (RANS) as time averaging: hui = limT T 0T u dt. Reynolds averaging has the properties hhuii = hui huhvii = huihvi: (3) According to the idea of LES a means is required to distinguish between small, unresolved, and larger, resolved structures. This is accomplished by the operation u 7! u de ned below. Unlike the above Reynolds time averaging it is an operation in space. The fact that RANS and LES methods employ averaging in di erent dimensions inhibits an easy link between them. Several attempts have been made to put both in a common framework (Speziale 1998, Germano 1999) but will not be discussed here. We now turn to the ways of de ning u and illustrate them in the one{dimensional case. 2.1 Schumann's approach
The `volume{balance approach' of Schumann (1975) starts from a given nite volume mesh. The integral of a continuous unknown u(x) in (1),(2) over one cell is denoted V u = 1 R u(x)dx as illustrated in Fig. 1 (indices referring to cells are dropped). Integrating xV the NSE over a cell and using Gauss' theorem relates these values to surface{averaged quantities denoted j , such as j uv. These need to be expressed in terms of the cell{ averages, which is done in two steps. If the discretization is su ciently ne replacing j uv by j uj v as is usual in nite volume methods is possible with only a minor approximation error. This is done in DNS. If the grid is not ne enough, however,the di erence can be signi cant and the unresolved momentum ux j uv ; j uj v has to be accounted for by a model, the so{called subgrid{scale (SGS) model. Subsequently, j u are related to the V u either by setting them equal to cell averaged quantities if a staggered arrangement is used or by interpolating from neighbouring values. The nal SGS contribution to be modelled therefore also depends on the expressions used for j u , i.e. on the discretization scheme. To sum up, the equations are discretized and with discretizing them the splitting into large and small scales is performed since the latter cannot be resolved by the discrete system. Observe that the operations u 7! V u and u 7! j u map an integrable function onto discrete values, a continuous function u(x) is not constructed. Thus, with Schumann's approach, scale separation, discretization, and SGS model are not separated conceptually but intimately tied together. This has advantages in that anisotropies and inhomogeneities of the grid can easily be incorporated. However, it renders the analysis of the various contributions to the solution relatively di cult and hence is considered too restrictive by many workers in the eld. 2.2 Filtering
Leonard (1974) proposed to de ne u by u(x) = Z+ 1 ;1 G (x ; x ) u (x ) dx :
0 0 0 (4) An integral of this kind is called a convolution. Here, G is a compactly supported or at least R rapidly decaying lter function with qG(x) dx = 1 and width . The latter can be de ned by theq second moment of G as = 12 R x2G(x) dx. Fig. 2 displays the Gaussian Filter GG = 6= 1= exp(;6x2= 2) and the box lter de ned by GB = 1= if jxj =2 and GB = 0 elsewhere. In fact, already Deardor (1966) used (4) in the special case G = GB . Figs. 3a and 3b illustrate the ltering with smaller or larger lter width: the larger , the smoother is u. According to (4) u is a continuous smooth function as displayed in Fig. 3 which can subsequently be discretized by any numerical method. This has the advantage that one can separate conceptually the ltering from the discretization issue. It is helpful to transfer eq. (4) to Fourier space by means of the de nition u(!) = ^ R i!x dx, since in Fourier space, where the spatial frequency ! is the independent u(x) e variable, a convolution integral turns into a simple product. Eq. (4) then reads
; b b b u(!) = G(!) u(!) : (5) Figure 4 illustrates the ltering in Fourier space. Equation (5) allows the de nition of b another lter, the Fourier cuto lter with GF (!) = 1 if j!j = and 0 elsewhere. From b b (5) it is obvious that only this lter yields u = u since (GF )2 = GF . In all other cases the identity is not ful lled. This can be appreciated by comparing u and u for the box lter in Figs. 3 and 4. The second relation in (3) is never ful lled except in trivial cases, so that for general ltering we have u 6= u uv 6= u v (6) which distinguishes clearly the ltering in LES from Reynolds averaging (see Germano (1992) for a detailed discussion). The vertical line in Fig. 4 represents the nominal cuto d at = related to the grid. The Fourier cuto lter GF would yield a spectrum of u which is equal to the one of u left of this line and zero right of it. Eq. (5) and Fig. 4 therefore demonstrate that when a general lter is applied, such as the box lter, this does not yield a neat cut through the energy spectrum but rather some smoother decay to zero. This is important since SGS modelling often assumes that the spectrum of the resolved scales near the cuto follows an inertial spectrum with a particular slope and a particular amount of energy transported from the coarse to the ne scales on the average. We see that even if u ful lls this property this can be altered by the ltering (for further remarks see Section 4.3). Nevertheless, it is convenient and common to use the notion of a simple cuto as a model in qualitative discussions. Eq. (5) is also helpful to illustrate that derivative and lter commute, i.e. @ [email protected] = (@[email protected]). Any convolution lter (4) can be written as in (5) regardless of the choice of G. Di erentiation appears as multiplication by i! in Fourier space, eq. (28) below, which is commutative. Applying the three{dimensional equivalent of the lter (4) to the NSE (1) and (2), the following equations for the ltered velocity components ui result where Sij and @ ui = 0 @xi @ ui + @ (ui uj ) + @ = @ ( 2S ij ) ; @ ij @t @xj @ xi @xj @xj (7) (8) (9) are de ned analogous to the un ltered case. The term ij = ui uj ; ui uj represents the impact of the unresolved velocity components on the resolved ones and has to be modelled. In mathematical terms it arises from the nonlinearity of the convection term which does not commute with the linear ltering operation. An important property of ui is that it depends on time. Hence, an LES necessarily is an unsteady computation. Furthermore, ui always depends on all three space{dimensions (except for very special cases). Symmetries of the boundary conditions generally produce the same symmetries for the RANS variable hui i, e.g. vanishing dependence on a homogeneous direction. However, due to the very nature of turbulence this does not hold for ui since the instantaneous turbulent motion is always three{dimensional. The fact that a three{ dimensional unsteady ow is to be computed makes LES a computationally demanding approach. We nally note that for any lter the term in (9) vanishes in the limit ! 0, since then u ! u according to (4), and all scales are resolved so that the LES turns into a DNS. 2.3 Variable lter size
It should be mentioned here that ltering as de ned by (4) is not easily compatible with boundary conditions. Applying a box lter of constant size for instance yields u 6= 0 within a distance =2 outside of the computational domain and raises the question of how to impose boundary conditions for u. This issue is removed by supposing G to be x{dependent and locally asymmetric. However, if G(x ; x ) is generalized to some G(x x ) or if the prolongation of u from a nite domain to the real axis induces discontinuities, the commutation property is lost and additional commutator terms arise in (7), (8) (Ghosal & Moin 1995). In contrast to the usual SGS term ij , which is generated by the nonlinearity of the convection term, the commutator also appears for linear expressions (see the discussion by Geurts (1999) and Section 4.3). This issue is relevant for pronounced grid stretching in the interiour of the domain and close to walls but has been disregarded until recently. Studies for a channel ow are reported in (Frohlich et al. 1998, 2000).
0 0 2.4 Implicit versus explicit ltering
The ltering approach relaxes the link between the size of the computed scales and the size of the grid since the lter can be coarser than the employed grid. Consequently, the modelled motion should be called sub lter{ rather than subgrid{scale motion. The latter labelling results from the Schumann{type approach and is frequently used for historical reasons to designate the former. In practice, however, the lter G does not appear explicitly at all in many LES codes 2 so that in fact the Schumann approach is followed. Due to the conceptual advantages of the ltering approach reconciliation of both is generally attempted in two ways. The rst observation is that a nite di erence method for (7), (8) with a box lter employs the same discrete unknowns as Schumann's approach, such as u(xk ) = V u with k referring to a grid point. Choosing appropriate nite di erence formulae the same or very similar discretization matrices are obtained in both cases. Another argument is that the de nition of discrete unknowns amounts to an `implicit ltering' { i.e. ltering with some unknown lter ( but one that in principle exists ) { since any scale smaller than the grid is automatically discarded. In this way the lter is more or less used symbolically only to make the e ect of a later discretization appear in the continuous equations. This is easier in terms of notation and stimulates physical reasoning for the subsequent SGS modelling. In contrast to implicit ltering one can use a computational grid ner than the width of G and only retain the largest scales by some (explicit) ltering operation. This explicit ltering is recently being advocated by several authors such as Moin (1997) since it considerably reduces numerical discretization errors as the retained motion is always well resolved. On the other hand it increases the modelling demands since for the same number of grid points more scales of turbulent motion have to be modelled and it is up to now not fully clear which approach is more advantageous (Lund & Kaltenbach 1995). The ltering approach of Leonard is almost exclusively introduced today in papers on LES and has triggered substantial development, e.g. in subgrid{scale modelling. In practice, however, it is most often used rather as a concept than as a precise algorithmic construction.
k 3 Subgrid{scale modelling
3.1 Introduction
Subgridscale modelling is a particular feature of LES and distinguishes it from all other approaches. It is well{known that in three{dimensional turbulent ows energy cascades in the mean from large to small scales. The primary task of the SGS model therefore is to ensure that the energy drain in the LES is the same as obtained with the cascade fully resolved as in a DNS. The cascading, however, is an average process. Locally and instantaneously the transfer of energy can be much larger or much smaller than the average and can also occur in the opposite direction ("backscatter") (Piomelli et al. 1996). Hence, ideally, the SGS model should also account for this local, instantaneous transfer. If the grid scale is much ner than the dominant scales of the ow, even a crude model will su ce to yield the right behaviour of the dominant scales. This is due to two reasons. First, the larger the distance in wavenumber space between di erent contributions the looser is
2 Apart from some ltering operations for the dynamic model, discussed below, which is of a somewhat di erent nature their coupling. Second, as a consequence of this as well as of the energy cascading, the ner scales exhibit a more universal character which is more amenable to modelling. On the other hand, if the grid scale is coarse and close to the most energetic, anisotropic, and inhomogeneous scales the SGS model should be of better quality. Obviously, there exist two possible approaches, one is to improve the SGS model and the other is to re ne the grid. In the limit, the SGS contribution vanishes and the LES turns into a DNS. Re ning the grid, however is restricted due to the rapid increase of the computational cost. The alternative strategy, for example solving an additional transport equation in a more elaborate SGS model, can be comparatively inexpensive. Another aspect results from the numerical discretization scheme which introduces a difference between the continuous di erential operators and their discrete equivalents. This di erence is particularly large close to the cuto scale. For DNS this is not so disturbing, but with LES precisely these scales have a substantial in uence on the modelled SGS contribution as will be illustrated below. Hence, in LES the discretization scheme and the SGS model have to be viewed together. Indeed, some schemes such as low order upwind discretizations generate a considerable amount of numerical dissipation as discussed in Section 5.2 below. Therefore certain authors perform LES without any explicit SGS model (Tamura, Ohta & Kuwahara 1990, Meinke et al. 1998). The grid is re ned as much as possible to decrease the importance of the SGS terms, and the energy drain is in one way or another accomplished by the nu...
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 Numerical Analysis, ows, SGS model

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