Pope, Stephen B.

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Ten Questions Concerning the Large-Eddy Simulationof Turbulent FlowsStephen B. PopeSibley School of Mechanical & Aerospace EngineeringCornell UniversityIthaca NY 14853September 2003Workshop on: LES & SGS Modeling For Turbulent Mixing and Reactive FlowsCaltech December 8-9, 2003ABSTRACTThere have been many and substantial advances in LES since the pioneering works ofSmagorinsky (1963), Lilly (1967), Deardorﬀ (1974), Schumann (1975) and others. Advanceshave been made in: modelling the unresolved processes; accurate numerical methods onstructured and unstructured grids; detailed comparison of LES calculations with DNS andexperimental data in canonical ﬂows; extensions to include additional phenomena, e.g.,turbulent combustion; and in computational power, which has increased by about fourorders of magnitude since the 1970’s.In spite of these advances, there remain fundamental questions about the conceptualfoundations of LES, and about the methodologies and protocols used in its application. Thepurpose of this talk is to raise and to discuss some of these questions.Before posing the questions to be addressed, it is necessary to introduce the terminologyused to describe LES. This needs to be done with some care in order to include existingdivergent views on LES, and to avoid pre-judging some of the questions raised. The fun-damental quantity considered in LES is a three-dimensional unsteady velocity ﬁeld whichis intended to represent the larger-scale motions of the turbulent ﬂow under consideration.We refer to this as the resolved velocity field and denote it by W(x, t). In the “ﬁlteringapproach” introduced by Leonard (1974), W(x, t) is identiﬁed as the filtered velocity field ,denoted by U(x, t), obtained by applying a low-pass spatial ﬁlter of characteristic width∆ to the underlying turbulent velocity ﬁeld, U(x, t). The eﬀects of the sub-ﬁlter scalesare modelled, and the resulting evolution equation for W(x, t) is solved numerically on amesh of spacing h. In contrast, in the “MILES approach” advocated by Boris et al. (1992),the Navier-Stokes equations are written forW(x, t) and are solved on a mesh of spacing hwhich is insuﬃciently ﬁne to resolve the smaller-scale motions, using a numerical methoddesigned to respond appropriately in regions of inadequate spatial resolution. In order toaccommodate all viewpoints we refer to W(x, t) as the resolved velocity field, and to ∆ asthe turbulence-resolution lengthscale, which for MILES we deﬁne as ∆ = h. Turbulent mo-tions that are not resolved are referred to as residual motions, and we use the term residualstress for the quantity often referred to as the sub-grid scale (SGS) stress, or the sub-ﬁlterscale stress.The fundamental quantity in LES–namely the resolved velocity ﬁeld W(x, t)—is anextremely complex object. It is a three-dimensional, time-dependent random ﬁeld, whichhas a fundamental dependence on the artiﬁcial (i.e., non-physical) parameter ∆, and which(in some approaches) depends also on the mesh spacing h and on the numerical methodused. It is not surprising, therefore, that LES raises non-trivial conceptual questions.We address the following ten questions:1. Is LES the right approach?2. Can the resolution of all scales be made tractable?3. Do we have suﬃcient computer power for LES?4. Is LES a physical model, a numerical procedure, or a combination of both?5. How can LES by made complete?6. What is the relationship between U andW?7. How do predicted ﬂow statistics depend on ∆?8. What is the goal of an LES calculation?9. How are diﬀerent LES models to be appraised?10. Why is the dynamic procedure successful?There is insuﬃcient space here to discuss these questions fully. Prior to the Workshop,in November 2003, a paper addressing these questions will be available athttp://mae.cornell.edu/~pope/Reports.For ﬂows in which rate-controlling processes occur below the resolved scales (e.g., near-wall ﬂows and combustion), LES calculations have a ﬁrst-order dependence on the modellingof these processes. Approaches that include a statistical resolution of all scales provide amore fundamental description of the rate-controlling processes; but it remains a challengeto devise such approaches that are computationally tractable and free of empiricism.The relationship between the resolved LES velocity ﬁeld W(x, t) and the turbulentvelocity ﬁeld U(x, t) can only be statistical. Corresponding to a turbulence statistic Q, theLES provides a model Qm for Q of the formQm = Qw + Qr, (1)where Qw is the contribution from the resolved motions (which is obtained directly fromW) and Qr is the modelled contribution from the residual motions. In LES, the turbulenceresolution lengthscale ∆(x) is an artiﬁcial parameter of prime importance. As a rule,as ∆ decreases, Qw increases and Qr decreases. Unless demonstrated otherwise, there isevery reason to suppose that LES predictions Qm depend (maybe strongly) on ∆. As aconsequence, characterizing the dependence of predictions on ∆ must be part of the overallLES methodology.As currently practiced, LES is incomplete because the turbulence resolution lengthscale∆(x) is speciﬁed subjectively in a ﬂow-dependent manner. It can be made complete throughadaptive LES. The variation of ∆(x, t) is controlled (by grid adaption) so that a measureM(x, t) of turbulence resolution (e.g., the fraction of the kinetic energy in the resolvedmotions) is everywhere below a speciﬁed tolerance M .An alternative principle is advanced to justify the dynamic procedure, namely: the LESmodel coeﬃcients should be chosen to minimize the diﬀerence between Qm(∆) and Qm( ˜∆)(where ∆ is the value of ∆ used in the LES, and ˜∆ is somewhat larger). It is shown that thisprinciple applied to the Smagorinsky model results in essentially the same formula for thecoeﬃcient cs as the standard dynamic model. Rather than depending on scale similarity,the procedure selects cs to minimize the dependence of Qm on ∆ in regions where scalesimilarity does not apply.AcknowledgmentThis work is supported by the Air Force Oﬃce of Scientiﬁc Research under Grant No.F-49620-00-1-0171.

Ten Questions Concerning the Large-Eddy Simulation of Turbulent Flows

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