Grinstein, Fernando F.

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ON IMPLICIT SUBGRID SCALE MODELING FOR TURBULENT FLOWSFernando F. GrinsteinLCP&FD, Naval Research Laboratory, Washington, DC 20375-5344Large Eddy Simulation (LES) is an effective intermediate approach between DNS andRANS, capable of simulating flow features which cannot be handled with RANS such as signifi-cant flow unsteadiness and strong vortex-acoustic couplings, and providing higher accuracy thanRANS at reasonable cost but still typically an order of magnitude more expensive. Desirablemodeling choices involve selecting an appropriate discretization of the flow problem at hand,such that the LES cut-off lies within the inertial sub-range, and ensuring that a smooth transitioncan be enforced at the cut-off. The main assumptions of LES are: (i) that transport is largelygoverned by large-scale unsteady features and that such dominant features of the flow can beresolved, (ii) that the less-demanding accounting of the small-scale flow features can be un-dertaken by using suitable Sub Grid Scale (SGS) models. In the absence of an accepted universaltheory of turbulence, the development and improvement of SGS models are unavoidablypragmatic and based on the rational use of empirical information. Classical approaches haveincluded many proposals ranging from, inherently-limited eddy-viscosity formulations, to moresophisticated and accurate mixed models, e.g., [1].For the sake of the discussion, we restrict ourselves to the incompressible regime, andfocus on the crucial LES closure issue of modeling the SGS stress tensor, † B = v ƒ v - v N ƒ v N ,which needs to be prescribed in terms of the represented-filtered velocity v N , where the upper baris denotes filtered quantities. The SGS stress tensor can be conveniently recast in the form† B = (v ƒ v - vN ƒ vN ) + (vN ƒ vN - v N ƒ v N ) = B1 + B2 , where B1 denotes the interaction betweenrepresented and non-represented scales – which is not known a priori – and therefore must bemodeled, whereas B2 relates to the interaction between filtered and discretized represented scaleswhich can be approximated by prescribing an estimated vN  in the represented-velocity space(i.e., the solution to the so-called soft deconvolution problem), [2]. In this framework, a basicstructural SGS model such as the scale-similarity model provides B2, and the eventual need ofmixed models results from the recognition that  B 2  is not dissipative enough and a secondaryregularization through B1 is needed, i.e., an approximation to v in physical-velocity space must beprescribed (the hard deconvolution problem).The main drawback of mixed models relates to their computational complexity, andultimately, to the fact that well-resolved (discretization-independent) LES is prohibitively expen-sive for the practical flows of interest at moderate-to-high Re. In fact, because of the need todistinctly separate (i.e. resolve) the effects of explicit filtering and SGS reconstruction modelsfrom those due to discretization, carrying out such well-resolved LES can typically amount inpractice to performing a coarse DNS. As a consequence, it has been argued that the use of hybridRANS/LES models for realistic whole-domain complex configurations might be unavoidable inthe foreseeable future, e.g., [3]. This has recently led many researchers to abandoning theclassical LES formulations, shifting the focus directly to the SGS modeling implicitly provided bynon-linear stabilization achieved algorithmically, through use of a particular class of numericalschemes, or based on regularization of the discretization of the conservation laws, [4].The traditional approaches motivated by physical considerations on the energy transfermechanism from resolved to subgrid scales, express B1 in terms of an appropriate functional B1(e.g. an eddy-viscosity SGS model), and seek sufficiently high-order discretization and gridresolution to ensure that their effects are sufficiently small. However, it can also be argued that B1might be implicitly provided by discretization if non-linear stabilization can be achieved al-gorithmically, through use of a particular class of numerical schemes, or based on regularizationof the discretization of the conservation laws. In fact, most schemes can potentially provide built-in or implicit SGS models enforced by the discretization errors, provided that their leading orderterms are dissipative. We are thus lead to the natural question: to what extent can we avoid the(explicit) filtering and modeling phases of LES (i.e., B2≡0) and focus on the implicit B1  providedby a suitably-chosen discretization scheme?Not all implicitly implemented SGS models are expected to work for LES: the numericalscheme has to be constructed such that the leading order truncation errors satisfy physically re-quired SGS-model properties, and hence non-linear discretization procedures will be required.The analogy to be recalled is that of shock-capturing schemes designed under the requirements ofconvergence to weak solution while satisfying the entropy condition. Finite-volume versions ofsuch schemes can likewise be viewed as relevant for Implicit LES (ILES) if we focus on thesmall-scale characteristic features of turbulence. In the Monotonically Integrated LES (MILES)approach – see  Ref. [5] for a recent survey, the effects of the SGS physics on the resolved scalesare incorporated in the functional reconstruction of the convective fluxes using locally-monotonicmethods. Analysis based on the modified equations can be used to demonstrate an intriguing fea-ture of MILES, namely that when based on a particular class of flux-limiting schemes, theconvection discretization implicitly generates a non-linear tensor-valued eddy-viscosity that actsto stabilize the flow and suppress unphysical oscillations. The MILES performance has beendemonstrated in case studies including, 1) canonical flows (forced & decaying homogeneousisotropic turbulence and turbulent channel flows), 2) complex free and wall-constrained flows(mixing layers, jets, wakes, jets and flow past a prolate spheroid), and, 3) extremely complexflows at the frontiers of current unsteady flow simulation capabilities (e.g., submarinehydrodynamics and pollutant dispersion in urban areas).MILES seeks to emulate the flow features in the high-wave-number end of the inertialsubrange region of turbulent flows – characterized by thin filaments of intense vorticityembedded in a background of weak vorticity. We have proposed that emulation of the latterfeature be the requirement for implicit SGS models. MILES can thus extend to the more generalconcept of non-linear ILES in which the functional reconstruction of the convective fluxfunctions is carried out using high-resolution non-linear numerical schemes incorporating a sharpvelocity-gradient capturing capability operating at the smallest resolved scales. By focusing onthe inviscid inertial-range dynamics and on regularization of the under-resolved flow, ILES thusfollows up very naturally on the historical precedent of using this kind of numerical schemes forshock capturing. Challenges for ILES development include developing a common appropriatemathematical and physical framework for its analysis and development, further understanding theconnections between implicit SGS model and numerical scheme, and in particular, addressinghow to build physics into the numerical scheme to improve on the global ILES performance, i.e.,on the implicitly-implemented SGS dissipation & backscatter features. Moreover, additional(explicit) SGS modeling might be needed to address inherently small-scale physical phenomenasuch as scalar mixing and combustion – which are actually outside the realm of any LESapproach: how do we exploit the implicit SGS modeling provided by the numerics, to buildefficient "mixed" (explicit/implicit) SGS models ?[1] Sagaut P.; 2002, “Large Eddy Simulation for Incompressible Flows”, Springer, New York. [2] Adams N.A. & Stolz S.; 2001, “Deconvolution Methods for Subgrid-Scale Approximation inLES”, in Modern Simulation Strategies for Turbulent Flows, B.J. Geurts, ed., Edwards, Phila-delphia, p.21.[3] Spalart P.R., Jou, W.H., Strelets M. and All-maras S.R.; 1997, “Comments on the Feasibilityof LES for Wings, and on Hybrid RANS/LES Approach”, in Advances in DNS/LES, FirstAFOSR International Conference in DNS/LES (Greyden Press, Columbus).[4] Alternative LES and Hybrid RANS/LES; 2002, edited by F.F. Grinstein & G.Em Karniada-kis, J. Fluids Engineering, 124, pp. 821-942.[5] Grinstein, F.F. and Fureby; 2003 C., Implicit Large Eddy Simulation of High-Re Flows withFlux-Limiting Schemes, AIAA Paper 2003-4100, AIAA CFD Conference, Orlando, June 2003.

On Implicit Subgrid Scale Modeling for Turbulent Flows

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