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Approximate Wall Boundary Conditions in the Large-Eddy Simulation of High Reynolds Number Flow W. CABOT and P. MOIN

Center for Turbulence Research, Stanford University, Bldg. 500, 488 Escondido Mall, Stanford, CA 94305-3030, U.S.A.

Abstract. The near-wall regions of high Reynolds numbers turbulent ﬂows must be modelled to treat many practical engineering and aeronautical applications. In this review we examine results from simulations of both attached and separated ﬂows on coarse grids in which the near-wall regions are not resolved and are instead represented by approximate wall boundary conditions. The simulations use the dynamic Smagorinsky subgrid-scale model and a second-order ﬁnite-difference method. Typical results are found to be mixed, with acceptable results found in many cases in the core of the ﬂow far from the walls, provided there is adequate numerical resolution, but with poorer results generally found near the wall. Deﬁciencies in this approach are caused in part by both inaccuracies in subgrid-scale modelling and numerical errors in the low-order ﬁnite-difference method on coarse near-wall grids, which should be taken into account when constructing models and performing largeeddy simulation on coarse grids. A promising new method for developing wall models from optimal control theory is also discussed. Key words: turbulence, large-eddy simulation, wall models, channel ﬂow, separation. Abbreviations: DNS – direct numerical simulation; LES – large-eddy simulation; RANS – Reynolds-averaged Navier–Stokes; SGS – subgrid-scale; TBLE – thin boundary layer equations

Nomenclature

A+ B C Cf Cp h k L L M P Pm Reh = = = = = = = = = = = = = damping function parameter log law intercept dynamic coefﬁcient for the Smagorinsky model 2 friction coefﬁcient, 2τw /U∞ 2 relative wall pressure coefﬁcient, 2(Pw − Po )/U∞ step height turbulent kinetic energy inertial length scale, k/ε residual SGS stress between test and grid ﬁlter levels residual SGS model strain between test and grid ﬁlter levels mean pressure matching pressure at ym step Reynolds number, U∞ h/ν

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Reτ S T U u uj U∞ Um Umi uτ x y y+ ym δ ε κ ν νs νt τ τw τwi = = = = = = = = = = = = = = = = = = = = = = = = friction Reynolds number, uτ δ/ν strain rate tensor residual SGS stress at test ﬁlter level mean streamwise velocity streamwise rms velocity ﬂuctuation intensity velocity in the inner layer free stream velocity streamwise matching velocity at ym matching velocity at ym in the ith direction friction speed streamwise coordinate wall-normal coordinate wall-normal coordinate in wall units, yuτ /ν matching height channel half-width or boundary layer thickness effective ﬁlter width turbulent dissipation rate von Kármán constant, inverse slope of log law kinematic coefﬁcient of molecular viscosity SGS eddy viscosity eddy viscosity in the inner layer residual SGS stress at grid ﬁlter level streamwise wall stress wall stress in the ith direction

W. CABOT AND P. MOIN

1. Introduction The near-wall region in high Reynolds number turbulent ﬂow contains small vortical structures (streaks) that are dynamically important to the ﬂow, but which have dimensions that scale with the viscous scale, making it impractical to resolve them in numerical simulations at very high Reynolds numbers. Thus there is a crucial need to approximate the overall dynamical effects of the streaks on the larger outer scales through appropriate boundary conditions without resolving the inner viscous regions. On the other hand, the need for approximate wall boundary conditions has not been so well established in separated ﬂow regions, which do not exhibit this streak-like structure, and which behave effectively like low Reynolds numbers ﬂows with lower resolution requirements. In 1970, Deardorff [17], constrained by the limited computing power of his time,...