TOWARDS WALL-ADAPTION OF TURBULENCE MODELS WITHIN THE LATTICE BOLTZMANN FRAMEWORK

Abstract

This paper presents the development towards wall adaptive explicit filters for the simulation of turbulent wall bounded flows in the framework of the lattice Boltzmann method (LBM). First, we show the effect of different collision models on the characteristics of turbulent flow simulations by employing the Taylor-Green vortex as a numerical testcase. Second, an extension of the approximate deconvolution method (ADM), see Malaspinas & Sagaut (2012), Malaspinas & Sagaut (2011) and Sagaut (2010) for the simulation of wall-bounded turbulent flows is presented. A temporal dissipation relaxation is applied for explicit filtering, in order to suppress filtering in regions, where the flow is resolved and to adapt filtering in underresolved regions in such way, that the energy drain in the scales is physically motivated and consistent with the kinetic theory of turbulence. We apply the extended ADM for the simulation of a turbulent channel flow at Reτ = 180 and Reτ = 395 to demonstrate, that the ADM method of Malaspinas & Sagaut (2011) with selective viscosity filters is strictly dissipative for low-order filters. Hence, especially for wall-bounded flows the application of the proposed adaptive relaxation of the filter can be beneficial. The lattice-Boltzmann method LBM solves a set of kinetic equations in terms of discrete velocity distribution functions fα (t,x) numerically. The discrete Boltzmann equations can be written as fα (t +∆t,x+ cα ∆t) = fα (t,x)+Ωα ( fα (t,x)) (1) where Ωα ( fα (t,x)) is the collision operator, which represents non-linear and viscous effects of the Navier Stokes equations and cα is the discrete velocity set of the lattice applied. Macroscopic moments are reconstructed with a Gauss-Hermite quadrature based on the Hermite Polynomial expansion on a discrete lattice. The first two moments 1daniel.gaudlitz@aer.mw.tum.de 2nikolaus.adams@tum.de of the velocity distribution functions are the conserved moments ρ and the momentum ρu, which read ρ = ∑ α fα , ρu = ∑ α cα fα (2) while the momentum flux is the second-order offequilibrium moment of the velocity distribution functions Π = ∑ α f neq α cα cα (3) In order to reconstruct the macroscopic equations of fluid motion, a Chapman Enskog expansion is used. The interested reader can refer to Chen & Doolen (1998) among others. To close equation (1) the collision term needs to be modeled. One well-known approach is the linearization around small perturbations of the thermodynamic equilibrium. This approach is called the Bhatnagar-Gross-Krook (BGK) ansatz, see He & Luo (1997); Guo et al. (2000); Guo & Shu (2013) or Sukop & Thorne (2006) among others, which represents the collision term as a linear relaxation towards a maxwellian equilibrium Ωα ( fα (t,x)) = fα (t +∆t,x+ cα ∆t)− fα (t,x) =− τ ( fα (t,x)− f eq α (t,x) ) . (4) f eq α (t,x) is a low Mach number truncated MaxwellBoltzmann distribution, which is adjusted in such a way, that equation (3) is fulfilled and mass and momentum are conserved. A widely used formulation for f eq α is given by f eq α = ρωα [ 1+ cα u cs + 1 2cs (uu− cs δ )uu ]