We include Smagorinsky's algebraic eddy viscosity approach into the multiple-relaxationtime (MRT) lattice Boltzmann equation (LBE) for large-eddy simulations (LES) of turbulent flows. The main advantage of the MRT-LBE model over the popular lattice BGK model is a significant improvement of numerical stability which leads to a substantial reduction of oscillations in the pressure field, especially for turbulent flow simulations near the numerical stability limit. The MRT-LBE model for LES is validated with a benchmark case of a surface mounted cube in a channel at Re = 40 000. Our preliminary results agree well with experimental data.

This paper presents an approach for adaptive flow simulations based on Lattice-Boltzmann (LB) models in the sense, that in addition to an a priori distribution of degrees of freedom (DOF) a dynamic modification of the computational grid (coarsening and/or refinement) in the course of the transient computation is initiated, based on the local evaluation of error indicators in order to optimize the ratio of accuracy versus absolute number of DOF. Using quadtree-based data structures, both arbitrarily formed grid interfaces of different resolutions as well as their dynamic modification are facilitated. Efficiency aspects will be discussed based on laminar two-dimensional flow simulations. The obtained results compare well to reference solutions cited in the literature and indicate the usefulness and computational efficiency of the approach.

This article provides a concise survey of the lattice Boltzmann equation: its mathematical theory and its capabilities for applications in computational fluid dynamics (CFD). The lattice Boltzmann method stems from the Boltzmann equation, and thus differentiates from any conventional method for CFD based on direct discretizations of the Navier–Stokes equations. The lattice Boltzmann method is formulated for near incompressible flows. We will show some examples to demonstrate the state of the art and capabilities of the lattice Boltzmann method. The examples include direct numerical simulations of decaying homogeneous turbulence, large eddy simulations of flows past a sphere, suspensions in fluid, a droplet moving on a surface, and multi‐component flows through porous media. The chapter is concluded with an outlook of future work.

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We present a comprehensive analysis of the cumulant lattice Boltzmann model with the three-dimensional Taylor–Green vortex benchmark at Reynolds number 1600. The cumulant model is investigated in several different variants, using regularization, fourth-order convergent diffusion and fourth-order convergent advection with and without limiters. In addition, a cumulant model combined with a WALE sub-grid scale model is being evaluated. The turbulence model is found to filter out the high wave number contributions from the energy spectrum and the enstrophy, while the non-filtered cumulant methods show good correspondence to spectral simulations even for the high wave numbers. The application of the WALE turbulence model appears to be counter productive for the Taylor–Green vortex at a Reynolds number of 1600. At much higher Reynolds numbers ($${\hbox {Re}}=160{,}000$$ Re = 160 , 000 ) a deviation from the ideal Kolmogorov theory can be observed in the absence of an explicit turbulence model. Cumulant models with fourth-order convergent diffusion show much better results than single relaxation time methods.