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The possibilities of simulation of viscous incompressible fluid flows with lattice Boltzmann method are considered. Unlike the traditional discretization approach based on the use of Navier–Stokes equations, the lattice Boltzmann method uses a mesoscopic model to simulate incompressible fluid flows. Macroscopic parameters of a fluid, such as density and velocity, are expressed through the moments of the discrete probability distribution function. Discretization of the lattice Boltzmann equation is carried out using schemes D2Q9 (two-dimensional case) and D3Q19 (three-dimensional case). To simulate collisions between pseudo-particles, the Bhatnagar–Gross–Crooke approximation with one relaxation time is used. The specification of initial and boundary conditions (no penetration and no-slip conditions, outflow conditions, periodic conditions) is discussed. The patterns of formation and development of vortical flows in a square cavity and cubic cavities are computed. The results of calculations of flow characteristics in a square and cubic cavity at various Reynolds numbers are compared with data available in the literature and obtained based on the finite difference method and the finite volume method. The dependence of the numerical solution and location of critical points on faces of cubic cavity on the lattice size is studied. Computational time is compared with performance of fine difference and finite volume methods. The developed implementation of the lattice Boltzmann method is of interest for the transition to further modeling non-isothermal and high-speed compressible flows.
Brykov et al. (Mon,) studied this question.