A lattice Boltzmann model for the Navier-Stokes equation

Abstract

The lattice Boltzmann model is a vital numerical calculation tool, which has become a useful solution to fluid problems because of the limitations of the Navier-Stokes equation and other macroscopic models in solving such problems. Here, the effectiveness of this model in the nearly incompressible Navier-Stokes equation is investigated as follows. First, the principle and excellent characteristics of the lattice Boltzmann model in numerical calculations are described and analyzed. Second, the approximate incompressible Navier-Stokes equations of the BGK (Bhatnagar-Gross-Krook) model in lattice Boltzmann method are analyzed. Finally, the two-dimensional five discrete velocity (D2N5) model in BGK model is taken as an example, and the vector rebound boundary scheme and the single point second-order boundary scheme based on the Dirichlet boundary condition are derived. Numerical results based on the Poiseuille flow and Taylar-Green vortex show that the BGK-based vector rebound boundary scheme has second-order accuracy only when the proportional distance is 0.5, and in other cases it cannot reach the accuracy. For the single-point second-order boundary scheme, the second-order accuracy can be achieved both in the Poiseuille flow based on the straight boundary region and in the Taylar-Green vortex based on the curved boundary. Through Maxwell iteration, the vector BGK model based on discretization is stable. When the boundary position is in the middle of two lattice points, the vector rebound boundary scheme has second order accuracy, and the single point second order boundary scheme has better performance in second order accuracy. This provides an idea for dealing with boundary conditions in incompressible Navier-Stokes equations.