Lattice Boltzmann simulation of natural convection in a square cavity

In this paper, we describe the development of a lattice Boltzmann scheme for incompressible thermohydrodynamics. Being based on kinetic theory, the scheme simulates macroscopic fluid flows and heat transfers with the use of distribution functions. A systematic derivation of the lattice Boltzmann scheme from the continuous Boltzmann equation is discussed in details. We find that a 5-velocity model can be employed to simulate heat transfer in such a case where the viscous and compressive heating effects are negligible. As a benchmark, numerical simulations of natural convection in a square cavity are carried out. Through the results, the scheme is found to have a second-order convergence rate. In addition, the scheme is verified to be as accurate as conventional methods over a wide range of Rayleigh numbers.