A Comparative Study of 3d Cumulant and Central Moments Lattice Boltzmann Schemes with Interpolated Boundary Conditions for the Simulation of Thermal flows in High Prandtl Number Regime

Abstract

Thermal flows characterized by high Prandtl number are numerically challenging as the transfer of momentum and heat occurs at different time scales. To account for very low thermal conductivity and obey the Courant-Friedrichs-Lewy condition, the numerical diffusion of the scheme has to be reduced. As a consequence, the numerical artefacts are dominated by dispersion errors commonly known as wiggles. In this study, we explore possible remedies in the framework of the lattice Boltzmann method by applying novel collision kernels, lattices with a large number of discrete velocities, namely D3Q27, and second-order boundary conditions. For the first time, the cumulant-based collision operator is utilised to simulate both the hydrodynamic and the thermal field. Alternatively, the advected field is computed using the central moments' collision operator. Different relaxation strategies have been examined to account for additional degrees of freedom introduced by a higher-order lattice. To validate the proposed kernels for a pure advection-diffusion problem, the numerical simulations are compared against an analytical solution of a Gaussian hill. The structure of the numerical dispersion is shown by simulating advection and diffusion of a square indicator function. Finally, a study of steady forced heat convection from a confined cylinder is performed and compared against a FEM solution. It has been found, that the relaxation scheme of the advected field must be adjusted to profit from lattice with a larger number of discrete velocities. Obtained results show clearly that it is not sufficient to assume that only the first-order central moments/cumulants contribute to solving the macroscopic advection-diffusion equation. In the case of central moments, the beneficial effect of the two relaxation time approach is presented.