A lattice Boltzmann model for thermal compressible flows at low Mach numbers beyond the Boussinesq approximation

Abstract

A modified weakly compressible lattice Boltzmann model is developed for simulating low Mach number thermal compressible flows. Using the low Mach number approximation (LMNA) and splitting the pressure into two parts, namely thermodynamic and hydrodynamic pressure, the Navier–Stokes equations at low Mach number limits with variable density are recovered from the kinetic model. The fluid density is governed by the chosen equation of state. In order to satisfy the continuity equation at large temperature variations, which leads to large density differences, a suitable compensation for the finite divergence of the velocity field is introduced via a forcing term. This relationship correlates the total derivatives of the temperature and thermodynamic pressure to the velocity field divergence. The present model is successfully validated against well-known benchmarks. Numerical experiments are conducted to predict the critical Rayleigh number in a compressible Rayleigh–Benard convection test case, where the Boussinesq approximation is no longer valid due to the large temperature differences across the domain. The results are in good agreement with theory and previously published works. Moreover, two-dimensional natural convection heat transfer in a square cavity with large temperature differences is simulated and the results agree with available literature data. One interesting aspect of the presented algorithm is its capability to be extended to two-phase liquid-gas flows to model the gas phase as thermally compressible for low Mach numbers flows.