Fokker-Planck based Central Moment Lattice Boltzmann Method for Simulations of Thermal Convective Flows at High Rayleigh Numbers

Fokker-Planck (FP) equation represents the drift and diffusive processes in kinetic models. It can also be regarded as a model for the collision integral of the Boltzmann equation that retains the quadratic nonlinearity of the latter. The lattice Boltzmann method (LBM) is a drastically simplified discretization of the Boltzmann equation. We construct two FP-based LBMs, one for recovering the Navier-Stokes equations and the other for simulating the energy equation, where, in each case, the effect of collisions is represented as relaxations of different central moments to their respective attractors. Such attractors are obtained by matching the changes in various discrete central moments due to collision with the continuous central moments prescribed by the FP model. As such, the resulting central moment attractors depend on the lower order moments and the diffusion coefficient tensor, and significantly differ from those based on the Maxwell distribution. The use of such central moment formulations in modeling the collision step offers significant improvements in numerical stability especially for simulating flows in the turbulent regime. We demonstrate the utility of our approach for a case study involving thermal convective buoyancy-driven flows at high Rayleigh numbers.