Fokker-Planck Model Based Central Moment Lattice Boltzmann Method for Effective Simulations of Thermal Convective Flows

Abstract

The Fokker-Planck (FP) equation represents the drift-diffusive processes in kinetic models. It can also be regarded as a model for the collision integral of the Boltzmann-type equation to represent thermo-hydrodynamic processes in fluids. The lattice Boltzmann method (LBM) is a drastically simplified discretization of the Boltzmann equation. We construct two new 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 under 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 tensor parameters and significantly differ from those based on the Maxwell distribution. The diffusion tensor parameters for evolving higher moments in simulating fluid motions at relatively low viscosities are chosen based on a renormalization principle. The use of such central moment formulations in modeling the collision step offers significant improvements in numerical stability, especially for simulations of thermal convective flows with a wide range of variations in the transport coefficients. We develop new FP central moment LBMs for thermo-hydrodynamics in both two- and three-dimensions and demonstrate the ability of our approach to accurately simulate various cases involving thermal convective buoyancy-driven flows, especially at high Rayleigh numbers. Moreover, we show significant improvements in numerical stability of our FP central moment LBMs when compared to other existing central moment LBMs using the Maxwell distribution in achieving higher Peclet numbers for mixed convection flows.