Fokker-Planck Central Moment Lattice Boltzmann Method for Effective Simulations of Fluid Dynamics

Abstract

We present a new formulation of the central moment lattice Boltzmann (LB) method based on a continuous Fokker-Planck (FP) kinetic model, originally proposed for stochastic diffusive-drift processes (e.g., Brownian dynamics), by adapting it as a collision model for the continuous Boltzmann equation (CBE) for fluid dynamics. The FP collision model has several desirable properties, including its ability to preserve the quadratic nonlinearity of the CBE, unlike that based on the common Bhatnagar-Gross-Krook model. Rather than using an equivalent Langevin equation as a proxy, we construct our approach by directly matching the changes in different discrete central moments independently supported by the lattice under collision to those given by the CBE under the FP-guided collision model. This can be interpreted as a new path for the collision process in terms of the relaxation of the various central moments to 'equilibria', which we term as the Markovian central moment attractors that depend on a diffusion coefficient tensor. The construction of the method using central moments rather than via distribution functions facilitates its numerical implementation and analysis. We show its consistency to the Navier-Stokes equations via a Chapman-Enskog analysis and elucidate the choice of the diffusion coefficient based on the second order moments in accurately representing flows at relatively low viscosities. We will demonstrate the accuracy and robustness of our new central moment FP-LB formulation, termed as the FPC-LBM, using the D3Q27 lattice for simulations of a variety of flows, including wall-bounded turbulent flows. We show that the FPC-LBM is more stable than other existing LB schemes based on central moments, while avoiding numerical hyperviscosity effects in flow simulations at relatively very low physical fluid viscosities through a refinement to a model founded on kinetic theory.

Figures (17)

• Figure 1: Routes for the derivation of the Boltzmann equation and the modeling of its collision term via BGK or FP approach under appropriate approximations, and their applications to representing the dynamics in fluids and plasmas (inspired from liboff2003kinetic).
• Figure 2: Representation of collision processes at different levels of modeling description and the associated mathematical nature of the collision operator.
• Figure 3: Streamlines for two-dimensional lid-driven square cavity flow computed using the FPC-LBM at Reynolds numbers of $\hbox{Re}$ = 1000, 3200, 5000, and 7500. The formation of secondary and tertiary vortices is consistent with those in the benchmark results of Ghia et al (1982) ghia1982high for each Reynolds number shown here.
• Figure 4: Comparisons of the horizontal velocity component along the vertical centerline in a two-dimensional lid-driven square cavity flow at different Reynolds numbers computed using the FPC-LBM with the reference results of Ghia et al (1982) ghia1982high. (a) $\hbox{Re}=1000$, (b) $\hbox{Re}=3200$, (c) $\hbox{Re}=5000$, (d) $\hbox{Re}=7500$.
• Figure 5: Comparisons of the vertical velocity component along the horizontal centerline in a two-dimensional lid-driven square cavity flow at different Reynolds numbers computed using the FPC-LBM with the reference results of Ghia et al (1982) ghia1982high. (a) $\hbox{Re}=1000$, (b) $\hbox{Re}=3200$, (c) $\hbox{Re}=5000$, (d) $\hbox{Re}=7500$.