A Multi-Relaxation-Time Finite Volume Discrete Boltzmann Method for Viscous Flows

Abstract

Conventional constitutive law-based fluid dynamic models solve the conservation equations of mass and momentum, while kinetic models, such as the well-known lattice Boltzmann method (LBM), solve the propagation and collision processes of the Boltzmann equation-governed particle distribution function (PDF). Such models can provide an a priori modeling platform on a more fundamental level while easily reconstructing macroscopic variables such as velocity and pressure from the PDF. While the LBM requires a rigid and uniform grid for spatial discretization, another similar unique kinetic model known as the finite volume discrete Boltzmann method (FVDBM) has the ability to solve the discrete Boltzmann equation (DBE) on unstructured grids. The FVDBM can easily and accurately capture curved and more complicated fluid flow boundaries (usually solid boundaries), which cannot be satisfactorily realized in the LBM framework. As a result, the FVDBM preserves the physical advantages of the LBM over the constitutive law-based model approach, but also incorporates a better boundary treatment. However, the FVDBM suffers larger diffusion errors compared to the LBM approach. Building on our previous work, the FVDBM is further developed by integrating the multi-relaxation-time (MRT) collision model into the existing framework. Compared to the existing FVDBM approach that uses the Bhatnagar–Gross–Krook (BGK) collision model, which is also known as the single-relaxation-time (SRT) model, the new model can significantly reduce diffusion error or numerical viscosity, which is essential in the simulation of viscous flows. After testing the new model, the MRT-FVDBM, and the old model, the BGK-FVDBM, on Taylor-Green vortex flow, which can quantify the diffusion error of the applied model, it is found that the MRT-FVDBM can reduce the diffusion error at a faster rate as the mesh resolution increases, which renders the MRT-FVDBM a higher-order model than the BGK-FVDBM. At the highest mesh resolution tested in this paper, the reduction of the diffusion error by the MRT-FVDBM can be up to 30%.