Volumetric Lattice Boltzmann Models in General Curvilinear Coordinates: Theoretical Formulation

Hudong Chen

doi:10.3389/fams.2021.691582

Abstract

A theoretical formulation of lattice Boltzmann models on a general curvilinear coordinate system is presented. It is based on a volumetric representation so that mass and momentum are exactly conserved as in the conventional lattice Boltzmann on a Cartesian lattice. In contrast to some previously existing approaches for arbitrary meshes involving interpolation approximations among multiple neighboring cells, the current formulation preserves the fundamental one-to-one advection feature of a standard lattice Boltzmann method on a uniform Cartesian lattice. The new approach is built on the concept that a particle is moving along a curved path. A discrete space-time inertial force is derived so that the momentum conservation is exactly ensured for the underlying Euclidean space. We theoretically show that the new scheme recovers the Navier-Stokes equation in general curvilinear coordinates in the hydrodynamic limit, along with the correct mass continuity equation.

Highlights

Lattice Boltzmann Methods (LBM) have been developed as an advantageous method for computational fluid dynamics during past few decades [1, 2]
Our goal is to theoretically demonstrate how discrete kinetic theories can be formulated in curvilinear coordinates in a way that mass and momentum are exactly conserved, and the correct hydrodynamics is recovered in the microscopic limit
We present a theoretical formulation of lattice Boltzmann models in a general curvilinear coordinate system

Summary

INTRODUCTION

Lattice Boltzmann Methods (LBM) have been developed as an advantageous method for computational fluid dynamics during past few decades [1, 2]. A possible way to accomplish this is to construct the micro-dynamic process on a non-Euclidean space represented by a general curvilinear coordinate system based on Riemann geometry [13] In such a non-Euclidean space, a constant particle velocity corresponds to a curved and spatially varying path in the underlying Euclidean space. The key difference between a constant particle velocity on a Euclidean and a non-Euclidean space is that the latter is accompanied with an inertial force, due to the 1st law of Newton in Euclidean space According to this concept, it is entirely conceivable to formulate LBM models on a cubic Cartesian lattice in non-Euclidean space that corresponds to a general curvilinear mesh in the Euclidean space. In order for the paper to be more self-contained, we provide, in Supplementary Appendices SA1–SA3 basic theoretical description of the continuum Boltzmann kinetic theory in curvilinear coordinates as in the literature [15], some fundamental properties [13], as well as a derivation of the Navier-Stokes hydrodynamic equation on a general curvilinear coordinate system

Construction of Geometric Quantities on a Curvilinear Mesh

Volumetric Lattice Boltzmann Model on a Curvilinear Mesh

DERIVATION OF THE NAVIER-STOKES HYDRODYNAMICS IN CURVILINEAR COORDINATES

DISCUSSION

DATA AVAILABILITY STATEMENT