A Method for Analytical Solutions in the Lattice Boltzmann Method

TL;DR

This paper introduces a method for deriving analytical solutions to the lattice Boltzmann equation for known flow profiles, enabling better understanding and validation of the method's properties and boundary conditions.

Contribution

The paper presents a novel approach to find analytical solutions for specific flow profiles in the lattice Boltzmann method, including for quadratic and entropic distributions, with insights into their accuracy and limitations.

Findings

Analytical solutions are obtained for bulk Couette flow with quadratic distribution.

Entropic distribution provides solutions aligned with the lattice for certain shear rates.

The entropic lattice Boltzmann method is incompatible with inclined Couette flow due to momentum conservation issues.

Abstract

Analytical solutions to the lattice Boltzmann Equation make it possible to study the method itself, explore the properties of its collision operator, and identify implementations of boundary conditions. In this paper, we propose a method to find analytical solutions where the macroscopic flow profile is known. We test this method on bulk Couette flow aligned and inclined to the simulation lattice with the quadratic and entropic equilibrium distributions. Our method indeed provides an analytical solution to these flows when using the quadratic distribution. When the flow is aligned to the lattice, our method provides an analytical solution using the entropic distribution for practical relaxation times and shear rates. We show that a small even order truncation of the formal solution is optimal for accuracy-compute-time trade-off. In the inclined case, our method does not conserve momentum, by a small relative error, when using the entropic distribution. We also discover that entropic lattice Boltzmann method is not compatible with the angled Couette flow. We discuss the application of our method to more complicated flows.