Shear stress in lattice Boltzmann simulations

TL;DR

This paper thoroughly analyzes shear stress accuracy in lattice Boltzmann simulations, revealing second-order convergence affected by boundary conditions and deriving an analytic expression for density variation in Poiseuille flow.

Contribution

It provides a detailed investigation of shear stress errors, boundary condition effects, and an analytic model for density changes in lattice Boltzmann methods.

Findings

Shear stress in lattice Boltzmann is second order accurate.

Boundary conditions often spoil convergence.

Derived an analytic expression for density evolution.

Abstract

A thorough study of shear stress within the lattice Boltzmann method is provided. Via standard multiscale Chapman-Enskog expansion we investigate the dependence of the error in shear stress on grid resolution showing that the shear stress obtained by the lattice Boltzmann method is second order accurate. This convergence, however, is usually spoiled by the boundary conditions. It is also investigated which value of the relaxation parameter minimizes the error. Furthermore, for simulations using velocity boundary conditions, an artificial mass increase is often observed. This is a consequence of the compressibility of the lattice Boltzmann fluid. We investigate this issue and derive an analytic expression for the time-dependence of the fluid density in terms of the Reynolds number, Mach number and a geometric factor for the case of a Poiseuille flow through a rectangular channel in three dimensions. Comparison of the analytic expression with results of lattice Boltzmann simulations shows excellent agreement.