Shear Stress and Fluid-Wall Interaction Force in LBM Simulations of Hydrodynamic and MHD Flows

TL;DR

This work clarifies how external and Lorentz forces are implemented in lattice Boltzmann simulations of hydrodynamic and magnetohydrodynamic flows. Through a Chapman-Enskog analysis, it shows artefact terms in force schemes are negligible for computing the shear-stress tensor and validates a half-step distribution $\overline{f}_{\alpha}$ for consistent fluid-wall interaction via momentum exchange. It demonstrates how Maxwell stress contributions arise in MHD and how to properly separate hydrodynamic pressure and viscous stress from moving-boundary and Lorentz-related terms in the surface force. The results provide a principled basis for accurate wall-interaction forces and stress calculations in LES and moving-boundary MHD simulations, improving global force balance and Galilean invariance in LBM. The approach unifies the treatment of hydrodynamic and MHD stresses and clarifies the role of artefact terms across force models.

Abstract

Chapman-Enskog analysis of the lattice Boltzmann method (LBM) is adopted to recover the Navier-Stokes (N-S) equation for the magnetohydrodynamic (MHD) flows driven by external body forces other than the induced Lorentz force. Various numerical schemes are discussed for the implementation of external body forces, leading to different artefact terms. An order-of-magnitude analysis is provided to demonstrate that the artefact terms are negligible in calculating the shear stress and strain rate tensors, which keeps the study of fluid-wall interaction and the implementation of large eddy simulation (LES) simple. This clarifies the confusion in the literature, where both considering and not considering the artefact terms have been adopted without discussion. Additionally, the fluid-wall interaction force can be computed as the intuitive momentum exchange rate using a special distribution function after a half-step propagation, and consistency is obtained with the formula derived using the Chapman-Enskog analysis. The momentum flux tensor for moving boundaries is uncovered as the cause of violating Galilean invariance and the Maxwell stress tensor for Lorentz force is also contained in this intuitively calculated interaction force for MHD flows due to the modification of equilibrium distribution function. Both extra terms should be removed for an interaction force associated with the hydrodynamic pressure and viscous shear stress. The implication of extra terms for the global balance of different physical forces is discussed.