Effective boundary conditions for magnetohydrodynamic flows with thin Hartmann layers

TL;DR

This paper develops effective boundary conditions for magnetohydrodynamic flows with thin Hartmann layers to simplify numerical simulations, accounting for various wall conductivities and inertial effects.

Contribution

It introduces novel boundary conditions that eliminate the need for fine meshing in MHD simulations involving Hartmann layers, including conditions for normal velocity with inertial effects.

Findings

Derived boundary conditions for tangential and normal electric currents and velocities.

Validated conditions for different wall conductivities including perfect insulators and conductors.

Included inertial effects in the boundary condition for normal velocity.

Abstract

Here we build some effective boundary conditions to be used in numerical calculations in order to avoid the thin meshing usually required in problems involving Hartmann layers near a locally plane wall. Wall model are provided for both tangential and normal electric current density and velocity. In particular, a condition on the normal derivative of the tangential velocity is derived. A wide variety of problems is covered as the only restriction is that the magnetic Reynolds number has to be large at the scale of the Hartmann layer. The cases of perfectly conducting or insulating wall are examined, as well as the case of a thin conducting wall. The newest result is a condition on the normal velocity accounting for inertial effects in the Hartmann layer.