An effective two-dimensional model for MHD flows with transverse magnetic field

TL;DR

This paper develops a quasi two-dimensional model for MHD flows with a transverse magnetic field, incorporating 3D effects via a transverse velocity profile to better understand recirculating flows and inertial effects.

Contribution

It introduces a novel 2D model that accounts for 3D effects in MHD flows using a double perturbation asymptotic approach, capturing inertial effects in Hartmann layers and the core flow.

Findings

Model agrees well with experimental data

Recirculating flows explain vortex spreading

Inertial effects influence dissipative behavior

Abstract

This paper presents a model for quasi two-dimensional MHD flows between two planes with small magnetic Reynolds number and constant transverse magnetic field orthogonal to the planes. A method is presented that allows to take 3D effects into account in a 2D equation of motion thanks to a model for the transverse velocity profile. The latter is obtained by using a double perturbation asymptotic development both in the core flow and in the Hartmann layers arising along the planes. A new model is thus built that describes inertial effects in these two regions. Two separate classes of phenomena are thus pointed out : the one related to inertial effects in the Hartmann layer gives a model for recirculating flows and the other introduces the possibility of having a transverse dependence of the velocity profile in the core flow. The ''recirculating'' velocity profile is then introduced in the transversally averaged equation of motion in order to provide an effective 2D equation of motion. Analytical solutions of this model are obtained for two experimental configurations : isolated vortices aroused by a point electrode and axisymmetric parallel layers occurring in the MATUR (MAgneticTURbulence) experiment. The theory is found to give a satisfactory agreement with the experiment so that it can be concluded that recirculating flows are actually responsible for both vortices core spreading and excessive dissipative behavior of the axisymmetric side wall layers.