Quasi-two-dimensional perturbations in duct flows under transverse magnetic field

Abstract

Inspired by the experiment from Moresco and Alboussiere [J. Fluid Mech. 504, 167 (2004)], we study the stability of a flow of liquid metal in a rectangular, electrically insulating duct with a steady homogeneous magnetic field perpendicular to two of the walls. In this configuration, the Lorentz force tends to eliminate the velocity variations in the direction of the magnetic field. This leads to a quasi-two-dimensional base flow with Hartmann boundary layers near the walls perpendicular to the magnetic field, and so-called Shercliff layers in the vicinity of the walls parallel to the field. Also, the Lorentz force tends to strongly oppose the growth of perturbations with a dependence along the magnetic field direction. On these grounds, we represent the flow using the model from Sommeria and Moreau [J. Fluid Mech. 118, 507 (1982)], which essentially consists of two-dimensional (2D) motion equations with a linear friction term accounting for the effect of the Hartmann layers. The simplicity of this quasi-2D model makes it possible to study the stability and transient growth of quasi-two-dimensional perturbations over an extensive range of nondimensional parameters and reach the limit of high magnetic fields. In this asymptotic case, the Reynolds number based on the Shercliff layer thickness Re∕H1∕2 becomes the only relevant parameter. Tollmien-Schlichting waves are the most linearly unstable mode as for the Poiseuille flow, but for H≳42, a second unstable mode, symmetric about the duct axis, appears with a lower growth rate. We find that these layers are linearly unstable for Re∕H1∕2≳48350 and energetically stable for Re∕H1∕2≲65.32. Between these two bounds, some nonmodal quasi-two-dimensional perturbations undergo some significant transient growth (between two and seven times more than in the case of a purely 2D Poiseuille flow, and for much more subcritical values of Re). In the limit of a high magnetic field, the maximum gain Gmax associated with this transient growth is found to vary as Gmax∼(Re∕Rec)2∕3 and occur at time tGmax∼(Re∕Rec)1∕3 for streamwise wavenumbers of the same order of magnitude as the critical wavenumber for the linear stability.