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

Inspired by the experiment from Moresco \& Alboussi\`ere (2004, J. Fluid Mech.), we study the stability of a liquid metal flow in a rectangular, electrically insulating duct with a steady homogeneous transverse magnetic field. The Lorentz force tends to eliminate velocity variations along the magnetic field, leading to a quasi-two dimensional base flow with Hartmann boundary layers near the walls perpendicular to the magnetic field, and Shercliff layers near the walls parallel to the field. Since the Lorentz force strongly opposes the growth of perturbations with a dependence along the magnetic field direction too, we represent the flow with Sommeria \& Moreau's (1982, J. Fluid Mech.) model, a two-dimensional shallow water model with linear friction accounting for the effect of the Hartmann layer. The simplicity of this model makes it possible to study the stability and transient growth of quasi-two dimensional perturbations over an extensive range of parameters up to the limit of high magnetic fields, where the Reynolds number based on the Shercliff layer thickness $Re/H^{1/2}$ becomes the only relevant parameter. Tollmien-Schlichting waves are the most linearly unstable mode, with a further unstable mode $H \gtrsim 42$. The flow is linearly unstable for $Re/H^{1/2}\gtrsim 48350$ and energetically stable for $Re/H^{1/2}\lesssim 65.32$. Between these two bounds, non-modal quasi-two dimensional perturbations undergo significant transient growth (between 2 and 7 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 $G_{max}$ associated to this transient growth varies as $G_{max} \sim (Re/Re_c)^{2/3}$ and occurs at time $t_{Gmax}\sim(Re/Re_c)^{1/3}$ for streamwise wavenumbers of the same order of magnitude as the critical wavenumber for the linear stability.