Edge states in MHD duct flows with electrically insulating walls

The study of laminar-turbulent transition from a dynamical systems perspective has significantly advanced our understanding of subcritical transition in hydrodynamic (HD) shear flows. In spite of this, little attention has been paid to these developments in the magnetohydrodynamic (MHD) community, where subcritical transition occurs in wall-bounded flows relevant for liquid metal applications. In a duct subject to a transverse magnetic field, Hartmann and Shercliff layers form on the walls orthogonal and parallel to the field direction, respectively. Traditionally, transition to turbulence in such flows has been characterized by a critical Reynolds number based on the Hartmann layer thickness, whereas modern direct numerical simulations (DNS) suggest that transition first takes place in the Shercliff layers. Motivated by this contradiction, the MHD duct flow is revisited using the quasi-static MHD approximation, by means of DNS and dynamical system concepts developed in the HD community. Emphasis is placed on the periodic square duct for which different transition routes and edge states, which correspond to relative attractors on the basin boundary of the laminar flow, are studied.