Instabilities in quasi-two-dimensional magnetohydrodynamic flows

Abstract

The improvement of heat transfer conditions in liquid-metal magnetohydrodynamic (MHD) flows is of prime importance for self-cooled fusion blanket design concepts. For many years the research was based on stationary inertialess assumptions since it was expected that time-dependent inertial flows would be suppressed by strong electromagnetic damping, especially in the extreme range of fusion relevant parameters. In the present analysis the stationary inertialess assumptions are abandoned. Nevertheless, the classical ideas usually used to obtain inertialess asymptotic solutions are drawn on. The basic inertial equations are reduced to a coupled two-dimensional problem by analytical integration along magnetic field lines. The magnetic field is responsible for a quasi-two-dimensional flow; the non-uniform distribution of the wall conductivity creates a wake-type profile, the MHD effect reducing to a particular forcing and friction. The solution for the two-dimensional variables, the field aligned component of vorticity, the stream function, and the electric potential are obtained by numerical methods. In a flat channel with non-uniform electrical wall conductivity, time-dependent solutions similar to the Kármán vortex street behind bluff bodies are possible. The onset of the vortex motion, i.e. the critical Reynolds number depends strongly on the strength of the magnetic field expressed by the Hartmann number. Stability analyses in viscous hydrodynamic wakes often use the approximation of a unidirectional flow which does not take into account the spatial evolution of the wake. The present problem exhibits a wake-type basic flow, which does not change along the flow path. It represents, therefore, an excellent example to which the simple linear analysis on the basis of Orr-Sommerfeld-type equations applies exactly. Once unstable, the flow first exhibits a regular time periodic vortex pattern which is rearranged further downstream. One can observe an elongation, pairing, or sometimes more complex merging of vortices. All these effects lead to larger flow structures with lower frequencies. The possibility for a creation and maintenance of time-dependent vortex-type flow pattern in MHD flows is demonstrated.