Instability of the Kolmogorov flow in a wall-bounded domain

Abstract

In the magnetohydrodynamics (MHD) experiment performed by Bondarenko and his co-workers in 1979, the Kolmogorov flow loses stability and transits into a secondary steady state flow at the Reynolds number R = O(103), as Hartmann boundary layer friction dominates the energy dissipation of the fluid motion. This problem is modelled as a two-dimensional MHD flow bounded between horizontal walls under slip wall boundary condition. However, the rigorous analysis showing the existence of the secondary steady state flow was missing in earlier investigations and has now been established. The theoretical solution has a very good agreement with the flow measured in laboratory experiment at R = O(103). Further transition of the secondary flow is observed numerically. Especially, well developed turbulence arises at R = O(104) via an inverse energy cascade scenario.