Instability of two-dimensional square eddy flows

Abstract

Plane nonparallel flow in a square fluid domain satisfying free-slip boundary condition is examined. The energy dissipation of the flow is controlled by viscosity and linear friction, which is from the friction effect of Hartmann bottom boundary layer in three-dimensional magnetohydrodynamic experiment in a cell bottomed with the square domain. For the four eddy basic flow of the problem, there exist two bicritical parameters corresponding to the existence of two neutral eigenfunction spaces, respectively. The first neutral eigenfunction is one-dimensional and gives rise to the bifurcation of the basic flows into a pair of secondary flow, while the second one is two-dimensional and leads to the occurrence of a circle of secondary flows. These results are obtained numerically and can be approximated by elementary functions in a simple form. The secondary flows with respect to the first bicritical parameter exhibits the merging of diagonal eddies observed by Sommeria’s experiments on an inverse energy cascade to turbulence. More instability phenomena are displayed from the secondary flows with respect to the second bicritical parameter.