Instability of a potential swirling flow with a free surface

Abstract

Rotating flows with free surface are commonly observed in everyday-life, the most popular example being the drain vortex observed in bathtubs and sinks. These flows are known to support spectacular patterns arising from breaking of the axial symmetry. A situation which has been particularly investigated through experiments is the case of a partially filled container with a rotating bottom. In this situation, an instability leads to the appearance of rotating polygons along the free surface. The axisymmetric base flow preexisting the instability is rather complex, involving strong recirculations, but includes a region where the rotation is approximately potential, suggesting that the potential vortex may be a relevant model to investigate the instability mechanism. Consequently, we investigate here the linear inviscid stability of the potential swirling flow with a free surface; namely, the flow defined Vθ = Γ/(2πr), with a free surface given by h0 (r) ∼ R2 − r12 , where R1 corresponds to the radius of the dry area. For the study, we use both a global stability approach based on the finite element software Freefem++ and a shallow water approach. Our results show that unstable modes exist in narrow ranges of the Froude number (defined by F r = Γ/2π (gR0 ), R0 being the radius of the cylinder), for all non-zero values of the azimuthal wavenumber m (see figure 1). The instability seems to be related to the ”strato-rotational” instability mechanism existing in stratified rotating fluids.