Shallow-water approach to the circular hydraulic jump

Abstract

We show that the circular hydraulic jump can be qualitatively understood using simplified equations of the shallow-water type which include viscosity. We find that the outer solutions become singular at a finite radius and that this lack of asymptotic states is a general phenomenon associated with radial flow with a free surface. By connecting inner and outer solutions through a shock, we obtain a scaling relation for the radius R j of the jump, R j ∼ Q ⅝ v ⅜ g ⅛ , where Q is the volume flux, v is the kinematic viscosity and g is the gravitational acceleration. This scaling relation is valid asymptotically for large Q . We discuss the corrections appearing at smaller Q and compare with experiments.