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 radiusRjof the jump,Rj∼Q⅝v⅜g⅛, whereQis the volume flux,vis the kinematic viscosity andgis the gravitational acceleration. This scaling relation is valid asymptotically for largeQ. We discuss the corrections appearing at smallerQand compare with experiments.