SHALLOW-WATER APPROACH TO THE CIRCULAR HYDRAULIC JUMP

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) approximately Q5/8v-3/8g-1/8, 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.