Nonsimilar solutions of the viscous shallow water equations governing weak hydraulic jumps

Abstract

The steady viscous shallow water equations are often used for the study of hydraulic jumps. We cast these as a single parametric ordinary differential equation with global continuity as a constraint. The solution provides both the local velocity profile and the downstream evolution of the film height. Moreover this is an exact approach, in contrast with existing approaches which encounter a closure problem and need modeling. There is only one solution which is supercritical initially. This shows a jumplike behavior at a Froude number close to unity, in consonance with predictions of inviscid theory. At low Froude number, it is shown that two solutions are possible, one with a separated profile and one without. Flow downstream of a real hydraulic jump must switch to the second solution, calling into question the validity of the shallow water approach in resolving the region of the switch. A series solution of the velocity profile shows that the first correction to a streamwise-varying parabolic profile is a quartic term. Circular and planar solutions are qualitatively similar.