The early stages of shallow flows in an inclined flume

The motion of an initially quiescent shallow layer of fluid within an impulsively tilted flume is modelled using the nonlinear shallow water equations. Analytical solutions for the two-dimensional flow are constructed using the method of characteristics and, in regions where neither of the characteristic variables is constant, by adopting hodograph variables and using the Riemann construction for the solution. These solutions reveal that the motion is strongly influenced by the impermeable endwalls of the flume. They show that discontinuous solutions emerge after some period following the initiation of the flow and that for sufficiently long flumes there is a moving interface between wetted and dry regions. Using the hodograph variables we are able to track the evolution of the flow analytically. After the discontinuities develop, we also calculate the velocity and height fields by using jump conditions to express conservation of mass and momentum across the shock and thus we show how the hydraulic jump moves within the domain and how its magnitude grows. In addition to providing the behaviour of the flow in this physical scenario, this unsteady solution also provides an important test case for numerical algorithms designed to integrate the shallow water equations.