Two dimensional shallow water flow through a valley
The two-dimensional structure of shallow water flow through a valley with vertical sidewalls is investigated. First, the linear theory of steady flow through a contraction is presented. The related normal modes are decaying exponentially with distance to the walls of the valley for subcritical flow. The complete linear solution for "witch of Agnesi" corrugations exhibits maximum along-valley flow speed and minimum fluid depth at that wall point where the valley width is smallest. The eigenmodes are harmonic for supercritical flow. A rather general solution is presented in this case where a wave train forms downstream of the contraction. In particular, momentum is transferred to the walls in contrast to the prediction of the one-dimensional theory of valley flows. A numerical model with wall-following coordinates is used to extend the results beyond linearity. Flow solutions are found which exhibit the characteristics of a transition from sub- to supercritical flow near the walls with hydraulic jumps downstream of the constriction while the flow is subcritical at the valley's axis. Linear theory is found to be quite useful in predicting the 'critical' parameter sets for which the Froude number first becomes unity at the walls of the valley while being smaller elsewhere.