St. Venant–Exner Equations for Near-Critical and Transcritical Flows

The solution of the St. Venant–Exner equations as a model for bed evolution is studied under conditions when the Froude number, F, approaches unity, and the quasi-steady model becomes singular. It is confirmed that the strict criterion for critical flow, the vanishing of a water surface disturbance celerity, is not met, yet the direction of propagation of a bed wave apparently changes depending on whether $F>1$ or $F<1\text{.}$ An analysis of the linearized model problem for an infinitesimal bed wave under near-uniform conditions is performed, and qualitative features of the solution are brought out. Under appropriate sediment transport conditions, when $F^2\to 1\text{,}$ two bed waves, one traveling upstream and the other traveling downstream, are found to develop from an initially single localized bed perturbation. Simulations of the full unsteady problem were performed with the Preissmann scheme to confirm the linear analysis and to study the effects of nonlinearity and friction. A transcritical case, in which a region where $F^2<1$ is succeeded by a region where $F^2>1\text{,}$ is also investigated, and the solution exhibits an apparently different behavior than cases where the flow is everywhere sub- or supercritical, but can be understood as a hybrid of the latter cases.