Supercritical flow in sharp bends

The paper deals with the behaviour of supercritical free surface flows in bends, when the usual hypothesis of small width/curvature radius ratio does not hold. In such a case, one dimensional models are not suitable to reproduce the physics of the phenomenon, and two dimensional modelling is required. Here, simplified hypothesis on specific energy distribution or longitudinal velocity component distribution are removed, and a complete two dimensional shallow water mathematical model is adopted. The solution that is proposed here is a numerical one, which is an application of a finite volume method (FVM) computer code, independently developed by the authors. A second order accurate (in time and space) Godunov cell-centered method is adopted, using the Harten, Lax and van Leer (HLL) Riemann solver in order to approximately estimate the momentum fluxes passing through the boundaries of each cell. The scheme is a high resolution one, satisfying the Total Variation Diminishing (TVD) condition. An original new treatment of the source terms relative to the bottom slope is applied, whereas a stabilizing semi-implicit treatment of source terms relative to the friction slope is used. Several aspects concerning the physics of the phenomenon are well reproduced, such as the blocking of the stream when the Froude number of the undisturbed flow is not large enough and the bend is sufficiently sharp. Notwithstanding that, maximum water depth in the bend is systematically underestimated. Such an underestimation seems quite surprising, taking into account that the integration method does not introduce any diffusive effect and that it is fully designed and tested for reproducing sharp discontinuous solutions.