Different approximations of shallow fluid flow over an obstacle

Three different sets of shallow water equations, representing different levels of approximation are considered. The numerical solutions of these different equations for flow past bottom topography in several different flow regimes are compared. For several cases the full Euler solutions are computed as a reference, allowing the assessment of the relative accuracies of the different approximations. Further, the differences between the dispersive shallow water (DSW) solutions and those of the highly simplified, hyperbolic shallow water (SW) equations is studied as a guide to determining what level of approximation is required for a particular flow. First, the Green-Naghdi (GN) equations are derived as a vertically-integrated rational approximation of the Euler equations, and then the generalized Boussinesq (gB) equations are obtained under the further assumption of weak nonlinearity. A series of calculations, each assuming different values of a set of parameters emdash undisturbed upstream Froude number, and the height and width of the obstacle, are then presented and discussed. In almost all regions of the parameter space, the SW and DSW theories yield different results; it is only when the flows are entirely subcritical or entirely supercritical and when the obstacles are very wide compared to the depth of the fluid that the SW and DSW theories are in qualitative and quantitative agreement. It is also found that while the gB solutions are accurate only for small bottom topographies (less than 20% of the undisturbed fluid depth), the GN solutions are accurate for much larger topographies (up to 65% of the undisturbed fluid depth). The limitation of the gB approximation to small topographies is primarily due to the generation of large amplitude upstream propagating solitary waves at transcritical Froude numbers, and is consistent with previous analysis. (Abstract Truncated)