V-waves, bow shocks, and wakes in supercritical hydrostatic flow

Abstract

The structure of the bow shock, V-wave, and the related wave drag and wake in supercritical ambient flow are investigated for homogeneous hydrostatic single-layer flow with a free surface over an isolated two-dimensional (i.e. h(x, y)) obstacle. The two control parameters for this physical system are the ratio of obstacle height to fluid depth and the Froude number F = U/√gH. Based on theoretical analysis and numerical modelling, a steady-state regime diagram is constructed for supercritical flow. This study suggests that supercritical flow may have an upstream bow shock with a transition from the supercritical state to the subcritical state near the centreline, and a V-shock in the lee without a state transition. Unlike subcritical flow, neither a flank shock nor a normal lee shock is observed, due to the local supercritical environment. Both the bow shock and V-shock are dissipative and reduce the Bernoulli constant, but the vorticity generation is very weak in comparison with subcritical ambient flow. Thus, in supercritical flow, wakes are weak and eddy shedding is absent.

Formulae for V-wave shape and V-wave drag are given using linear theory. Both formulae compare well with numerical model runs for small obstacles.

These results can be applied to air flow over mountains, river hydraulics and coastal ocean currents with bottom topographies.