A smoothly curved body skimming on shallow water

We investigate the dynamics of a smoothly curved convex body skimming on a layer of shallow water, with the phenomenon of skipping stones as a primary modelling motivation but also industrial applications in mind. The skimming process of such an object may consist of two successive stages: an impact stage and a conditional planing stage. The focus here is on explaining the change from one stage to the other, with the body movement responding freely to the fluid flow pressures and vice versa. We first introduce a water impact model and analyse the conditions under which a smooth body may rapidly transit to a planing motion via small time asymptotics. A planing model is then introduced, and in particular we investigate the effects of pressure conditions in the separation flow; we demonstrate under certain weak adverse pressure gradient conditions a smooth body’s planing motion at early times can be seen as undergoing three successive transitory phases