Exact solutions for the initial stage of dam-break flow on a plane hillside or beach

Abstract

Inviscid, incompressible liquid is released from rest by a sudden dam break, accelerating under gravity over a uniformly sloping impermeable plane bed. The liquid flows downhill or up a beach. A linearised model is derived from Euler's equations for the early stage of motion, of duration $2\sqrt {H/g}$ , where H is the depth scale and $g$ is the acceleration due to gravity. Initial pressure and acceleration fields are calculated in closed form, first for an isosceles right-angled triangle on a slope of $45^{\circ }$ . Second, the triangle belongs to a class of finite-domain solutions with a curved front face. Third, an unbounded domain is treated, with a curved face resembling a steep-fronted breaking water wave flowing up a beach. The fluid goes uphill due to a nearshore pressure gradient. In all cases the free-surface-bed contact point is the most accelerated particle, exceeding the acceleration due to gravity. Physical consequences are discussed, and the pressure approximation of shallow water theory is found poor during this early stage, near the steep free surface exposed by a dam break.