Two-layer critical flow over a semi-circular obstruction

Abstract

Steady, trans-critical flow of a two-fluid system over a semi-circular cylinder on the bottom of a channel is considered. Each fluid is assumed to be inviscid and incompressible and to flow irrotationally, but the fluids have different densities, so that one flows on top of the other. Consequently, a sharp interface exists between the fluids, in addition to a free surface at the top of the upper fluid. Trans-critical flow is investigated, in which waves are absent from the system, but the upstream and downstream fluid depths differ in each fluid layer. The problem is formulated using conformal mapping and a system of three integrodifferential equations, and solved numerically with the aid of Newton's method. The free-surface shape and that of the interface are obtained along with the Froude numbers in each fluid layer. Results of computation are presented and discussed.