Upstream influence and the form of standing hydraulic jumps in liquid-layer flows on favourable slopes

Abstract

Steady planar flow of a liquid layer over an obstacle is studied for favourable slopes. First, half-plane Poiseuille flow is found to be a non-unique solution on a uniformly sloping surface since eigensolutions exist which are initially exponentially small far upstream. These have their origin in a viscous–inviscid interaction between the retarding action of viscosity and the hydrostatic pressure from the free surface. The cross-stream pressure gradient caused by the curvature of the streamlines also comes into play as the slope increases. As the interaction becomes nonlinear, separation of the liquid layer can occur, of a breakaway type if the slope is sufficiently large. The breakaway represents a hydraulic jump in the sense of a localized relatively short-scaled increase in layer thickness, e.g. far upstream of a large obstacle. The solution properties give predictions for the shape and structure of hydraulic jumps on various slopes. Secondly, the possibility of standing waves downstream of the jump is addressed for various slope magnitudes. A limiting case of small gradient, governed by lubrication theory, allows the downstream boundary condition to be included explicitly. Numerical solutions showing the free-surface flow over an obstacle confirm the analytical conclusions. In addition the predictions are compared with the experimental and computational results of Pritchardet al.(1992), yielding good qualitative and quantitative agreement. The effects of surface tension on the jump are also discussed and in particular the free interaction on small slopes is examined for large Bond numbers.