Solutions near the hydraulic control point in a gravity current model

Abstract

A steady-state gravity current model that incorporates entrainment and friction is used to describe large-scale gravity currents and channel flows. When the model includes pressure effects from varying current thickness, critical points occur when the current velocity is equal to the phase velocity of waves on the interface. Some solutions have the possibility to pass from super to subcritical flow, or vice versa. These solutions pass through a hydraulic control point and the objective is to analyze the behavior of the solutions in the vicinity of such points. Using a phase space in which the hydraulic control points occur as equilibrium points, and performing Taylor expansion to the first order, the result is a system of autonomous differential equations with constant coefficients that can describe the behavior of the solutions for different parameter regimes near a hydraulic control point. If an equilibrium point in phase space represents a saddle point, it is distinguished between three different solution classes; solutions that approach the critical velocity but never reach it, solutions that reach the critical velocity and obtain infinitely large derivative, and the solutions (one from subcritical and one from supercritical) that reach the critical velocity exactly in the equilibrium point. The method gives a way to tell whether hydraulically controlled solutions exist and in special cases an algorithm for finding these solutions.