Self-similar solutions of the axisymmetric shallow-water equations governing converging inviscid gravity currents

Abstract

A phase-plane approach is used to determine similarity solutions of the axisymmetric shallow-water equations which represent inwardly propagating, inviscid gravity currents. A Froude number condition characterizes the movement of the front. The unique similarity exponent is found numerically as a function of the frontal Froude number and the height and velocity profiles are presented for three different Froude numbers. The fluid speed and height are seen to increase monotonically towards the front except very close to the front where the height decreases. The maxima in both height and speed increase as the Froude number increases, reflecting the change in ambient resistance.

For the Froude number that has been obtained experimentally for lock-exchange Boussinesq flows ($Fr\,{=}\,1.19$) for which the similarity exponent is 0.859094, the similarity solution is compared to the numerical solution of the initial value problem, obtained recently by Hallworth, Huppert & Ungarish (2003). Our similarity solution compares reasonably well with their integration of the shallow-water equations in the neighbourhood of the front and at times close to collapse (when the front reaches the origin); however, near this point their numerics begin to fail. The solution at collapse and the similarity solution after collapse are also found for $Fr\,{=}\,1.19$. This similarity solution describes the formation of a shock, as well as its initial propagation.