Nonlinear evolution of small disturbances into roll waves in an inclined open channel

Abstract

This paper concerns the asymptotic behaviour (for the limiting case of small amplitudes) of small disturbances as they evolve in time to produce the quasi-steady pattern of roll waves first discussed by Dressler in 1949. Roll waves exist if F, the undisturbed Froude number (dimensionless speed) of the flow, exceeds 2, and consist of a periodic pattern of bores separating two special continuous solutions of the governing equations in a uniformly translating frame. The mathematical problem is rather interesting as solutions of the linearized equations are unstable for F > 2. Thus, it is crucial to account for the cumulative effect of small nonlinearities to obtain a correct description of the flow over long times. We concentrate on the weakly unstable problem (0 < F - 2 < 1) and use multiple scale expansions to derive the dominant evolution equation that governs the solution behaviour for long times. This turns out to be an integro-partial differential equation of first order that we solve numerically in conjunction with the jump condition that follows from the exact bore conditions. We present asymptotic and numerical results for periodic as well as isolated initial disturbances, and show that our theory predicts the solution accurately for both the transient and quasi-steady phases.