Channel confinement mollifies roll wave instabilities

Abstract

Under certain conditions, gravity currents spontaneously develop large-amplitude streamwise undulations on their free surfaces ('roll waves'). The effect of channel geometry on this instability is investigated herein. Via a section-averaged stability analysis conducted independently of the rheology of the flowing material, convex open channels are found to be stabilising, relative to unconfined flows. This is shown to agree with experimental observations of laterally shallow currents of water and dry granular material in trapezoidal channels. For such flows, the influence of the geometry can be characterised by a single dimensionless parameter, which predicts stabilisation as the channel width narrows. Including the cross-stream velocity profile in the analysis further stabilises predictions and can render steady flows in triangular channels unconditionally stable - a finding borne out by our experiments. Consequently, laterally tilting a trapezoidal channel can also stabilise flows by adjusting the wetted region towards a triangular one. Implications for observations in natural channels are considered by adapting an existing model of roll waves in the Illgraben, Switzerland, to incorporate channel geometry. The resulting simulations produce suitable waves for a realistic cross-section, but are stabilised by a modest narrowing of the channel. Complementary nonlinear wave solutions are then constructed and used to show that channel confinement also diminishes amplitudes of observed waves. Finally, exceptions to the analysis are discussed, including the possibility for static material to shield waves from the effects of geometry. We demonstrate this using observations of undamped granular avalanches travelling through an arrested deposit in a triangular chute.