Multi-train roll waves, antidunes and bars in inclined flumes

This article offers fresh theoretical insights into the phenomenon of bedforms and roll waves, highlighting their emergence from shallow water flows in a fixed-width inclined channel. Our theory is based on the Saint–Venant–Exner equations and linear stability analysis. We examine both bed and free-surface instabilities. Bed instabilities include alternate bars, antidunes, and multiple bars, while free-surface instabilities involve roll waves. We conduct a thoughtful analysis of how the transverse dimension affects flow dynamics, proving that our theoretical framework is sufficiently flexible to encompass both two-dimensional and three-dimensional bedforms, regardless of whether they are arranged in a single row or multiple rows. We compare our theoretical predictions with over 800 laboratory experiments. One significant result is the ability to predict the type of instability using three independent dimensionless parameters: the Froude number, the grain roughness relative to the water depth and the width-to-depth ratio. For Froude numbers in excess of 2, roll waves are the dominant mode, a consequence of convective instability along the free surface. For Froude numbers less than 2, two types of instability can occur along the water–sediment interface: (1) an absolute instability leading to the formation of antidunes, and (2) a convective instability giving rise to two-dimensional dunes or three-dimensional bars. Antidunes and alternate bars coexist within a specific range of the dimensionless parameters. Furthermore, a small width-to-depth ratio favors the development of two-dimensional antidunes and roll waves or three-dimensional alternate bars, whereas a large ratio leads to multi-train wave patterns. We conduct model validation by examining real-world scenarios, such as the Oroville spillway’s multi-row roll waves and classic experiments on bedforms. We find that transverse flow and bed organization are sensitive to experimental and boundary conditions for convective instabilities. This means that different bedform configurations are possible within a certain range of water discharge, bed slope, channel width and grain size.