Stationary single waves in turbulent open‐channel flow

Abstract

Steady two-dimensional turbulent open-channel flow is considered. Stationary single-wave solutions are investigated. The fully-developed oncoming flow is slightly supercritical. The Reynolds number is very large. The analysis is kept free of turbulence modelling. As stationary solitary waves cannot exist in turbulent flow for a plane bottom with constant roughness [1], two particular perturbations of the conditions at the channel bottom are examined: 1) We revisit the case [1] where the friction coefficient locally differs slightly by a constant from the reference value upstream; 2) An unevenness of very small height in the channel bottom (bump, ramp) is admitted, with the bottom roughness taken constant. An analogy between these cases is presented. In both cases, three stationary solutions for the surface elevation are found: A stable and an unstable solitary wave, respectively, and a single wave of a second kind with smaller amplitude. For the latter, an analysis for weak dissipation yields a uniformly valid solution that is in good agreement with the numerical results for various parameters.