Near‐critical turbulent open‐channel flow over wavy bottom

Abstract

AbstractSteady two‐dimensional turbulent free‐surface flow in a channel with mild baseline slope is considered. The shape of the channel bottom is assumed to be undular with a very small amplitude. Asymptotic expansions for large Reynolds numbers and Froude numbers slightly above the critical value 1, respectively, give for the surface elevation a differential equation of KdV‐type, with the additional terms representing turbulent dissipation and forcing due to the wavy bottom, respectively. No turbulence modelling is required. This asymptotic approach was used in [1] to describe stationary solitary waves in a channel with plain bottom and small variations in the bottom friction coefficient. It was shown recently [2,3] that there exist stationary single‐wave solutions of a different kind that are characterized by smaller wave amplitudes. In [4], both kinds of single stationary waves above single obstacles (bumps, ramps) are investigated theoretically and experimentally. In this paper, stationary space‐periodic surface waves for channel bottoms with undular shape are studied. First, a one‐parametric family of exact solutions for particular bottom shapes is derived. Remarkably, these particular solutions exist only in a narrow parameter range. The solutions are reproduced with a numerical solver to verify that the solver gives correct results. Secondly, a different type of asymptotic expansion is performed in order to describe solutions characterized by smaller wave amplitudes. The resulting linear differential equation is solved numerically. Both solutions are briefly discussed.