Abstract
AbstractTwo-dimensional turbulent free-surface flow is considered. The ensemble-averaged flow quantities may depend on time. The slope of the plane bottom of the channel is assumed to be small. The roughness of the bottom is allowed to vary with the space coordinate, leading to small variations in the bottom friction coefficient. An asymptotic analysis, which is free of turbulence modelling, is performed for large Reynolds numbers and Froude numbers close to the critical value 1. As a result, an extended Korteweg–deVries (KdV) equation for the surface elevation is obtained. Other flow quantities, such as pressure, flow velocity components, and bottom shear stress, are expressed in terms of the surface elevation. The steady-state version of the extended KdV equation has eigensolutions that describe stationary solitary waves. Time-dependent solutions of the extended KdV equation provide a means for discriminating between stable and unstable stationary solitary waves. Solutions of initial value problems show that there are transient solutions that approach asymptotically the stable stationary solitary wave, whereas other transient solutions decay asymptotically with increasing time.
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