Near-critical turbulent free-surface flow over a wavy bottom

Abstract

Steady plane turbulent free-surface flow over a slightly wavy bottom is considered for very large Reynolds numbers, very small bottom slopes, and Froude numbers close to the critical value 1. As in previous works, the slope and the deviation from the critical Froude number are assumed to be coupled such that turbulence modeling is not required. The amplitudes of the periodic bottom elevations, however, are assumed to be half an order of magnitude larger than in the previous case of bumps or ramps of finite length. Asymptotic expansions give a steady-state version of an extended Korteweg–deVries (KdV) equation for the surface elevation. The extension consists of a forcing term due to the unevenness of the bottom and a damping term due to friction at the bottom. Other flow quantities, such as pressure, flow velocity components, local Froude number and bottom friction force, can be expressed in terms of the surface elevation. Exact solutions of the extended KdV equation, describing stationary cnoidal waves, are obtained for bottoms of particular periodic shapes. As a limiting case, the solitary waves over a bottom ramp are re-obtained in accord with previous results.