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The basic equations for near‐critical stationary turbulent open‐channel flow over horizontal surfaces are solved by means of a multiple scales analysis. The multiple scales solution describing the first‐order perturbation of the free‐surface elevation is compared with the numerical solution of an extended steady‐state version of the Korteweg–de Vries (KdV) equation, confirming the uniform validity of this non‐linear third‐order ODE. In comparison with experimental data the numerical solution of the extended KdV equation shows strong dependence on the initial curvature in terms of wavelength and amplitude.

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.

Stationary near-critical turbulent open-channel flow is considered. Allowing for locally enlarged roughness of the inclined bottom leads to the development of a stationary single wave. The waveform of the free surface is the key interest of this work. The flow is assumed to be fully developed far upstream and far downstream. In order to obtain the shape of the surface elevation as a result of the full equations of motion, an iteration procedure is used, which was successfully applied to the related problem of the undular hydraulic jump [1]. In addition to the classical solitary wave, the so-called single wave of the second kind is studied numerically. The numerical results are compared with theoretical predictions and experimental data.