Solutions of an extended KdV equation describing single stationary waves with strong or weak downstream decay in turbulent open-channel flow

Abstract

The present problem was first studied by Schneider, JFM 726 (2013). An asymptotic analysis was performed for small slope of the plane channel bottom and slightly supercritical, fully-developed flow far upstream and far downstream. Turbulence modelling can be circumvented. The asymptotic analysis yields an extended Korteweg-de Vries (KdV) equation for the surface elevation. The stationary leading-order solutions for weak dissipation have the shape of solitons in inviscid flow. Stationary solitary waves are permitted only if the surface roughness is varied along the bottom. The particular example was examined where the bottom roughness in a region is larger by a constant in comparison with the roughness in fully-developed flow. This enlargement was considered as an eigenvalue to obtain solitary waves with strongly decaying elevations. In the present paper, the analysis is extended up to the second order to describe how the shape and the structure of the solitary waves is affected by the dissipation terms. In particular, the cases are considered when the enlargement of the bottom roughness differs from the eigenvalue. The surface elevation far downstream is then characterized by a long shallow tail that is tackled by means of matched asymptotic expansions. The required minimum enlargement of the roughness for the existence of solitary waves is given. The asymptotic solutions for the shape of the surface elevation, the height and location of the wave crest, and the eigenvalue are in good agreement with numerical results. The existence and shape of a novel kind of stationary solutions is also shown.