Solution to the steady, wave-free, fully nonlinear, free surface flow of an incompressible fluid over a topography by Legendre polynomial expansion

Abstract A new semi-analytical approach relying on Legendre finite expansions of the free surface elevation and boundary collocation is proposed to solve the fully nonlinear two-dimensional, steady free-surface wave-free gravity flow in an infinite channel with topography. The fluid is inviscid with constant density and the flow is irrotational. The unknown coefficients of the expansions are determined straightforward by solving a nonlinear system of algebraic equations using computer package. Two types of solutions are considered according to the upstream and downstream values of Froude number. Results are plotted for two shapes of the bottom topography. The results and the limitations of the method are discussed. These may be of interest in designing drag-free bottom flows.