An Exact Solution to the Inverse Problem of Steady Free-Surface Flow over Topography

Abstract

AbstractA simple exact solution is presented to the inverse problem in steady, two-dimensional idealised flow over topography that seeks the bottom profile given knowledge of the free-surface data. Attention is focused on the case when a uniform stream flows over a localised obstacle, although the solution is not restricted to this case. The inverse problem is formulated as a Stieltjes integral equation which is solved exactly using a Fourier transform. The solution requires the analytic continuation of two real functions representing the surface speed and the angle between the surface velocity vector and the horizontal. Some example surface profiles and their corresponding bottom topographies are discussed. Although the solution requires the prescription of the surface as a function of the velocity potential, it is shown to closely resemble the corresponding profile in physical space, even for quite large surface displacements, while significant discrepancy occurs at the bottom. Inference of the bottom profile from discrete surface data is accomplished by way of polynomial interpolation and rational approximation in the complex plane for the sample case of a hydraulic fall.