A boundary integral method for gravitational fluid flow over a semi-circular obstruction using an interpolative technique

Abstract

A boundary integral technique is developed to study the behaviour of a steady, two-dimensional free surface flow of an incompressible, irrotational and inviscid fluid over a submerged semi-circular obstruction in the presence of gravity. The solution technique is different to that employed by many of the previous research workers since it involves the application of the Riemann–Hilbert problem in the derivation of the nonlinear boundary integral–differential equations. The boundary integral equations are solved using piecewise constant and linear interpolative techniques for the fluid velocity on both the solid boundary and the free surface for various values of the upstream Froude number and the radius of the semi-circular obstruction. An investigation into the numerical accuracy of the interpolation techniques is employed. It is found that it is difficult to obtain a solution when the non-dimensional radius of the semi-circular obstruction is large and hence a hybrid technique is developed which is capable of computing the free surface profiles for all values of the radius of the semi-circular obstruction. Also by considering the local Froude number we have found that the fluid flow can become subcritical, i.e. the local Froude number is less than unity, in the vicinity just above the obstacle but no waves are found to be present on the free surface.