A Schwarz-christoffel technique for inverse free surface flow problems

Abstract

This project investigates the behaviour of open channel flow ofan inviscid, steady incompressible and irrotational fluid. The investigation focuses on the instant when a fast~flowing stream of the fluid encounters an unknown obstacle, for example, a rock at the nverbed. To facilitate this, equations of motion, namely the Euler and Bernoulli's equations are established. Conformal mapping techniques are then used to map the fluid on the 50- plane whence a simple complex potential is obtained using the generalized Schwarz- Christoffel transformation. After obtaining an integro-differential equation for both the free and the bottom surfaces (satisfying the boundary conditions), the ensuing equations are solved using Fourier transforms. The resulting solution is further evaluated numerically using the Trapezioidal Rule. The basis of the inverse method in this case is the specification of the known surface in terms of the intrinsic coordinates, Later, the free surface is specified using canesian coordinates (x, )1) allowing an explicit selection of free surface equations. A new modified integro-differential equation is obtained for the free surface, and resolved by method of Fourier integrals. The results obmins are presented and discussed. A Taylor series approach to the integro-differential equation is then employed, and various examples computed.