Some complex numerical integrations of the Schwarz-Christoffel transformation and application to potential flow

Graduation date: 1966The Schwarz-Christoffel transformation is used to map simply connected polygons onto the upper half plane and can be applied to problems in which the validity of Laplace's equation is assumed. However the direct integration of the resulting complex integral is often not possible and some approximate method must be used. By applying real numerical integration jointly on the real and imaginary parts of the complex integral, this thesis extends the principles and formulas of real numerical integration to include complex analytic functions with a finite number of singularities. This extension includes a theorem on the integral path of least error and a discussion of error cancellation along paths which have an accumulated directional change of more than ninety degrees. The principles of complex numerical integration are then shown to be applicable to problems of potential flow which require a Schwarz-Christoffel mapping. A method of computing the fixed points of the transformation is presented with a procedure for determining the value of the integral when the integrand becomes infinite. This thesis also contains a generalization of the Schwarz-Christoffel transformation to include curved corners along with the formula for the radius of curvature of these corners. Computational examples of the various transformations are shown in the form of streamline plots of potential flow over irregular boundaries with velocity components at selected points