Numerical Streamline Methods for Solving Steady Flow Problems

A streamline coordinate method is investigated as a means for solving problems of steady flow. In the streamline coordinate method, attention is focused on the streamline geometry rather than field variables. This is accomplished by considering a transformation to streamline coordinates. When the conservation equations are transformed, the result is a hybrid set of equations for the field variables and the components of the transformation. These equations are advantageous for studying free surface problems or problems with complicated solid wall geometry. Three different types of flow are investigated: 2-d, inviscid, incompressible flow; 2-d, inviscid, compressible flow; 3-d, inviscid, incompressible flow. In the first two cases, free surface problems are studied and it is shown that nonlinear elliptic boundary value problems result. The physical boundary conditions are found to couple to the transformed equations in an unusual way. Based upon this formulation, a fast numerical algorithm is developed. This algorithm, which is based on solving the finite difference discretizations via a relaxation-marching scheme, is an order of magnitude faster than a previous algorithm. The investigations in the 3-d case are not as complete. The transformed equations are a coupled, symmetric system and fall within no general theory. A simple nozzle flow is investigated and an algorithm is developed which appears to be fast. This is one of the few extensions of a streamline method to a three-dimensional setting.