The numerical solution of two-dimensional fluid motion in the neighbourhood of stagnation points and sharp corners

Abstract

Methods are given in this paper of dealing with singularities of functions satisfying certain two dimensional partial differential equations. For a numerical solution the differential equations are replaced by difference equations on a square mesh. Log (I/q) where q is the Velocity, becomes infinite at stagnation points, sharp corners, sinks, etc., while the conjugate function 0 (flow direction) becomes multi-valued. The method consists in finding a series expansion for the function (log 1/q or 0) in the neighbourhood of the singularity. This expansion is then used to find relationships between the function values at points of the mesh adjacent to the singularity. A method of working directly in the transformed flow plane (in which the aerofoil is a slit), and thus avoiding irregular squares on the boundary, is also given. The method is developed for incompressible flow, but an approximation suitable for compressible flow is given.