Two-dimensional aerofoil design in compressible flow

Abstract

This paper deals with the following two-dimensional problem:-- 'The design of an aerofoil to give a specified velocity against chord curve at a given free-stream Mach number.' A 'relaxation' method is adopted, based on the differential equations for incompressible and compressible flow. An essential feature of the method is that the calculations are carried out in the (φ, ψ) or w-plane, in which the aerofoil is represented by a slit along ψ = 0. The square mesh in this plane is formed by the streamlines (ψ = constant), and equipotentials (φ = constant) for incompressible flow about the aerofoil. The method is developed for a symmetrical aerofoil at zero incidence, but the modifications necessary for the more general case are indicated. A worked example is given, from which some idea of the accuracy of the method can be gained. The compressible velocity distribution about a known aerofoil was taken as the initial data. This aerofoil was actually 12 per cent thick at 30 per cent of the chord distance from the leading edge. Using a mesh giving only fourteen mesh points on the aerofoil, we find that the calculations yield a 12.06 per cent aerofoil at 28.2 per cent of the chord distance from the leading edge.