Aerofoil design in two-dimensional subsonic compressible flow

Abstract

In Part I of this paper the method of two-dimensional aerofoil design in incompressible flow due to Lighthill (1945) is extended to compressible subsonic flow. Lighthill's equations are derived as special cases of more general equations due to tile author, and some advances are made in tile application of these equations to aerofoil design. It is shown, for example, how the designer can control the nose radius of curvature. The method of Ref. 1 requires that the velocity distribution be prescribed analytically, whereas this paper deals with distributions defined numerically, a development especially important for compressible flow design. The compressible flow theory is based on an approximation to the equation of flow not unlike, and with at least the same accuracy, as the Kabrman - Tsien approximation for calculating the flow about a given aerofoil. A method of estimating the effects of a modification to the designed aerofoil shape, on the velocity distribution is also given. In Part 2 five examples have been calculated. Aerofoil I is symmetrical, with a 'roof-top' distribution at a given angle of incidence, showing how a given nose radius can be achieved; Aerofoil II is symmetrical, designed for M∞ = 0, and α = 0 (M∞being the Mach number at infinity, and α the absolute angle of incidence),while Aerofoil III has been designed for the same distribution but at M∞ = 0.7. A comparison is made between Aerofoil III and that obtained from Aerofoil II by linear pertubation theory. It is shown, as would be expected, that this theory underestimates the reduction in thickness necessary to produce a compressible flow aerofoil from one designed for incompressible flow and the same velocity distribution. Aerofoils IV and V are asymmetric aerofoils designed for M∞ = 0.7, the former being designed to have a given distribution over each surface at incidence, while the latter is designed so that the upper surface has a given distribution at incidence and the lower surface has a given distribution at zero incidence. The design of an asymmetric aerofoil by the author's method is about two days' work for one computer.