Generalized aerofoil theory

Abstract

A theory is developed enabling the flow of an inviscid compressible gas at subsonic speeds past a porous aerofoil, with a given pressure on the inside of the porous surface, to be calculated. The assumptions made in the paper are ( a ) that the ideal gas can be replaced by a Kármán-Tsien tangent gas, ( b ) that the mass flow through the porous wall is linearly related to the pressure difference across the wall, and ( c ) that the mass of air sucked into the aerofoil is relatively small; no restrictions are placed on the magnitude of the aerofoil incidence or thickness. Generalized forms of Blasius’s well-known formulae for the lift and moment in incompressible flow are obtained for the tangent gas, and applied to the porous aerofoil problem. The theory is shown to yield as special cases ( a ) the flow about a given aerofoil, ( b ) the design of an aerofoil for a given pressure distribution, ( c ) the flow about a given aerofoil with point sources or sinks on the surface, or with a surface distribution of these (distributed suction), ( d ) the flow about a thin aerofoil with a limited region of flow separation, such as occurs in 'thin-aerofoil' stall, and ( e ) the flow about bluff bodies (two-dimensional) to which finite bubbles or cavities adhere. By 'flow’ here is meant, of course, the inviscid flow external to the bubble or boundary later. The paper contains incidentally a new treatment of the solid lifting aerofoil in a tangent gas; this treatment does not follow the usual method of first finding the incompressible flow about a profile—which must be distorted from the original shape in a manner initially unknown—but is based on an integral equation directly applicable to the given circulation and profile shape. As an integral equation must be solved in any case, to determine the in­compressible flow about the distorted profile, the direct treatment of the compressible flow problem, given in the paper, is simpler than the usual treatment. The paper includes a discussion of the problems of 'thin-aerofoil' and 'leading-edge' stall.