The Ideal Drag Due to a Shock Wave. Parts I and II

Abstract

Part 1. The subject of the present note is the increase of the drag of an aerofoil which arises from the presence of limited shock waves when the forward speed lies between the so-called shock stalling speed and the velocity of sound. Evidence from photographs and other sources shows that under certain conditions a single limited shock wave exists on one or both surfaces of an aerofoil when the local speed at some point of the surface exceeds the velocity of sound. It is therefore suggested that ideal two-dimensional motions about an aerofoil may exist, which satisfy the equations of motion of a non-viscous non-conducting compressible fluid, at all points of the field outside a limited shock wave attached to one or both surfaces of the aerofoil. The shock wave is to be considered merely as a surface of discontinuity across which the usual conditions of continuity of flow, momentum, total energy and velocity parallel to the surface, are satisfied; it forms the rear boundary of a limited region in which the flow is supersonic and its intensity falls to zero at its outer edge where the velocity is equal to the local velocity of sound. As a result of the increase of entropy on passing through the shock wave, the density and velocity at a large distance behind the aerofoil of a particle of fluid that has passed through the shock wave will both be reduced below their free-stream values; the pressure will have regained its free-stream value. The ordinary momentum integrals taken across lines far in front of, and far behind, the aerofoil thus determine the drag in the ideal case; this may be considered as the ideal (lowest possible) drag due to an actual shock wave. Part 2. The method given in Part I of calculating a first approximation to the ideal theoretical drag rise due to a shock wave on an aerofoil is extended to cover the use of the Karman-Tsien solution in place of the Glauert relation. It is shown how the drag rise can be calculated from the thooretical critical Mach number and the geometrical curvature of the surface at the point of maximum suction. In particular for two aerofoils having the same critical Mach number the drag rise is proportional to the radius of curvature; thus the drag rise on a 17.3% ellipse at zero incidence will be three times that for an NACA 0012 section having the same critical Mach number. Comparison with experiment in the N.P.L. 20 in x 8 in High.Speed Tunnel shows that for a number of aerofoil shapes the theoretical rise occurs consistently from M = 0.1 to 0.13 later than the observed rise. The present method is simple and should give at least a better indication of the relative merits of different aerofoil shapes than a knowledge of the theoretical critical speed alone.