Second Approximation to Supersonic Conical Flows

The method of expansion of the velocity potential in powers of a small geometrical parameter is applied with a view to finding a second-order correction to the linear theory for steady, isentropic, supersonic flow of an inviscid perfect fluid about a body having conical symmetry. A body lying entirely within the tip Mach cone is treated in detail. A method is developed for formulating, at a mean plane, boundary conditions at a thin body, and the particular integrals of the differential equations applicable to the second-order velocity perturbations are developed in integral form. The technique given by Lighthill is applied to determine the upstream boundary conditions consistent with the presence of an attached conical shock wave. The second approximation to the pressure coefficient on an arrow-head airfoil at zero angles of attack and yaw and which is infinite in downstream extent is computed by this technique.