Solution of the transonic airfoil problem through parametric differentiation.

Abstract

The difficulties associated with the nonlinearity of the transonic small disturbance equation are avoided by considering the linear problem governing the rate of change of the flow velocity with respect to the airfoil thickness ratio parameter and integrating solutions of this problem over the thickness ratio. This procedure is equivalent to summing a series of perturbations in airfoil thickness and has the advantage of moving all the nonlinearity to a firstorder ordinary differential equation, where it causes little difficulty. Solutions for nonlifting airfoils exhibit all the commonly observed features of transonic flow, such as the formation and movement of a shock on the airfoil surface and the presence of a detached shock upstream of the airfoil for low supersonic Mach numbers. Numerical results are given for subsonic and supersonic flow and for flows at Mach number unity which are in general agreement with other theoretical results and with experiment. Preliminary results for mixed flows with subsonic freestream Mach numbers indicate a rearward movement of the shock impinging on the airfoil surface with increasing airfoil thickness and exhibit the commonly observed rapid acceleration behind the shock. Extensions to lifting and unsteady flows are also discussed, as well as possible applications of the method to other problems in fluid mechanics.