Integration of the equations of transonic flow in two dimensions

Abstract

The equation for the velocity potential ϕ of compressible flow is transformed to α, β coordinates, where α + i β is the corresponding complex potential of the incompressible flow; the equation is then integrated on the following assumptions: (1) Compressibility in the case of subsonic and transonic flows can be considered as a perturbation effect, and the equation for the velocity potential ϕ can be treated as a form of the non-linear Poisson equation. (2) To a first approximation ∂ ϕ /∂ β is disregarded; ∂ ϕ /∂ α , which only appears in the equation multiplied by a rapidly varying function, is taken to be constant in differentiation and integration; ∂ 2 ϕ /∂ α 2 and ∂ 2 ϕ /∂ β 2 are, however, retained in the equation. The integration leads to an algebraic equation of fifth degree in the unknown quantity ∂ ϕ /∂ α , which can thus be evaluated at any given point of the fluid. The condition for the occurrence of double roots in this equation determines the characteristic Mach number at which, it is assumed, a shock wave first appears. The method is applied to the motion past a circular cylinder, and the results are compared with Imai’s solution based on Janzen–Rayleigh’s method, and with Woods’s calculations by the relaxation method. The agreement is satisfactory in both cases. The Mach number first reaches its characteristic value due to conditions at the point θ = ½π, r = 1, and is equal to M 1 = 0.414; the characteristic velocity is q = 2.75 (corresponding to q = 2 for incompressible flow) and the characteristic local Mach number is M e = 1·29. The method is also applied to an aerofoil of thickness 1/10, which consists of two arcs with cusps at leading and trailing edges; and when the results are compared with Kaplan’s calculations, the agreement is close. The characteristic Mach number is M 1 = 0.839, the increase in the velocity over that of incompressible flow is 32%, and the local characteristic Mach number is M e = 1·44.