Similarity solutions of phi(3)(x)phi(xx)=phi((y)over-tilde(y)over-tilde) with applications to transonic aerodynamics of dense gases

摘要

Similarity solutions of the nonlinear small-disturbance equation phi(x)(3)phi(xx) = phi (x) over tilde (x) over tilde which describes a two-dimensional near-sonic potential flow of dense gases are studied. The analysis is based on a coordinate transformation, from the Cartesian (physical) plane to the hodograph (velocity) plane at which the governing equation becomes linear. The relations between the similarity parameters in these two planes are constructed. The first-order similarity solution of the velocity potential, phi, in the physical plane is described analytically by a parametric representation in terms of the hodograph-similarity variable. The new solutions are applied to the problem of a near-sonic small-disturbance flow of dense gases in the surroundings of two-dimensional, slender, semi-infinite bodies with x(n) generators, where 2/7 less than or equal to n < 1 and x is the distance along the body axis. In this problem, the similarity term describes a symmetric flow about the body axis. In the case when n = 2/7, the body constitutes a surface over which the flow is sonic at every point and the pressure distribution is constant. When 2/7 < n < 1, the pressure distribution along the body shows a singular behavior near the body leading edge. The analysis also indicates that the slender blunt body y = +/-epsilon x(2/7) may be the optimal shape to use in a near-sonic stream of dense gases, since it may have a minimal pressure drag. [References: 7]