Boundary Singularities in Steady Potential Compressible Flow Through Plane Two-Dimensional Channels

Abstract

Boundary singularity solutions are reviewed, which relate to the steady, transonic potential flow of a perfect gas, through plane, two-dimensional convergent channels. Two singularities occur in the physical plane, one at upstream infinity and the other at the concave corner in channels that have a parallel entry section followed by a convergent section that ends at the throat plane. Solutions of these singularities are obtained from the general solution of Chaplygin's stream function equation in the hodograph system of coordinates. Uniform, uni-directional subsonic entry flow, in a finite-length channel, gives rise to a boundary singularity in the hodograph plane. The stream function is multi-valued at this singularity, where it varies between one and zero, while the flow direction is zero. This condition is removed by introducing circular polar coordinates (r, α) with origin at the singular point. For a sufficiently small value of r, a second-order partial differential equation of the stream function is obtained in the new coordinates. A particular solution of this equation is given, when r is small, rs say, in which the stream function gradient in the radial direction is taken as zero for all values of α at rs. The full series solution of the new equation is given in a paper which is to follow. Finally, consideration is given to the problem of matching the subsonic and supersonic flow regions at the sonic surface. A new system of equations is given, which determine the stream function gradient at sonic points from values of the stream function in the expansion region. Computed results are given that relate to the steady, choked flow of a perfect gas through a 15° convergent channel. The overall solution is shown to converge.