One‐Dimensional High‐Speed Flows

Abstract

The paper discusses the types of singular points occurring in the first‐order ordinary differential equation which describes compressible viscous flow in a channel or stream tube of varying cross‐sectional area. The treatment is one‐dimensional, viscosity being allowed for by assuming a tangential stress acting on the circumference. The resulting patterns of the integral curves arc examined. It is shown that for convergent‐divergent channels whose profile has no point of inflexion, the singular point is a saddle point, as is the case in frictionlcss flow. However, the sonic section or the section of highest or lowest Mach number do not coincide with the throat but arc situated downstream of it in the divergent portion. The slopes of the integral curves which pass through the sonic section arc evaluated. When the convergent‐divergent channel has a point of inflexion in its profile there may be two singular points, the first being a saddle point and the second cither a spiral point or a nodal point. It is shown that spiral points are more likely to occur than nodal points and that, when they occur, there is no radical change in the Mach number variation along the channel due to friction. On the other hand, the existence of a nodal point admits the possibility of a continuous transition from supersonic to subsonic How in which the Mach number at exit may vary within certain limits, the Mach number in the second sonic section remaining always equal to unity. In all types of flow there arc portions of the channel over which the influence of friction outweighs that of area change.