On the Curvature of Boundary Streamlines of an Ideal Gas at Separation and Reattachment Points

The paper considers the stationary flows of an ideal (nonviscous and nonheat-conducting) gas with streamlines, i.e., boundaries of flowing and stationary media. In the 19th century such boundaries appeared in the problems of the outflow of jets into a flooded space. Until 1903 only jets of incompressible fluid were considered; the main contribution was made by Zhukovsky. In 1903 Chaplygin began studying flat subsonic jets of an ideal gas. In 1949 Ovsyannikov, having solved the problem of the outflow of a “critical” jet, discovered the fascinating properties of a flow with a sonic boundary streamline. Soon, segments of such streamlines, which arose mostly in problems of jet-flow theory, appeared in the construction of bodies subjected to subsonic flows with the largest “critical” Mach numbers M*. For an incident flow with M₀ < M* M < 1 in the entire flow, there are no shock waves and wave drag. At M₀ > M* supersonic zones appear, shock waves arise as well as wave drag, increasing with increasing M₀. It turned out that M* is achieved by bodies subject to a flow in which with M₀ = M* some of the contours are segments of sonic streamlines. It is useful to know their curvature at the separation and reattachment points. Zhukovsky states it to be infinite for a fluid at separation points. The infinity of the curvature of such streamlines in an ideal gas has been established only after 100 years. The following shows how the flow parameters and their derivatives, including the curvature of the streamlines, behave when approaching separation and reattachment points. The curvature of the boundary streamlines at these points is infinite, while the curvature of sonic streamlines when they intersect with a straight sonic transition line is zero.