Compressible Supersonic Flow in Jets under the Kármán‐Tsien Pressure‐Volume Relation

Abstract

The two‐dimensional supersonic irrotational flow of a gas in a jet is studied by use of the Kármán‐Tsien pressure‐volume law. There are two limitations to such a study: (1) since the fluid flow is not continued from the subsonic range, arbitrary boundary conditions must be prescribed; (2) use of the Kármán‐Tsien pressure‐volume relation implies a restriction on the permissible range of pressure, density, and velocity. On the other hand, use of the Kármán‐Tsien law furnishes several advantages: (1) the velocity potential and stream function satisfy the wave equation in the hodograph plane and hence these functions can be easily determined; (2) the mappings between the physical and hodograph planes may be completely characterized and studied in detail. This gain in information should be valuable in the qualitative understanding of phenomena as well as in obtaining first approximations to quantitative solutions. In the case of jets, with free stream lines as boundaries, it is shown that two functions possessing certain desired properties completely determine the Kármán‐Tsien flow. Further, the phenomenon of the periodic recurrence of the free stream jet boundary is explained by a folding property of the map of the flow in the hodograph plane.