Axially Symmetrical Jet Mixing of a Compressible Fluid*

The corresponding laminar problem has been theoretically analyzed by Schlichting3 in 1933. Neither of these solutions, however, holds for finite opening, i.e. at a short distance from the opening. Kuethe,3 in 1935, extended Tollmien's results to the case of a two-dimensional jet issuing into a medium not at rest and also worked out an approximate method for'the computation of the velocity profile in the initial part of a round jet issuing into medium at rest. Squire and Trouncer,4 in 1944, extended Kuethe's results to the case of a round jet issuing into a uniform stream by assuming certain velocity profiles across the jet. Abramovich,5 in 1939, extended Tollmien's problem, i.e. the mixing of a parallel stream with the adjacent medium at rest, to the case of a compressible fluid. He considered the effects of compressibility due to high temperature and those due to high subsonic speed separately and obtained some approximate solutions. Reichardt,0 in 1941, from the experimental data, suggested the constant exchange coefficient over each cross-section of the mixing zone for the free turbulence problem and Gortler,7 in 1942, reexamined Tollmien's problems by the application of Reichardt's theory and obtained some improvement in the velocity profiles. In 1949, the present author,8 investigated the problems of the flow of a two-dimensional jet from a finite opening of a compressible fluid exhausting into uniform stream and of the mixing of two uniform streams of compressible fluid. Both the laminar and the turbulent cases were considered. The effects of compressibility due to large temperature difference and those due to high velocity were considered simultaneously. In the present paper, this analysis is extended to the case of an axially symmetrical jet of a compressible fluid exhausting into a uniform stream. The flow of the jet is assumed to be under full expansion from a nozzle, i.e. the pressure of the flow at the exit of the nozzle is exactly equal to that of the surrounding stream. The pressure gradient in the jet is assumed to be negligible. Both the laminar and the turbulent cases will be considered. The first part of this paper is concerned with the laminar flow. The usual assumptions of boundary layer theory adopted to simplify the Navier-Stokes equations. A solution by the method of small perturbations is first obtained. Then the exact solution is examined. A numerical integration method is used to compute the velocity and the temperature distributions in the jet for the exact solution.