Example of solving transonic equations for a shock-free flow past a symmetric profile

Abstract

A parametric method /1/ of solving the transonic Kármán—Fal'kovich equations is developed. The nozzle solution is generalized to the case of the flows not symmetric about the longitudinal axis of the nozzle. A procedure of passing from this solution to the case of a flow past a profile discussed in /2/ is then shown. This in fact means that the real and imaginary part of the complex function describing this flow have been obtained. The resulting solution depends on three constants determining the dimensions of the profile (length of chord and the maximum thickness) and also the flow rate at infinity. Numerical analysis is used to obtain the condition for the flow to be shock-free, and a continuous velocity field is constructed under the conditions close to the limiting state. Setting up a flow chart for the cases when the condition of no shock is violated shows that a three-sheeted fold appears, the top of which lies within the supersonic region. This confirms the conclusion made in /3–5/ that in a typical case of a flow past a profile, the shock wave originates not at the sonic stream line, but within the zone. The example constructed can be used as the basis for the theory of flow past a profile of a sufficiently general form, of a gas stream subsonic at infinity. Below the transonic Tricomi model is used to show the corresponding generalization.