Regular subsonic-sonic flows in general nozzles

Abstract

This paper concerns subsonic-sonic potential flows in general two dimensional nozzles. For finitely long symmetric nozzles, we formulate the subsonic-sonic flow problem by prescribing the flow angle at the inlet and the outlet. It is shown that this problem admits a unique Lipschitz continuous subsonic-sonic flow, and the sonic points of the flow must occur at the wall or the throat. This is the first result on the well-posedness for general subsonic-sonic flow problems. More importantly, the location of sonic points is classified completely. Indeed, it is shown that there exists a critical value depending only on the length and the geometry of the nozzle such that the flow is sonic on the whole throat if the height of the nozzle is not greater than this critical value, while the sonic points must be located at the wall if the height is greater than this value. Furthermore, the critical height is positive iff the nozzle is suitably flat near the throat. As a direct application of this theory, we can obtain conditions on whether there is a smooth transonic flow of Meyer type whose sonic points are all exceptional in de Laval nozzles.