Mathematical Modeling, Numerical Methods and Software Complexes

On the place of sonic points in a critical flow

A. I. Besportochnyy, A. N. Burmistrov

Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Region, 141701, Russian Federation

Abstract: Stationary irrotational barotropic gas flows are investigated on the basis of the analysis of three-dimensional Euler equations. Critical flows in the article are those in which the Mach number is everywhere less than or equal to one, and at least at one point the Mach number reaches one. In 1954, D. Gilbarg and M. Shiffman showed that if an internal (not lying on the streamlined surface) sonic point exists in a critical flow, then it lies on a flat sonic surface, which at all its points is perpendicular to the gas velocity vector and cannot end inside the flow (theorem about the sonic point). Using this theorem, D. Gilbarg and M. Shiffman obtained a conclusion that is important for the problems of maximizing the critical Mach number. It consists in the fact that in a critical flow for a wide class of bodies in flow, sonic points can be located only on its surface. This conclusion is essentially used in constructing the shapes of streamlined bodies with the maximum value of the critical Mach number (for given isoperimetric conditions).
In this paper, the question of the curvature of streamlines at the internal sonic points of critical flows is considered. It is shown that this curvature is zero. The result is a new necessary condition for the existence of an interior sonic point (and sonic surface). It consists in the fact that at the point of intersection with the sonic surface, the normal curvature of the streamlined surface in the direction normal to the sonic surface should be equal to zero. Examples of streamlined bodies are given for which the theorem by D. Gilbarg and M. Shiffman (on the sonic point) does not answer the question of the location of the sonic points, at the same time a new necessary condition makes it possible to prove that the existence of internal sonic points in a critical flow around these bodies is impossible.

Keywords: Euler's equations, subsonic maximum principle, critical flow, sonic point, first sonic point, sonic line, sonic plane.

UDC: 533.6.011

MSC: 76H05

Received: March 25, 2021
Revised: May 7, 2021
Accepted: May 11, 2021
First online: September 30, 2021

DOI: 10.14498/vsgtu1856

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