On a paradoxical property of solving the problem of~stationary flow around a~body by a~subsonic stratified flow of an ideal gas

G. B. Sizykh

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 27 (2023) no. 1, pp. 142-158

Résumé

The problem of flow around a smooth convex body moving horizontally at a constant subsonic velocity in a stratified atmosphere at rest consisting of an ideal gas is considered. By the condition of the problem, the (vertical) gradient of the Bernoulli function (taking into account the potential energy of a uniform gravity field) in the atmosphere at rest at all altitudes is nonzero (as is the case in the Earth's standard atmosphere at altitudes up to 51 km), and the flight altitude does not exceed a value equal to the square of the body's flight speed divided by twice the acceleration of gravity. The surface of the earth is considered flat. The coordinate system associated with the body is used. The general spatial case is considered (an asymmetric body or a symmetric body at an angle of attack). We use the generally accepted assumption that in some neighborhood of the stagnation streamline (streamline that ends on the body at the forward stagnation point) there is no second stagnation point, the flow parameters in this neighborhood are twice continuously differentiable, and the stagnation point is spreading point (i.e. in some neighborhood of it, all streamlines on the surface of the body start at this point). Based on a rigorous analysis of the Euler equations, it is shown that the existence of a stationary solution to the problem contradicts this generally accepted (but not strictly proven) idea of the stagnation streamline. This property of the solution of the problem is called paradoxical and casts doubt on the existence of the solution.

Détail

Keywords: stratified atmosphere, existence of a stationary solution, stagnation streamline, forward stagnation point.
Mots-clés : Euler equations