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

Bibliographie

[1] Sizykh G. B., “Helical vortex lines in axisymmetric viscous incompressible fluid flows”, Fluid Dyn., 54:8 (2019), 1038–1042 | DOI | DOI

[2] Mironyuk I. Yu., Usov L. A., “The invariant of stagnation streamline for a stationary vortex flow of an ideal incompressible fluid around a body”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 24:4 (2020), 780–789 (In Russian) | DOI

[3] Markov V. V., Sizykh G. B., “Existence criterion for the equations solution of ideal gas motion at given helical velocity”, Izvestiya VUZ. Applied Nonlinear Dynamics, 28:6 (2020), 643–652 (In Russian) | DOI

[4] Van Dyke M., An Album of Fluid Motion, Parabolic Press, Stanford, California, 1982, 176 pp.

[5] Matyash E. S., Savelyev A. A., Troshin A. I., Ustinov M. V., “Allowance for gas compressibility in the $\gamma$-model of the laminar–turbulent transition”, Comput. Math. Math. Phys., 59:6 (2019), 1720–1731 | DOI | DOI

[6] Korolev G. L., “Steady viscous flow past an elliptic cylinder up to Reynolds number 900”, TsAGI Science Journal, 43:5 (2012), 615–635 | DOI

[7] Egorov I. V., Fedorov A. V., Palchekovskaya N., Obraz A. O., “Effects of injection on heat transfer and the boundary-layer instability for a hypersonic blunt body configuration”, Int. J. Heat Mass Transfer, 149 (2020), 119197 | DOI

[8] Chuvakhov P. V., Egorov I. V., “Numerical simulation of disturbance evolution in the supersonic boundary layer over an expansion corner”, Fluid Dyn., 56:5 (2021), 645–656 | DOI | DOI

[9] Loytsyansky L. G., Mechanics of Liquids and Gases, International Series of Monographs in Aeronautics and Astronautics, 6, Pergamon Press, Oxford, 1966, xii+804 pp.

[10] Ovsyannikov L. V., Lektsii po osnovam gazovoi dinamiki [Lectures on Basic Gas Dynamics], Institute of Computer Studies, Moscow, Izhevsk, 2003, 336 pp. (In Russian)

[11] Ladyzhenskii M. D., Prostranstvennye giperzvukovye techeniia gaza [Spatial Hypersonic Gas Flows], Mashinostroenie, Moscow, 1968, 120 pp. (In Russian)

[12] Nikol'skii A. A., Teoreticheskie issledovaniia po mekhanike zhidkosti i gaza [Theoretical Studies in Fluid and Gas Mechanics], Tr. TsAGI, 2122, TsAGI, Moscow, 1981, 286 pp. (In Russian)

[13] Mironyuk I. Yu., Usov L. A., “Stagnation points on vortex lines in flows of an ideal gas”, Proc. of MIPT, 12:4 (2020), 171–176 (In Russian) | DOI

[14] Sizykh G. B., “Integral invariant of ideal gas flows behind a detached bow shock”, Fluid Dyn., 56:8 (2021), 1027–1030 | DOI | DOI

[15] Ankudinov A. L., “A kinetic shock layer in the spreading plane of a lifting-body apparatus”, Fluid Dyn., 56:7 (2021), 967–974 | DOI | DOI

[16] Sizykh G. B., “Second integral generalization of the Crocco invariant for 3D flows behind detached bow shock wave”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 25:3 (2021), 588–595 (In Russian) | DOI | Zbl

[17] Batchelor G. K., An Introduction to Fluid Dynamics, University Press, Cambridge, 1970, xviii+615 pp.

[18] Sizykh G. B., “The stagnation line behind a detached bow shock wave in plane flows”, Proc. of MIPT, 14:4 (2022), 84–94 (In Russian)

[19] Sedov L. I., A Course in Continuum Mechanics, v. 1, Basic Equations and Analytical Techniques, Wolters-Noordhoff Publ., Groningen, The Netherlands, 1971, xix+242 pp. | Zbl | Zbl

[20] Mises R., Mathematical Theory of Compressible Fluid Flow, Applied Mathematics and Mechanics, 3, Academic Press, New York, London, 1958, xiii+514 pp. | Zbl

[21] Pontryagin L. S., Ordinary Differential Equations, Adiwes International Series in Mathematics, Pergamon Press, London, Paris, 1962, vi+298 pp. | Zbl | Zbl

[22] Sizykh G. B., “Solution of the Dorodnitsin problem”, Proc. of MIPT, 14:4 (2022), 95–107 (In Russian)

[23] Prim R., Truesdell C., “A derivation of Zorawski's criterion for permanent vector-lines”, Proc. Amer. Math. Soc., 1:1 (1950), 32–34 | DOI

[24] Truesdell C., The Kinematics of Vorticity, Indiana University Publications Science Seres, 14, Indiana University Press, Bloomington, 1954, xvii+232 pp. | Zbl