Solution of the Dorodnitsin--Ladyzhensky problem

G. B. Sizykh

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 4, pp. 764-776.

Résumé

The article is devoted to a rigorous proof of the statement that entropy takes the maximum value on the surface of a body with a blunted nose, streamlined by a supersonic flow, in the presence of a plane of symmetry of the flow. This is obvious for bodies of rotation at zero angle of attack, and it is established by numerical calculations and experimentally at non-zero angles of attack. The proof boils down to the justification that the leading streamline (the current line crossing the shock along the normal) ends on the body. In other words, that the leading streamline and the stagnation line are coincide. Such a proof was obtained by G. B. Sizykh in 2019 for the general spatial case (not only for flows with a plane of symmetry). This rather complicated proof is based on the Zoravsky criterion, which only a narrow circle of specialists has experience using, and is based on the assumption of the continuity of the second derivatives of density and pressure. In this paper, for the practically important case of flows with a plane of symmetry (in particular, the flow around bodies of rotation at a non-zero angle of attack), an original simple proof is proposed, for which the continuity of only the first derivatives of the density and pressure fields is sufficient and it is not necessary to use the Zoravsky criterion.

Détail

Mots-clés : Euler equations
Keywords: isenthalpic flows, vorticity, stagnation streamline, leading streamline, detached bow shock.

Bibliographie

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

[2] Kraiko A. N., Kratkii kurs teoreticheskoi gazovoi dinamiki [Short Course of Theoretical Gas Dynamics], MIPT, Moscow, 2007, 300 pp. (In Russian)

[3] 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

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

[5] Sizykh G. B., “Entropy value on the surface of a non-symmetric convex bow part of a body in the supersonic flow”, Fluid Dyn., 54:7 (2019), 907–911 | DOI | DOI

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

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

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

[9] Friedman A. A., Opyt gidromekhaniki szhimaemoi zhidkosti [Experience of Hydromechanics of Compressible Fluid], ONTI, Leningrad, Moscow, 1934, 368 pp. (In Russian)

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

[11] Bibikov Yu. N., Kurs obyknovennykh differentsial'nykh uravnenii [A Course in Ordinary Differential Equations], Vyssh. Shk., Moscow, 1991, 303 pp. (In Russian)

[12] Petrovsky I. G., Lektsii po teorii obyknovennykh differentsial'nykh uravnenii [Lectures on the Theory of Ordinary Differential Equations], Moscow Univ. Press, Moscow, 1984, 296 pp. (In Russian)

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