Closed vortex lines in fluid and gas

G. B. Sizykh

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 3, pp. 407-416.

Résumé

Continuous fluid and gas flows with closed vortex tubes are investigated. The circulation along the vortex line of the ratio of the density of the resultant of all forces (applied to the fluid or gas) to the density of the fluid or gas is considered. It coincides with the circulation (along the same vortex line) of the partial derivative of the velocity vector with respect to time and, therefore, for stationary flows, it is equal to zero on any closed vortex line. For non-stationary flows, vortex tubes are considered, which remain closed for at least a certain time interval. A previously unknown regularity has been discovered, consisting in the fact that at, each fixed moment of time, such circulation is the same for all closed vortex lines that make up the vortex tube. This regularity is true for compressible and incompressible, viscous (various rheologies) and non-viscous fluids in a field of potential and non-potential external mass forces. Since this regularity is not embedded in modern numerical algorithms, it can be used to verify the numerical calculations of unsteady flows with closed vortex tubes by checking the equality of circulations on different closed vortex lines (in a tube). The expression for the distribution density of the resultant of all forces applied to fluid or gas may contain higher-order derivatives. At the same time, the expression for the partial derivative of the velocity vector with respect to time and the expression for the vector of vorticity (which is necessary for constructing the vortex line) contain only the first derivatives; which makes it possible to use new regularity for verifying the calculations made by methods of high and low orders simaltaniously.

Détail

Keywords: closed vortex tubes, verification of calculations of fluid and gas flows, vortex theorems, Zorawski’s criterion.

Bibliographie

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

[2] Kochin N. E., Kibel I. A., Roze I. V., Theoretical Hydromechanics, Wiley, New York, 1964, v+577 pp. | MR | Zbl

[3] Golubinskii A. I., Sychev V. V., “Some conservation properties of turbulent gas flows”, Dokl. Akad. Nauk SSSR, 237:4 (1977), 798–799 (In Russian) | MR | Zbl

[4] Mobbs S., “Some vorticity theorems and conservation laws for non-barotropic fluids”, J. Fluid Mech., 108 (1981), 475–483 | DOI | MR | Zbl

[5] Golubinskii A. I., Golubkin V. N., “On certain conservation properties in gas dynamics”, J. Appl. Math. Mech., 49:1 (1985), 88–95 | DOI | MR

[6] Markov V. V., Sizykh G. B., “Vorticity evolution in liquids and gases”, Fluid Dyn., 50:2 (2015), 186–192 | DOI | MR | Zbl

[7] Krocco L., “Eine neue Stromfunktion für die Erforschung der Bewegung der Gase mit Rotation”, Z. Angew. Math. Mech., 17:1 (1937), 1–7 (In German) | DOI

[8] Truesdell C., “On curved shocks in steady plane flow of an ideal fluid”, J. Aeronaut. Sci., 1952, no. 19, 826–828 | DOI | MR | Zbl

[9] Hayes W. D., “The vortycity jump across a gasdynamic discontinuities”, J. Fluid Mech., 1957, no. 2, 595–600 | Zbl

[10] Levin V. A., Markov V. V., Sizykh G. B., “Vorticity on the Surface of an Axially Symmetric Body behind a Detached Shock Wave”, Doklady Physics, 63:12 (2018), 530–532 | DOI

[11] Sizykh G. B., “Entropy Value on the Surface of a Non-Symmetric Convex Bow Part at Supersonic Streamlining”, Fluid Dyn., 54 (2019) (to appear) | DOI

[12] Beltrami E., “Considerazioni idrodinamiche”, Il Nuovo Cimento Series 3, 25:1 (1889), 212–222 | DOI | Zbl

[13] Biushgens S. S., “On Helical Flow”, Nauchn. Zapiski Mosk. Gidrom. Inst. (MGMI), 17 (1948), 73–90 (In Russian)

[14] Sizykh G. B., “Axisymmetric Helical Flows of Viscous Fluid”, Russian Mathematics, 63:2 (2019), 44–50 | DOI

[15] Sizykh G. B., “Helical Vortex Lines in Axisymmetric Viscous Incompressible Fluid Flows”, Fluid Dyn., 54 (2019) (to appear) | DOI

[16] Kotsur O. S., “On the existence of local formulae of the transfer velocity of local tubes that conserve their strengths”, Proceedings of MIPT, 11:1 (2019), 76–85 (In Russian)

[17] Rowland H., “On the Motion of a Perfect Incompressible Fluid When no Solid Bodies are Present”, Am. J. Math, 3:3 (1880), 226–268 | DOI | MR

[18] Lamb H., Hydrodynamics, Cambridge Univ. Press, Cambridge, 1895, xvii+604 pp. | DOI | MR | Zbl

[19] Hamel G., “Ein allgemeiner Satz über den Druck bei der Bewegung volumbeständiger Flüssigkeiten”, Monatsh. Math. Phys., 43:1 (1936), 345–363 (In German) | DOI | MR

[20] Truesdell C., “Two measures of vorticity”, Indiana Univ. Math. J., 2:2 (1953), 173–217 | DOI | MR | Zbl

[21] Vyshinsky V. V., Sizykh G. B., “The verification of the calculation of stationary subsonic flows and the presentation of the results”, Mathematical Models and Computer Simulations, 11:1 (2019), 97–106 | DOI | MR

[22] Golubkin V. N., Sizykh G. B., “On the vorticity behind 3-D detached bow shock wave”, Advances in Aerodynamics, 1:1 (2019), 15 | DOI

[23] Troshin A., Shiryaeva A., Vlasenko V., Sabelnikov V., “Large-Eddy Simulation of Helium and Argon Supersonic Jets in Supersonic Air Co-flow”, Progress in Turbulence VIII. iTi 2018, Springer Proceedings in Physics, 226, 2019, 253–258 | DOI

[24] Vyshinsky V. V., Sizykh G. B., “Verification of the Calculation of Stationary Subsonic Flows and Presentation of Results”, Smart Modeling for Engineering Systems. GCM50 2018, Smart Modeling for Engineering Systems, 133, Springer, Cham, 2019, 530–532 | DOI | MR

[25] Afonina N. E., Gromov V. G., Levin V. A., Manuilovich I. S., Markov V. V., Smekhov G. D., Khmelevskii A. N., “Investigation of the annular nozzle start in actual and virtual intermittent aerodynamic setups”, Fluid Dyn., 51:2 (2016), 281–287 | DOI | MR | Zbl

[26] Dergachev S. A., Marchevsky I. K., Scheglov G. A., “Flow simulation around 3D bodies by using Lagrangian vortex loops method with boundary condition satisfaction with respect to tangential velocity components”, Aerospace Science and Technology, 2019, 105374 (to appear) | DOI

[27] Borovoy V. Y., Egorov I. V., Skuratov A. S., Struminskaya I. V., “Two-Dimensional Shock-Wave/Boundary-Layer Interaction in the Presence of Entropy Layer”, AIAA Journal, 51:1 (2013), 80–93 | DOI

[28] Egorov I. V., Novikov A. V., “Direct numerical simulation of laminar–turbulent flow over a flat plate at hypersonic flow speeds”, Comput. Math. Math. Phys., 56:6 (2016), 1048–1064 | DOI | MR | Zbl

[29] Pontryagin L. S., Ordinary differential equations, Adiwes International Series in Mathematics, Pergamon Press, London, Paris, 1962, vi+298 pp. | MR | Zbl