LOCALIZED VORTICES IN TWO-DIMENSIONAL HYDRODYNAMIC MEDIA

  • Vladimir V. Yanovsky Institute for Single Cristals, Nat. Academy of Science Ukraine, Lenine Ave.60, Kharkov 31001,UkraineV.N. Karazin Kharkiv National UniversitySq. Svobody 4, Kharkiv, 61022 Ukraine https://orcid.org/0000-0003-0461-749X
  • A. V. Tur Universite Toulouse [UPS], CNRS, Institute of Research for Astrophysics and Planetology 9 avenue du Colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France
  • K. N. Kulik Institute for Single Cristals, Nat. Academy of Science Ukraine, Lenine Ave.60, Kharkov 31001,Ukraine

Abstract

The review covers the main achievements in the field of the dynamics of strongly localized vortices in two-dimensional hydrodynamic media. The types of point vortices and their motion equations are discussed. The consideration is focused on dipole point vortices, a new type of point vortices. Investigated is the influence of the boundaries of the vortices on the character of their motion, and some new stationary fluid flows. The impact of potential waves on the interacting point vortices leads to new effects also considered in the review.

Downloads

Download data is not yet available.

Author Biographies

Vladimir V. Yanovsky, Institute for Single Cristals, Nat. Academy of Science Ukraine, Lenine Ave.60, Kharkov 31001,UkraineV.N. Karazin Kharkiv National UniversitySq. Svobody 4, Kharkiv, 61022 Ukraine
A. V. Tur, Universite Toulouse [UPS], CNRS, Institute of Research for Astrophysics and Planetology 9 avenue du Colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France
  
K. N. Kulik, Institute for Single Cristals, Nat. Academy of Science Ukraine, Lenine Ave.60, Kharkov 31001,Ukraine
  

References

Meleshko V.V., Konstantinov M.Ju. Dinamika vihrevyh struktur. – Kiev: Naukova dumka, 1993.-279s.

Tur A.V., Janovskij V.V. Gidrodinamicheskie vihrevye struktury. – Har'kov: Institut monokristallov, 2012. - 290s.

Seffmen F.Dzh. Dinamika vihrej. - M.: Nauchnyj mir, 2000. - 376s.

Ekkart K. Gidrodinamika okeana i atmosfery. - Moskva-Izhevsk: NIC «Reguljarnaja i haoticheskaja dinamika», 2004. – 328s.

Matveev L.T., Matveev Ju.L. Oblaka i vihri - osnova kolebanij pogody i klimata. - SPb.: izd.RGGMU, 2005. - 327s.

Monin A.S., Seidov D.G. Pogoda i klimat okeana // Priroda. – 1983. – S.34-43.

Nalivkin D.V. Uragany, buri i smerchi. Geograficheskie osobennosti i geologicheskaja dejatel'nost'.- L.: Nauka, 1969. - 487s.

Mamedov E.S., Pavlov N.I. Tajfuny.- L. Gidrometeoizdat, 1974.- 139s.

Nalivkin D.V. Smerchi.- M.: Nauka, 1984.- 111 s.

Getling A.V. Konvekcija Releja-Benara. Struktury i dinamika.- M.: Editorial URSS, 1999.- 248s.

Gershuni G.Z., Zhuhovickij E.M., Nepomnjashhij A.A. Ustojchivost' konvektivnyh techenij.- M.: Nauka, 1989.- 320s.

Monin A.S. Vvedenie v teoriju klimata.- L.: Gidrometeoizdat, 1982.- 296 s.

Merkulov V.I. Gidrodinamika znakomaja i neznakomaja.- M.: Nauka, 1989.- 136s.

Lavrent'ev M.A., Shabat B.V. Problemy gidrodinamiki i ih matematicheskie modeli.- M.: Nauka, 1973.- 416s.

Uizem D. Linejnye i nelinejnye volny.- M.: Mir, 1977.- 624s.

Lajthill Dzh. Volny v zhidkostjah.- M.: Mir, 1981.- 598s.

Vil'hel'mson H., Vejland Ja. Kogerentnoe nelinejnoe vzaimodejstvie voln v plazme.- M.: Energoizdat, 1981.- 224s.

Helmholtz H. Uber Integrate hydrodinamischen Gleichungen weiche den Wirbelbewegungen entsprechen // J. rein, angew. Math. 1858. – Vol.55. – P.25-55., sm. takzhe russkij perevod s kommentarijami S.A. Chaplygina v knige Gel'mgol'c G. Osnovy vihrevoj teorii. – M.-Izh.: IKI, 2002. - 82s.

Puankare A. Teorija vihrej. Izhevsk: Izd-vo RHD. 2000.- 160 s. Per. s fr. Poincare H.Theorie des tourbillions. Paris, Carre. 1893.

Yanovsky V.V., Tur A.V., Kulik K.N. Singularities Motion Equations in 2-Dimensional Ideal Hydrodynamics of Incompressible Fluid // Phys.Lett. A. – 2009. –Vol. 373. – P.2484-2487.

Gel'fand I.M., Shilov G.E. Obobshhennye funkcii i dejstvija nad nimi.- M.: Gos. izd. fiziko-matematicheskoj literatury, 1959.- 470s.

Vladimirov B.S. Obobshhennye funkcii v matematicheskoj fizike. - M.: Nauka, 1979.- 320 s.

Lamb G. Gidrodinamika.-M.: OGIZ, Gostehizdat, 1947:- 929s. Per. s ang. Lamb H. Hydrodynamics, Ed. 6-th., N. Y. Dover publ. 1945.

Glaz H.M. Two attempts at modeling two-dimensional turbulence, in “Turbulence Seminare”, ed.P.Bernard, T.Ratio, Lecture Notes in Mathematics.- Berlin-Heidelberg-New York: Springer-Verlag. 1977.

Novikov E.A., Sedov Ju.B. Stohasticheskie svojstva sistemy chetyreh vihrej // ZhETF. – 1978. - T.75. - Vyp. 3. - S.868-876.

Ziglin S.L. Neintegriruemost' zadachi o dvizhenie chetyreh tochechnyh vihrej // DAN SSSR. – 1979. – T.250. – No.6.- C.1296-1300.

Synge J.L. On the motion of three vortices // Can. J. Math. – 1949. – Vol.1. – P.257-270.

Novikov E.A. Dinamika i statistika sistemy vihrej // ZhETF. - 1975. - T.68. - Vyp.5. - S.1868-1882.

Aref H. Motion of three vortices // Phys. Fluids.- 1988.- Vol. 31.- No.6.- P. 1392-1409.

Kulik K.N., Tur A.V., Janovskij V.V. Vzaimodejstvie tochechnogo i dipol'nogo vihrja v neszhimaemoj zhidkosti // TMF.- 2010. - T.162. - No.3. - S.459-480.

Stefan G. Llewellyn Smith, How do singularities move in potential flow? // Physica D.- 2011.-Vol.240. - P.1644-1651.

Lewis T.C. Some cases of vortex motion // Messenger of Math. - 1879. - Vol.9. - P. 93–95.

Borisov A.V., Mamaev I.S. Matematicheskie metody dinamiki vihrevyh struktur.- Moskva-Izhevsk: Institut komp'juternyh issledovanij, 2005.- 368s.

Greenhill A.G. Plane vortex motion // Quart. J. Pure Appl. Math. – 1877/78. – Vol. 15. – No.58. – P.10–27.

Tur A.V., Yanovsky V.V. Interaction of a dipole point vortex with flat boundary // arXiv:1204.4557v1 (physics.flu-dyn) 20 Apr 2012.

Kulik K.N., Tur A.V., Janovskij V.V. Evoljucija tochechnogo vihrja dipol'nogo tipa v krugovoj oblasti // Nelinejnaja dinamika. – 2013. - T.9. - No.4. - S.559-669.

Fridman A.A., Polubarinova P.Ja. O peremeshhajushhihsja osobennostjah ploskogo dvizhenija neszhimaemoj zhidkosti // Geofiz. sb. - 1928. - T.5. – No.2. - S.9–23.

Novikov A.E., Novikov E.A. Vortex-sink dynamics // Phys. Rev. E.- 1996.- Vol. 54.- No. 4.- P. 3681– 3686.

Bogomolov V.A. Dvizhenie ideal'noj zhidkosti postojannoj plotnosti pri nalichii stokov // Izv. AN SSSR. Mehanika zhidkosti i gaza.- 1976.- No. 4.- S. 21–27.

Borisov A.V., Mamaev I. S. On the problem of motion vortex sources on a plane // Regul. Chaotic Dyn.- 2006.- Vol. 11.- No.4.-P. 455–466.

Borisov A.V., Mamaev I.S. Matematicheskie metody dinamiki vihrevyh struktur.- Moskva – Izhevsk: Institut komp'juternyh issledovanij, 2005.- 368 s.

Bizjaev I.A., Borisov A.V., Mamaev I.S. Dinamika treh vihreistochnikov // Nelinejnaja dinamika.- 2014.- T. 10.- No 3.- S.319–327.

Noguchi T., Yukimoto S., Kimura R., Niino H. Structure and instability of a sink vortex // Proc. PSFVIP-4 (Chamonix, France).- 2003.-P.1–7.

Routh E.J. Some applications of conjugate functions // Proc. Lond. Math. Soc.- 1991.- Vol.12.- No.170/171. - P. 73–89.

Patterman S. Gidrodinamika sverhtekuchej zhidkosti.- M.: Mir, 1978. - 520 s.

Feynman R. P. Progress in Low Temperature Physics. v.1 (ed. C. J. Cortner), North-Holland, Amsterdam. - 1955.- P.17-53.

Halatnikov I. M. Teorija sverhtekuchesti.- M.: Nauka, 1971.- 320 s.

Volovik G.E., Mineev V.P. Issledovanie osobennostej v sverhtekuchem Ne3 i zhidkih kristallah metodami gomotopicheskoj topologii // ZhETF. - 1977.- T. 72. - Vyp. 6.- S.2256-2259.

Fejnman R. Satisticheskaja mehanika.- M.: Mir, 1975.- 407s.

Abrikosov A.A. Sverhprovodniki vtorogo roda i vihrevaja reshetka // UFN.- 2004.- T. 174.- Vyp. 11.- S.1234–1239.

Gor'kov L.P., Kopnin N.B. Dvizhenie vihrej i elektrosoprotivlenie sverhprovodnikov vtorogo roda v magnitnom pole // UFN. - 1975.- T.116.- S.413-448.

Tinkham M. Vvedenie v sverhprovodimost'. - M.: Mir, 1980.- 312s.

Abrikocov A.A. Osnovy teorii metallov.- M.: Fizmatlit, 1987.- 520s.

Vinen W.F. The detection of a single quantum of circulation in liquid helium II // Proc. Roy. Soc.- 1961.- A260.-P. 218-236.

Rayfield G.W., Reif F. Evidence for The Creation and Motion of Quantized Vortex Rings in Superfluid Helium // Phys. Rev. Lett.- 1963. - Vol.11. - P.305-308.

Rayfield G.W., Reif F. Quantized Vortex Rings in Superfluid Helium // Phys. Rev. - 1964. -Vol. 136. - P. A1194–A1208.

Tur A.V., Yanovsky V.V. Point vortices in 2D-plasma hydrodynamics // Physics of Plasmas. - 2010. - Vol.17. - No.11.- P.112308.

Novikov E.A. Obobshhennaja dinamika trehmernyh vihrevyh osobennostej (vortonov) // ZhETF. - 1983. - T.84. - Vyp.3. - S.975 981.

Safman P.G., Meiron D.I. Difficulties with three-dimensional weak so- solutions for inviscid incompressible flow // Phys. Fluids.- 1986. - Vol. 29(8). - P.2373-2375.

Chefranov S.G. Dinamika tochechnyh vihrevyh dipolej i spontannye singuljarnosti v trehmernyh turbulentnyh potokah // ZhETF. - 1987. - Vyp. 1. - S.151-158.

Tur A.V., Kulik K.N., Yanovsky V.V. Vortex structures with complex points singularities in the two-dimensional Euler equation. New exact solutions // Physica D. - 2011. - Vol.240. - P.1069-1079.

Hartman P. Ordinary differetial equations. - N.Y.:J.Wiley Sons, 1964. – 314p.

Nemitsky V.V., Stepanov V.V. Qualitative theory of differential equations. - N.J. : Princeton, 1960. – 523p.

Tur A.V., Yanovsky V.V. Point vortices with a rational necklace: New exact stationary solutions of the two-dimensional Euler equation // Physycs of Fluids. - 2004. - Vol.16. - No.8. - P.2877-2885.

Crowdy G.D. A class of exact multipolar vortices // Phys.Fluids. - 1999. - Vol.11(9). - P.2556-2564.

Crowdy D.G. The construction of exact multipolar equilibria of the two-dimensional Euler equations // Phys.Fluids. - 2002. - Vol.14(1). - P.257.

Crowdy D. Polygonal N-vortex arraya: Stuart model // Phys.Fluids. - 2003. - Vol.15(12). - P.3710-3717.

Newton P.K. The N-Vortex Problem: Analytical Techniques. - New-York: Springer-Verlag, 2001.-420p.

Lighthill M.J. On sound generated aerodynamically. I. General theory // Proc. Roy. Soc., ser. A. - 1952. - Vol.211. - No.1107.- P.564-587.

Lighthill M.J. On sound generated aerodynamically, II. Turbulence as a source of sound // Proc. Roy. Soc., ser. A. - 1954. - Vol.222. - No.1148. - P.1-32.

Gonchar V.Ju., Ostapchuk P.N., Tur A.V., Janovskij V.V. Dinamika i stohastichnost' v obratimoj sisteme, opisyvajushhej vzaimodejstvie dvuh tochechnyh vihrej v pole potencial'noj volny // Pr.1550, Institut Kosmicheskih issledovanij. - 1989. - 70s.

Gonchar V.Yu., Ostapchuk P.N., Tur A.V., Yanovsky V.V. Dynamics and stochasticity in reversible system describing interaction of point vortices with a potential wave // Phys. Lett. A. - 1991. - Vol.152. - No.5,6. - P.287-292.

Tur A.V., Yanovsky V.V. Invariants in Dissipationless hydrodynamics media // J.Fluid.Mech. - 1993. - Vol.248. - P.67-106.

Volkov D.V., Tur A.V., Yanovsky V.V. Hidden supersymmetry of classical systems (hydrodynamics and conservatijn laws) // Phys.Lett. A. - 1995. - Vol.203. - P.357-361.

Sagdeev R.Z., Moiseev S.S., Tur A.V., Yanovsky V.V. Problem of the strong turbulence and topological solitons. «Nonlinear Phenomena in plasma and Hydrodynamics». - M.: Mir, 1986. - P.137-182.

Zaharov V.E. Ob algebre integralov dvizhenija dvumernoj gidrodinamiki v peremennyh Klebsha // Funkc. analiz i ego pril. - 1989. – T.23. – Vyp.3. - S.24–31.

Yates J.E. Interaction with and production of sound by vortex flows // AIAA Paper 77-1352. - 1977.

Kulik K.N., Tur A.V., Janovskij V.V. Evoljucija vihrja u tverdoj stenki pod vozdejstviem potencial'noj volny // UFZh.- 2006. - T.51. - No.10. - S.1008-1017.

Arnold V.I., Sevryuk M.B. Oscillations and bifurcations in reversible systems, in Nonlinear Phenomena in Plasma Physics and Hydrodynamics. - M.: Mir Publishers, 1986. - P.31-64.

Sevryuk M.B. Reversible systems, Lect.Notes in Math., vol.1211. - Berlin: Springer – Verlag, 1986. - 319p.

Politi A., Oppo G.L., Badii R. Coexistence of conservative and dissipative behaviour in reversible dynamical systems // Phys. Rev. A - 1986. -Vol.33. - P.4055-4060.

Gonchar V.Y., Svirinovskaya E.Y., Tur A.V., Yanovsky V.V. The interection beetwin resonances and attractor in a reversible dynamic system // Phys. Lett. A. - 1993. - Vol.174. - P.241-246.

Gonchar V.Ju., Tur A.V., Janovskij V.V., Bifurkacii haoticheskih rezhimov obratimyh otobrazhenij pri vzaimodejstvii rezonansov i attraktorov // Zhurnal Elektromagnitnye javlenija. - 1998. - No.1. - S.3-20.

Gonchar V.Ju., Ostapchuk P.N., Tur A.V., Janovskij V.V. Evoljucija tochechnyh vihrej v pole shirokogo paketa voln / Preprint HFTI, HFTI 95-5. - 1995. - 17s.

Vetchanin E.V., Kazakov A.O. Bifurkacii i haos v zadache o dvizhenii dvuh tochechnyh vihrej v akusticheskoj volne // Nelinejnaja dinamika. - 2014. - T. 10. - No. 3. - S.329–343.

Published
2015-06-13
Cited
0 article
How to Cite
Yanovsky, V., Tur, A., & Kulik, K. (2015). LOCALIZED VORTICES IN TWO-DIMENSIONAL HYDRODYNAMIC MEDIA. East European Journal of Physics, 2(1), 4-14. https://doi.org/10.26565/2312-4334-2015-1-01