Abstract
A Hamiltonian version of contour dynamics is formulated for models of constant-vorticity plane flows with interfaces. The proposed approach is used as a framework for a nonlinear scenario for instability development. Localized vortex blobs are analyzed as structural elements of a strongly perturbed wall layer of a vorticity-carrying fluid with free boundary in gravity field. Gravity and vorticity effects on the geometry and velocity of vortex structures are examined. It is shown that compactly supported nonlinear solutions (compactons) are candidates for the role of particle-like vortex structures in models of flow breakdown. An analysis of the instability mechanism demonstrates the possibility of a self-similar collapse. It is found that the vortex shape stabilizes at the final stage of the collapse, while the vortex sheet strength on its boundary increases as (t 0 − t)−1, where t 0 is the collapse time.
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References
N. A. Inogamov, A. Yu. Dem’yanov, and É. E. Son, Hydrodynamics of Mixing (Mosk. Fiz.-Tekh. Inst., Moscow, 1999) [in Russian].
E. Fermi, in Scientific Works (Nauka, Moscow, 1972), Vol. 2, pp. 490–498 [in Russian].
N. J. Zabusky, M. H. Hughes, and K. V. Roberts, J. Comput. Phys. 30, 96 (1979).
D. G. Dritschel, J. Comput. Phys. 77, 240 (1988).
V. P. Goncharov, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 65 (1986).
V. P. Goncharov and V. I. Pavlov, Eur. J. Mech. B/Fluids 19, 831 (2000).
V. P. Goncharov and V. I. Pavlov, Zh. Éksp. Teor. Fiz. 119, 685 (2001) [JETP 92, 594 (2001)].
V. P. Goncharov and V. I. Pavlov, in Fundamental and Applied Problems in the Vortex Theory, Ed. by A. V. Borisov, I. S. Mamaev, and M. A. Sokolovskiĭ (Inst. Komp’yut. Issled., Moscow-Izhevsk, 2003), p. 179 [in Russian].
V. P. Goncharov and V. I. Pavlov, Pis’ma Zh. Éksp. Teor. Fiz. 84, 459 (2006) [JETP Lett. 84, 384 (2006)].
P. Rosenau, Am. Math. Soc. Not. 55, 738 (2005).
Y. A. Li and P. J. Olver, Discrete Contin. Dyn. Syst. 3, 419 (1997).
V. E. Zakharov and E. A. Kuznetsov, Zh. Éksp. Teor. Fiz. 91, 1310 (1986) [Sov. Phys. JETP 64, 773 (1986)].
E. A. Kuznetsov, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 56, 342 (2003).
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Original Russian Text © V.P. Goncharov, 2007, published in Zhurnal Éksperimental’noĭ i Teoreticheskoĭ Fiziki, 2007, Vol. 132, No. 5, pp. 1222–1234.
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Goncharov, V.P. A nonlinear scenario for development of vortex layer instability in gravity field. J. Exp. Theor. Phys. 105, 1075–1084 (2007). https://doi.org/10.1134/S1063776107110192
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DOI: https://doi.org/10.1134/S1063776107110192