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Interaction of a Vortex Induced by a Rotating Cylinder with a Plane

  • Daozhi Han (a1), Yifeng Hou (a1) and Roger Temam (a1)

In this article, we study theoretically and numerically the interaction of a vortex induced by a rotating cylinder with a perpendicular plane. We show the existence of weak solutions to the swirling vortex models by using the Hopf extension method, and by an elegant contradiction argument, respectively. We demonstrate numerically that the model could produce phenomena of swirling vortex including boundary layer pumping and two-celled vortex that are observed in potential line vortex interacting with a plane and in a tornado.

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*Corresponding author. Email addresses: (D. Han), (Y. Hou) (R. Temam)
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[1] Alexandre R., Wang Y.-G., Xu C.-J., and Yang T.. Well-posedness of the Prandtl equation in Sobolev spaces. J. Amer. Math. Soc., 28(3):745784, 2015.
[2] Amick Charles J.. Existence of solutions to the nonhomogeneous steady Navier-Stokes equations. Indiana Univ. Math. J., 33(6):817830, 1984.
[3] Brown R., Mitrea I., Mitrea M., and Wright M.. Mixed boundary value problems for the Stokes system. Trans. Amer. Math. Soc., 362(3):12111230, 2010.
[4] Bělík Pavel, Dokken Douglas P., Scholz Kurt, and Shvartsman Mikhail M.. Fractal powers in serrin's swirling vortex solutions. Asymptotic Analysis, 90(1-2):105132, 2014.
[5] Weinan E and Engquist Bjorn. Blowup of solutions of the unsteady Prandtl's equation. Comm. Pure Appl. Math., 50(12):12871293, 1997.
[6] Fabes E. B., Kenig C. E., and Verchota G. C.. The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J., 57(3):769793, 1988.
[7] Gérard-Varet David and Dormy Emmanuel. On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc., 23(2):591609, 2010.
[8] Gol'dshtik M. A.. Viscous-flow paradoxes. In Annual review of fluid mechanics, Vol. 22, pages 441472. Annual Reviews, Palo Alto, CA, 1990.
[9] Gol'dshtik M.A.. A paradoxical solution of the Navier-Stokes equations. Journal of Applied Mathematics and Mechanics, 24(4):913929, 1960.
[10] Hamouda Makram, Han Daozhi, Jung Chang-Yeol, and Temam Roger. Boundary layers for the 3D primitive equations in a cube: the subcritical modes. Submitted, 2016.
[11] Hamouda Makram, Jung Chang-Yeol, and Temam Roger. Boundary layers for the 3D primitive equations in a cube: the supercritical modes. Nonlinear Anal., 132:288317, 2016.
[12] Hecht F.. New development in FreeFem++. J. Numer. Math., 20(3-4):251265, 2012.
[13] Holton J.R.. An Introduction to Dynamic Meteorology. Number v. 1 in An Introduction to Dynamic Meteorology. Elsevier Academic Press, 2004.
[14] Hopf E.. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr., 4:213231, 1951.
[15] Korobkov Mikhail V., Pileckas Konstantin, and Russo Remigio. On the flux problem in the theory of steady Navier-Stokes equations with nonhomogeneous boundary conditions. Arch. Ration. Mech. Anal., 207(1):185213, 2013.
[16] Korobkov Mikhail V., Pileckas Konstantin, and Russo Remigio. Solution of Leray's problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains. Ann. of Math. (2), 181(2):769807, 2015.
[17] Ladyzhenskaya O. A.. The mathematical theory of viscous incompressible flow. Second English edition, revised and enlarged. Translated from the Russian by Silverman Richard A. and Chu John. Mathematics and its Applications, Vol. 2. Gordon and Breach Science Publishers, New York, 1969.
[18] Lanzani Loredana, Capogna Luca, and Brown Russell M.. The mixed problem in L p for some two-dimensional Lipschitz domains. Mathematische Annalen, 342(1):91124, 2008.
[19] Maz'ya Vladimir and Shaposhnikova Tatyana. Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains. Bull. Math. Sci., 1(1):3377, 2011.
[20] Mitrea Marius and Taylor Michael. Navier-Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Ann., 321(4):955987, 2001.
[21] Oleinik O. A. and Samokhin V. N.. Mathematical models in boundary layer theory, volume 15 of Applied Mathematics and Mathematical Computation. Chapman & Hall/CRC, Boca Raton, FL, 1999.
[22] Rotunno Richard. The fluid dynamics of tornadoes. In Annual review of fluid mechanics. Volume 45, 2013, volume 45 of Annu. Rev. Fluid Mech., pages 5984. Annual Reviews, Palo Alto, CA, 2013.
[23] Sammartino Marco and Caflisch Russel E.. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution. Comm. Math. Phys., 192(2):463491, 1998.
[24] Serrin J.. The swirling vortex. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 271(1214):325360, 1972.
[25] Shen Zhong Wei. A note on the Dirichlet problem for the Stokes system in Lipschitz domains. Proc. Amer. Math. Soc., 123(3):801811, 1995.
[26] Shih Shagi-Di and Kellogg R. Bruce. Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal., 18(5):14671511, 1987.
[27] Taylor J. L., Ott K. A., and Brown R. M.. The mixed problem in Lipschitz domains with general decompositions of the boundary. Trans. Amer. Math. Soc., 365(6):28952930, 2013.
[28] Temam R.. Navier-Stokes Equations: Theory and Numerical Analysis. AMS/Chelsea publication. AMS Chelsea Pub., 2001.
[29] Trapp R. Jeffrey. A clarification of vortex breakdown and tornadogenesis. Monthly Weather Review, 128(3):888895, 3 2000.
[30] Xin Zhouping and Zhang Liqun. On the global existence of solutions to the Prandtl's system. Adv. Math., 181(1):88133, 2004.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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