Skip to main content
×
Home
    • Aa
    • Aa

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

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

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.

Copyright
Corresponding author
*Corresponding author. Email addresses: djhan@iu.edu (D. Han), houyifeng1005@hotmail.com (Y. Hou) temam@indiana.edu (R. Temam)
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] R. Alexandre , Y.-G. Wang , C.-J. Xu , and T. Yang . Well-posedness of the Prandtl equation in Sobolev spaces. J. Amer. Math. Soc., 28(3):745784, 2015.

[2] Charles J. Amick . Existence of solutions to the nonhomogeneous steady Navier-Stokes equations. Indiana Univ. Math. J., 33(6):817830, 1984.

[3] R. Brown , I. Mitrea , M. Mitrea , and M. Wright . Mixed boundary value problems for the Stokes system. Trans. Amer. Math. Soc., 362(3):12111230, 2010.

[6] E. B. Fabes , C. E. Kenig , and G. C. Verchota . The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J., 57(3):769793, 1988.

[7] David Gérard-Varet and Emmanuel Dormy . On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc., 23(2):591609, 2010.

[8] M. A. Gol'dshtik . Viscous-flow paradoxes. In Annual review of fluid mechanics, Vol. 22, pages 441472. Annual Reviews, Palo Alto, CA, 1990.

[9] M.A. Gol'dshtik . A paradoxical solution of the Navier-Stokes equations. Journal of Applied Mathematics and Mechanics, 24(4):913929, 1960.

[11] Makram Hamouda , Chang-Yeol Jung , and Roger Temam . Boundary layers for the 3D primitive equations in a cube: the supercritical modes. Nonlinear Anal., 132:288317, 2016.

[12] F. Hecht . New development in FreeFem++. J. Numer. Math., 20(3-4):251265, 2012.

[14] E. Hopf . Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr., 4:213231, 1951.

[15] Mikhail V. Korobkov , Konstantin Pileckas , and Remigio Russo . On the flux problem in the theory of steady Navier-Stokes equations with nonhomogeneous boundary conditions. Arch. Ration. Mech. Anal., 207(1):185213, 2013.

[18] Loredana Lanzani , Luca Capogna , and Russell M. Brown . The mixed problem in L p for some two-dimensional Lipschitz domains. Mathematische Annalen, 342(1):91124, 2008.

[19] Vladimir Maz'ya and Tatyana Shaposhnikova . Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains. Bull. Math. Sci., 1(1):3377, 2011.

[20] Marius Mitrea and Michael Taylor . Navier-Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Ann., 321(4):955987, 2001.

[23] Marco Sammartino and Russel E. Caflisch . 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] J. Serrin . The swirling vortex. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 271(1214):325360, 1972.

[25] Zhong Wei Shen . A note on the Dirichlet problem for the Stokes system in Lipschitz domains. Proc. Amer. Math. Soc., 123(3):801811, 1995.

[26] Shagi-Di Shih and R. Bruce Kellogg . Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal., 18(5):14671511, 1987.

[27] J. L. Taylor , K. A. Ott , and R. M. Brown . The mixed problem in Lipschitz domains with general decompositions of the boundary. Trans. Amer. Math. Soc., 365(6):28952930, 2013.

[29] R. Jeffrey Trapp . A clarification of vortex breakdown and tornadogenesis. Monthly Weather Review, 128(3):888895, 3 2000.

[30] Zhouping Xin and Liqun Zhang . On the global existence of solutions to the Prandtl's system. Adv. Math., 181(1):88133, 2004.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 7 *
Loading metrics...

Abstract views

Total abstract views: 18 *
Loading metrics...

* Views captured on Cambridge Core between 6th July 2017 - 21st July 2017. This data will be updated every 24 hours.