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

Part of: Rotating fluids Incompressible viscous fluids

Published online by Cambridge University Press:  06 July 2017

Daozhi Han ,
Yifeng Hou  and
Roger Temam
Daozhi Han*
Affiliation:
Department of Mathematics & Institute for Scientific Computing and Applied Mathematics, Indiana University at Bloomington, 47405, USA
Yifeng Hou*
Affiliation:
Department of Mathematics & Institute for Scientific Computing and Applied Mathematics, Indiana University at Bloomington, 47405, USA
Roger Temam*
Affiliation:
Department of Mathematics & Institute for Scientific Computing and Applied Mathematics, Indiana University at Bloomington, 47405, USA
*
*Corresponding author. Email addresses:djhan@iu.edu (D. Han), houyifeng1005@hotmail.com (Y. Hou) temam@indiana.edu (R. Temam)
*Corresponding author. Email addresses:djhan@iu.edu (D. Han), houyifeng1005@hotmail.com (Y. Hou) temam@indiana.edu (R. Temam)
*Corresponding author. Email addresses:djhan@iu.edu (D. Han), houyifeng1005@hotmail.com (Y. Hou) temam@indiana.edu (R. Temam)
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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.

Keywords

Tornadoswirling vortexNavier-Stokesaxisymmetric flowboundary layer

MSC classification

Secondary: 76U05: Rotating fluids 76D03: Existence, uniqueness, and regularity theory 76D17: Viscous vortex flows 65Z05: Applications to physics
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017 , pp. 599 - 619
Copyright
Copyright © Global-Science Press 2017 

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References

[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):745–784, 2015.Google Scholar
[2] Amick, Charles J.. Existence of solutions to the nonhomogeneous steady Navier-Stokes equations. Indiana Univ. Math. J., 33(6):817–830, 1984.Google Scholar
[3] Brown, R., Mitrea, I., Mitrea, M., and Wright, M.. Mixed boundary value problems for the Stokes system. Trans. Amer. Math. Soc., 362(3):1211–1230, 2010.Google Scholar
[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):105–132, 2014.Google Scholar
[5] Weinan, E and Engquist, Bjorn. Blowup of solutions of the unsteady Prandtl's equation. Comm. Pure Appl. Math., 50(12):1287–1293, 1997.Google Scholar
[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):769–793, 1988.Google Scholar
[7] Gérard-Varet, David and Dormy, Emmanuel. On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc., 23(2):591–609, 2010.Google Scholar
[8] Gol'dshtik, M. A.. Viscous-flow paradoxes. In Annual review of fluid mechanics, Vol. 22, pages 441–472. Annual Reviews, Palo Alto, CA, 1990.Google Scholar
[9] Gol'dshtik, M.A.. A paradoxical solution of the Navier-Stokes equations. Journal of Applied Mathematics and Mechanics, 24(4):913–929, 1960.Google Scholar
[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.Google Scholar
[11] Hamouda, Makram, Jung, Chang-Yeol, and Temam, Roger. Boundary layers for the 3D primitive equations in a cube: the supercritical modes. Nonlinear Anal., 132:288–317, 2016.Google Scholar
[12] Hecht, F.. New development in FreeFem++. J. Numer. Math., 20(3-4):251–265, 2012.Google Scholar
[13] Holton, J.R.. An Introduction to Dynamic Meteorology. Number v. 1 in An Introduction to Dynamic Meteorology. Elsevier Academic Press, 2004.Google Scholar
[14] Hopf, E.. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr., 4:213–231, 1951.Google Scholar
[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):185–213, 2013.Google Scholar
[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):769–807, 2015.Google Scholar
[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.Google Scholar
[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):91–124, 2008.Google Scholar
[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):33–77, 2011.Google Scholar
[20] Mitrea, Marius and Taylor, Michael. Navier-Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Ann., 321(4):955–987, 2001.Google Scholar
[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.Google Scholar
[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 59–84. Annual Reviews, Palo Alto, CA, 2013.Google Scholar
[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):463–491, 1998.Google Scholar
[24] Serrin, J.. The swirling vortex. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 271(1214):325–360, 1972.Google Scholar
[25] Shen, Zhong Wei. A note on the Dirichlet problem for the Stokes system in Lipschitz domains. Proc. Amer. Math. Soc., 123(3):801–811, 1995.Google Scholar
[26] Shih, Shagi-Di and Kellogg, R. Bruce. Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal., 18(5):1467–1511, 1987.Google Scholar
[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):2895–2930, 2013.Google Scholar
[28] Temam, R.. Navier-Stokes Equations: Theory and Numerical Analysis. AMS/Chelsea publication. AMS Chelsea Pub., 2001.Google Scholar
[29] Trapp, R. Jeffrey. A clarification of vortex breakdown and tornadogenesis. Monthly Weather Review, 128(3):888–895, 3 2000.Google Scholar
[30] Xin, Zhouping and Zhang, Liqun. On the global existence of solutions to the Prandtl's system. Adv. Math., 181(1):88–133, 2004.Google Scholar