Advertisement

Journal of Nonlinear Science

, Volume 10, Issue 1, pp 23–47 | Cite as

On the Evolution of an Angle in a Vortex Patch

  • J.A. Carrillo
  • S. Soler
Article

Summary.

The inviscid incompressible two-dimensional motion of some initially convex singular vortex patches is examined. The angles evolution of a tangent-slope discontinuity on a singular contour is studied from a numerical and theoretical point of view. Different numerical examples show that the angle shrinks for initial angle less than 90o, and the angle widens when the initial angle is greater than 90o or is “approximately” preserved for initial angle 90o for small time evolution. An asymptotic expansion of the initial velocity field near a singularity for a class of singular vortex patches is performed to reinforce this result analytically. Some initially nonconvex singular patches in which the evolution does not follow this rule are shown.

Keywords

Euler Equation Lipschitz Function Initial Angle Initial Contour Vortex Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Baker, G. R., and Shelley, M. J., On the connection between thin vortex layers and vortex sheets, J. Fluid Mech., 215 (1990), 161–194.Google Scholar
  2. [2]
    Berk, H. L., and Roberts, K. V., The water-bag model, Method Comput. Phys. (B. Alder, S. Fernbach, and M. Rotenberg, Eds.), 9, 88–97, Academic Press, New York, 1970.Google Scholar
  3. [3]
    Bertozzi, A. L., and Constantin, P., Global regularity for vortex patches, Commun. Math. Phys., 152 (1993), 19–28.Google Scholar
  4. [4]
    Buttke, T. F., The observation of singularities in the boundary of patches of constant vorticity, Phys. Fluids A, 1 (1989), 1283–1285.Google Scholar
  5. [5]
    Chemin, J. Y., Persistence de structures geometriques dans les fluides incompressibles bidimensionels, Ann. Sci. Ec. Norm. Sup., 26 (1993), 1–26.CrossRefGoogle Scholar
  6. [6]
    Chemin, J. Y., Fluides Parfaits Incompressibles, Astérisque 230 (1995), 1–177.MathSciNetzbMATHGoogle Scholar
  7. [7]
    Constantin, P., and Titi, E., On the evolution of nearly circular vortex patches, Commun. Math. Phys., 119 (1988), 177–198.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Deem, G. S., and Zabusky, N. J., Vortexwaves: Stationary “V-states,” interactions, recurrence and breaking, Phys. Rev. Lett., 40 (1978), 859–862.CrossRefGoogle Scholar
  9. [9]
    Dritschel, D. G., The repeated filamentation of two-dimensional vorticity interfaces. J. Fluid Mech., 194 (1988), 511–547.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Dritschel, D. G., and McIntyre, M. E., Does contour dynamics go singular? Phys. Fluids A, 2 (1990), 748–753.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Leonard, A., Vortex methods for flow simulation, J. Comp. Phys., 37 (1980), 289–335.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Lopes Filho, M. C., and Nussenzveig Lopes, H. J., An extension of C. Marchioros bound on the growth of a vortex patch to flows with L p vorticity, SIAM J. Math. Anal., 29 (1998), 596–599.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Majda, A., Vorticity and the mathematical theory of incompressible fluid flow, Commun. Pure Appl. Math., 39 (1986), 5187–5220.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Marchioro, C., Bounds on the growth of the support of a vortex patch, Comun. Math. Phys., 164 (1994]), 507–524MathSciNetCrossRefGoogle Scholar
  15. [15]
    Marchioro, C., and Pulvirenti, M., The Mathematical Theory of Incompressible Nonviscous Fluids, Springer Series in Applied Mathematical Sciences 96, Springer-Verlag, New York, 1994.Google Scholar
  16. [16]
    Meyer, K.R., Counterexamples in dynamical systems via normal form theory, SIAM Rev., 28 (1986), 41–51.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Overman, E. A., Steady-state solutions for the Euler equation in two dimensions II. Local analysis of the the limiting V-states, SIAM J. Appl. Math., 46 (1986), 765–800.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Pullin, D., and Moore, D. W., Remark on a result of D. Dritschel, Phys. Fluids A, 2 (1990), 1039–1041.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Pullin, D. I., Contour dynamics methods. Ann. Rev. Fluid Mech., 24 (1992), 89–115.MathSciNetCrossRefGoogle Scholar
  20. [20]
    Roberts, K. V., and Christiansen, J. P., Topics in computational fluid mechanics, Comput. Phys. Commun. Suppl., 3 (1972), 14–32.CrossRefGoogle Scholar
  21. [21]
    Saffman, P. G., Vortex Dynamics, Cambridge monograph on mechanics and applied mathematics, Cambridge University Press, New York, 1992.Google Scholar
  22. [22]
    Schochet, S., The point-vortex method for periodic weak solutions of the 2-D Euler equations, Commun. Pure Appl. Math., 49 (1996), 911–965.MathSciNetCrossRefGoogle Scholar
  23. [23]
    Shub, M., Global Stability and Dynamical Systems, Springer-Verlag, New York, 1987.CrossRefGoogle Scholar
  24. [24]
    Soler, J., Convergence of the contour dynamics method, Num. Meth.PDE, 7 (1991), 261–276.MathSciNetCrossRefGoogle Scholar
  25. [25]
    Wan, Y. H., and Pulvirenti, M., Nonlinear stability of circular vortex patches, Commun. Math. Phys., 99 (1985), 435–450.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Welington de Melo, J. P.., Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York, 1982.zbMATHGoogle Scholar
  27. [27]
    Yudovitch, V. I., Non-stationary flow of an ideal incompressible liquid, Zh. Vych. Mat., 3 (1963), 1032–1066.Google Scholar
  28. [28]
    Zabusky, N. J., Hughes, M. H., and Roberts, K. V., Contour dynamics for the Euler equations in two dimensions, J. Comp. Phys., 30 (1979), 96–106.MathSciNetCrossRefGoogle Scholar
  29. [29]
    Zou, Q., Overman, E. A., Wu, H. M., and Zabusky, N. J., Contour dynamics for the Euler equations: Curvature controlled initial node placement and accuracy, J. Comp. Phys., 78 (1988), 350–368.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 2000

Authors and Affiliations

  • J.A. Carrillo
    • 1
  • S. Soler
    • 1
  1. 1.Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, SpainSpain

Personalised recommendations