Skip to main content
SpringerLink
Log in

Stability and capillary dynamics of circular vortex sheets

  • Original Article
  • Published:
Theoretical and Computational Fluid Dynamics Aims and scope Submit manuscript

Abstract

We investigate the motion of circular vortex sheets with surface tension. A linear stability analysis shows that high modes of the circular vortex sheet are stabilized by surface tension, and the sheet is stable if surface tension is larger than a critical value. The modes of perturbations, n = 1 and 2, are always stable, regardless of surface tension, and the mode n = 3 is also stable for large surface tension. The numerical results show that a stable vortex sheet rotates and oscillates weakly. The oscillations of each mode of the interface mainly consist of two travelling waves of different frequencies in time. The amplitude and the period of the oscillation depend on the mode of the perturbation and surface tension. We also perform long-time computations for the unstable evolution of circular sheets. For a high Weber number, ripples are produced on the sheets, as well as pinching and self-intersection. It is found that the appearance of ripples is associated with the growth of noise. For an intermediate Weber number, the sheet evolves to an exotic structure with small spikes on the fingers, while for a low Weber number, it is nonlinearly stable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Perlin M., Schultz W.W.: Capillary effects on surface waves. Annu. Rev. Fluid Mech. 32, 241–274 (2000)

    Article  MathSciNet  Google Scholar 

  2. Stone H.A.: Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26, 65–102 (1994)

    Article  Google Scholar 

  3. Hou T.Y., Lowengrub J.S., Shelley M.J.: The long-time motion of vortex sheets with surface tension. Phys. Fluids 9, 1933–1954 (1997)

    Article  MathSciNet  Google Scholar 

  4. Birkhoff, G.: Helmholtz and Taylor instability. Proceedings of Symposia in Applied Mathematics, vol. XIII. pp. 55–76. American Mathematical Society, Providence (1962)

  5. Moore D.W.: The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. Lond. Ser. A 365, 105–119 (1979)

    Article  Google Scholar 

  6. Krasny R.: A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167, 65–93 (1986)

    Article  MathSciNet  Google Scholar 

  7. Cowley S.J., Baker G.R., Tanveer S.: On the formation of Moore curvature singularities in vortex sheets. J. Fluid Mech. 378, 233–267 (1999)

    Article  MathSciNet  Google Scholar 

  8. Sohn S.-I.: Singularity formation and nonlinear evolution of a viscous vortex sheet model. Phys. Fluids 25, 014106 (2013)

    Article  Google Scholar 

  9. Krasny R.: Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65, 292–313 (1986)

    Article  Google Scholar 

  10. Sohn S.-I., Yoon D., Hwang W.: Long-time simulations of the Kelvin–Helmholtz instability using an adaptive vortex method. Phys. Rev. E 82, 046711 (2010)

    Article  Google Scholar 

  11. Siegel M.: A study of singualarity formation in the Kelvin–Helmholtz instability with surface tension. SIAM J. Appl. Math. 55, 865–891 (1995)

    Article  MathSciNet  Google Scholar 

  12. Ambrose D.W.: Well-posedness of vortex sheets with surface tension. SIAM J. Math. Anal. 35, 211–244 (2003)

    Article  MathSciNet  Google Scholar 

  13. Hoz F., Fontelos M.A., Vega L.: The effect of surface tension on the Moore singularity of vortex sheet dynamics. J. Nonlinear Sci. 18, 463–484 (2008)

    Article  MathSciNet  Google Scholar 

  14. Fontelos M.A., Hoz F.: Singularities in water waves and the Rayleigh–Taylor problem. J. Fluid Mech. 651, 211–239 (2010)

    Article  MathSciNet  Google Scholar 

  15. Chomaz J.M., Rabaud M., Basdevant C., Couder Y.: Experimental and numerical investigation of a forced circular shear layer. J. Fluid Mech. 187, 115–140 (1988)

    Article  Google Scholar 

  16. Poncet S., Chauve M.P.: Shear-layer instability in a rotating system. J. Flow Vis. Image Proc. 14, 85–105 (2007)

    Article  Google Scholar 

  17. Humphrey J.A.C., Schuler C.A., Webster D.R.: Unsteady laminar flow between a pair of disks corotathg in a fixed cylindrical enclosure. Phys. Fluids 7, 1225–1240 (1995)

    Article  Google Scholar 

  18. Moore D.W., Griffith-Jones R.: The stability of an expanding circular vortex sheet. Mathematika 23, 128–133 (1976)

    Article  Google Scholar 

  19. Love A.E.H.: On the stability of certain vortex motion. Proc. Lond. Math. Soc. 25, 18–43 (1893)

    Article  MathSciNet  Google Scholar 

  20. Wan Y.H.: The stability of rotating vortex patches. Commun. Math. Phys. 107, 1–20 (1986)

    Article  Google Scholar 

  21. Mitchell T.B., Rossi L.F.: The evolution of Kirchhoff elliptic vortices. Phys. Fluids 20, 054103 (2008)

    Article  Google Scholar 

  22. Moore D.W.: On the point vortex method. SIAM J. Sci. Stat. Comput. 2, 65–84 (1981)

    Article  Google Scholar 

  23. Sohn S.I.: Numerical simulations of circular vortex sheets by using two regularization models. J. Korean Phys. Soc. 63, 1583–1590 (2013)

    Article  Google Scholar 

  24. Pullin D.I.: Numerical studies of surface-tension effects in nonlinear Kelvin–Helmholtz and Rayleigh–Taylor instabilites. J. Fluid Mech. 119, 507–532 (1982)

    Article  MathSciNet  Google Scholar 

  25. Hou T.Y., Lowengrub J.S., Shelley M.J.: Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114, 312–338 (1994)

    Article  MathSciNet  Google Scholar 

  26. Baker G., Nachbin A.: Stable methods for vortex sheet motion in the presence of surface tension. SIAM J. Sci. Comput. 19, 1737–1766 (1998)

    Article  MathSciNet  Google Scholar 

  27. Shin S., Sohn S.I., Hwang W.: Simple and efficient numerical methods for vortex sheet motion with surface tension. Int. J. Numer. Methods Fluids 74, 422–438 (2014)

    Article  MathSciNet  Google Scholar 

  28. Vooren A.I.: A numerical investigation of the rolling up of vortex sheets. Proc. R. Soc. Lond. Ser. A 373, 67–91 (1980)

    Article  Google Scholar 

  29. Hou T.Y., Li R.: Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226, 379–397 (2007)

    Article  MathSciNet  Google Scholar 

  30. Ceniceros H., Hou T.Y.: Dynamic generation of capillary waves. Phys. Fluids 11, 1042–1050 (1999)

    Article  MathSciNet  Google Scholar 

  31. Duncan J.H., Qiao H., Philomin V., Wenz A.: Gentle spilling breakers: crest profile evolution. J. Fluid Mech. 379, 191–222 (1999)

    Article  Google Scholar 

  32. Meiburg E., Homsy G.M.: Nonlinear unstable viscous fingers in Hele-Shaw flows. II. Numerical simulation. Phys. Fluids 31, 429–439 (1988)

    Article  Google Scholar 

  33. Dai W.S., Shelley M.J.: A numerical study of the effect of surface tension and noise of an expanding Hele-Shaw bubble. Phys. Fluids A 5, 2131–2146 (1993)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Gangneung-Wonju National University, Gangneung, 210-702, Korea

    Sung-Ik Sohn

Authors
  1. Sung-Ik Sohn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sung-Ik Sohn.

Additional information

Communicated by Peter Duck.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sohn, SI. Stability and capillary dynamics of circular vortex sheets. Theor. Comput. Fluid Dyn. 29, 291–310 (2015). https://doi.org/10.1007/s00162-015-0354-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-015-0354-9

Keywords

Search

Navigation