Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-09T21:40:52.660Z Has data issue: false hasContentIssue false
  • English
  • Français

Absence of singular stretching of interacting vortex filaments

Published online by Cambridge University Press:  10 August 2012

Sahand Hormoz  and
Michael P. Brenner
Sahand Hormoz*
Affiliation:
School of Engineering and Applied Sciences, and Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, MA 02138, USA
Michael P. Brenner
Affiliation:
School of Engineering and Applied Sciences, and Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, MA 02138, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A promising mechanism for generating a finite-time singularity in the incompressible Euler equations is the stretching of vortex filaments. Here, we argue that interacting vortex filaments cannot generate a singularity by analysing the asymptotic dynamics of their collapse. We use the separation of the dynamics of the filament shape, from that of its core, to derive constraints that must be satisfied for a singular solution to remain self-consistent uniformly in time. Our only assumption is that the length scales characterizing filament shape obey scaling laws set by the dimension of circulation as the singularity is approached. The core radius necessarily evolves on a different length scale. We show that a self-similar ansatz for the filament shapes cannot induce singular stretching, due to the logarithmic prefactor in the self-interaction term for the filaments. More generally, there is an antagonistic relationship between the stretching rate of the filaments and the requirement that the radius of curvature of filament shape obeys the dimensional scaling laws. This suggests that it is unlikely that solutions in which the core radii vanish sufficiently fast to maintain the filament approximation exist.

JFM classification

Mathematical Foundations: Mathematical Foundations Vortex Flows: Vortex dynamics Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 707 , 25 September 2012 , pp. 191 - 204
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Aref, H. 1979 Motion of three vortices. Phys. Fluids 22, 393–400.CrossRefGoogle Scholar
2. Beale, J., Kato, T. & Majda, A. 1984 Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66.CrossRefGoogle Scholar
3. Chae, D. 2007 Nonexistence of self-similar singularities for the 3d incompressible Euler equations. Commun. Math. Phys. 273, 203–215.CrossRefGoogle Scholar
4. Chae, D. 2010 On the generalized self-similar singularities for the Euler and the Navier–Stokes equations. J. Funct. Anal. 258, 2865–2883.CrossRefGoogle Scholar
5. Childress, S. 2008 Growth of anti-parallel vorticity in Euler flows. Physica D: Nonlinear Phenomena 237, 1921–1925.CrossRefGoogle Scholar
6. Constantin, P. 2008 Singular, weak and absent: solutions of the Euler equations. Physica D: Nonlinear Phenomena 237, 1926–1931.CrossRefGoogle Scholar
7. Constantin, P., Fefferman, C. & Majda, A. 1996 Geometric constraints on potentially singular solutions for the 3-D Euler equations. Commun. Part. Diff. Equ. 21, 559–571.Google Scholar
8. Deng, J., Hou, T. & Yu, X. 2006 Improved geometric conditions for non-blowup of the 3D incompressible Euler equation. Commun. Part. Diff. Equ. 31, 293–306.CrossRefGoogle Scholar
9. Eggers, J. & Fontelos, A. 2009 The role of self-similarity in singularities of partial differential equations. Nonlinearity 22, R1–R44.CrossRefGoogle Scholar
10. Gibbon, J. 2008 The three-dimensional Euler equations: Where do we stand? Physica D: Nonlinear Phenomena 237, 1894–1904.CrossRefGoogle Scholar
11. Goldstein, R. E. & Langer, S. 1995 Nonlinear dynamics of stiff polymers. Phys. Rev. Lett. 75, 1094–1097.CrossRefGoogle ScholarPubMed
12. Gutierrez, S., Rivas, J. & Vega, L. 2003 Formation of singularities and self-similar vortex motion under the localized induction approximation. Commun. Part. Diff. Equ. 28, 927–968.CrossRefGoogle Scholar
13. Hou, T. Y. & Li, R. 2006 Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations. J. Nonlinear Sci. 16, 639–664.CrossRefGoogle Scholar
14. Kerr, R. 1993 Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids A: Fluid Dynamics 5, 1725–1746.CrossRefGoogle Scholar
15. Kimura, Y. 2010 Self-similar collapse of 2D and 3D vortex filament models. Theor. Comput. Fluid Dyn. 24, 389–394.CrossRefGoogle Scholar
16. Klein, R. & Majda, A. 1993 An asymptotic theory for the nonlinear instability of antiparallel pairs of vortex filaments. Phys. Fluids A: Fluid Dynamics 5, 369–379.CrossRefGoogle Scholar
17. Klein, R., Majda, A. & Damodaran, K. 1995 Simplified equations for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288, 201–248.CrossRefGoogle Scholar
18. Leray, J. 1934 On the motion of a viscous liquid filling space. Acta Mathematica 63, 193–248.CrossRefGoogle Scholar
19. Majda, A. & Bertozzi, A. L. 2001 Vorticity and Incompressible Flow. Cambridge University Press.CrossRefGoogle Scholar
20. Meiron, D., Orszag, S., Nickel, B., Morf, R. & Frisch, U. 1983 Small-scale structure of the Taylor–Green vortex. J. Fluid Mech. 130, 411–452.Google Scholar
21. Moffatt, H. 2000 The interaction of skewed vortex pairs: a model for blow-up of the Navier–Stokes equations. J. Fluid Mech. 49, 51–68.CrossRefGoogle Scholar
22. Morf, R., Orszag, S. & Frisch, U. 1980 Spontaneous singularity in three-dimensional inviscid, incompressible flow. Phys. Rev. Lett. 44, 572–575.CrossRefGoogle Scholar
23. Nakayama, K., Segur, H. & Wadati, M. 1992 Integrability and the motion of curves. Phys. Rev. Lett. 69, 2603–2606.CrossRefGoogle ScholarPubMed
24. Paoletti, M. S., Fisher, M. E. & Lathrop, D. P. 2010 Reconnection dynamics for quantized vortices. Physica D: Nonlinear Phenomena 239, 1367–1377.CrossRefGoogle Scholar
25. Paoletti, M. S., Fisher, M. E., Sreenivasan, K. R. & Lathrop, D. P. 2008 Velocity statistics distinguish quantum turbulence from classical turbulence. Phys. Rev. Lett. 101, 154501.CrossRefGoogle ScholarPubMed
26. Pelz, R. 1997 Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. 55, 1617–1626.Google Scholar
27. Pomeau, Y. & Sciamarella, D. 2005 An unfinished tale of nonlinear PDEs: Do solutions of 3D incompressible Euler equations blow-up in finite time? Physica D: Nonlinear Phenomena 205, 215–221.CrossRefGoogle Scholar
28. Pumir, A., Shraiman, B. I. & Siggia, E. 1992 Vortex morphology and Kelvin’s theorem. Phys. Rev. 45, R5351–R5354.CrossRefGoogle ScholarPubMed
29. Pumir, A. & Siggia, E. 1987 Vortex dynamics and the existence of solutions to the Navier–Stokes equations. Phys. Fluids 30, 1606–1626.CrossRefGoogle Scholar
30. Pumir, A. & Siggia, E. 1990 Collapsing solutions to the 3-D Euler equations. Phys. Fluids A: Fluid Dynamics 2, 220–241.CrossRefGoogle Scholar
31. Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
32. Saffman, P. & Baker, G. 1979 Vortex interactions. Annu. Rev. Fluid Mech. 11, 95–122.CrossRefGoogle Scholar
33. Schwarz, K. W. 1985 Three-dimensional vortex dynamics in superfluid : line–line and line–boundary interactions. Phys. Rev. 31, 5782–5804.CrossRefGoogle Scholar
34. Siggia, E. 1985 Collapse and amplification of a vortex filament. Phys. Fluids 28, 794–805.CrossRefGoogle Scholar
35. Taylor, G. & Green, A. 1937 Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. 158, 499–521.Google Scholar
36. de Waele, A. & Aarts, R. 1994 Route to vortex reconnection. Phys. Rev. Lett. 72, 482–485.CrossRefGoogle Scholar
Supplementary material: PDF

Hormoz and Brenner

Supporting information

Download Hormoz and Brenner(PDF)
PDF 295.8 KB