- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Potential singularity mechanism for the Euler equations

Michael P. Brenner1, Sahand Hormoz2,3, and Alain Pumir4

  • 1School of Engineering and Applied Sciences and Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, Massachusetts 02138, USA
  • 2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, Santa Barbara, California 93106, USA
  • 3Division of Biology and Biological Engineering, California Institute of Technology, Pasadena, California 91125, USA
  • 4Université Lyon, Ecole Normale Supérieure de Lyon, Université Claude Bernard Lyon 1, and CNRS, 69007 Lyon, France

PDF

Phys. Rev. Fluids 1, 084503 – Published 28 December, 2016

DOI: https://doi.org/10.1103/PhysRevFluids.1.084503

Abstract

Singular solutions to the Euler equations could provide essential insight into the formation of very small scales in highly turbulent flows. Previous attempts to find singular flow structures have proven inconclusive. We reconsider the problem of interacting vortex tubes, for which it has long been observed that the flattening of the vortices inhibits sustained self-amplification of velocity gradients. Here we consider an iterative mechanism, based on the transformation of vortex filaments into sheets and their subsequent instability back into filaments. Elementary fluid mechanical arguments are provided to support the formation of a singular structure via this iterated mechanism, which we analyze based on a simplified model of filament interactions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (42)

  1. G. I. Taylor and A. E. Green, Mechanism of the production of small eddies from large ones, Proc. R. Soc. London Ser. A 158, 499 (1937).
  2. C. L. Fefferman, The Millennium Prize Problems (Clay Mathematics Institute, Cambridge, 2006), p. 57.
  3. J. Leray, Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math. 63, 193 (1934).
  4. L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure Appl. Math. 35, 771 (1982).
  5. J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys. 94, 61 (1984).
  6. P. Constantin, Singular, weak and absent: Solutions of the Euler equations, Physica D 237, 1926 (2008).
  7. D. Chae, Non-existence of self-similar singularities for the 3D incompressible Euler equations, Commun. Math. Phys. 273, 203 (2007).
  8. M. E. Brachet, D. I. Meiron, S. A. Orszag, B. G. Nickel, R. H. Morf, and U. Frisch, Small-scale structures on the Taylor-Green vortex, J. Fluid Mech. 130, 411 (1983).
  9. A. Pumir and R. M. Kerr, Numerical Simulation of Interacting Vortex Tubes, Phys. Rev. Lett. 58, 1636 (1987).
  10. A. Pumir and E. D. Siggia, Collapsing solutions of the 3-D Euler equations, Phys. Fluids A 2, 220 (1990).
  11. R. M. Kerr, Evidence for a singularity of the three-dimensional, incompressible euler equations, Phys. Fluids A 5, 1725 (1993).
  12. R. Pelz, Locally self-similar, finite-time collapse in a high-symmetry vortex filament model, Phys. Rev. E 55, 1617 (1997).
  13. Y. T. Hou and R. Li, Absence of singular stretching of interacting vortex filaments, J. Nonlinear Sci. 16, 639 (2006).
  14. G. Luo and T. Y. Hou, Potentially singular solutions of the 3D axisymmetric Euler equations, Proc. Natl. Acad. Sci. USA 111, 12968 (2014).
  15. T. Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation, arXiv:1402.0290.
  16. E. D. Siggia, Origin of intermittency in fully developed turbulence, Phys. Rev. A 15, 1730 (1977).
  17. E. D. Siggia, Collapse and amplification of a vortex filament, Phys. Fluids 28, 794 (1985).
  18. A. Pumir and E. D. Siggia, Vortex dynamics and the existence of solutions to the Navier-Stokes equations, Phys. Fluids 30, 1606 (1987).
  19. S. Childress, Growth of anti-parallel vorticity in euler flows, Physica D 237, 1921 (2008).
  20. S. Hormoz and M. P Brenner, Absence of singular stretching of interacting vortex filaments, J. Fluid Mech. 707, 191 (2012).
  21. A. Pumir, A numerical study of pressure fluctuations in three-dimensional, incompressible, homogeneous, isotropic turbulence, Phys. Fluids 6, 2071 (1994).
  22. T. S. Lundgren and W. T. Ashurst, Area-varying waves on curved vortex tubes with application to vortex breakdown, J. Fluid Mech. 200, 283 (1989).
  23. R. Klein, A. Majda, and K. Damodaran, Simplified equations for the interaction of nearly parallel vortex filaments, J. Fluid Mech. 288, 201 (1995).
  24. S. Kida, Motion of an elliptic vortex in a uniform shear flow, J. Phys. Soc. Jpn. 50, 3517 (1981).
  25. J. C. Neu, The dynamics of stretched vortices, J. Fluid Mech. 143, 253 (1984).
  26. J. C. Neu, The dynamics of a columnar vortex in an imposed strain, Phys. Fluids 27, 2397 (1984).
  27. L. Rayleigh, On the stability, or instability, of certain fluid motions, Proc. London Math. Soc. s1-11, 57 (1879).
  28. T. Passot, H. Politano, P. L. Sulem, J. R. Angilella, and M. Meneguzzi, Instability of strained vortex layers and vortex tube formation in homogeneous turbulence, J. Fluid Mech. 282, 313 (1995).
  29. A. T. A. M. de Waele and R. G. K. M. Aarts, Route to Vortex Reconnection, Phys. Rev. Lett. 72, 482 (1994).
  30. L. Boué, D. Khomenko, V. S. L'vov, and I. Procaccia, Analytic Solution of the Approach of Quantum Vortices Towards Reconnection, Phys. Rev. Lett. 111, 145302 (2013).
  31. A. Pumir, B. I. Shraiman, and E. D. Siggia, Vortex morphology and Kelvin's theorem, Phys. Rev. A 45, R5351 (1992).
  32. Y. Pomeau and D. Sciamarella, An unfinished tale of nonlinear PDEs: Do solutions of 3D incompressible euler equations blow-up in finite time? Physica D 205, 215 (2005).
  33. P. Cvitanović, Recurrent flows: The clockwork behind turbulence, J. Fluid Mech. 726, 1 (2013).
  34. W. T. Ashurst and D. I. Meiron, Numerical Study of Vortex Reconnection, Phys. Rev. Lett. 58, 1632 (1987).
  35. Y. Oshima, T. Kambe, and S. Asaka, Interaction of two vortex rings moving along a common axis of symmetry, J. Phys. Soc. Jpn. 38, 1159 (1975).
  36. T. Fohl and J. S. Turner, Colliding vortex rings, Phys. Fluids 18, 433 (1975).
  37. T. T. Lim and T. B. Nickels, Instability and reconnection in the head-on collision of vortex rings, Nature (London) 357, 225 (1992).
  38. D. Kleckner and W. T. M. Irvine, Creation and dynamics of knotted vortices, Nat. Phys. 9, 253 (2013).
  39. Y. Kimura and H. K. Moffatt, Reconnection of skewed vortices, J. Fluid Mech. 751, 329 (2014).
  40. P. G. Saffman, A model of vortex reconnection, J. Fluid Mech. 212, 395 (1990).
  41. M. J. Shelley, D. I. Meiron, and S. A. Orszag, Dynamical aspects of vortex reconnection of perturbed anti-parallel vortex tubes, J. Fluid Mech. 246, 613 (1993).
  42. A. Pumir and E. D. Siggia, Finite-Time Singularities in the Axisymmetric Three-Dimension Euler Equations, Phys. Rev. Lett. 68, 1511 (1992).

Outline

Information

Phys. Rev. Fluids 1, 084503– Published 28 December, 2016
  • Received 22 June 2016
Article part of CHORUS

©2016 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation