Skip to main content
SpringerLink
Log in

Vorticity intensification and transition to turbulence in three-dimensional euler equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The evolution of a perturbed vortex tube is studied by means of a second-order projection method for the incompressible Euler equations. We observe, to the limits of grid resolution, a nonintegrable blowup in vorticity. The onset of the intensification is accompanied by a decay in the mean kinetic energy. Locally, the intensification is characterized by tightly curved regions of alternating-sign vorticity in a 2n-pole structure. After the firstL ∞ peak, the enstrophy and entropy continue to increase, and we observe reconnection events, continued decay of the mean kinetic energy, and the emergence of a Kolmogorov (k −5/3) range in the energy spectrum.

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.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Ashurst, W., Meiron, D.: Numerical study of vortex reconnection. Phys. Rev. Lett.58, 1632 (1987)

    Google Scholar 

  2. Batchelor, G.K., Townsend, A.A.: Decay of vorticity in isotropic turbulence. Proc. R. Soc. A191, 534–550 (1947)

    Google Scholar 

  3. Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys.94, 61–66 (1989)

    Google Scholar 

  4. Bell, J.B., Colella, P., Glaz, H.: A second-order projection method for the incompressible Navier-Stokes equations. J. Comp. Phys. 257–283 (1989)

  5. Bell, J.B., Solomon, J.M., Szymczak, W.G.: A second-order projection method for the three-dimensional Euler and Navier-Stokes equations. Preprint 1990

  6. Brachet, M., Meiron, D., Orszag, S., Nickel, B., Morf, R., Frisch, U.: Small-scale structure of the Taylor-Green vortex. J. Fluid Mech.130, 411 (1983)

    Google Scholar 

  7. Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comp.22, 745–762 (1968)

    Google Scholar 

  8. Chorin, A.J.: The evolution of a turbulent vortex. Commun. Math. Phys.83, 517–535 (1982)

    Google Scholar 

  9. Chorin, A.J.: Turbulence and vortex stretching on a lattice. Commun. Pure Appl. Math.39, S47-S65 (1986)

    Google Scholar 

  10. Chorin, A.J.: Hairpin removal in vortex interaction. LBL-26173, Lawrence Berkeley Laboratory Report, 1988

  11. Chorin, A.J., Akao, J.: Vortex equilibria in turbulence and quantum analogues. Physica D, 1991 (in press)

  12. Chorin, A.J.: Statistical mechanics and vortex motion. PAM-500, Center for Pure and Applied Mathematics Report, University of California, Berkeley 1990

    Google Scholar 

  13. Chorin, A.J.: Equilibrium statistics of a vortex filament with applications. LBL-30419, Lawrence Berkeley Laboratory Report, 1991

  14. Colella, P.: Multidimensional upwind methods for hyperbolic conservation laws. J. Comp. Phys.94, 61–66 (1990)

    Google Scholar 

  15. Falco, R.E.: Phys. Fluids20, S124 (1977)

    Google Scholar 

  16. de Gennes, P.G.: Scaling Concepts in Polymer Physics, Ithaca, NY: Cornell University Press 1971

    Google Scholar 

  17. Kerr, R.M., Hussain, F.: Simulation of vortex reconnection. Physica D37, 474–484 (1989)

    Google Scholar 

  18. Kim, J., Moin, P.: The structure of the vorticity field in turbulent channel flow. Part 2. Study of ensemble averaged fields. J. Fluid Mech.162, 339 (1986)

    Google Scholar 

  19. Moin, P., Leonard, A., Kim, J.: Evolution of a curved vortex filament into a vortex ring. Phys. Fluids29, 955–963 (1986)

    Google Scholar 

  20. Moin, P., Rogers, M.M., Moser, R.D.: Proceedings of the Fifth Symposium on Turbulent Shear Flows, Cornell University, Ithaca, NY, 1985

  21. Numerical simulations of turbulence. In: Rosenblatt, M., Van Atta, C. (eds.). Statistical Models of Turbulence, pp. 127–147. Berlin, Heidelberg, New York: Springer 1972

  22. Pumir, A., Kerr, R.M.: Numerical simulations of interacting vortex tubes. Phys. Rev. Lett.58, 1636 (1987)

    Google Scholar 

  23. Pumir, A., Siggia, E.D.: Vortex dynamics and the existence of solutions to the Navier-Stokes equations. Phys. Fluids30, 1606–1626 (1987)

    Google Scholar 

  24. Pumir, A., Siggia, E.D.: Collapsing solutions to the 3-D Euler equations. Phys. Fluids A2 (3), 220–241 (1990)

    Google Scholar 

  25. Shelley, M.J., Meiron, D.I.: Vortex reconnection and smoothness of the Euler equations. In: Anderson, C., Greengard, C. (eds.). Lectures in Applied Mathematics, AMS, 1991

  26. Siggia, E.D.: Collapse and amplification of a vortex filament. Phys. Fluids28, 794–805 (1985)

    Google Scholar 

  27. Zabusky, N.J., Melander, M.V.: Three-dimensional vortex tube reconnection: Morphology for orthogonally offset tubes. Physica D37, 555–562 (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lawrence Livermore National Laboratory, 94551, Livermore, CA, USA

    John B. Bell & Daniel L. Marcus

Authors
  1. John B. Bell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel L. Marcus
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by S.-T. Yau

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bell, J.B., Marcus, D.L. Vorticity intensification and transition to turbulence in three-dimensional euler equations. Commun.Math. Phys. 147, 371–394 (1992). https://doi.org/10.1007/BF02096593

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02096593

Keywords

Search

Navigation