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Archive for Rational Mechanics and Analysis

, Volume 177, Issue 1, pp 115–150 | Cite as

Intermittency and Regularity Issues in 3D Navier-Stokes Turbulence

  • J. D. Gibbon
  • Charles R. Doering
Article

Abstract

Two related open problems in the theory of 3D Navier-Stokes turbulence are discussed in this paper. The first is the phenomenon of intermittency in the dissipation field. Dissipation-range intermittency was first discovered experimentally by Batchelor and Townsend over fifty years ago. It is characterized by spatio-temporal binary behaviour in which long, quiescent periods in the velocity signal are interrupted by short, active ‘events’ during which there are violent fluctuations away from the average. The second and related problem is whether solutions of the 3D Navier-Stokes equations develop finite time singularities during these events. This paper shows that Leray’s weak solutions of the three-dimensional incompressible Navier-Stokes equations can have a binary character in time. The time-axis is split into ‘good’ and ‘bad’ intervals: on the ‘good’ intervals solutions are bounded and regular, whereas singularities are still possible within the ‘bad’ intervals. An estimate for the width of the latter is very small and decreases with increasing Reynolds number. It also decreases relative to the lengths of the good intervals as the Reynolds number increases. Within these ‘bad’ intervals, lower bounds on the local energy dissipation rate and other quantities, such as ||u(·, t)|| and ||∇u(·, t)||, are very large, resulting in strong dynamics at sub-Kolmogorov scales. Intersections of bad intervals for n≧1 are related to the potentially singular set in time. It is also proved that the Navier-Stokes equations are conditionally regular provided, in a given ‘bad’ interval, the energy has a lower bound that is decaying exponentially in time.

Keywords

Reynolds Number Energy Dissipation Rate Increase Reynolds Number Quiescent Period Interval Solution 
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.

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References

  1. 1.
    Batchelor, G.K., Townsend, A.A.: The nature of turbulent flow at large wave-numbers. Proc R. Soc. Lond. A. 199, 238–255 (1949)Google Scholar
  2. 2.
    Leray, J.: Essai sur le mouvement d’un liquide visquex emplissant l’espace. Acta Math. 63, 193–248 (1934)Google Scholar
  3. 3.
    Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, 1963Google Scholar
  4. 4.
    Serrin, J.: The initial value problem for the Navier-Stokes equations. In: Nonlinear Problems. University of Wisconsin Press, Madison, R E Langer edition, 1963Google Scholar
  5. 5.
    Foias, C.: Statistical study of Navier-Stokes equations I. Rend. Sem. Mat. Univ. Padova 48, 219–348 (1972) and 49, 9–123 (1973)Google Scholar
  6. 6.
    Foias, C., Prodi, G.: Sur les solutions statistique des équations de Navier-Stokes. Ann. Mat. Pura Appl. 111, 307–330 (1976)Google Scholar
  7. 7.
    Foias, C., Guillopé, C., Temam, R.: New a priori estimates for Navier-Stokes equations in Dimension 3. Comm. Partial Diff. Equat. 6, 329–359 (1981)Google Scholar
  8. 8.
    Temam, R.: Navier-Stokes Equations and Non-linear Functional Analysis. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM Press, Philadelphia, 1983Google Scholar
  9. 9.
    Constantin, P., Foias, C.: Navier-Stokes Equations. The University of Chicago Press, Chicago, 1988Google Scholar
  10. 10.
    Foias, C., Manley, O., Rosa, R., Temam, R.: Navier-Stokes equations and Turbulence. Cambridge: Cambridge University Press, 2001Google Scholar
  11. 11.
    Majda, A.J., Bertozzi, A.: Vorticity and incompressible flow. Cambridge: Cambridge University Press, 2002Google Scholar
  12. 12.
    Kerr, R.: Evidence for a singularity of the 3-dimensional, incompressible Euler equations. Phys. Fluids A 5, 1725–1746 (1993)CrossRefGoogle Scholar
  13. 13.
    Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)CrossRefGoogle Scholar
  14. 14.
    Constantin, P., Fefferman, Ch., Majda, A.: Geometric constraints on potentially singular solutions for the 3D Euler equations. Comm. Partial. Diff. Equations 21, 559–571 (1996)Google Scholar
  15. 15.
    Scheffer, V.: Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Maths 66, 535–552 (1976)Google Scholar
  16. 16.
    Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure & Appl. Math. 35, 771–831 (1982)Google Scholar
  17. 17.
    Babin, A., Mahalov, A., Nicolaenko, B.: Regularity and Integrability of 3D Euler and Navier-Stokes Equations for Uniformly Rotating Fluids. Asympt. Anal. 15, 103–150 (1997)Google Scholar
  18. 18.
    Babin, A., Mahalov, A., Nicolaenko, B.: Global Regularity of 3D Rotating Navier-Stokes Equations for Resonant Domains. Indiana Univ. Math. J. 48, 1133–1176 (1999)Google Scholar
  19. 19.
    Babin, A., Mahalov, A., Nicolaenko, B.: 3D Navier-Stokes and Euler Equations with Initial Data Characterized by Uniformly Large Vorticity. Indiana Univ. Math. J. 50, 1–35 (2001)Google Scholar
  20. 20.
    Kuo, A. Y.-S., Corrsin, S.: Experiments on internal intermittency and fine-structure distribution functions in fully turbulent fluid. J. Fluid Mech. 50, 285–320 (1971)Google Scholar
  21. 21.
    Emmons H.W.: The laminar-turbulent transition in boundary layers. J. Aero Sci. 18, 490–498 (1951)Google Scholar
  22. 22.
    Frisch, U.: Turbulence: The legacy of A N Kolmogorov. Cambridge: Cambridge University Press, 1995Google Scholar
  23. 23.
    Mandelbrot, B.B.: Some fractal aspects of turbulence: intermittency, dimension, kurtosis, and the spectral exponent 5/3+B. Proc. Journées Mathématiques sur la Turbulence Orsay (ed. R. Temam). Springer, Berlin, 1975Google Scholar
  24. 24.
    Sreenivasan, K., Meneveau, C.: Singularities of the equations of fluid motion. Phys. Rev. A 38, 6287–6295 (1988)CrossRefPubMedGoogle Scholar
  25. 25.
    Meneveau, C., Sreenivasan, K.: The multifractal nature of turbulent energy dissipation. J. Fluid Mech. 224, 429–484 (1991)Google Scholar
  26. 26.
    Sreenivasan, K.: Fractals and multifractals in fluid turbulence. Ann. Rev. Fluid Mech. 23, 539–600 (1991)CrossRefGoogle Scholar
  27. 27.
    Zeff, B.W., Lanterman, D.D., McAllister, R., Roy, R., Kostelich, E.J., Lathrop, D.P.: Measuring intense rotation and dissipation in turbulent flows. Nature 421, 146–149 (2003)Google Scholar
  28. 28.
    Hosokawa, I., Yamatoto, K.: Intermittency of dissipation in fully developed isotropic turbulence. J. Phys. Soc. Japan 59, 401–404 (1990)Google Scholar
  29. 29.
    Douady, S., Couder, Y., Brachet, M.E.: Direct observation of the intermittency of intense vortex filaments in turbulence. Phys. Rev. Letts. 67, 983 (1991)CrossRefGoogle Scholar
  30. 30.
    Vincent, A., Meneguzzi, M.: The dynamics of vorticity tubes of homogeneous turbulence. J. Fluid Mech. 258, 245–254 (1994)Google Scholar
  31. 31.
    Galanti, B., Tsinober, A.: Self-amplification of the field of velocity derivatives in quasi-isotropic turbulence. Phys. Fluids 12 3097–3099 (2000); erratum Phys. Fluids 13 1063 (2001)Google Scholar
  32. 32.
    Tsinober, A.: Vortex stretching versus production of strain/dissipation (ed: Hunt J C R and Vassilicos J C). In: Turbulence Structure and Vortex Dynamics. Cambridge: Cambridge University Press, 2000, pp. 164–191Google Scholar
  33. 33.
    Lundgren, T.: Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25, 2193–2203 (1982)CrossRefGoogle Scholar
  34. 34.
    Vassilicos, J.C., Hunt, J.C.R.: Fractal Dimensions and Spectra of Interfaces with Application to Turbulence. Proc. R. Soc. Lond. A 435, 505–534 (1991)Google Scholar
  35. 35.
    Flohr, P., Vassilicos, J.C.: Accelerated scalar dissipation in a vortex. J. Fluid Mech. 348, 295–317 (1997)CrossRefGoogle Scholar
  36. 36.
    Angilella, J.R., Vassilicos, J.C.: Spectra and dynamical properties of fractal and spiral fields. Physica D 124, 23–57 (1998)Google Scholar
  37. 37.
    Angilella, J.R., Vassilicos, J.C.: Time-dependent geometry and energy distribution in a spiral vortex layer. Phys. Rev. E 59, 5427–5439 (1999)CrossRefGoogle Scholar
  38. 38.
    Frisch, U., Morf, R.: Intermittency in nonlinear dynamics and singularities at complex times. Phys. Rev. A 23, 2673–2705 (1981)CrossRefGoogle Scholar
  39. 39.
    Batchelor, G.K.: The theory of homogeneous turbulence. Cambridge: Cambridge University Press, 2001Google Scholar
  40. 40.
    Zakharov, V.E., L’vov, V.S., Falkovich, G.: Kolmogorov spectra of weak turbulence V1. Springer-Verlag, Berlin, 1992Google Scholar
  41. 41.
    Zakharov, V.E.: Statistical theory of gravity and capillary waves on the surface of finite-depth fluid. Eur. J. Mech B/Fluids 18, 327–344 (1999)CrossRefGoogle Scholar
  42. 42.
    Doering, C.R., Gibbon, J.D.: Applied analysis of the Navier-Stokes equations. Cambridge: Cambridge University Press, 1995Google Scholar
  43. 43.
    Lin, F.: A new proof of the Caffarelli, Kohn and Nirenberg theorem. Commun. Pure Appl. Maths 51, 241–257 (1998)CrossRefGoogle Scholar
  44. 44.
    Choe, H.J., Lewis, J.L.: On the singular set in the Navier-Stokes equations. J. Funct. Anal. 175, 348–369 (2000)CrossRefGoogle Scholar
  45. 45.
    Bartuccelli, M.V., Doering, C.R., Gibbon, J.D., Malham, S.: Length scales in solutions of the Navier-Stokes equations on a finite periodic domain. Nonlinearity 6, 549–568 (1993)CrossRefGoogle Scholar
  46. 46.
    Doering, C.R., Gibbon, J.D.: Bounds on moments of the energy spectrum for weak solutions of the three-dimensional Navier-Stokes equations. Physica D 165, 163–175 (2002)Google Scholar
  47. 47.
    Gibbon, J.D., Doering, C.R.: Intermittency in solutions of the three-dimensional Navier-Stokes equations. J. Fluid Mech. 478, 227–235 (2003)CrossRefGoogle Scholar
  48. 48.
    Doering, C.R., Foias, C.: Energy dissipation in body-forced turbulence. J. Fluid Mech. 467, 289–306 (2002)CrossRefGoogle Scholar
  49. 49.
    Tsinober, A.: An informal introduction to turbulence. Kluwer, Amsterdam, 2001Google Scholar
  50. 50.
    Bartuccelli, M.V., Doering, C.R., Gibbon, J.D.: Ladder theorems for the 2d and 3d Navier-Stokes equations on a finite periodic domain. Nonlinearity 4, 531–542 (1991)CrossRefGoogle Scholar
  51. 51.
    Constantin, P., Foias, C., Temam, R.: On the dimension of the attractors in two-dimensional turbulence. Physica D 30, 284–296 (1998)Google Scholar
  52. 52.
    Jones, D.S., Titi, E.T.: Upper bounds for the number of determining modes, nodes and volume elements for the Navier-Stokes equations. Indiana Univ. Math. J. 42, 875–887 (1993)CrossRefGoogle Scholar
  53. 53.
    Foias, C., Holm, D.D., Titi, E.S.: The Navier-Stokes α-model of fluid turbulence. Physica D (Special Issue in Honor of V. E. Zakharov on the Occasion of His 60th Birthday) D152, 505–519 (2001)Google Scholar
  54. 54.
    Foias, C., Holm, D.D., Titi, E.S.: The three dimensional viscous Camassa-Holm equations and their relation to the Navier–Stokes equations and turbulence theory. J. Dyn. Diff. Eqns 14, 1–35 (2002)CrossRefGoogle Scholar
  55. 55.
    Cheskidov, A., Holm, D.D., Olson, E., Titi, E.S.: On a Leray-α Model of Turbulence. Preprint 2003.Google Scholar
  56. 56.
    Bartuccelli, M., Constantin, P., Doering, C.R., Gibbon, J.D., Gisselfalt, M.: On the possibility of soft and hard turbulence in the complex Ginzburg Landau equation. Physica D 44, 421–444 (1990)Google Scholar
  57. 57.
    Bartuccelli, M., Gibbon, J.D., Oliver, M.: Length scales in solutions of the complex Ginzburg Landau equation. Physica D 89, 267–286 (1996)Google Scholar
  58. 58.
    Sulem, P.-L., Frisch, U.: Bounds on energy flux for finite energy turbulence. J. Fluid Mech. 72, 417–424 (1975)Google Scholar
  59. 59.
    Frisch, U., Sulem, P.-L., Nelkin, M.: A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech. 87, 719–736 (1978)Google Scholar
  60. 60.
    Sreenivasan, K.: On the fine-scale intermittency of turbulence. J. Fluid Mech. 151, 81–103 (1985)Google Scholar
  61. 61.
    Arad, I., L’vov, V.S., Procaccia, I.: Correlation Functions in Isotropic and Anisotropic Turbulence: the Role of the Symmetry Group. Phys. Rev. E 59, 6753–6765 (1999)CrossRefGoogle Scholar
  62. 62.
    Arad, I., Biferale, L., Mazzitelli, I., Procaccia, I.: Disentangling Scaling Properties in Anisotropic and Inhomogeneous Turbulence. Phys. Rev. Lett 82, 5040 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Department of Mathematics & Michigan Center for Theoretical PhysicsUniversity of MichiganAnn Arbor, MichiganUSA

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