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Prog. Theor. Phys. Vol. 73 No. 3 (1985) pp. 629-637

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Singularities in the Navier-Stokes Equation

Tohru Nakano

Department of Physics, Chuo University, Tokyo 112

(Received September 6, 1984)

Abstract:

We consider the singularities of the Navier-Stokes equation in the framework of our previous turbulence model. With the aid of the values obtained by Morf, Orszag and Frisch for an inviscid Taylor-Green flow, we find that (1) the singular exponent γp depends on the topological dimensions of the singular structure and the amplification of the singular structure occurs roughly in the form of a sheet, and that (2) the helicity cascade solution plays and important role for large p. The model predicts γp as 2p-1 ≥γp ≥3p-5 / 2 for p ≤3/2, and 6p-7 ≥γp ≥4p-4 for p ≥3/2. MOF's results are located in these allowable regions.


URL : http://ptp.ipap.jp/link?PTP/73/629/
DOI : 10.1143/PTP.73.629

[ Full Text PDF : FREE ACCESS (500K) ] Citation:

References:

  1. S. A. Orszag, in Fluid Dynamics, ed. R. Balian and J. L. Peube (Gordon and Breach, London, 1977).
  2. A. Brissaud, U. Frisch, J. Leorat, M. Lesieur, A. Mazure, A. Pouquet, R. Sadourny and P. L. Sulem, Ann. Geophys. 29 (1973), 539.
  3. P. Vieillefosse, J. de Phys. 43 (1982), 837.
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  5. M. E. Brachet, D. Meiron, S. A. Orszag, B. Nickel, R. Morf and U. Frisch, J. Fluid Mech. 130 (1983), 411[CrossRef].
  6. T. Nakano and M. Nelkin, to be published in Phys. Rev. A.
  7. A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30 (1941), 301.
  8. See A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, vol. 2 (MIT Press, Cambridge, Mass., 1975), chap. 22.
  9. K. Takayoshi and H. Mori, Prog. Theor. Phys. 70 (1983), 961[PTP].
  10. U. Frisch, P. L. Sulem and M. Nelkin, J. Fluid Mech. 87 (1978), 719[CrossRef].
  11. A. Brissaud, U. Frisch, J. Leorat, M. Lesieur and A. Mazure, Phys. Fluids 16 (1973), 1366[AIP Scitation].
  12. J. D. Andre and M. Lesieur, J. Fluid. Mech. 81 (1977), 187[CrossRef].
  13. K. G. Wilson, Phys. Rev. B4 (1971), 3184[APS].
    See also E. D. Siggia, Phys. Rev. A15 (1977), 1730[APS].

Citing Article(s) :

  1. Journal of the Physical Society of Japan 64 (1995) pp. 685-689 :
    Time Correlation between Entropy and/or Energy Distributed into Scales by 2D Wavelet in 2D Free-Convective Turbulence
    Sadayoshi Toh
  2. Progress of Theoretical Physics Vol. 75 No. 6 (1986) pp. 1295-1303 :
    The Scaling Exponents of Intermittent Fully Developed Turbulence Based on a New Model
    Tohru Nakano
  3. Progress of Theoretical Physics Vol. 79 No. 3 (1988) pp. 569-580 :
    Determination of a Dynamical Scaling Function in a Cascade Model of Turbulence
    Tohru Nakano
  4. Progress of Theoretical Physics Vol. 81 No. 2 (1989) pp. 329-341 :
    Temporal Intermittency in the Energy Cascade Process and Local Lyapunov Analysis in Fully-Developed Model Turbulence
    Koji Ohkitani and Michio Yamada

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