Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-31T10:40:51.223Z Has data issue: false hasContentIssue false
  • English
  • Français

The modified cuinulant expansion for two-dimensional isotropic turbulence

Published online by Cambridge University Press:  20 April 2006

Tomomasa Tatsumi  and
Shinichiro Yanase
Tomomasa Tatsumi
Affiliation:
Department of Physics, Faculty of Science, University of Kyoto, Kyoto 606, Japan
Shinichiro Yanase
Affiliation:
Department of Physics, Faculty of Science, University of Kyoto, Kyoto 606, Japan Engineering Mathematics, School of Engineering, Okayama University, Okayama 700, Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

The two-dimensional isotropic turbulence in an incompressible fluid is investigated using the modified zero fourth-order cumulant approximation. The dynamical equation for the energy spectrum obtained under this approximation is solved numerically and the similarity laws governing the solution in the energy-containing and enstrophy-dissipation ranges are derived analytically. At large Reynolds numbers the numerical solutions yield the k−3 inertial subrange spectrum which was predicted by Kraichnan (1967), Leith (1968) and Batchelor (1969) assuming a finite enstrophy dissipation in the inviscid limit. The energy-containing range is found to satisfy an inviscid similarity while the enstrophy-dissipation range is governed by the quasi-equilibrium similarity with respect to the enstrophy dissipation as proposed by Batchelor (1969). There exists a critical time tc which separates the initial period (t < tc) and the similarity period (t > tc) in which the enstrophy dissipation vanishes and remains non-zero respectively in the inviscid limit. Unlike the case of three-dimensional turbulence, tc is not fixed but increases indefinitely as the viscosity tends to zero.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 110 , September 1981 , pp. 475 - 496
Copyright
© 1981 Cambridge University Press

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

André, J. C. & Lesieur, M. 1977 Influence of helicity on the evolution of isotropic turbulence at high Reynolds number. J. Fluid Mech. 81, 187–207.Google Scholar
Basdevant, C., Lesieur, M. & Sadourny, R. 1978 Subgrid modelling of enstrophy transfer in two-dimensional turbulence. J. Atmos. Sci. 6, 1028–1042.Google Scholar
Basdevant, C. & Sadourny, R. 1975 Ergodic properties of inviscid truncated models of two-dimensional incompressible flows. J. Fluid Mech. 69, 673–688.Google Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids Suppl. 12, II 233–239.Google Scholar
Brissaud, A., Frisch, U., Léorat, J., Lesieur, M., Mazure, A., Pouquet, A., Sadourny, R. & Sulem, P. L. 1973 Catastrophe énergétique et nature de la turbulence. Annales de Géophysique 29, 539–546.Google Scholar
Deem, G. S. & Zabusky, N. J. 1971 Ergodic boundary in numerical simulations of two-dimensional turbulence. Phys. Rev. Lett. 27, 396–399.Google Scholar
Fjørtoft, R. 1953 On the changes in the spectral distribution of kinetic energy for two-dimensional, non-divergent flow. Tellus 5, 225–230.Google Scholar
Fornberg, B. 1977 A numerical study for 2-D turbulence. J. Comp. Phys. 25, 1–31.Google Scholar
Fox, D. G. & Orszag, S. A. 1973 Inviscid dynamics of two-dimensional turbulence. Phys. Fluids 16, 169–171.Google Scholar
Herring, J. R., Orszag, S. A., Kraichnan, R. H. & Fox, D. G. 1974 Decay of two-dimensional turbulence. J. Fluid Mech. 66, 417–444.Google Scholar
Kato, T. 1967 On the classical solutions of the two-dimensional non-stationary Euler equation. Arch. Rat. Mech. Anal. 25, 188–200.Google Scholar
Kells, L. C. & Orszag, S. A. 1978 Randomness of low-order models of two-dimensional inviscid dynamics. Phys. Fluids 21, 162–168.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.Google Scholar
Kraichnan, R. H. 1971 Inertial-range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47, 525–535.Google Scholar
Kraichnan, R. H. 1975 Statistical dynamics of two-dimensional flow. J. Fluid Mech. 67, 155–175.Google Scholar
Kraichnan, R. H. & Montgomery, D. 1979 Two-dimensional turbulence. Preprint.
Leith, C. E. 1968 Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11, 671–673.Google Scholar
Leith, C. E. 1971 Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci. 28, 145–161.Google Scholar
Leith, C. E. & Kraichnan, R. H. 1972 Predictability of turbulent flows. J. Atmos. Sci. 29, 1041–1058.Google Scholar
Lesieur, M. & Schertzer, D. 1978 Amortissement autosimilaire d'une turbulence à grand nombre de Reynolds. Mécanique 17, 609–646.Google Scholar
Lilly, D. K. 1969 Numerical simulation of two-dimensional turbulence. Phys. Fluids Suppl. 12, II 240–249.Google Scholar
Lilly, D. K. 1971 Numerical simulation of developing and decaying two-dimensional turbulence. J. Fluid Mech. 45, 395–415.Google Scholar
Ogura, Y. 1962 Energy transfer in a normally distributed and isotropic turbulent velocity field in two dimensions. Phys. Fluids 5, 395–401.Google Scholar
Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento Suppl. 6, 279–287.Google Scholar
Orszag, S. A. 1977 Lectures on the statistical theory of turbulence. Fluid Dynamics: Les Houches 1973 (eds. R. Balian & J. L. Peube). Gordon & Breach.
Orszag, S. A. 1978 Turbulence and transition: A progress report. Private communication.
Pouquet, A., Lesieur, M., André, J. C. & Basdevant, C. 1975 Evolution of high Reynolds number two-dimensional turbulence. J. Fluid Mech. 72, 305–319.Google Scholar
Proudman, I. & Reid, W. H. 1954 On the decay of normally distributed and homogeneous turbulent velocity field. Phil. Trans. Roy. Soc. A 247, 163–189.Google Scholar
Saffman, P. G. 1971 On the spectrum and decay of random two dimensional vorticity distributions. Studies in Appl. Math. 50, 377–383.Google Scholar
Seyler, C. E., Salu, Y., Montgomery, D. & Knorr, G. 1975 Two-dimensional turbulence in inviscid fluids or guiding center plasmas. Phys. Fluids 18, 803–813.Google Scholar
Sulem, C. 1978 Régularite globale d'un fluide parfait occupant une bande de R2, pour des données initiales non nulles a l'infini. Private communication.
Tatsumi, T. & Kida, S. 1980 The modified cumulant expansion for isotropic turbulence at large Reynolds numbers. J. Phys. Soc. Japan 49, 2014–2025.Google Scholar
Tatsumi, T., Kida, S. & Mizushima, J. 1978 The multiple-scale cumulant expansion for isotropic turbulence. J. Fluid Mech. 85, 97–142.Google Scholar
Van Atta, C. W. & Chen, W. Y. 1968 Correlation measurements in grid turbulence using digital harmonic analysis. J. Fluid Mech. 34, 497–515.Google Scholar