Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-19T07:03:56.207Z Has data issue: false hasContentIssue false
  • English
  • Français

The multiple-scale cumulant expansion for isotropic turbulence

Published online by Cambridge University Press:  12 April 2006

Tomomasa Tatsumi ,
Shigeo Kida  and
Jiro Mizushima
Tomomasa Tatsumi
Affiliation:
Department of Physics, Faculty of Science, University of Kyoto, Kyoto 606, Japan
Shigeo Kida
Affiliation:
Research Institute for Mathematical Sciences, University of Kyoto, Kyoto 606, Japan
Jiro Mizushima
Affiliation:
Department of Physics, Faculty of Science, University of Kyoto, Kyoto 606, Japan Present address: Department of Mathematical Engineering, Sagami Institute of Technology, Fujisawa 251, Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

A method of multiple-scale expansion is applied to the theory of incompressible isotropic turbulence in order to close the infinite system of cumulant equations. The dynamical equation for the energy spectrum derived from this method is found to give positive-definite solutions at all Reynolds numbers. At large Reynolds numbers the spectrum takes the form of Kolmogorov's $-\frac{5}{3}$ power spectrum in the inertial subrange, whose extent increases indefinitely with Reynolds number. The spectrum in the energy-containing range satisfies an inviscid similarity law, so that the rate of energy decay or of viscous dissipation is also independent of the viscosity. In the higher wavenumber region beyond the inertial subrange the spectrum takes a universal form which is independent of its structure at lower wavenumbers. The universal spectrum is composed of three different subspectra, which are, in order of increasing wavenumber, the $k^{-\frac{5}{3}}$ spectrum, the k−1 spectrum and the exp [−σk1·5] spectrum, σ being a constant. Various statistical quantities such as the energy, the skewness of the velocity derivative, the microscale and the microscale Reynolds number are calculated from the numerical data for the energy spectrum. Theoretical results are discussed in detail in comparison with experimental results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 85 , Issue 1 , 7 March 1978 , pp. 97 - 142
Copyright
© 1978 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

Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Batchelor, G. K. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid. 1. General discussion and the case of small conductivity. J. Fluid Mech. 5, 113–133.Google Scholar
Batchelor, G. K. & Proudman, I. 1956 The large-scale structure of homogeneous turbulence. Phil. Trans. Roy. Soc. A 248, 369–405.Google Scholar
Batchelor, G. K. & Townsend, A. A. 1947 Decay of vorticity in isotropic turbulence. Proc. Roy. Soc. A 191, 534–550.Google Scholar
Batchelor, G. K. & Townsend, A. A. 1948 Decay of turbulence in the final period. Proc. Roy. Soc. A 194, 527–543.Google Scholar
Batchelor, G. K. & Townsend, A. A. 1949 The nature of turbulent motion at large wave-numbers. Proc. Roy. Soc. A 199, 238–255.Google Scholar
Birkhoff, G. 1954 Fourier synthesis of homogeneous turbulence. Comm. Pure Appl. Math. 7, 19–44.Google Scholar
Bogoliubov, N. N. 1962 Studies in Statistical Mechanics, vol. 1 (ed. J. de Boer & G. E. Uhlenbeck). North-Holland.
COMTE-BELLOT, G. & Corrsin, S. 1966 The use of a contraction to improve the isotropy of grid-generated turbulence. J. Fluid Mech. 25, 657–682.Google Scholar
Edwards, S. F. 1964 The statistical dynamics of homogeneous turbulence. J. Fluid Mech. 18, 239–273.Google Scholar
Frenkiel, F. N. & Klebanoff, P. S. 1967 Higher-order correlations in a turbulent field. Phys. Fluids 10, 507–520.Google Scholar
Gad-El-Hak, M. & Corrsin, S. 1974 Measurements of the nearly isotropic turbulence behind a uniform jet grid. J. Fluid Mech. 62, 115–143.Google Scholar
Gibson, M. M. 1962 Spectra of turbulence at high Reynolds number. Nature, 195, 1281–1283.Google Scholar
Gibson, M. M. 1963 Spectra of turbulence in a round jet. J. Fluid Mech. 15, 161–173.Google Scholar
Gibson, C. H. & Schwarz, W. H. 1963 The universal equilibrium spectra of turbulent velocity and scalar fields. J. Fluid Mech. 16, 365–384.Google Scholar
Grant, H. L., Hughes, B. A., Vogel, W. M. & Moilliet, A. 1968 The spectrum of temperature fluctuations in turbulent flow. J. Fluid Mech. 34, 423–442.Google Scholar
Grant, H. L. & Moilliet, A. 1962 The spectrum of a cross-stream component of turbulence in a tidal stream. J. Fluid Mech. 13, 237–240.Google Scholar
Grant, H. L., Stewart, R. W. & Moilliet, A. 1962 Turbulence spectra from a tidal channel. J. Fluid Mech. 12, 241–268.Google Scholar
Helland, K. N., Van atta, C. W. & Stegen, G. R. 1977 Spectral energy transfer in high Reynolds number turbulence. J. Fluid Mech. 79, 337–359.Google Scholar
Herring, J. R. 1965 Self-consistent-field approach to turbulence theory. Phys. Fluids 8, 2219–2225.Google Scholar
Herring, J. R. 1966 Self-consistent-field approach to nonstationary turbulence. Phys. Fluids 9, 2106–2110.Google Scholar
Herring, J. R. & Kraichnan, R. H. 1972 Comparison of some approximations for isotropic turbulence. In Statistical Models and Turbulence. Lecture Notes in Physics, vol. 12 (ed. M. Rosenblatt & C. Van Atta), pp. 148–194. Springer.
Hopf, E. 1952 Statistical hydromechanics and functional calculus. J. Rat. Mech. Anal. 1, 87–123.Google Scholar
KarmÁn, T. Von & Howarth, L. 1938 On the statistical theory of isotropic turbulence. Proc. Roy. Soc. A 164, 192–215.Google Scholar
Kawahara, T. 1968 A successive approximation for turbulence in the Burgers model fluid. J. Phys. Soc. Japan 25, 892–900.Google Scholar
Kistler, A. L. & Vrebalovich, T. 1961 Grid turbulence at high Reynolds numbers. Bull. Am. Phys. Soc. 6, 207.Google Scholar
Kistler, A. L. & Vrebalovich, T. 1966 Grid turbulence at large Reynolds numbers. J. Fluid Mech. 26, 37–47.Google Scholar
Kolmogorov, A. N. 1941a The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. U.R.S.S. 30, 301–305.Google Scholar
Kolmogorov, A. N. 1941b On degeneration of isotropic turbulence in an incompressible viscous liquid. C. R. Acad. Sci. U.R.S.S. 31, 538–540.Google Scholar
Kolmogorov, A. N. 1941c Dissipation of energy in the locally isotropic turbulence. C. R. Acad. Sci. U.R.S.S. 32, 16–18.Google Scholar
Kraichnan, R. H. 1964 Approximations for steady-state isotropic turbulence. Phys. Fluids 7, 1163–1168.Google Scholar
Kraichnan, R. H. 1971 An almost-Markovian Galilean-invariant turbulence model. J. Fluid Mech. 47, 513–524.Google Scholar
Leith, C. E. 1971 Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci. 28, 145–161.Google Scholar
Ling, S. C. & Wan, C. A. 1972 Decay of isotropic turbulence generated by a mechanically agitated grid. Phys. Fluids 15, 1363–1369.Google Scholar
Malfliet, W. P. M. 1969 A theory of homogeneous turbulence deduced from the Burgers equation. Physica 45, 257–271.Google Scholar
Millionshtchikov, M. 1941 On the theory of homogeneous isotropic turbulence. C. R. Acad. Sci. U.R.S.S. 32, 615–618.Google Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics, vol. 2. M.I.T. Press.
Ogura, Y. 1963 A consequence of the zero-fourth-cumulant approximation in the decay of isotropic turbulence. J. Fluid Mech. 16, 38–41.Google Scholar
Obszag, S. A. 1970 Analytical theories of turbulence. J. Fluid Mech. 41, 363–386.Google Scholar
Orszag, S. & Patterson, G. S. 1972 Numerical simulation of turbulence. In Statistical Models and Turbulence. Lecture Notes on Physics, vol. 12 (ed. M. Rosenblatt & C. Van Atta), pp. 127–147. Springer.
Pao, Y. H. 1965 Structure of turbulent velocity and scalar fields at large wavenumbers. Phys. Fluids 8, 1063–1075.Google Scholar
Pao, Y. H. 1968 Transfer of turbulent energy and scalar quantities at large wavenumbers. Phys. Fluids 11, 1371–1372.Google Scholar
Proudman, I. & Reid, W. H. 1954 On the decay of normally distributed and homogeneous turbulent velocity fields. Phil. Trans. Roy. Soc. A 247, 163–189.Google Scholar
Saffman, P. G. 1967 Note on decay of homogeneous turbulence. Phys. Fluids 10, 1349.Google Scholar
Saffman, P. G. 1968 In Lectures on Homogeneous Turbulence. Topics in Nonlinear Physics (ed. N. J. Zabusky), pp. 485–614. Springer.
Schedvin, J., Stegen, G. R. & Gibson, C. H. 1974 Universal similarity at high grid Reynolds numbers. J. Fluid Mech. 56, 561–579.Google Scholar
Stewart, R. W. 1951 Triple velocity correlations in isotropic turbulence. Proc. Camb. Phil. Soc. 47, 146–147.Google Scholar
Stewart, R. W. & Townsend, A. A. 1951 Similarity and self-preservation in isotropic turbulence. Phil. Trans. Roy. Soc. A 243, 359–386.Google Scholar
Tanaka, H. 1969 0–5th cumulant approximation of inviscid Burgers turbulence. J. Met. Soc. Japan 47, 373–383.Google Scholar
Tanaka, H. 1973 Higher order successive expansion of inviscid Burgers turbulence. J. Phys. Soc. Japan 34, 1390–1395.Google Scholar
Tassa, Y. & Kamotani, Y. 1975 Experiments on turbulence behind a grid with jet injection in downstream and upstream direction. Phys. Fluids 18, 411–414.Google Scholar
Tatsumi, T. 1955 Theory of isotropic turbulence with the normal joint-probability distribution of velocity. Proc. 4th Japan Nat. Cong. App. Mech. pp. 307–311.Google Scholar
Tatsumi, T. 1957 The theory of decay process of incompressible isotropic turbulence. Proc. Roy. Soc. A 239, 16–45.Google Scholar
Tatsumi, T. 1960 Energy spectra in magneto-fluid dynamic turbulence. Rev. Mod. Phys. 32, 807–812.Google Scholar
Uberoi, M. S. 1953 Quadruple velocity correlations and pressure fluctuations in homogeneous turbulence. J. Aero. Sci. 20, 197–204 (corrigendum 21 (1954), 142).Google Scholar
Uberoi, M. S. 1963 Energy transfer in isotropic turbulence. Phys. Fluids 6, 1048–1056.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
Van Atta, C. W. & Chen, W. Y. 1969 Correlation measurements in turbulence using digital Fourier analysis. Phys. Fluids Suppl. 12, II 264–269.Google Scholar
Van Atta, C. W. & Yeh, T. T. 1970 Some measurements of multi-point time correlations in grid turbulence. J. Fluid Mech. 41, 169–178.Google Scholar
Wyngaard, J. C. & Pao, Y. H. 1972 Some measurements of the fine structure of large Reynolds number turbulence. In Statistical Models and Turbulence. Lecture Notes in Physics, vol. 12 (ed. M. Rosenblatt & C. Van Atta), pp. 384–401. Springer.