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Numerical calculation of decaying isotropic turbulence using the LET theory

Published online by Cambridge University Press:  20 April 2006

W. D. Mccomb  and
V. Shanmugasundaram
W. D. Mccomb
Affiliation:
School of Engineering, University of Edinburgh, King's Buildings, Mayfield Road, Edinburgh EH9 3JL, U.K.
V. Shanmugasundaram
Affiliation:
School of Engineering, University of Edinburgh, King's Buildings, Mayfield Road, Edinburgh EH9 3JL, U.K.
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Abstract

The local-energy-transfer (LET) theory (McComb 1978) was used to calculate freely decaying turbulence for four different initial spectra at low-to-moderate values of microscale Reynolds numbers (Rλ up to about 40). The results for energy, dissipation and energy-transfer spectra and for skewness factor all agreed quite closely with the predictions of the well-known direct-interaction approximation (DIA: Kraichnan 1964). However, LET gave higher values of energy transfer and of evolved skewness factor than DIA. This may be related to the fact that LET yields the k−5/3 law for the energy spectrum at infinite Reynolds number.

The LET equations were then integrated numerically for decaying isotropic turbulence at high Reynolds number. Values were obtained for the wavenumber spectra of energy, dissipation rate and inertial-transfer rate, along with the associated integral parameters, at an evolved microscale Reynolds number Rλ of 533. The predictions of LET agreed well with experimental results and with the Lagrangian-history theories (Herring & Kraichnan 1979). In particular, the purely Eulerian LET theory was found to agree rather closely with the strain-based Lagrangian-history approximation; and further comparisons suggested that this agreement extended to low Reynolds numbers as well.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 143 , June 1984 , pp. 95 - 123
Copyright
© 1984 Cambridge University Press

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References

Balescu, R. & Senatorski, A. 1970 A new approach to the theory of fully developed turbulence. Ann. Phys. 58, 587.Google Scholar
Batchelor, G. K. 1971 The Theory of Homogeneous Turbulence, 2nd edn. Cambridge University Press.
Bennett, J. C. & Corrsin, S. 1978 Small Reynolds number nearly isotropic turbulence in a straight duct and a contraction. Phys. Fluids 21, 2129.Google Scholar
Brachet, M. E., Meiron, D. I., Orszag, S. A., Nickel, B. G., Morf, R. H. & Frisch, U. 1983 Small-scale structure of the Taylor—Green vortex. J. Fluid Mech. 130, 411.Google Scholar
Champagne, F. H. 1978 The fine-scale structure of the turbulent velocity field. J. Fluid Mech. 86, 67.Google Scholar
Chen, W. Y. 1968 Spectral energy transfer and higher-order correlations in grid turbulence. Ph.D. thesis. University of California, San Diego.
Coantic, M. & Favre, A. 1974 Activities in, and preliminary results of, air—sea interaction research at I.M.S.T. Adv. Geophys. 18A, 391.Google Scholar
Comte-Bellot, G. & Corrsin, S. 1971 Simple Eulerian time correlation of full- and narrow-band velocity signals in grid-generated, ‘isotropic’ turbulence. J. Fluid Mech. 48, 273.Google Scholar
Edwards, S. F. 1964 The statistical dynamics of homogeneous turbulence. J. Fluid Mech. 18, 239.Google Scholar
Edwards, S. F. & McComb, W. D. 1969 Statistical mechanics far from equilibrium. J. Phys. A: Gen. Phys. 2, 157.Google Scholar
Edwards, S. F. & McComb, W. D. 1972 Local transport equations for turbulent shear flow. Proc. R. Soc. Lond. A 330, 495.Google Scholar
Forster, D., Nelson, D. R. & Stephen, M. J. 1977 Long-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A 16, 732.Google Scholar
Frenkiel, N. F. & Klebanoff, P. S. 1967 Higher-order correlations in a turbulent field. Phys. Fluids 10, 507.Google Scholar
Frenkiel, N. F. & Klebanoff, P. S. 1971 Statistical properties of velocity derivatives in a turbulent field. J. Fluid Mech. 48, 183.Google Scholar
Frisch, U., Sulem, P. L. & Nelkin, M. 1978 A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech. 87, 719.Google Scholar
Grant, H. L., Stewart, R. W. & Moilliet, A. 1962 Turbulence spectra from a tidal channel. J. Fluid Mech. 12, 241.Google Scholar
Herring, J. R. 1965 Self-consistent field approach to turbulence theory. Phys. Fluids 8, 2219.Google Scholar
Herring, J. R. 1966 Self-consistent field approach to non-stationary turbulence. Phys. Fluids 11, 2106.Google Scholar
Herring, J. R. & Kraichnan, R. H. 1972 Comparison of some approximations for isotropic turbulence. In Statistical Models and Turbulence (ed. M. Rosenblatt & C. Van Atta). Lecture Notes in Physics, vol. 12 p. 148. Springer.
Herring, J. R. & Kraichnan, R. H. 1979 A numerical comparison of velocity-based and strain-based Lagrangian-history turbulence approximations. J. Fluid Mech. 91, 581.Google Scholar
Kistler, A. L. & Vrebalovich, T. 1966 Grid turbulence at large Reynolds numbers. J. Fluid Mech. 26, 37.Google Scholar
Kolmogorov, A. N. 1941 Local structure of turbulence in an incompressible fluid at very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299.Google Scholar
Kraichnan, R. H. 1959 The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5, 497.Google Scholar
Kraichnan, R. H. 1964 Decay of isotropic turbulence in the direct-interaction approximation. Phys. Fluids 7, 1030.Google Scholar
Kraichnan, R. H. 1965 Lagrangian-history closure approximation for turbulence. Phys. Fluids 8, 575.Google Scholar
Kraichnan, R. H. 1966 Isotropic turbulence and inertial-range structure. Phys. Fluids 9, 1728.Google Scholar
Kraichnan, R. H. 1974 On Kolmogorov's inertial-range theories. J. Fluid Mech. 62, 305.Google Scholar
Kraichnan, R. H. 1982 Hydrodynamic turbulence and the renormalization group. Phys. Rev. A 25, 3281.Google Scholar
Kraichnan, R. H. & Herring, J. R. 1978 A strain-based Lagrangian-history turbulence theory. J. Fluid Mech. 88, 355.Google Scholar
Kuo, A. Y. & Corrsin, S. 1971 Experiments on internal intermittency and fine-structure distribution functions in fully turbulent fluid. J. Fluid Mech. 50, 285.Google Scholar
Lee, L. L. 1965 A formulation of the theory of isotropic hydro-magnetic turbulence in an incompressible fluid. Ann. Phys. 32, 292.Google Scholar
Leslie, D. C. 1973 Developments in the Theory of Turbulence. Oxford University Press.
McComb, W. D. 1974 A local energy-transfer theory of isotropic turbulence. J. Phys. A: Math. Nucl. Gen. 7, 632.Google Scholar
McComb, W. D. 1976 The inertial-range spectrum from a local energy-transfer theory of isotropic turbulence. J. Phys. A: Math. Gen. 9, 179.Google Scholar
McComb, W. D. 1978 A theory of time-dependent, isotropic turbulence. J. Phys. A: Math. Gen. 11, 613.Google Scholar
McComb, W. D. 1982 Reformulaton of the statistical equations for turbulent shear flow. Phys. Rev. A 26, 1078.Google Scholar
McComb, W. D. & Shanmugasundaram, V. 1983 Fluid turbulence and the renormalisation group: a preliminary calculation of the eddy viscosity. Phys. Rev. A 28, 2588.Google Scholar
Martin, P. C., Siggia, E. D. & Rose, H. A. 1973 Statistical dynamics of classical systems. Phys. Rev. A 8, 423.Google Scholar
Nakano, T. 1972 A theory of homogeneous, isotropic turbulence of incompressible fluids. Ann. Phys. 73, 326.Google Scholar
Nelkin, M. 1974 Turbulence, critical fluctuations and intermittency. Phys. Rev. A 9, 388.Google Scholar
Nelkin, M. 1975 Scaling theory of hydrodynamic turbulence. Phys. Rev. A 11, 1737.Google Scholar
Orszag, S. A. & Patterson, G. S. 1972 Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett. 28, 76.Google Scholar
Pao, Y. H. 1965 Structure of turbulent velocity and scalar field at large Reynolds numbers. Phys. Fluids 8, 1063.Google Scholar
Phythian, R. 1969 Self-consistent perturbation series for stationary homogeneous turbulence. J. Phys. A: Gen. Phys. 2, 181.Google Scholar
Phythian, R. 1975 The operator formalism of classical statistical dynamics. J. Phys. A: Math. Nucl. Gen. 8, 1423.Google Scholar
Phythian, R. 1976 Further application of the Martin, Siggia, Rose formalism. J. Phys. A: Math. Nucl. Gen. 9, 269.Google Scholar
Stewart, R. W. & Townsend, A. A. 1951 Similarity and self-preservation in isotropic turbulence. Phil. Trans. R. Soc. Lond. A 243, 359.Google Scholar
Tavoularis, S., Bennett, J. C. & Corrsin, S. 1978 Velocity-derivative skewness in small Reynolds number nearly isotropic turbulence. J. Fluid Mech. 88, 63.Google Scholar
Uberoi, M. S. & Freymuth, P. 1969 Spectra of turbulence in wakes behind circular cylinders. Phys. Fluids 12, 1359.Google Scholar
Ueda, H. & Hinze, J. O. 1975 Fine-structure turbulence in the wall region of a turbulent boundary layer. J. Fluid Mech. 67, 125.Google Scholar
Van Atta, C. W. & Chen, W. Y. 1969 Measurements of spectral energy transfer in grid turbulence. J. Fluid Mech. 38, 743.Google Scholar
Wilson, K. G. 1975 The renormalisation group: critical phenomena and the Kondo problem. Rev. Mod. Phys. 47, 773.Google Scholar
Wyld, H. W. 1961 Formulation of the theory of turbulence in an incompressible fluid. Ann. Phys. 14, 143.Google Scholar
Wyngaard, J. C. & Tennekes, H. 1970 Measurements of the small-scale structure of turbulence at moderate Reynolds numbers. Phys. Fluids 8, 1962.Google Scholar