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Rederivation and further assessment of the LET theory of isotropic turbulence, as applied to passive scalar convection

Published online by Cambridge University Press:  26 April 2006

W. D. Mccomb
Affiliation:
Department of Physics, University of Edinburgh, King's Buildings, Mayfield Road. Edinburgh EH9 3JZ, UK
M. J. Filipiak
Affiliation:
Department of Physics, University of Edinburgh, King's Buildings, Mayfield Road. Edinburgh EH9 3JZ, UK
V. Shanmugasundaram
Affiliation:
Department of Physics, University of Edinburgh, King's Buildings, Mayfield Road. Edinburgh EH9 3JZ, UK

Abstract

A simpler and more rigorous derivation is presented for the LET (Local Energy Transfer) theory, which generalizes the theory to the non-stationary case and which corrects some minor errors in the original formulation (McComb 1978), Previously, ad hoc generalizations of the LET theory (McComb & Shanmugasundaram 1984) gave good numerical results for the free decay of isotropic turbulence. The quantitative aspects of these previous computations are not significantly affected by the present corrections, although there are some important qualitative improvements.

The revised LET theory is also extended to the problem of passive scalar convection, and numerical results have been obtained for freely decaying isotropic turbulence, with Taylor–Reynolds numbers in the range 5 [les ] Rλ [les ] 1060, and for Prandtl numbers of 0.1, 0.5 and 1.0. At sufficiently high values of the Reynolds number, both velocity and scalar spectra are found to exhibit Kolmogorov-type power laws, with the Kolmogorov spectral constant taking the value α = 2.5 and the Corrsin–Oboukhov constant taking a value of β = 1.1.

Type
Research Article
Copyright
© 1992 Cambridge University Press

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References

Antonia, R. A. & Chambers C. W. 1980 On the correlation between turbulent velocity and temperature derivatives in the atmospheric surface layer. Boundary-Layer Met. 18, 399.Google Scholar
Bradshaw P. 1967 Conditions for the existence of an inertial sub-range in turbulent flow. Natl Phys. Lab. Aero. Rep. 1220.Google Scholar
Champagne F. H., Friehe, C. A. & LaRue J. C. 1977 Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land. J. Atmos. Sci. 35, 515.Google Scholar
Filipiak M. J. 1991 Further assessment of the LET theory. Ph.D. thesis, University of Edinburgh.
Helland K. N., Van Atta, C. W. & Stegen, G. R. 1977 Spectral energy transfer in high Reynolds number turbulence. J. Fluid Mech. 79, 337.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. 1971 Comparisons 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.
Kaneda Y. 1981 Renormalised expansion in the theory of turbulence with the use of the Lagrangian position function. J. Fluid Mech. 107, 131.Google Scholar
Kerr R. M. 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 31.Google Scholar
Kraichnan R. H. 1959 The structure of isotropic turbulence at very large 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. 1977 Eulerian and Lagrangian renormalisation in turbulence theory. J. Fluid Mech. 83, 349.Google Scholar
Kubo R. 1966 The fluctuation-dissipation theorem. Rep. Prog. Phys. XXIX, 255.Google Scholar
Lee J. 1965 Decay of scalar quantity fluctuations in a stationary isotropic turbulent velocity field. Phys. Fluids 8, 1647.Google Scholar
Lumley J. L. 1964 The spectrum of nearly inertial turbulence in a stably stratified fluid. J. Atmos. Sci. 21, 99.Google Scholar
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 (referred to herein as I).Google Scholar
McComb, W. D. & Shanmugasundaram V. 1984 Numerical calculation of decaying isotropic turbulence using the LET theory. J. Fluid Mech. 143, 95 (referred to herein as II).Google Scholar
McComb W. D., Shanmugasundaram, V. & Hutchinson P. 1989 Velocity-derivative skewness and two-time correlations of isotropic turbulence as predicted by the LET theory. J. Fluid Mech. 208, 91 (referred to herein as III).Google Scholar
Moeng, C.-H. & Wyngaard J. C. 1988 Spectral analysis of large-eddy simulations of the convective boundary layer. J. Atmos. Sci. 45, 3573.Google Scholar
Nakano T. 1988 Direct interaction approximation of turbulence in the wave packet representation. Phys. Fluids 31, 1420.Google Scholar
Orszag, S. A. & Patterson G. S. 1972 Numerical simulation of three-dimensional homogeneous, isotropic turbulence. Phys. Rev. Lett. 28, 76.Google Scholar
Uberoi M. S. 1963 Energy transfer in isotropic turbulence. Phys. Fluids 6, 1048.Google Scholar
Wyld H. W. 1961 Formulation of the theory of turbulence in an incompressible fluid. Ann. Phys. 14, 143.Google Scholar