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On the spectral turbulent diffusivity theory for homogeneous turbulence

Published online by Cambridge University Press:  19 April 2006

Ruwim Berkowicz  and
Lars P. Prahm
Ruwim Berkowicz
Affiliation:
National Agency of Environmental Protection, Air Pollution Laboratory, Risø National Laboratory, DK-4000 Roskilde, Denmark
Lars P. Prahm
Affiliation:
National Agency of Environmental Protection, Air Pollution Laboratory, Risø National Laboratory, DK-4000 Roskilde, Denmark
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Abstract

The spectral turbulent diffusivity (STD) theory, originally deduced from a spectral generalization of the gradient-transfer theory (Berkowicz & Prahm 1979), is here derived from a basic concept of turbulent mixing for the case of homogeneous turbulence. The turbulent mixing is treated in a way similar to Prandtl's mixing-length concept. The contribution to the turbulent flux from eddies of different length is represented by a linear superposition. The spatial variation of the concentration distribution is described in terms of Fourier series. This procedure results in the spectral diffusivity formulation, which is Eulerian and scale dependent. If the concentration distribution is approximated by a truncated Taylor expansion instead of an exact representation by the Fourier series, the gradient-transfer approximation is retrieved.

The turbulent energy density, as function of the eddy length, is related to the eddy transport velocity and a probability of the occurrence of the eddies. The eddy transport velocity, derived from the relation between the energy spectrum and the Lagrangian correlation function, is used for computation of the spectral turbulent diffusivity. The turbulent energy spectrum is approximated by the inertial sub-range form $(-\frac{5}{3}$ law). The STD coefficient obtained here has, for large wavenumbers, a slope of $k^{-\frac{4}{3}}$ as predicted previously.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 100 , Issue 2 , 25 September 1980 , pp. 433 - 448
Copyright
© 1980 Cambridge University Press

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