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Homogeneous, Isotropic TurbulencePhenomenology, Renormalization and Statistical Closures$
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W. David McComb

Print publication date: 2014

Print ISBN-13: 9780199689385

Published to Oxford Scholarship Online: April 2014

DOI: 10.1093/acprof:oso/9780199689385.001.0001

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Kolmogorov’s (1941) theory revisited

Kolmogorov’s (1941) theory revisited

(p.143) 6 Kolmogorov’s (1941) theory revisited
Homogeneous, Isotropic Turbulence

W. David McComb

Oxford University Press

By restricting our attention in this book to HIT, we have ruled out effects due to mean shear, system rotation, density stratification; and so on. This leaves us with a stark choice: deviation from Kolmogorov’s (1941) predictions for the energy spectrum (or second-order structure function) in stationary flows must be due to either the Reynolds number being finite (K41 is based on an assumption of very large Reynolds numbers) or internal intermittency, as was suggested later on by Kolmogorov, in 1962. Over the last few decades a veritable industry has grown up, based on the search for so-called intermittency corrections. Currently it is dominated by multiscale or multifractal models of turbulence. This activity tends to find a receptive audience, because many people seem to regard the K41 picture as being counter-intuitive, when one considers aspects of turbulence such as vortex-stretching, localness, and intermittency. Running counter to this belief in ‘intermittency corrections’ (or, increasingly, ‘anomalous exponents’) which has been dominant in recent times, there is a growing view that K41is an asymptotic theory, valid in the limit of infinite Reynolds number. In this school of thought, any deviations from K41 are due to finite viscosity. As a result, opinion in the turbulence community is deeply divided on this fundamental issue This chapter begins by considering the various criticisms of, or sources of unease about, K41. Then it reformulates the K41 spectral theory in terms of scale-invariance, the resolution of the scale-invariance paradox, and conservation of energy. This is followed by a discussion of the various theories which advocate finite-viscosity effects in explaining deviations from K41. We conclude with a brief discussion of the current situation.

Keywords:   intermittency corrections, anomalous exponents, scale invariance, inertial range, Kolmogorov prefactor, infinite Reynolds number limit, finite Reynolds number effects

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