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Since about 1997, the realisation that the finite Reynolds number (FRN) effect needs to be carefully taken into account when assessing the behaviour of small-scale statistics came to the fore. The FRN effect can be analysed either in the real domain or in the spectral domain via the scale-by-scale energy budget equation or the transport equation for the energy spectrum. This analysis indicates that the inertial range (IR) is established only when the Taylor microscale Reynolds number R e λ is infinitely large, thus raising doubts about published power-law exponents at finite values of R e λ , for either the second-order velocity structure function (δ u) 2 ¯ or the energy spectrum. Here, we focus on the transport equation of (δ u) 2 ¯ in decaying grid turbulence, which represents a close approximation to homogeneous isotropic turbulence. The effect on the small-scales of the large-scale forcing term associated with the streamwise advection decreases as R e λ increases and finally disappears when R e λ is sufficiently large. An approach based on the dual scaling of (δ u) 2 ¯ , i.e., a scaling based on the Kolmogorov scales (when the separation r is small) and another based on the integral scales (when r is large), yields (δ u) 2 ¯ ∼ r 2 / 3 when R e λ is infinitely large. This approach also yields (δ u) n ¯ ∼ r n / 3 when R e λ is infinitely large. These results seem to be supported by the trend, as R e λ increases, of available experimental data. Overall, the results for decaying grid turbulence strongly suggest that a tendency towards the predictions of K41 cannot be dismissed at least at Reynolds numbers which are currently beyond the reach of experiments and direct numerical simulations. [ABSTRACT FROM AUTHOR]

A re-examination of some early classic turbulence literature, mainly of isotropic turbulence, is given in this selective review. Some early studies, including original concepts and points, are reviewed or highlighted. Two earliest studies and six major original concepts are found: (i) Lord Kelvin's pioneering elementary studies of homogeneous, isotropic turbulence; (ii) Kelvin's early introduction of Fourier Principles into turbulence studies; (iii) the Kelvin elementary concept of the direct energy cascade; (iv) the Kelvin early concept of the symmetry of turbulence; (v) the Taylor concept of the coefficient of eddy viscosity; (vi) the Taylor concept of the 'age' of the eddy; (vii) the Taylor–Fage–Townend concept of small eddies or microturbulence or small scale turbulence; and (viii) the Obukhov concept of a function of the inner Reynolds number (i.e., R e dependent coefficient) in both the balance equation and the energy distribution equation (the two-thirds law). Both Kelvin and Taylor should be regarded as the co-founders of the statistical theory of homogeneous, isotropic turbulence. The notion, 'the Maxwell–Reynolds decomposition of turbulent flow velocity', should be used. The Kolmogorov–Obukhov scaling laws are reviewed in detail. Heisenberg's inverse seventh power spectrum is briefly reviewed. The implications or significances of these early studies, original concepts and points are briefly discussed, with special reference to their possible links with modern approaches and theories. [ABSTRACT FROM AUTHOR]

The pioneering applications of the methods of theoretical physics to the turbulence statistical closure problem are summarised. These are: the direct-interaction approximation (DIA) of Kraichnan, the self-consistent-field theory of Edwards, and the self-consistent-field theory of Herring. Particular attention is given to the latter, in terms of its elegance and its pedagogical value. We then concentrate on the assessment of these theories and take the historical route of Kraichnan's diagnosis of the failure of DIA, followed by Edwards's analysis of the failure of his self-consistent theory, when compared to the Kolmogorov spectrum. As all three theories are closely related, these analyses also shed light on Herring's theory. The second-generation theories that grew out of this assessment are then discussed. First, there were the Lagrangian theories, initially stemming from the work of Kraichnan and Herring, and later the purely Eulerian local energy-transfer (LET) theory. The latter is significant because its development exposes the underlying problems with the pioneering theories in terms of the basic physics of the inertial energy transfer. In particular, later work allows us to assign a unified explanation of the incompatibility of all three pioneering theories with the Kolmororov spectrum, in that they are all Markovian approximations (in wavenumber) to the non-Markovian phenomenon of fluid turbulence. In the interests of completeness, we briefly review the formalisms of Wyld and Martin, Siggia, and Rose. More recent developments are also discussed, in order to bring the subject up to the present day. [ABSTRACT FROM AUTHOR]