Results from full numerical simulations of the Navier–Stokes equations for three-dimensional, stationary, and isotropic turbulence are used (1) to study the scaling properties of two-time modal velocity correlations and (2) to determine the dominant process in the decorrelation of the small-scale velocities. The main results obtained are (1) that the convective sweeping of the small scales by the energy-containing large scales determines the scaling property of the inertial- and near-dissipation ranges and (2) that the sweeping effect is dominant in the decorrelation of the small scales. Reynolds-number dependence of the decorrelation is also considered.

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.

The local-energy-transfer (LET) theory was used to calculate freely decaying turbulence for arbitrary initial conditions over a range of microscale-based Reynolds numbers 0.5 [les ] Rλ(tf) [les ] 1009, where tf is the final time of computation. The predicted skewness factor S(Rλ) agreed closely with the results of numerical simulations at low-to-moderate Reynolds numbers and followed the same general trend at larger values of Rλ. It was also found that, for Rλ(tf) [les ] 5, the LET calculation was almost indistinguishable from that of the direct-interaction approximation (DIA), with the difference between the two theories tending to zero as Rλ(tf)∞ 0.Two-time correlation and propagator (or response) functions were also obtained. Tests of their scaling behaviour suggest that, contrary to general belief, the convective sweeping of the energy-containing range is much less important than the Kolmogorov timescale in determining inertial-range behaviour. This result raises questions about the accepted explanation for the failure of the direct-interaction approximation, thus motivating a discussion about the relevance of random Galilean invariance (RGI). It is argued that, for a properly constructed ensemble of transformations to inertial frames, invariance in every realization necessarily implies RGI. It is suggested that the defects of the direct-interaction approximation can be understood in terms of a failure to renormalize the stirring forces.

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.

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