Velocity-derivative skewness and two-time velocity correlations of isotropic turbulence as predicted by the LET theory

Abstract

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.

On space and time correlations of isotropic and rotating turbulence

“…Note, however, that the inertial eddy-turnover time has been observed in Ref. 11 as the dominant time scale.…”

Eulerian field-theoretic closure formalisms for fluid turbulence

“…Notable ones are those of Nakano [19], and the Local Energy Transfer (LET) theory of McComb [9,[20][21][22][23][24][25][26][27][28][29][30]; the latter being the only purely Eulerian theory which is compatible with the Kolmogorov inertial range. Convinced of the perceived intrinsic failings of the DIA based on an Eulerian framework, Kraichnan reformulated fluid dynamics to use Lagrangian variables and produced the Lagrangian-DIA [31].…”

On modal time correlations of turbulent velocity and scalar fields

“…McComb et al [9] studied numerical solutions to the DIA and local energy transfer (LET) theories and, in contrast to Kraichnan's asymptotic prediction for the DIA, found that while neither the 'sweeping' time scale nor the inertial eddy-turnover time scale were completely effective in collapsing the modal time-correlation data, the inertial scaling became more dominant for both theories as R λ was increased. Finally, Gotoh et al [3] studied the modal time correlation using both DNS and DIA at R λ ∼ 35 and found that the sweeping time scale gave a poor collapse of the correlation data from the DIA relative to the data from the DNS.…”