Numerical calculation of decaying isotropic turbulence using the LET theory

Abstract

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.

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].…”

A Comparison of Statistical Dynamical and Ensemble Prediction Methods during Blocking

“…Closely related non-Markovian closures, such as Herring's self-consistent field theory (SCFT; Herring 1965) and McComb's local energy transfer theory (LET; McComb 1974;McComb and Shanmugasundaram 1984), are essentially variants of Kraichnan's original DIA, which differ only in their application of the fluctuation dissipation theorem (Frederiksen and Davies 2000;Kiyani and McComb 2004). In the DIA the (direct) interactions between wavenumber components of the second-order cumulants and response functions are represented in terms of the same original (or "bare") interaction coefficients that appear in the DNS equations from which the closures are derived.…”

Closure model for homogeneous isotropic turbulence in the Lagrangian specification of the flow field

“…However, the second is incompatible with the characteristic time scale of the Eulerian velocity correlation, unlike the Eulerian DIA. The abridged Lagrangian DIA and the LET model yield Kolmogorov constants of 1.77 (Kraichnan 1966) and2.3 (McComb &Shanmugasundaram 1984), respectively. Regarding the former, which is very complicated to derive, two Lagrangian DIAs have been proposed: the Lagrangian renormalized approximation (Kaneda 1981), which introduces the Lagrangian position function to simplify the Lagrangian treatment, and the sparse direct-interaction perturbation method (Kida & Goto 1997), which applies the original Eulerian DIA presented by Kraichnan (1959) to the Lagrangian framework.…”