Rederivation and further assessment of the LET theory of isotropic turbulence, as applied to passive scalar convection

Abstract

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.

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].…”

Entropy, Closures and Subgrid Modeling

“…The SCFT closure of Herring (1965Herring ( , 1966 [43,44] and the LET closure of McComb (1974McComb ( , 1992 [45,46] are non-Markovian Eulerian closure theories of homogeneous turbulence that are closely related to Kraichnan's (1959) DIA [42]. They were formulated independently but in retrospect the SCFT and LET closure equations may be formally obtained from the DIA by invoking the fluctuation-dissipation theorem (FDT, Kraichnan (1959b) [48], Carnevale & Frederiksen (1983) [51]) out of strict statistical mechanical equilibrium as noted by Frederiksen, Davies & Bell (1994) [52].…”

“…The development of modern turbulence closures based on renormalized perturbation theory was pioneered by Kraichnan (1958 [41,42] who formulated the Eulerian direct interaction approximation (DIA) for homogeneous turbulence. Herring's self consistent field theory (SCFT, Herring (1965Herring ( , 1966 [43,44]) and McComb's local energy transfer theory (LET, McComb (1974) [45], McComb et al (1992) [46], McComb & Quinn (2003) [47]) were independently developed. The SCFT has the same equations for the single-time cumulant and response function as the DIA but obtains the two-time two-point cumulant from a fluctuation dissipation theorem (FDT; [48], Leith (1975) [49], Deker & Haake (1975) [50], Carnevale & Frederiksen (1983) [51]).…”

“…These closures and related Markovianized versions such as the eddy-damped quasi-normal Markovian model (EDQNM, Orszag (1970) [54], Leith (1971) [55], Bowman, Krommes & Ottaviani (1993) [56], Frederiksen & Davies (1997) [57]), test field model (TFM, [58], [59], Leith & Kraichnan (1972) [60]) and realizable TFM (Bowman & Krommes (1997) [61]) have been successfully applied to a variety of important problems. Their performance has been compared with direct numerical simulations and experimental data (e.g., Herring et al (1974) [62], Pouquet et al (1975) [63], Kraichnan & Herring (1978) [64], McComb (1990) [65], McComb et al (1992) [46], Frederiksen, Davies & Bell (1994) [52], Frederiksen & Davies (2000, 2004 [66,67]). Importantly turbulence closures allow the study of the statistics of the predictability of homogeneous turbulent flows (Kraichnan (1970) [68], Leith (1971Leith ( , 1974 [55,69], Leith & Kraichnan (1972) [60], Métais & Lesieur (1986) [70]) and most recently of inhomogeneous turbulen...…”

On the spectrum of mid-latitude sporadic-<i>E</i> irregularities

“…Since the neutral atmosphere turbulence in the altitude range of interest is approximately isotropic with a large scale of about 5 km (Justus, 1967) and the sporadic-E thickness is usually not larger than this scale, then crosscorrelations of the velocity and scalar ®elds, scalar 3 velocity and velocity 3 scalar, are ruled out by the restriction to passive convection and by the symmetry requirements of isotropic turbulence (e.g., see McComb et al, 1992). Hence, hdn Á u 1 i 0, and Eq.…”