Langevin equation of big structure dynamics in turbulence: Landau's invariant in the decay of homogeneous isotropic turbulence

As empirically observed over some fifty years, the steepness of the infrared spectrum of velocity fluctuations in a turbulent flow-E(kappa) alpha kappa(mu) at scales D similar to 1/kappa >> l larger than energy-containing eddies-constrains the decay of turbulent kinetic energy. However, the theoretical understanding of the physical process controlling decay is still patchy, even in the simplest case of homogeneous isotropic turbulence (HIT). Here, HIT decay laws are derived from angular momentum invariance at big scales D-an approach first mentioned by Landau in 1944, but unduly dismissed later. It is restated in terms of a stochastic equation similar to a Langevin equation, to which usual investigation techniques can be adapted. By deriving two forms of closed equations for the variance of angular momentum fluctuations, various new results on HIT are established: (i) the clear physical difference of the present approach with respect to previous investigations based on the Karman-Howarth equation-or its various integrated forms such as Loitsyanskii's integral-due to the separation of fast (noise) and slow (friction) components of the fluctuating fields; (ii) under the appropriate closure assumptions and necessary conditions, a proof of the so-far conjectured permanence of big structures: (iii) novel relationships relating mu and the velocity correlation function f(s), obtained from the higher order terms of the expansion in D; (iv) the relationship between the exponent of the infrared spectrum kappa(mu) and the decay invariant in self-similar regimes, both for mu <= 3 and 3 < mu <= 4 which display different behavior; (v) the stringent conditions required to reach Kolmogorov's decay exponent n = 10/7 in ideal HIT; (vi) a closed relationship between the velocity correlation function f (s) at big scales and the turbulent viscosity coefficient C-mu. Beyond an improved theoretical understanding and the present results, this framework will lead to new predictions on turbulence decay and dissipation in numerous other types of flows with practical impact. (C) 2011 Elsevier Masson SAS. All rights reserved.