3D isotropic turbulence at very high Reynolds numbers: EDQNM study

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Abstract
We present a detailed eddy-damped quasi-normal Markovian (EDQNM) analysis of three-dimensional isotropic turbulence (without helicity) at very high Reynolds number. The EDQNM equations are solved numerically using Lesieur and Schertzer's method (1978 J. Mécanique 17 609–46). We first consider the freely decaying case and look for behaviours at very high Reynolds numbers. We show that the skewness factor time evolution reaches an asymptotic limit when the Reynolds number goes to infinity. We show also a perfect self-similarity of the ultraviolet kinetic-energy spectrum in terms of Kolmogorov dissipative units, with the classical ‘bump’-shaped self-preserving spectrum, and a short k -5/3 plateau increasing extremely slowly. Still in the decaying case, but at lower Reynolds number, we concentrate on the infrared effects and kinetic-energy decay. We consider an initial kineticenergy spectrum scaling as k s for k → 0. Calculations are carried out up to 600 initial large-eddy turnover times, with integer values of s going from 1 to 8, and non-integer ones ranging from 3.2 to 3.9. Up to s = 3, no sensible spectral backscatter is observed, and the kinetic energy time-decay exponents α E are in good agreement with high-Reynolds-number asymptotic self-similar laws, namely 1, 6/5 and 4/3. Between s = 3 and s = 4, backscatter appears, and α E goes gradually from 4/3 to 1.38. We confirm also that complete self-similarity (i.e. at all scales) at finite Reynolds number can be achieved only for s = 1. In this case, the Reynolds numbers based upon the integral scale and the Taylor microscale are constant with time in the self-similar regime. We look also at finite-size effects, and show an extremely slow relaxation of α E towards 2, as was predicted theoretically and recovered experimentally in liquid helium by Stalp et al (1999 Phys. Rev. Lett. 82 4831–4). The last part of the work concerns stationary forced turbulence, for which we recover the bump-shaped energy spectrum, and the infrared k 2 equipartition kinetic-energy spectrum.