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

We present a detailed eddy-damped quasi-normal Markovian (EDQNM) analysis o f three-dimensional isotropic turbulence (without helicity) at very high Re ynolds number. The EDQNM equations are solved numerically using Lesieur and Schertzer's method (1978 J. Mecanique 17 609-46). We first consider the fr eely decaying case and look for behaviours at very high Reynolds numbers. W e show that the skewness factor time evolution reaches an asymptotic limit when the Reynolds number goes to infinity. We show also a perfect self-simi larity of the ultraviolet kinetic-energy spectrum in terms of Kolmogorov di ssipative units, with the classical 'bump'-shaped self-preserving spectrum, and a short k(-5/3) plateau increasing extremely slowly. Still in the deca ying case, but at lower Reynolds number, we concentrate on the infrared eff ects and kinetic-energy decay. We consider an initial kinetic-energy spectr um scaling as k(s) for k --> 0. Calculations are carried out up to 600 init ial 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 spec tral backscatter is observed, and the kinetic energy time-decay exponents a lpha (E) are in good agreement with high-Reynolds-number asymptotic self-si milar laws, namely 1, 6/5 and 4/3. Between s = 3 and s = 4, backscatter app ears, and aE 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 achi eved only for s = 1. In this case, the Reynolds numbers based upon the inte gral scale and the Taylor microscale are constant with time in the self-sim ilar regime. We look also at finite-size effects, and show an extremely slo w relaxation of alpha (E) towards 2, as was predicted theoretically and rec overed experimentally in liquid helium by Stalp et al (1999 Phys. Rev. Lett . 82 4831-4). The last part of the work concerns stationary forced turbulen ce, for which we recover the bump-shaped energy spectrum, and the infrared k(2) equipartition kinetic-energy spectrum.