The evolution of freely-decaying, isotropic, two-dimensional turbulence

Periodic boundary conditions impose mirror image long-range correlations of velocity and vorticity. Lilly [J. Fluid Mech. 45 (2) (1971) 395-415], and Davidson [Turbulence: An Introduction for Scientists and Engineers, Oxford University Press, 2004], note that these correlations have the potential to influence the dynamical behavior of the turbulence. Varying the energy-containing length-scale of the turbulence relative to the size of the periodic domain allows this effect to be investigated. To this end, we introduce the box-ratio, ldomain/lturbulence, as a measure of the number of energy containing eddies in our simulations. Over a wide range of box-ratios we show that l grows approximately as t1/2, the enstrophy decays as ∼t−1 (at least for large Re), and the l-normalised vorticity correlations more or less collapse onto a single, self-similar curve. We provide one possible explanation for the observed t1/2 growth of l.