Number Theory of Decaying Turbulence 3: Comparing with Physical and Numerical Experiments

Abstract

We compare the predictions of the Euler-ensemble solution of freely decaying incompressible turbulence with simulations and experiments: the scaling function of the energy spectrum, the index of the second-order structure function and the decay laws. The predictions have a fixed shape; only their normalization and scale are fitted. In the final $4096^3$ direct numerical simulations of Rodhiya and Sreenivasan, the turbulent attractor is selected by a parabola law for the growth of the bulk length. Along it the collapsed spectrum depends only on the instantaneous Reynolds number, identically for Saffman and Loitsyansky initial spectra; extrapolated to infinite Reynolds number, it agrees with the theory to 1% in the bulk of the enstrophy range, below the dissipation range. The energy balance holds exactly. The energy integral of the theory diverges in the infrared, so energy and enstrophy depend on how a finite flow cuts off the spectrum: with a sharp cutoff $k_0$, the decay laws, energy $\propto t^{-5/4}$ and enstrophy $\propto t^{-9/4}$, are reached only when the cutoff enters the $k^{-7/2}$ tail of the spectrum. Such a cutoff is only an estimate; the reliable test is the bulk of the spectrum. Fitted to the simulations, the cutoff lies at the lower edge of the bulk, where the spectrum still carries the initial data, and the decay exponents depend on the initial infrared spectrum. In the Max Planck wind tunnel at $Re_λ$ up to 5779, the longitudinal index extrapolated to infinite Reynolds number matches the odd ensemble with one scale parameter at large separations, where standard crossover forms fail. The freely decaying turbulent blob of Matsuzawa et al. is too small, too inhomogeneous and too weak to display the attractor. The theory is derived in the first two papers of this series.