The Asymptotic State of Decaying Turbulence

The long-time evolution of decaying homogeneous turbulence is a fundamental building block of the subject. We investigate the problem by using a comprehensive suite of Direct Numerical Simulations. The simulations cover initial Taylor microscale Reynolds numbers $Re_\lambda$ from $30 \text{ to } 145$, with multiple independent realizations obtained at each $Re_{\lambda}$ to ensure statistical robustness. The energy spectrum is initialized with the Birkhoff-Saffman (BS) form (with $E(k)\sim k^2$ for small $k$) in one case, and the Loitsianskii-Kolmogorov-Batchelor (LKB) form (with $E(k)\sim k^4$ for small $k$), in another. Simulations are performed for unprecedented durations, of the order of 200,000 initial eddy-turnover times in some instances. For both BS and LKB, the turbulent kinetic energy $En$ shows, after an initial transient, unambiguous power-law decay, $En\sim t^{-n}$, with nearly constant decay exponents $n$, whose values are consistent with past theoretical results (and thus not universal). We compute various length scales, second-order structure functions, and the spectral form at large wavenumbers; we note that an initially set $-5/3$ slope disappears quickly, while a perceptible $-1$ power region appears. In particular, we compare the present findings with predictions from the recent theory for decaying turbulence developed by Migdal 2026 Philos. Trans. R. Soc. A 384, 20250032. (doi:10.1098/rsta.2025.0032). The agreement for the BS case is excellent except for the large-wavenumber spectrum. A general discussion and assessment of results is provided in terms of the putative universality of energy decay. A main conclusion is that the energy decay is significantly influenced by ``boundary effects", and that universality likely manifests only when those effects are removed. Alternatively, it may be more useful to discuss the universality of enstrophy decay.