The asymptotic state of decaying turbulence

The long-time evolution of decaying homogeneous turbulence is a fundamental building block of the theory and modeling of turbulence. We investigate the problem by using a comprehensive suite of direct numerical simulations. The simulations cover initial Taylor microscale Reynolds numbers Reλ from 30 to 145, with multiple independent realizations obtained at each Reλ to ensure statistical robustness. The energy spectrum is initialized with the Birkhoff-Saffman (BS) form (with E(k)∼k2 for small k) in one case, and the Loitsianskii-Kolmogorov-Batchelor (LKB) form (with E(k)∼k4 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∼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 (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. This article is part of the theme issue 'Frontiers of turbulence and statistical physics'.