The Asymptotic State of Decaying Turbulence

TL;DR

This work investigates the asymptotic state of decaying homogeneous isotropic turbulence using unprecedented direct numerical simulations (DNS) with controlled BS ($E(k) \sim k^2$) and LKB ($E(k) \sim k^4$) low-k spectra. It rigorously tests Migdal's loop-space “Euler ensemble” theory, finding strong agreement for BS in energy decay, length-scale evolution, and the internal spectral structure, while the LKB regime shows nonuniversal energy decay despite similar internal features. The results emphasize the role of boundary effects in large-scale universality and suggest focusing on enstrophy decay as a more robust universal descriptor, while highlighting the need for higher Reynolds number studies to probe the predicted $E(k) \sim k^{-7/2}$ tail in the dissipation range.

Abstract

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 $Re_λ$ from $30 \text{ to } 145$, with multiple independent realizations obtained at each $Re_λ$ 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 \cite{migdal_this_issue}. 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.}

Figures (17)

• Figure 1: Initial energy spectra ($t=0$) used for the simulations. The spectra are normalized by the initial integral length $L$ and initial total energy $En(t=0)$. The wavenumber $k$ is normalized by $L$. (a) BS spectra with the prescribed $E(k) \sim k^2$ scaling at low wavenumbers. (b) LKB spectra with the prescribed $E(k) \sim k^4$ scaling at low wavenumbers. Different colored curves correspond to different initial Taylor-microscale Reynolds numbers ($Re_\lambda$) for each set of simulations. The dot-dashed lines indicate the theoretical slopes for the low-wavenumber range and the $k^{-5/3}$ inertial range.
• Figure 2: Validation of the grid modification method for the $Re_\lambda = 93$ simulation. The plot compares the time evolution of the decay exponent $n(t)$ for the standard simulation (solid black line) against simulations using grid modification with various thresholds (dashed lines). The curves for lower thresholds depart measurably, whereas the $k_{max}\eta = 6.0$ case (red dashed line) overlaps perfectly with the unmodified simulation, demonstrating no statistical modification.
• Figure 3: Quantification of fractional energy loss incurred due to spectral truncation at different regridding thresholds ($k_{max}\eta$) for $Re_{\lambda} = 93$.
• Figure 3: Time evolution of (a) total kinetic energy $En(t)$ and (b) its local decay exponent $n(t)$ for simulations with an initial BS spectrum ($E(k) \sim k^2$ for small $k$). The different-colored curves in both panels correspond to simulations with varying initial Taylor Reynolds numbers ($Re_\lambda$), as indicated in the legends. The time axis is normalized by the initial large-eddy turnover time, $T_{eddy,0}$. (a) $En$ is normalized by its initial value $En(t=0)$. The dot-dashed reference line shows Migdal's theoretical power-law decay slope of $-5/4$ (to be described later). High Reynolds number data asymptote to that slope. (b) The local kinetic energy decay exponent, defined in Eq. (\ref{['eq:decay_exp']}). The horizontal dot-dashed line marks $n = 5/4$ (which, to anticipate the results in Sec. 4, corresponds to Migdal's theory). The inset provides a magnified view of the early-time evolution, showing the approach towards 5/4. The index $n$ for the lowest Reynolds number slope does not touch 5/4 but it does so at the higher Reynolds numbers, with the range increasing correspondingly, as one may have expected.
• Figure 4: Time evolution of characteristic length scales and the Taylor Reynolds number for simulations with an initial BS spectrum ($E(k) \sim k^2$ for small $k$). All panels show results from simulations with the same initial $Re_\lambda$ (see legends), with time normalized by the initial large-eddy turnover time, $T_{eddy,0}$. The subplots show: (a) the integral length scale $L$, (b) the Taylor microscale $\lambda$, and (c) the Kolmogorov length scale $\eta$, each normalized by the edge length of the domain box ($L_{box}=2\pi$). (d) The unnormalized Taylor Reynolds number $Re_\lambda$. The dashed reference lines indicate the theoretical power-law behaviors: $L \sim t^{2/5}$ in (a), $\lambda \sim t^{1/2}$ in (b), $\eta \sim t^{9/16}$ in (c), and $Re_\lambda \sim t^{-1/8}$ in (d).