Is the dissipation anomaly strong or weak in wall-free turbulence?
In plain words
Experiments suggest that turbulent energy loss stays finite even as the fluid viscosity is made vanishingly small, a property called the dissipation anomaly. Recent large simulations without walls hint that the effect may instead fade very slowly as the flow speed grows.
Precise statement
Let $D = \epsilon L / U^{3}$ be the mean kinetic-energy dissipation rate $\epsilon$ normalized by rms velocity $U$ and box scale $L$, for body-forced Navier-Stokes turbulence in a periodic box. As $\mathrm{Re} = U L / \nu \to \infty$, does $D$ tend to a positive constant (strong anomaly), or decay as $\mathrm{Re}^{-\alpha}$ with small $\alpha > 0$ (weak anomaly)? A strong anomaly requires $\zeta_3 \le 1$ for absolute increments; $\zeta_3 > 1$ forces $D \to 0$ and bounds alpha from below. Answer: the limit of $D$, or $\alpha$ with error bar, shown to be independent of forcing details.
What would settle it
DNS of periodic-box turbulence over at least one further decade of Re, or a laboratory flow without walls, resolving whether D(Re) levels off or keeps decreasing.
Status in the literature
Unverified note
2025 periodic-box DNS (Iyer, Drivas, Eyink, Sreenivasan, arXiv:2504.13298) report $\alpha \sim 0.05$ and $\zeta_{3} \sim 1.07 \pm 0.07$, consistent with a weak anomaly; flows with walls or bluff bodies show $D \to \mathrm{constant}$.