Does the near-wall peak of streamwise fluctuations grow without bound?
In plain words
The strength of turbulent fluctuations just above a wall, measured relative to the wall friction, keeps rising as the flow gets faster in all experiments so far. Whether it rises forever or levels off is disputed, and the data cannot yet decide.
Precise statement
The inner peak of the streamwise variance $\langle u'^2\rangle+$ at $y+ \sim 15$ grows roughly as $3.7 + 0.64\operatorname{ln} \mathrm{Re}_{\tau}$ (approximately) in channel DNS for $\mathrm{Re}_{\tau}$ from 180 to about $10^4$. As $\mathrm{Re}_{\tau} \to \infty$, does it diverge logarithmically (attached-eddy prediction) or approach a finite limit with corrections of order $1/\operatorname{ln} \mathrm{Re}_{\tau}$ or $\mathrm{Re}_{\tau}^{-1/4}$? Answer: the asymptotic form, with evidence beyond $\mathrm{Re}_{\tau} \sim 10^5$ where the competing fits separate.
What would settle it
Well-resolved near-wall measurements or DNS at $\mathrm{Re}_{\tau} \sim 10^{5}$ or above distinguishing logarithmic growth from saturation.
Status in the literature
Unverified note
Chen and Sreenivasan (2021), Monkewitz (2022) and Hwang (2024) argue for a bound; Jimenez (2024, arXiv:2408.09259) argues the peak probably grows in wall units while vanishing relative to the bulk velocity.