Do high-order structure-function exponents saturate at large order?
In plain words
The statistics of the most extreme velocity differences appear to stop becoming more extreme beyond a certain order, as if the flow had a limiting roughness. Whether this saturation is real, and the same for differences measured along and across the separation, is not established.
Precise statement
In isotropic Navier-Stokes turbulence at $\mathrm{Re}_{\lambda} \to \infty$, do the transverse and longitudinal exponents $\zeta_p^T$ and $\zeta_p^L$ tend to a common finite limit $\zeta_{\mathrm{inf}}$ as $p \to \infty$? DNS up to $\mathrm{Re}_{\lambda} \sim 1300$ report saturation of transverse exponents near $\zeta_{\mathrm{inf}} \sim 2$. Answer: yes or no for each increment type, with $\zeta_{\mathrm{inf}}$ and error bars from statistically converged tails for $p$ up to about 12.
What would settle it
Converged DNS or experimental moments of both increment types at $Re_{\lambda}$ above about 2000 showing a common plateau, or a demonstrated difference.
Status in the literature
Unverified note
Transverse saturation near 2 in isotropic DNS (Iyer, Sreenivasan, Yeung 2020); longitudinal saturation near $2.2 \pm 0.1$ in turbulent shear layers at $\operatorname{Re}_{\lambda}$ up to 1400 (Gupta and Bewley 2026, arXiv:2605.01867); whether the two limits coincide in isotropic turbulence is untested.