Does small-scale anisotropy of a mixed scalar vanish at high Reynolds number?
In plain words
When turbulence stirs a dye or temperature that varies steadily in one direction, the small-scale ripples of the dye keep a preferred direction even at the highest speeds tested. This contradicts the textbook expectation that small scales forget large-scale directions, and whether the memory ever disappears is unknown.
Precise statement
For a passive scalar $\theta$ with uniform mean gradient $G$ in turbulence, the skewness $S = \langle(\mathrm{d}\theta/\mathrm{d}x)^3\rangle/\langle(\mathrm{d}\theta/\mathrm{d}x)^2\rangle^{3/2}$ of the scalar derivative along $G$ is of order 1 in experiments and DNS for $\mathrm{Re}_{\lambda}$ from about $10^2$ to $10^4$, whereas local isotropy requires $S \to 0$. Does $S \to 0$ as $\mathrm{Re}_{\lambda} \to \infty$ at Schmidt number $\mathrm{Sc} \sim 1$, and at what rate? Answer: the limit of $S$ and its $\mathrm{Re}_{\lambda}$ dependence.
What would settle it
Measurements or DNS of $S$ over $\mathrm{Re}_{\lambda}$ from $10^3$ to $10^5$ at $\mathrm{Sc}\sim 1$ showing a definite trend.
Status in the literature
At $Sc \sim 1$ the derivative skewness stays of order 1 up to the highest $Re_{\lambda}$ measured; DNS at $Re_{\lambda} 140 \text{ to } 650$ show isotropy is restored as Sc grows at fixed $Re_{\lambda}$ (Buaria, Clay, Sreenivasan, Yeung, PRL 2021), a ramp-cliff model fitting the trend.