Does spanwise wall forcing still save net power at very high Reynolds number?
In plain words
Moving a wall sideways back and forth, or in travelling waves, can cut turbulent friction by tens of percent. Whether a net energy saving survives at the very high speeds of ships and airliners is unknown.
Precise statement
For channel or boundary-layer flow with spanwise wall velocity $W(x,t) = A \sin(k_x x - \omega t)$, determine the maximum net power saving $S$ (drag reduction minus actuation power, as a fraction of uncontrolled pumping power) over $(A, k_x, \omega)$ as a function of $\mathrm{Re}_{\tau}$, and its limit as $\mathrm{Re}_{\tau} \to \infty$. At $\mathrm{Re}_{\tau} \sim 10^3$ drag reduction reaches about 30 percent; experiments at $\mathrm{Re}_{\tau}$ of order $10^4$ report net savings with actuation tuned to large scales.
What would settle it
Experiments or wall-resolved simulations at $\mathrm{Re}_{\tau}$ from $10^4$ to $10^5$ with measured actuation power, giving $S(\mathrm{Re}_{\tau})$ over that range.
Status in the literature
Predictions of a slow decline of drag reduction with Re (Gatti and Quadrio 2016) and 2021 high-Re experiments with large-scale actuation (Marusic et al.) leave the asymptotic trend open.