We address the Reynolds number dependence of the turbulent skin-friction drag reduction induced by streamwise-travelling waves of spanwise wall oscillations. The study relies on direct numerical simulations of drag-reduced flows in a plane open channel at friction Reynolds numbers in the range 1000 Re_ 6000, which is the widest range considered so far in simulations with spanwise forcing. Our results corroborate the validity of the predictive model proposed by Gatti & Quadrio (J. Fluid Mech. vol. 802, 2016, pp. 553–558): regardless of the control parameters, the drag reduction decreases monotonically with Re at a rate that depends on the drag reduction itself and on the skin-friction of the uncontrolled flow. We do not find evidence in support of the results of Marusic et al. (Nat. Commun. vol. 12, no. 1, 2021, pp. 5805), which instead report by experiments an increase of the drag reduction with Re in turbulent boundary layers, for control parameters that target low-frequency, outer-scaled motions. Possible explanations for this discrepancy are provided, including obvious differences between open channel flows and boundary layers, and possible limitations of laboratory experiments.
Gatti et al. (Sun,) studied this question.