  We investigate the role of inter-scale interactions in the high-Reynolds
number skin-friction drag reduction strategy reported by Marusic et al. (Nat.
Commun., vol. 12, 2021). The strategy involves imposing relatively
low-frequency streamwise travelling waves of spanwise velocity at the wall to
actuate the drag generating outer-scales. This approach has proven to be more
energy-efficient than the conventional method of directly targeting the drag
producing inner-scales, which typically requires actuation at higher
frequencies. Notably, it is observed that actuating the outer-scales at low
frequencies leads to a substantial attenuation of the major drag producing
inner-scales, suggesting that the actuations affect the non-linear inner-outer
coupling inherently existing in wall-bounded flows. In the present study, we
find that increased drag reduction, through imposition of spanwise wall
oscillations, is always associated with an increased coupling between the inner
and outer scales. This enhanced coupling emerges through manipulation of the
phase relationships between these triadically linked scales, with the actuation
forcing the entire range of energy-containing scales, from the inner (viscous)
to the outer (inertial) scales, to be more in-phase. We also find that a
similar enhancement of this non-linear coupling, via manipulation of the
inter-scale phase relationships, occurs with increasing Reynolds number for
canonical turbulent boundary layers. This indicates improved efficacy of the
energy-efficient drag reduction strategy at very high Reynolds numbers, where
the energised outer-scales are known to more strongly superimpose and modulate
the inner-scales. Leveraging the inter-scale interactions, therefore, offers a
plausible mechanism for achieving energy-efficient drag reduction at high
Reynolds numbers.
