  This paper examines how increasing the value of the Reynolds number $Re$
affects the ability of spanwise-forcing techniques to yield turbulent
skin-friction drag reduction. The considered forcing is based on the
streamwise-travelling waves of spanwise wall velocity (Quadrio {\em et al. J.
Fluid Mech.}, vol. 627, 2009, pp. 161--178). The study builds upon an extensive
drag-reduction database created with Direct Numerical Simulation of a turbulent
channel flow for two, 5-fold separated values of $Re$, namely $Re_\tau=200$ and
$Re_\tau=1000$. The sheer size of the database, which for the first time
systematically addresses the amplitude of the forcing, allows a comprehensive
view of the drag-reducing characteristics of the travelling waves, and enables
a detailed description of the changes occurring when $Re$ increases. The effect
of using a viscous scaling based on the friction velocity of either the
non-controlled flow or the drag-reduced flow is described. In analogy with
other wall-based drag reduction techniques, like for example riblets, the
performance of the travelling waves is well described by a vertical shift of
the logarithmic portion of the mean streamwise velocity profile. Except when
$Re$ is very low, this shift remains constant with $Re$, at odds with the
percentage reduction of the friction coefficient, which is known to present a
mild, logarithmic decline. Our new data agree with the available literature,
which is however mostly based on low-$Re$ information and hence predicts a
quick drop of maximum drag reduction with $Re$. The present study supports a
more optimistic scenario, where for an airplane at flight Reynolds numbers a
drag reduction of nearly 30\% would still be possible thanks to the travelling
waves.
