  Waves of spanwise velocity imposed at the walls of a plane turbulent channel
flow are studied by Direct Numerical Simulations. We consider sinusoidal waves
of spanwise velocity which vary in time and are modulated in space along the
streamwise direction. The phase speed may be null, positive or negative, so
that the waves may be either stationary or traveling forward or backward in the
direction of the mean flow. Such a forcing includes as particular cases two
known techniques for reducing friction drag: the oscillating wall technique (a
traveling wave with infinite phase speed) and the recently proposed steady
distribution of spanwise velocity (a wave with zero phase speed).
  The traveling waves alter the friction drag significantly. Waves which slowly
travel forward produce a large reduction of drag, that can relaminarize the
flow at low values of the Reynolds number. Faster waves yield a totally
different outcome, i.e. drag increase. Even faster waves produce a drag
reduction effect again. Backward-traveling waves instead lead to drag reduction
at any speed.
  The traveling waves, when they reduce drag, operate in similar fashion to the
oscillating wall, with an improved energetic efficiency. Drag increase is
observed when the waves travel at a speed comparable with that of the
convecting near-wall turbulence structures. A diagram illustrating the
different flow behaviors is presented.
