Drag reduction in turbulent pipe flow with feedback control applied partially to wall

Turbulent pipe flow controlled by the opposition control algorithm [J. Fluid Mech. 262 (1994) 75-110] is studied by means of direct numerical simulation. A special focus is laid upon a scheme in which the control input is applied only partially over a limited length in the streamwise direction, but not on the entire wall surface. The upstream control effect remains over a distance of about 11-14 times the pipe radius downstream of the point where the control is terminated. This results, however, in a simple relationship that the average drag reduction rate is nearly proportional to the control length. The recovery process after the control termination is quantitatively investigated by applying a recently proposed exact relation between the skin friction and the Reynolds stress distribution [Phys. Fluids 14 (11) (2002) L73-L76] and also by performing a budget analysis specially designed for that purpose.