A Periodic Motion Embedded in Plane Couette Turbulence

"A flow between two parallel plates which move with a constant velocity in opposite directions becomes turbulent at the Reynolds number above some critical value if it starts with a strongly disturbed state. [1,2]. This is called the plane Couette turbulence[3,4], the fluid motion in which is chaotic and never repeated. Nevertheless, it is known that the regeneration cycle[5,6] is present to sustain near-wall coherent structures such as streamwise vortices and low-velocity streaks though its theoretical description has not been established. Here we report a periodic motion, discovered by solving the Navier-Stokes equation iteratively, which describes a full cycle of repetition of a series of dynamical processes including the formation and breakdown of coherent structures. Since it is unstable, this periodic motion is not attained in reality. However, the turbulent state spends most of the time around it. As a result, the mean velocity profile as well as the root-mean-squares of velocity fluctuations of the Couette turbulence coincide remarkably well with the temporal averages of the corresponding quantities of the periodic motion.