Recurrent motions within plane Couette turbulence

The phenomenon of bursting, in which streaks in turbulent boundary layers oscillate and then eject low speed fluid away from the wall, has been studied experimentally, theoretically, and computationally for more than 50 years because of its importance to the three-dimensional structure of turbulent boundary layers. We produce five new three-dimensional solutions of turbulent plane Couette flow, one of which is periodic while four others are relative periodic. Each of these five solutions demonstrates the break-up and re-formation of near-wall coherent structures. Four of our solutions are periodic but with drifts in the streamwise direction. More surprisingly, two of our solutions are periodic but with drifts in the spanwise direction, a possibility that does not seem to have been considered in the literature. We argue that a considerable part of the streakiness observed experimentally in the near-wall region could be due to spanwise drifts that accompany the break-up and re-formation of coherent structures. We also compute a new periodic solution of plane Couette flow that could be related to transition to turbulence. The violent nature of the bursting phenomenon implies the need for good resolution in the computation of periodic and relative periodic solutions within turbulent shear flows. We address this computationally demanding requirement with a new algorithm for computing relative periodic solutions one of whose features is a combination of two well-known ideas — namely the Newton-Krylov iteration and the locally constrained optimal hook step. Each of our six solutions is accompanied by an error estimate. In the concluding discussion, we discuss dynamical principles that suggest that the bursting phenomenon, and more generally fluid turbulence, can be understood in terms of periodic and relative periodic solutions of the Navier-Stokes equation. ∗Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, U.S.A.