Exact Coherent States

Overview and database

Fabian Waleffe
Depts of Mathematics and Engr Physics, U. of Wisconsin-Madison

Exact Coherent States (ECS) are steady state or traveling wave solutions of the Navier-Stokes equations for incompressible flow, such as plane Couette flow, plane Poiseuille flow and pipe flow. These traveling waves are the order behind the disorder characteristic of turbulent shear flows. The traveling waves appear to capture the main statistical features of turbulent flows such as mean and rms velocity profiles and streak spacing. The physics underlying the exact coherent structures consists of a nonlinear, three-dimensional Self-Sustaining Process (SSP) that appears to be generic for shear flows. This self-sustaining process not only elucidates the underlying physics but also provides a method to calculate the exact coherent structures by continuation of solutions that bifurcate from a streaky flow as explained in: FW, Physical Review Letters, 81 , 4140 (Nov 1998) and FW, Physics of Fluids, 15, 1517-1534 (June 2003) . That same approach has been used by Faist and Eckhardt, PRL 91, 224502 (2003) and Wedin and Kerswell, JFM 508:333-371 (2004) to obtain traveling wave solutions in pipe flow.

Publications by FW on ECS and SSP .

Exact Coherent Structures in Viscoelastic Shear Flows Mike Graham and his students Philip Stone and Wei Li have been investigating the effect of polymers on the exact coherent structures in an effort to develop a solid theoretical understanding of the dramatic drag reduction that is seen when a small amount of long polymer chains or surfactants are added to the fluid.

Color figures appearing in Physics of Fluids, 15, June 2003 paper (and therefore now copyrighted by the American Institute of Physics)

Poiseuille ECS, Fig 15

Couette ECS, Fig 16

ECS database