# Exact Coherent States

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