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====== Symmetry of flows in channel geometries ======

The symmetry group of 3D fields in channel geometries is generated by 

\begin{eqnarray*}
  [ u,v,w ](x,y,z) & \rightarrow [-u, v, w](-x,y,z) \\
  [ u,v,w ](x,y,z) & \rightarrow [ u, -v, w](x,-y,z) \\
  [ u,v,w ](x,y,z) & \rightarrow [ u, v, -w](x, y,-z) \\
  [ u,v,w ](x,y,z) & \rightarrow [-u,-v,-w](x,y,z) \\
  [ u,v,w ](x,y,z) & \rightarrow [ u, v, w](x+\ell_x, y, z+\ell_z) 
\end{eqnarray*}

"Channel geometry" means a domain that is periodic or infinite in //x// and //z//
and bounded in //y//, with $-L_y/2 \leq y \leq L_y/2$ and Dirichlet or Neumann 
boundary conditions at the bounds in //y//. The symmetry groups of velocity 
fields for specific flows, with constraints such as incompressibility 
and specific boundary conditions, are subgroups of the group
generated by the above symmetries.
====== Symmetry of plane Couette flow=======

For the full description of 67 isotropy subgroups of plane Couette, see J. Halcrow, J. F. Gibson, and P. Cvitanović,
//Equilibrium and traveling-wave solutions of plane Couette flow//, [[http://arxiv.org/abs/0808.3375|arXiv:0808.3375]], J. Fluid Mech. (to appear, 2009), and [[http://chaosbook.org/projects/Halcrow/thesis.pdf|J. Halcrow, "Charting the state space of plane Couette flow: Equilibria, relative equilibria, and heteroclinic connections"]] (Georgia Tech Ph.D. thesis, Aug 2008). Here are some highlights.

===== Invariance =====

Plane Couette flow is invariant under the following symmetries

\begin{eqnarray*}
\sigma_x    \,       [u,v,w](x,y,z) &= [-u,-v,w](-x,-y,z) \\
\sigma_z    \,       [u,v,w](x,y,z) &= [u, v,-w](x,y,-z)  \\
\tau(\ell_x, \ell_z) [u,v,w](x,y,z) &= [u, v,-w](x+\ell_x, y, z+\ell_z)  \\
\end{eqnarray*}

That is, if f^t(u) is the time-t map of plane Couette flow, then

\begin{eqnarray*}
 f^t(s u) = s f^t(u) 
\end{eqnarray*}

for any s in group G generated by <latex> \{\sigma_x, \sigma_z,  \tau(\ell_x, \ell_z)\}. </latex>

Let u(t) be a solution of Navier Stokes with initial condition u(0),

\begin{eqnarray*}
  u(t) = f^t(u(0))
\end{eqnarray*}

then

\begin{eqnarray*}
  s u(t) = s f^t(u(0)) = f^t(s u(0))
\end{eqnarray*}

is a solution of Navier-Stokes with initial condition s u(0).


===== Isotropy =====

Suppose //u(0)// is invariant under a symmetry //s// in //G//, i.e. 

\begin{eqnarray*}
  s u(0) = u(0)
\end{eqnarray*}

Then //u(t)// satisfies that symmetry for all //t//, since

\begin{eqnarray*}
  s u(t) = s f^t(u(0)) = f^t(s u(0)) = f^t(u(0)) = u(t)
\end{eqnarray*}

The set of all symmetries //s// in //G// satisfied by u forms a subgroup //H ⊂ G//, 
called the //isotropy group// group of //u//. Isotropy groups are useful 
because they form invariant subspaces of the flow.


===== Isotropy groups of known solutions =====

The isotropy group most known equilibria and periodic orbits of plane Couette flow is 

\begin{eqnarray*}
  S = \{1, s_1, s_2, s_3 \}
\end{eqnarray*}

where

\begin{eqnarray*}
  s_1 \, [u, v, w](x,y,z) &= [u, v, -w](x+L_x/2, y, -z) \\
  s_2 \, [u, v, w](x,y,z) &= [-u, -v, w](-x+L_x/2,-y,z+L_z/2) \\
  s_3 \, [u, v, w](x,y,z) &= [-u,-v,-w](-x, -y, -z+L_z/2) \\
\end{eqnarray*}
It is helpful to express these symmetries in terms of //σ<sub>x</sub>, σ<sub>z</sub>,// and translations. Let 

\begin{eqnarray*}
  \tau_x    &= \tau(L_x/2, 0) \\
  \tau_z    &= \tau(0, L_z/2) \\
  \tau_{xz} &= \tau_x \tau_z 
\end{eqnarray*}

then 

\begin{eqnarray*}
  S = \{1, \, \tau_x \sigma_z, \, \tau_{xz} \sigma_x, \, \tau_z \sigma_{xz} \}
\end{eqnarray*}
===== Fun facts =====

1. If u has isotropy group S, then 

\begin{eqnarray*}
\tau_x u,  \, \tau_z u, \, \text{ and } \, \tau_{xz} u 
\end{eqnarray*}

also have isotropy group S. Thus for each equilibrium or periodic orbit with isotropy group S, there are four half-box shifted partners.

2. Since s^2 = 1 for s ∈ S, the eigenfunctions v of the linearized  dynamics 
about any solution u with isotropy group S are either symmetric or 
antisymmetric with respect each symmetry s in S. (I.e. sv = ±v)

3. σ<sub>x</sub> defines a center of symmetry in x, σ<sub>z</sub> in z,
and  σ<sub>xz</sub> in both. Therefore the presence of σ<sub>x</sub> in 
an isotropy group rules out traveling waves in x (similarly, z, and xz).

4. The S isotropy group admits of no traveling wave solutions and 
relative periodic orbits only of the form

\begin{eqnarray*}
  \tau f^t(u) - u = 0 \text{ for } \tau \in T = \{1, \, \tau_x, \, \tau_z, \, \tau_{xz} \}
\end{eqnarray*}

===== Isotropy groups and invariant solutions =====

So far we have restricted most of our attention to the
solutions with //S// isotropy. We have a few solutions with other isotropies.

One of the main simplifications of the restriction to //S// is that reduces 
the number of free parameters in the search for good initial guesses for
invariant solutions. E.g. we don't have to provide a guess for the wave 
speed of traveling waves, and for periodic orbits, there are only four 
choices for the symmetry //σ// in 

\begin{eqnarray*}
   \sigma f^t u - u = 0
\end{eqnarray*}
namely, <latex> \sigma = 1, \tau_x, \, \tau_z, \, \text{ or } \tau_{xz} </latex>, rather 
than the continuum <latex> \tau(\ell_x, \ell_z) </latex>.

To search for initial guesses for periodic orbits, we define a measure of
close recurrence within a trajectory u(t) by

\begin{eqnarray*}
  r(t,T) &= min_{\tau} \| \sigma f^T u(t) - u(t) \| \\
         &= min_{\tau} \| \sigma u(t+T) - u(t) \| 
\end{eqnarray*}
for $ \tau \in \{1, \, \tau_x, \, \tau_z,  \tau_{xz}\}$. 
We can compute r(t,T) from a time series of u(t) and look for places where r(t,T) << 1 for 
stretches of t and constant T. Those will be good guesses for periodic orbits.
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