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--- docs:math:symmetry [2009/05/04 05:42]
predrag
+++ docs:math:symmetry [2014/12/04 11:53] (current)
gibson [Symmetry of flows in channel geometries]
@@ Line 3: / Line 3: @@
 The symmetry group of 3D fields in channel geometries is generated by
-<latex> $ \begin{align*}
+\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, 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)
+  [ u,v,w ](x,y,z) & \rightarrow [ u, v, w](x+\ell_x, y, z+\ell_z)
-\end{align*} $ </latex>
+\end{eqnarray*}
-"Channel geometry" means a domain that is periodic or infinite in x and z
+"Channel geometry" means a domain that is periodic or infinite in //x// and //z//
-and bounded in y, with %%-Ly/2 <= y <= Ly/2%% and Dirichlet or Neumann
+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
+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 =====
@@ Line 24: / Line 26: @@
 Plane Couette flow is invariant under the following symmetries
-<latex>
+\begin{eqnarray*}
- $ \begin{align*}
 \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{align*}
+\end{eqnarray*}
-$
-</latex>
 That is, if f^t(u) is the time-t map of plane Couette flow, then
-<latex>
+\begin{eqnarray*}
  f^t(s u) = s f^t(u)
-</latex>
+\end{eqnarray*}
 for any s in group G generated by <latex> \{\sigma_x, \sigma_z,  \tau(\ell_x, \ell_z)\}. </latex>
@@ Line 43: / Line 42: @@
 Let u(t) be a solution of Navier Stokes with initial condition u(0),
-<latex>
+\begin{eqnarray*}
   u(t) = f^t(u(0))
-</latex>
+\end{eqnarray*}
 then
-<latex>
+\begin{eqnarray*}
   s u(t) = s f^t(u(0)) = f^t(s u(0))
-</latex>
+\end{eqnarray*}
 is a solution of Navier-Stokes with initial condition s u(0).
@@ Line 60: / Line 59: @@
 Suppose //u(0)// is invariant under a symmetry //s// in //G//, i.e.
-<latex>
+\begin{eqnarray*}
   s u(0) = u(0)
-</latex>
+\end{eqnarray*}
 Then //u(t)// satisfies that symmetry for all //t//, since
-<latex>
+\begin{eqnarray*}
   s u(t) = s f^t(u(0)) = f^t(s u(0)) = f^t(u(0)) = u(t)
-</latex>
+\end{eqnarray*}
 The set of all symmetries //s// in //G// satisfied by u forms a subgroup //H ⊂ G//,
@@ Line 79: / Line 78: @@
 The isotropy group most known equilibria and periodic orbits of plane Couette flow is
-<latex>
+\begin{eqnarray*}
   S = \{1, s_1, s_2, s_3 \}
-</latex>
+\end{eqnarray*}
 where
-<latex>
+\begin{eqnarray*}
- $ \begin{align*}
   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{align*}
+\end{eqnarray*}
-$
-</latex>
 It is helpful to express these symmetries in terms of //σ<sub>x</sub>, σ<sub>z</sub>,// and translations. Let
-<latex> $ \begin{align*}
+\begin{eqnarray*}
   \tau_x    &= \tau(L_x/2, 0) \\
   \tau_z    &= \tau(0, L_z/2) \\
   \tau_{xz} &= \tau_x \tau_z
-\end{align*} $ </latex>
+\end{eqnarray*}
 then
-<latex> $ \begin{align*}
+\begin{eqnarray*}
   S = \{1, \, \tau_x \sigma_z, \, \tau_{xz} \sigma_x, \, \tau_z \sigma_{xz} \}
-\end{align*} $ </latex>
+\end{eqnarray*}
 ===== Fun facts =====
 . If u has isotropy group S, then
-<latex>
+\begin{eqnarray*}
 \tau_x u,  \, \tau_z u, \, \text{ and } \, \tau_{xz} u
-</latex>
+\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.
@@ Line 129: / Line 123: @@
 relative periodic orbits only of the form
-<latex>
+\begin{eqnarray*}
   \tau f^t(u) - u = 0 \text{ for } \tau \in T = \{1, \, \tau_x, \, \tau_z, \, \tau_{xz} \}
-</latex>
+\end{eqnarray*}
-===== Isotropy groups and invariant solutions =====
-Our recent submission to JFM has a more complete treatment of the
+===== Isotropy groups and invariant solutions =====
-isotropy groups of plane Couette flow and the types of solutions they admit.
-For simplicity we have so far restricted most of our attention to the
+So far we have restricted most of our attention to the
-solutions with S isotropy. We have a few solutions with other isotropies.
+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
+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 the symmetry σ in
+choices for the symmetry //σ// in
-<latex>
+\begin{eqnarray*}
    \sigma f^t u - u = 0
-</latex>
+\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>.
@@ Line 156: / Line 147: @@
 close recurrence within a trajectory u(t) by
-<latex> $ \begin{align*}
+\begin{eqnarray*}
   r(t,T) &= min_{\tau} \| \sigma f^T u(t) - u(t) \| \\
          &= min_{\tau} \| \sigma u(t+T) - u(t) \|
-\end{align*} $ </latex>
+\end{eqnarray*}
+for $ \tau \in \{1, \, \tau_x, \, \tau_z,  \tau_{xz}\}$.
-for <latex> \tau \in \{1, \, \tau_x, \, \tau_z,  \tau_{xz}\}</latex>.
 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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