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====== Symmetry of flows in channel geometries ====== The symmetry group of 3D fields in channel geometries is generated by <latex> $ \begin{align*} [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{align*} $ </latex> "Channel geometry" means a domain that is periodic or infinite in //x// and //z// and bounded in //y//, with //-L_y/2 <= y <= 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 57 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 <latex> $ \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*} $ </latex> That is, if f^t(u) is the time-t map of plane Couette flow, then <latex> f^t(s u) = s f^t(u) </latex> 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), <latex> u(t) = f^t(u(0)) </latex> then <latex> s u(t) = s f^t(u(0)) = f^t(s u(0)) </latex> 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. <latex> s u(0) = u(0) </latex> Then //u(t)// satisfies that symmetry for all //t//, since <latex> s u(t) = s f^t(u(0)) = f^t(s u(0)) = f^t(u(0)) = u(t) </latex> 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 <latex> S = \{1, s_1, s_2, s_3 \} </latex> where <latex> $ \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*} $ </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*} \tau_x &= \tau(L_x/2, 0) \\ \tau_z &= \tau(0, L_z/2) \\ \tau_{xz} &= \tau_x \tau_z \end{align*} $ </latex> then <latex> $ \begin{align*} S = \{1, \, \tau_x \sigma_z, \, \tau_{xz} \sigma_x, \, \tau_z \sigma_{xz} \} \end{align*} $ </latex> ===== Fun facts ===== 1. If u has isotropy group S, then <latex> \tau_x u, \, \tau_z u, \, \text{ and } \, \tau_{xz} u </latex> 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 <latex> \tau f^t(u) - u = 0 \text{ for } \tau \in T = \{1, \, \tau_x, \, \tau_z, \, \tau_{xz} \} </latex> ===== 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 <latex> \sigma f^t u - u = 0 </latex> 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 <latex> $ \begin{align*} r(t,T) &= min_{\tau} \| \sigma f^T u(t) - u(t) \| \\ &= min_{\tau} \| \sigma u(t+T) - u(t) \| \end{align*} $ </latex> 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.
