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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 %%-Ly/2 <= y <= Ly/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======= ===== 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 ===== Most known equilibria and periodic orbits of plane Couette flow have the same isotropy group <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's helpful to express break these symmetries into these 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 ===== Our recent submission to JFM has a more complete treatment of the 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 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 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.
