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docs:math:symmetry [2009/02/11 08:47]
gibson
docs:math:symmetry [2014/12/04 11:53] (current)
gibson [Symmetry of flows in channel geometries]
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 ====== Symmetry of flows in channel geometries ====== ====== Symmetry of flows in channel geometries ======
  
-The following ​symmetry ​operations can be applied to 3D fields in channel geometries+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 <= <= Ly/2%% and Dirichlet or Neumann  +and bounded in //y//, with $-L_y/2 \leq \leq L_y/2and 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 ​ fields for specific flows, with constraints such as incompressibility ​
-and specific boundary conditions are subgroups of the group+and specific boundary conditionsare subgroups of the group
 generated by the above symmetries. generated by the above symmetries.
- 
 ====== Symmetry of plane Couette flow======= ====== 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 ===== ===== Invariance =====
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 Plane Couette flow is invariant under the following symmetries 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_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) ​ \\ \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) ​ \\ \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 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) ​  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>​ for any s in group G generated by <​latex>​ \{\sigma_x, \sigma_z, ​ \tau(\ell_x,​ \ell_z)\}. </​latex>​
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 Let u(t) be a solution of Navier Stokes with initial condition u(0), Let u(t) be a solution of Navier Stokes with initial condition u(0),
  
-<​latex>​+\begin{eqnarray*}
   u(t) = f^t(u(0))   u(t) = f^t(u(0))
-</​latex>​+\end{eqnarray*}
  
 then then
  
-<​latex>​+\begin{eqnarray*}
   s u(t) = s f^t(u(0)) = f^t(s u(0))   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). is a solution of Navier-Stokes with initial condition s u(0).
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 ===== Isotropy ===== ===== Isotropy =====
  
-Suppose u(0) satisfies ​a symmetry s in G, i.e. +Suppose ​//u(0)// is invariant under a symmetry ​//s// in //G//, i.e. 
  
-<​latex>​+\begin{eqnarray*}
   s u(0) = u(0)   s u(0) = u(0)
-</​latex>​+\end{eqnarray*}
  
-Then u(t) satisfies that symmetry for all t, since+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) +  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,  +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 ​+called the //isotropy group// group of //u//. Isotropy groups are useful ​
 because they form invariant subspaces of the flow. because they form invariant subspaces of the flow.
  
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 ===== Isotropy groups of known solutions ===== ===== Isotropy groups of known solutions =====
  
-Most known equilibria and periodic orbits of plane Couette flow have the  +The isotropy group most known equilibria and periodic orbits of plane Couette flow is 
-same isotropy group +
  
-<​latex>​+\begin{eqnarray*}
   S = \{1, s_1, s_2, s_3 \}   S = \{1, s_1, s_2, s_3 \}
-</​latex>​+\end{eqnarray*}
  
 where where
  
-<​latex>​ +\begin{eqnarray*}
- ​$ ​\begin{align*}+
   s_1 \, [u, v, w](x,y,z) &= [u, v, -w](x+L_x/​2,​ y, -z) \\   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_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) \\   s_3 \, [u, v, w](x,y,z) &= [-u,​-v,​-w](-x,​ -y, -z+L_z/2) \\
-\end{align*} +\end{eqnarray*} 
-$  +It is helpful to express these symmetries in terms of //​σ<​sub>​x</sub>, σ<​sub>​z</​sub>,//​ and translations. Let 
-</latex>+
  
-It's helpful to express break these symmetries into these σ<​sub>​x</​sub>,​ σ<​sub>​z</​sub>,​ +\begin{eqnarray*}
-and translations. Let  +
- +
-<​latex>​ $ \begin{align*}+
   \tau_x ​   &= \tau(L_x/2, 0) \\   \tau_x ​   &= \tau(L_x/2, 0) \\
   \tau_z ​   &= \tau(0, L_z/2) \\   \tau_z ​   &= \tau(0, L_z/2) \\
   \tau_{xz} &= \tau_x \tau_z ​   \tau_{xz} &= \tau_x \tau_z ​
-\end{align*} $ </​latex>​+\end{eqnarray*}
  
 then  then 
  
-<​latex>​ $ \begin{align*}+\begin{eqnarray*}
   S = \{1, \, \tau_x \sigma_z, \, \tau_{xz} \sigma_x, \, \tau_z \sigma_{xz} \}   S = \{1, \, \tau_x \sigma_z, \, \tau_{xz} \sigma_x, \, \tau_z \sigma_{xz} \}
-\end{align*} $ </​latex>​ +\end{eqnarray*}
 ===== Fun facts ===== ===== Fun facts =====
  
 1. If u has isotropy group S, then  1. If u has isotropy group S, then 
  
-<​latex> ​+\begin{eqnarray*}
 \tau_x u,  \, \tau_z u, \, \text{ and } \, \tau_{xz} u  \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. also have isotropy group S. Thus for each equilibrium or periodic orbit with isotropy group S, there are four half-box shifted partners.
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 relative periodic orbits only of the form 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} \}   \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 ===== ===== Isotropy groups and invariant solutions =====
  
-Our recent submission ​to JFM has a more complete treatment of the  +So far we have restricted most of our attention ​to the 
-isotropy ​groups of plane Couette flow and the types of solutions ​they admit.+solutions with //S// isotropy. We have a few solutions ​with other isotropies.
  
-For simplicity we have so far restricted most of our attention to the +One of the main simplifications of the restriction to //S// is that reduces ​
-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 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  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  speed of traveling waves, and for periodic orbits, there are only four 
-choices for the the symmetry ​in +choices for the symmetry ​//​σ// ​in 
  
-<​latex>​ +\begin{eqnarray*} 
-   f^t u - u = 0 +   \sigma ​f^t u - u = 0 
-</​latex>​ +\end{eqnarray*} 
- +namely, <​latex> ​\sigma ​= 1, \tau_x, \, \tau_z, \, \text{ or } \tau_{xz} ​</​latex>,​ rather ​ 
-namely, <​latex> ​= 1, \tau_x, \, \tau_z, \, \text{ or } \tau_xz ​</​latex>,​ rather than the continuum +than the continuum <​latex>​ \tau(\ell_x,​ \ell_z) </​latex>​.
-<​latex>​ \tau(\ell_x,​ \ell_z) </​latex>​.+
  
 To search for initial guesses for periodic orbits, we define a measure of To search for initial guesses for periodic orbits, we define a measure of
 close recurrence within a trajectory u(t) by close recurrence within a trajectory u(t) by
  
-<​latex>​ $ \begin{align*} +\begin{eqnarray*} 
-  r(t,T) &= min_{\tau} \| f^T u(t) - u(t) \| \\ +  r(t,T) &= min_{\tau} \| \sigma ​f^T u(t) - u(t) \| \\ 
-         &​= min_{\tau} \| u(t+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  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. stretches of t and constant T. Those will be good guesses for periodic orbits.
- 
- 
-// FieldSymmetry s(sx,​sy,​sz,​ax,​az,​sa) takes ax,az in [0,1] and su,sx,sy,sz in {-1,1} 
-// s : (u,​v,​w)(x,​y,​z) -> (su sx u, su sy v, su sz w)(sx x + ax Lx, sy y, sz z + az Lz) 
docs/math/symmetry.1234370839.txt.gz · Last modified: 2009/02/11 08:47 by gibson