User Tools

Site Tools


docs:math:symmetry

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
docs:math:symmetry [2010/02/02 07:55]
127.0.0.1 external edit
docs:math:symmetry [2014/12/04 11:53]
gibson [Symmetry of flows in channel geometries]
Line 3: Line 3:
 The symmetry group of 3D fields in channel geometries is generated by  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 //-L_y/​2 ​<= <= L_y/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 ​
Line 26: Line 26:
 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>​
Line 45: Line 42:
 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).
Line 62: Line 59:
 Suppose //u(0)// is invariant under 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//, 
Line 81: Line 78:
 The isotropy group most known equilibria and periodic orbits of plane Couette flow is  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 \}   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*}
-$  +
-</​latex>​ +
 It is helpful to express these symmetries in terms of //​σ<​sub>​x</​sub>,​ σ<​sub>​z</​sub>,//​ and translations. Let  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_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.
Line 131: Line 123:
 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 =====
  
Line 145: Line 138:
 choices for the symmetry //σ// in  choices for the symmetry //σ// in 
  
-<​latex>​+\begin{eqnarray*}
    ​\sigma 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>​ \sigma = 1, \tau_x, \, \tau_z, \, \text{ or } \tau_{xz} </​latex>,​ rather ​
 than the continuum <​latex>​ \tau(\ell_x,​ \ell_z) </​latex>​. than the continuum <​latex>​ \tau(\ell_x,​ \ell_z) </​latex>​.
Line 155: Line 147:
 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} \| \sigma f^T u(t) - u(t) \| \\   r(t,T) &= min_{\tau} \| \sigma f^T u(t) - u(t) \| \\
          &​= min_{\tau} \| \sigma 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.
docs/math/symmetry.txt · Last modified: 2014/12/04 11:53 by gibson