docs:math:symmetry
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docs:math:symmetry [2009/05/04 06:03] predrag bunch of minor edits. Added links to references - nothing substantial |
docs:math:symmetry [2014/12/04 11:46] gibson |
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| 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) \\ | ||
| Line 9: | Line 9: | ||
| [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// | ||
| Line 17: | Line 17: | ||
| and specific boundary conditions, are subgroups of the group | and specific boundary conditions, are 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 57 isotropy subgroups of plane Couette, see J. Halcrow, J. F. Gibson, and P. Cvitanović, | + | 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. | //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. | ||
| Line 27: | 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 46: | 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 63: | 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 82: | 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 132: | 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 146: | 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 156: | 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
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