docs:tutorials:findsoln
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docs:tutorials:findsoln [2014/12/04 12:32] gibson [periodic orbit of plane Couette flow] |
docs:tutorials:findsoln [2015/08/28 06:52] (current) gibson |
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| <code> | <code> | ||
| - | couette -T1 1000 -dt 0.05 -R 400 -symms sxyz-sxytxz.asc u0.h5 | + | couette -T1 1100 -dt 0.05 -R 400 -symms sxyz-sxytxz.asc u0.h5 |
| </code> | </code> | ||
| - | This will time-integrate the field u0 as a perturbation on top of a laminar base flow U(y) = y for 1000 time units, at Re=400, and with time integration step dt=0.05, and enforcing symmetries in the velocity field specified by the file ''sxyz-sxytxz.asc'' By default the velocity field will be saved into a ''data'' directory at intervals dT=1. See [[docs:tutorials:integration]] for more information on time integration, such as the base flow plus fluctuation decomposition, or changing from plane Couette to channel conditions. | + | This will time-integrate the field u0 as a perturbation on top of a laminar base flow U(y) = y for 1100 time units, at Re=400, and with time integration step dt=0.05, and enforcing symmetries in the velocity field specified by the file ''sxyz-sxytxz.asc'' By default the velocity field will be saved into a ''data'' directory at intervals dT=1. See [[docs:tutorials:integration]] for more information on time integration, such as the base flow plus fluctuation decomposition, or changing from plane Couette to channel conditions. |
| The symmetry file ''sxyz-sxytxz.asc'' has contents | The symmetry file ''sxyz-sxytxz.asc'' has contents | ||
| Line 33: | Line 33: | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| \sigma_{xyz} : [u,v,w] (x,y,z) &\rightarrow [-u,-v,-w] (-x,-y,-z) \\ | \sigma_{xyz} : [u,v,w] (x,y,z) &\rightarrow [-u,-v,-w] (-x,-y,-z) \\ | ||
| - | \sigma_{xy*} \tau_{xz} : [u,v,w] (x,y,z) &\rightarrow [-u,v,-w] (-x+L_x/2,y,-z+L_z/2) | + | \sigma_{xy} \tau_{xz} : [u,v,w] (x,y,z) &\rightarrow [-u,-v,w] (-x+L_x/2,-y,z+L_z/2) |
| - | \end{eqnarray} | + | \end{eqnarray*} |
| The pointwise inversion $\sigma_{xyz}$ of this group fixes the origin and prevents traveling waves | The pointwise inversion $\sigma_{xyz}$ of this group fixes the origin and prevents traveling waves | ||
| Line 55: | Line 55: | ||
| {{:docs:tutorials:2014-12-04-a.png?direct&600|}} | {{:docs:tutorials:2014-12-04-a.png?direct&600|}} | ||
| - | You can see a nice horizontal blue streak at $T \approx 60$ over the range $825 \leq t \leq 1000$ that indicates a long period over which $\|u(t) - u(t+60)\|$ is small. This suggests the turbulent trajectory is shadowing a periodic orbit with period $T \approx 60$ over this interval. The minimum of $\| u(t) - u(t+T) \|$ in this region occurs at $t=917$ and $T=63$. That's a good candidate for a search. | + | You can see strikingly periodic behavior over the range $825 \leq t \leq 1000$ with nice horizontal blue streak at $T \approx 65$. This suggests the turbulent trajectory is shadowing a periodic orbit with period $T \approx 65$. The minimum of $\| u(t) - u(t+T) \|$ in this region occurs at $t=917$ and $T=63$. That's an unusually promising initial guess for a periodic orbit. |
| === 4. Find the periodic orbit with ''findsoln'' === | === 4. Find the periodic orbit with ''findsoln'' === | ||
| Line 64: | Line 64: | ||
| findsoln -orb -T 63 -dt 0.05 -R 400 -symms ../sxyz_sxytxz.asc ../data/u917.h5 | findsoln -orb -T 63 -dt 0.05 -R 400 -symms ../sxyz_sxytxz.asc ../data/u917.h5 | ||
| </code> | </code> | ||
| + | |||
| Since I tend to run many searches for many initial guesses, I like to do each search in a subdirectory | Since I tend to run many searches for many initial guesses, I like to do each search in a subdirectory | ||
| - | named after the initial guess. The ''findsoln'' command runs a Newton-Krylov-hookstep search that finds a $u,T$ solution of the equation $u - f^T(u) = 0$ given the initial guess $T=63$ and $u = u(917)$. After about half an hour the search succeeds, finding a solution that satisfies the equation numerically to $O(10^{-13})$. The characteristics of the search routine are recorded in the file ''convergence.asc'' | + | named after the initial guess. The ''findsoln'' command runs a Newton-Krylov-hookstep search that finds a $u,T$ solution of the equation $u - f^T(u) = 0$ given the initial guess $T=63$ and $u = u(917)$. The search is restricted to the $\langle \sigma_{xyz}, \sigma_{xy} \tau_{xz} \rangle$ symmetry group. The symmetry restriction vastly reduces the search space and results in a faster and more robust search. |
| + | |||
| + | |||
| + | After about half an hour the search succeeds, finding a solution that satisfies the equation numerically to $O(10^{-13})$. The characteristics of the search routine are recorded in the file ''convergence.asc'' | ||
| <file> | <file> | ||
docs/tutorials/findsoln.1417725143.txt.gz · Last modified: 2014/12/04 12:32 by gibson
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