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====== How to find invariant solutions ======

The channelflow ''findsoln'' utility will compute unstable equilibria, traveling waves, and (relative) periodic orbits of plane Couette or channel flows. Here are a few examples of usage.

===== periodic orbit of plane Couette flow =====

=== 1. Generate a random velocity field with ''randomfield.'' ===


<code>
randomfield -m 0.50 -lx 0.875 -lz 0.6 -Nx 32 -Ny 33 -Nz 32 u0
</code>

This will construct a random velocity field in file ''u0.h5'', in a  Lx x [-1, 1] x Lz periodic box with Lx = 2 lx pi = 1.75pi and Lz = 2 lz pi = 1.2 pi (the plane Couette minimal flow unit of Hamilton, Kim, Waleffe JFM 1995) with a 32 x 33 x 32 collocation grid (which is a lower resolution than I like for these calculations, but will result in a fast-running demo.) The velocity field u0 will be incompressible and have u=0 BCs at the wall, and will have an L2Norm of 0.50. You can check this by running ''fieldprops -n u0''. I chose this magnitude for the field after testing some smaller fields and seeing that they died to laminar pretty quickly. 

=== 2. Integrate the random field forward in time. ===


<code>
couette -T1 1100 -dt 0.05 -R 400 -symms sxyz-sxytxz.asc u0.h5
</code>

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 
<code>
% 2
 1 -1 -1 -1 0.0 0.0
 1 -1 -1  1 0.5 0.5
</code>
That specifies a symmetry group with two generators

\begin{eqnarray*}
  \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)
\end{eqnarray*}

The pointwise inversion $\sigma_{xyz}$ of this group fixes the origin and prevents traveling waves
and arbitrary relative periodic orbits. For more on symmetry groups of plane Couette flow see [[docs:math:symmetry]] and {{:gibson:publications:gibson_jfm09b.pdf | Gibson, Halcrow, Cvitanovic JFM 2009}}. 

=== 3. Select a good initial guess from a recurrence plot. ===



The $\langle \sigma_{xyz}, \sigma_{xy} \tau_{xz} \rangle$ symmetry group allows periodic orbits of the form $\tau u - f^T(u) = 0$ for $\tau \in \{i, \tau_x, \tau_z, \tau_{xz} \}$, where $f^t$ is the forward-time evolution under Navier-Stokes, i.e. $u(t+T) = f^T(u(t))$, and where $\tau_x$ represents a half-box shift in the $x$ direction, etc. 

For this demo I'll just look for a solution of the form $u - f^T(u) = 0$. Thus local minima of $\| u(t) - u(t+T) \|$ should provide good initial guesses for the search.

<code>
seriesdist -T0 0 -T1 1000 -dT 1 -tmax 100 -da data/ -db data/ -as -kx 8 -kz 8
</code>

This reads in the saved velocity fields from the ''data'' directory, computes $\| u(t) - u(t+T) \|$ as a function of $t,T$, and saves this in the file ''dist.asc''. Here's the recurrence plot for the data in question. 

{{: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.

=== 4. Find the periodic orbit with ''findsoln'' ===

<code> 
mkdir findsoln-917-63
cd findsoln-917-63
findsoln -orb -T 63 -dt 0.05 -R 400 -symms ../sxyz_sxytxz.asc ../data/u917.h5
</code>

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 $u = u(917)$. 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>
%-L2Norm(G)   r             delta         dT            L2Norm(du)    L2Norm(u)     L2Norm(dxN)   dxNalign      L2Norm(dxH)   GMRESerr      ftotal    fnewt     fhook     CFL       
0.00888606    2.20611e-05   0.01          0             0             0.390225      0             0             0             0             1         0         1         0.609524  
0.00663114    1.19704e-05   0.01          -0.0640999    0.0116139     0.38873       0.00902435    0             0.00902435    0.00076865    14        12        2         0.609524  
0.000855143   2.14948e-07   0.01          0.190862      0.00429167    0.388192      0.0548914     -0.238276     0.01          0.000119926   29        25        4         0.608904  
0.000597692   9.06641e-08   0.01          0.194443      0.00326027    0.387135      0.035679      0.998803      0.01          0.000613148   42        37        5         0.581667  
0.000505451   6.2892e-08    0.01          0.18166       0.00364325    0.386258      0.00942818    0.997453      0.00942818    0.000289388   55        49        6         0.583459  
1.77901e-05   8.04837e-11   0.01          -0.0438562    0.000824778   0.386507      0.00227387    -0.99388      0.00227387    0.000612772   67        60        7         0.604789  
5.46682e-09   8.266e-18     0.01          -0.000571434  1.73752e-05   0.38651       3.13122e-05   0.981989      3.13122e-05   5.95861e-05   80        72        8         0.584728  
8.51125e-13   1.58971e-25   0.01          -2.67003e-07  1.49819e-08   0.38651       1.70279e-08   0.940857      1.70279e-08   0.000134271   93        84        9         0.584723  
9.72014e-14   2.358e-27     0.01          -1.08282e-10  2.94411e-12   0.38651       5.77394e-12   0.921371      5.77394e-12   0.000331508   106       96        10        0.584723  
</file>

The solution $u$ is stored in the file ''ubest.h5'' and the value of $T$ in ''Tbest.asc.''
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