====== 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.'' ===
randomfield -m 0.50 -lx 0.875 -lz 0.6 -Nx 32 -Ny 33 -Nz 32 u0
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. ===
couette -T1 1100 -dt 0.05 -R 400 -symms sxyz-sxytxz.asc u0.h5
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
% 2
1 -1 -1 -1 0.0 0.0
1 -1 -1 1 0.5 0.5
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.
seriesdist -T0 0 -T1 1000 -dT 1 -tmax 100 -da data/ -db data/ -as -kx 8 -kz 8
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 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'' ===
mkdir findsoln-917-63
cd findsoln-917-63
findsoln -orb -T 63 -dt 0.05 -R 400 -symms ../sxyz_sxytxz.asc ../data/u917.h5
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''
%-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
The solution $u$ is stored in the file ''ubest.h5'' and the value of $T$ in ''Tbest.asc.''