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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 1000 -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 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. 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. Compute
