# channelflow.org

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docs:utils:findsoln

# findsoln

find an equilibrium, traveling wave, periodic orbit, or relative periodic orbit of plane Couette or channel flow, using a Newton-Krylov-hookstep search algorithm.

# main options

```  -eqb      --equilibrium                                          search for equilibrium or relative equilibrium (trav wave)
-orb      --periodicorbit                                        search for periodic orbit or relative periodic orbit
-xrel     --xrelative                                            search over x phase shift for relative orbit or eqb
-zrel     --zrelative                                            search over z phase shift for relative orbit or eqb
-T        --maptime           <real>      default == 20          initial guess for orbit period or time of eqb/reqb map f^T(u)
-R        --Reynolds          <real>      default == 400         Reynolds number
-sigma    --sigma             <string>                           file containing symmetry of relative solution (default == identity)
-symms    --symmetries        <string>                           file containing generators of isotropy group for symmetry-constrained search
...
-o        --outdir            <string>    default == ./          output directory
-log      --logfile           <string>    default == stdout      output log (filename or "stdout")
<flowfield>      (trailing arg 1)                                initial guess for solution```

# solution type

`findsoln` finds solutions of σ f^T(u) - u = 0 where f^T is the time-T forward map of the Navier-Stokes equations plus boundary conditions and σ is a symmetry of the flow. This equation has several kinds of solutions

• equilibrium: σ = 1/, all T
• relative equilibrium: σ = a translation or phase shift that scales linearly in T, all T (also called a traveling wave)
• periodic orbit: σ = 1 and some finite value of T (and integer multiples of it)
• preperiodic orbit: σ ≠ 1 but σ^n = 1 for some n>1, and some finite value of T (and integer multiples)
• relative periodic orbit: some element σ of the appropriate symmetry group, and some finite value of T (and integer multiples)

You specify which kind of solution to find by telling the search algorithm which variables to treat as unknowns, with the `-eqb, -orb, -xrel, -zrel` options. The `-eqb` option specifies that T is held fixed; the `-orb` option specifies that is unknown. The `-xrel` and `-zrel` options specifying that the x and z phase shifts in σ are unknown. The appropriate combinations for each solution type are best shown by example.

commandsearch type
`findsoln -eqb uguess.h5` equilibrium solution
`findsoln -eqb -xrel uguess.h5` traveling wave in x
`findsoln -eqb -xrel -zrel uguess.h5` traveling wave in x,z
`findsoln -eqb -zrel -sigma sigma.asc -T 20 uguess.h5` traveling wave in z with an initial guess for wavespeed
`findsoln -orb -T 64.3 uguess.asc` periodic orbit with initial guess for period T
`findsoln -orb -T 34.8 -sigma sigma.asc uguess.h5` pre-periodic orbit
`findsoln -orb -xrel -T 49.1 -sigma sigma.asc uguess.h5` relative periodic orbit with phase shift in x 