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

-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

`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.

command | search 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 |

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