====== How to find a new solution of plane Couette flow ====== For now this discussion is limited to finding equilibria. See also * [[docs:math:symmetry]] * [[docs:utils:findorbit]] * [[http://www.cns.gatech.edu/~gibson/publications/HalcrowARX08b.pdf]] ===== Generate an initial guess ===== We expect to find equilibria in symmetry groups that contain sign changes in both x and z. See [[http://www.cns.gatech.edu/~gibson/publications/HalcrowARX08b.pdf|our PCF equilibrium paper]] for details. Chose any symmetry group containing σxz, for example, <latex> \{e, \sigma_{xz}\} </latex> Generate an initial condition in this symmetry group. One way to get a decent initial condition is to randomly perturb a known solution and then project onto the right symmetry group. For example, download the [[http://www.channelflow.org/dokuwiki/lib/exe/fetch.php/database:w03:eq2.ff|Nagata upper branch]] and then perturbfield --magnitude 0.01 eq2 eq2perturb symmetryop -sx -sy -sz -az 0.5 eq2perturb s3eq2pertub addfields 0.5 eq2perturb 0.5 s3eq2perturb eq2perturbsymm This sequence of commands constructs eq2perturb = eq2 + 0.01 (random perturbations) s3eq2perturb = s3(eq2perturb) eq2perturbsymm = 1/2 (eq2perturb + s3eq2perturbsymm) The final field %%eq2perturbsymm%% will be %%eq2%% plus some s3-symmetric perturbations. ===== Integrate ===== Next generate a sequence of s3-symmetric data by integrating the initial condition couette -symms sxz.asc -T0 0 -T1 1000 eq2perturbsymm The ''-symms sxz.asc'' option restricts the integration to a symmetric subspace specified by the file ''sxyz.asc''. Here there is just one generator and the file should be % 1 1 -1 -1 -1 0 0 See the [[docs:classes:fieldsymmetry#ascii_io|FieldSymmetry docs]] for more on the file formats. ===== Search for solutions ===== Now search for a new solution using random samples of the turbulent simulation data mkdir findorbit-u500 cd findorbit-u500 findorbit -eqb -symms sxz.asc ../data/u500 Keep an eye on the file convergence.asc. If the residual gets to 1e-08 or so you're on your way to a solution. ===== Check against known solutions =====
