# channelflow.org

### Site Tools

gtspring2009:howto:findsoln

# Differences

This shows you the differences between two versions of the page.

 — gtspring2009:howto:findsoln [2010/02/02 07:55] (current) 2009/03/20 17:50 gibson created 2009/03/20 17:50 gibson created Line 1: Line 1: + ====== 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}\} + ​ + + + 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 =====