User Tools

Site Tools


How to find a new solution of plane Couette flow

For now this discussion is limited to finding equilibria. See also

Generate an initial guess

We expect to find equilibria in symmetry groups that contain sign changes in both x and z. See our PCF equilibrium paper for details. Chose any symmetry group containing σxz, for example,

\{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 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.


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 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

gtspring2009/howto/findsoln.txt · Last modified: 2010/02/02 07:55 (external edit)