# channelflow.org

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gtspring2009:howto:findsoln

# How to find a new solution of plane Couette flow

## 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,

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.

## 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 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.