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====== How to produce a Poincare section of plane Couette flow ====== around the Nagata EQ2 upper branch. This is procedure is too special-case and kludgy to put in channelflow documentation. ===== Integrate perturbations ===== I'll assume you have the Nagata upper-branch eqb EQ2.ff and the eigenfunctions of the complex instability, ef2.ff and ef3.ff. These fields are actually real and imaginary parts of the complex eigenfunctions, and the evolution of perturbations goes like <latex> u(t) = \text{EQ}_2 + e^{\text{Re} \lambda t} [\text{ef}_2 \cos (\text{Im } \lambda t) - \text{ef}_3 \sin (\text{Im } \lambda t)] </latex> The eigenfunctions are non-orthogonal so the first thing to do is to make an orthogonal basis from them. Throw in the next leading S-symmetric eigenfunctions for good measure. makebasis ef2 ef3 ef11 ef12 The produces e0, e1, e2, e3. The first two will span ef2, ef3. Construct perturbations of the form <latex> u(0) = \text{EQ}_2 + \epsilon \; \Lambda^{n/N} e_0 </latex> where Λ = exp(Re λ * 2 π / Im λ) = 6.7549 is the expansion multiplier for one period of oscillation. This will produce trajectories uniformly distributed under the iterated unstable oscillation. I set ε = 1e-05 and started with N=16, and named my initial condition fields after the digits in Λ^(n/N) (using digits rather than integer labels will scale if I later increase N to 32 or 64). E.g Λ^(n/N) for n/N = 1/16 is 1.1268, and for 2/16 is 12696. addfields 1 EQ2.ff 1.1267e-05 e0.ff eq2_11268e0.ff addfields 1 EQ2.ff 1.2696e-05 e0.ff eq2_12696e0.ff ... Integrate these 16 fields for a few hundred time units and save. couette -T0 0 -T1 400 -o data-11268 eq2_11268e0.ff couette -T0 0 -T1 400 -o data-12696 eq2_11268e0.ff ===== Compute the Poincare crossings ===== [[gtspring2009:howto:eq2poincare.cpp]] is a special program I wrote to compute crossings of a Poincare section defined by (u(t) - EQ2, e(θ)) == 0 where e(θ) = e0 cos θ + e1 sin θ, and where u(t) is always mapped into a canonical 1st quadrant
