gtspring2009:gibson:hkw

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{{pod_e1e4.png?250}}{{p87pod_e1e3.png?250}} | {{pod_e1e4.png?250}}{{p87pod_e1e3.png?250}} | ||

- | ====== W03 cell ====== | ||

- | ===== 2009-02-05 Intro ===== | ||

- | In which I write at length of my long personal quest for periodic enlightenment. | ||

- | |||

- | Ok, at long last I have starting putting effort into the (alpha, gamma) = (1.14, 2.5) a.k.a. W03 ([[:references|Waleffe Phys Fluids 2003]]) or narrow cell again. | ||

- | This was prompted by a comment by Roman in class a couple weeks ago about relative ease of finding EQBs and POs in | ||

- | HKW ([[:references|Hamilton et al JFM 1995]]) vs W03 cells and a recommendation from Predrag to begin our periodic orbit paper (in development) with the W03 cell, | ||

- | whose state-space structure we understand a little better. Plus, previous results suggested that the two orbits we had | ||

- | for W03 were perturbations of each other (a short one P35.86 and a long one P97.08 that was like the short one plus | ||

- | an extra wiggle). | ||

- | |||

- | I computed these orbits way back in summer 2007 just before going to India. Yikes! That was the very beginning of my | ||

- | orbit-computing days. The 97.08 orbit was not very well converged (|f^t(u) - u| = 5e-3 compared to 1e-8 for the other), | ||

- | and I didn't have a good enough sense of these computations to know what the problem was and whether I really had an | ||

- | orbit or not. So I went on to hone my orbit-finding chops on easier problems (HKW, where good initial guesses are | ||

- | plentiful). And doing this, in India and more recently, I uncovered several problems in my implementation of trust-region | ||

- | heuristics that might have prevented P97.08 from converging. | ||

- | |||

- | Anyway back to the present. Over the last couple weeks I have run maybe thirty orbit searches in the WO3 box on | ||

- | initial guesses from moderately long-lived trajectories from perturbations of equilibria, plus the pre-India | ||

- | P35.86 and P97.08 orbits. P35.86 converged more precisely and is now renamed P35.77. PACE is still pounding on | ||

- | P97.08 and has the residual down to 1e-3, after about a hundred Newton-hooksteps. But I have found several other | ||

- | orbits to double precision (for a given discretization). Here are some pics. | ||

- | |||

- | {{p35p77e1e2.png?220}}{{p35p77e1e3.png?220}}{{p35p77e2e3.png?220}}{{p35p77e1e2e3.png?220}} | ||

- | |||

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- | {{p50p16e1e2.png?220}}{{p50p16e1e3.png?220}}{{p50p16e2e3.png?220}}{{p50p16e1e2e3.png?220}} | ||

- | |||

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- | {{p82p36e1e2.png?220}}{{p82p36e1e3.png?220}}{{p82p36e2e3.png?220}}{{p82p36e1e2e3.png?220}} | ||

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- | {{p83p60e1e2.png?220}}{{p83p60e1e3.png?220}}{{p83p60e2e3.png?220}}{{p83p60e1e2e3.png?220}} | ||

- | |||

- | In the pics: | ||

- | |||

- | * The black dot is laminar flow | ||

- | * The blue (green) dots are Nagata lower (upper) branch eqbs ([[:references|Nagata JFM 1990]]) | ||

- | * Red dots are EQ4 a.k.a "newbie" equilibrium from our recent JFM papers | ||

- | * Thin (blue, green, red) lines are trajectories in low-d unstable manifolds of (Nagata LB, Nagata UB, EQ4) | ||

- | * The thick lines are the periodic orbits | ||

- | * Projection is same as in most figures in [[:references|Gibson et al. JFM 2008]] | ||

- | |||

- | And some properties. Will add eigenvalues when they're done cooking. | ||

- | |||

- | ^ T ^ σ ^ %%<D>%% ^ %%|Λ|>1%% ^ max %%|Λ|%% ^ max Re λ ^ %%|Λ|%%<sub>S</sub> >1 ^ max %%|Λ|%%<sub>S</sub> ^ max Re λ<sub>S</sub> ^ | ||

- | | 35.77 | Lx/2 | 2.78 | 19 | 13.0 | 0.072 | 5 | 2.5 | 0.026 | | ||

- | | 50.16 | | 2.32 | 19 | 26.9 | 0.040 | 5 | 26.9 | 0.040 | | ||

- | | 82.36 | Lx/2 | 1.90 | 5 | 63.7 | 0.050 | 2 | 63.7| 0.050 | | ||

- | | 83.60 | | 2.21 | 8 | 57.6 | 0.049 | 3 | 9.1 | 0.026 | | ||

- | |||

- | * Λ is the characteristic multiplier of the map u → σ f<sup>T</sup>(u), computed with Arnoldi iteration. | ||

- | * The %%|Λ|>1%% column gives the number of unstable characteristic multipliers. | ||

- | * Re λ = 1/T ln(%%|Λ|%%) is a continuous-time measure of the orbit's instability. | ||

- | * Quantities with S subscripts are restricted to the set of eigenvalues with S = {1, s1, s2, s3} symmetry. See [[:docs:math:symmetry]]. | ||

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