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--- gtspring2009:gibson:hkw [2009/02/11 09:53]
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+++ gtspring2009:gibson:hkw [2010/02/02 07:55] (current)
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 {{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}}
-{{p50p16e1e2.png?220}}{{p50p16e1e3.png?220}}{{p50p16e2e3.png?220}}{{p50p16e1e2e3.png?220}}
-{{p82p36e1e2.png?220}}{{p82p36e1e3.png?220}}{{p82p36e2e3.png?220}}{{p82p36e1e2e3.png?220}}
-{{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]].

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