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gtspring2009:gibson:hkw [2009/02/11 09:53]
gibson created
gtspring2009:gibson:hkw [2010/02/02 07:55] (current)
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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}} 
- 
- 
-{{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]]. 
  
gtspring2009/gibson/hkw.txt · Last modified: 2010/02/02 07:55 (external edit)