This is an old revision of the document!
Let us see how this format works. For now I will just start writing. I hope that greater flexibility in the structure (multiple pages and links) will make it easy to reorganize as the material gathers.
2008-12-30 through 2009-01-06
I have been dissatisfied for quite a while with the projection basis sets I've used for plotting periodic orbits in the HKW cell. We don't have good equilibria there, so we can't easily do the translational-basis set we used for the narrow cell in the JFM08 paper. I have previously used a POD basis from the data of a single periodic orbit (P87p89). The plots were good, in that they showed turbulent trajectories lying in the same regions of state space as the periodic orbits, and forming similar pretty-well-defined shapes in whatever projection we looked at. But Predrag never liked this since it involved POD and averages, and to me, it was rhetorically weak to use a basis based on periodic orbits to argue that the periodic orbits live where turbulence is.
So, I decided to compare the orbit-POD basis to a translational basis based on a single velocity field (for Predrag) and a turbulent-data POD basis (for me). It turns out they're all remarkably similar.
Some key points:
The mean of the P87p89 orbit and its half-cell shift in x, , with and without the laminar flow.
Four basis sets
Main things to note:
The basis sets are very similar. That means that P87.89 explores the same region of state space as turbulence, in the crude let's-pretend-everything-is-a-Gaussian sense of the POD. Still, that is something, beyond the 1d plots showing that some orbits capture the mean and the Reynolds stresses within a few percent. Those 1d plots are averaged over x and z as well as time. Here we preserve the symmetry that breaks x,z homogeneity and find that the 3d average and multidimensional variances are very similar.
(Correction, 2009-01-05: The P87 POD basis elements shown above are corrected since the original posting. I think I had mistakenly copied the turbulent POD basis elements. The correspondence between the two sets is weaker but still strong. Note that the POD modes are ordered differently between the turbulent POD and the P87. E.g. turbulent POD mode e4 matches P87 POD mode e3. The sets are ordered by the variances of the imaginary Gaussian.)
It is not surprising that the translational basis based on the P87p89 mean is close to the P87p89 POD basis. The means of the P87p89 and P87p89 orbit are far from the origin and relatively close together. Therefor it is not surprising that the first two POD modes of the full orbit are close to of the means of the two orbits.
The P87p89 POD basis, however, continues on to e0, e1, e2, e3, … whereas the translational basis is just 2d. And the P87p89 POD basis is well-defined and easy to compute compared to the turbulent POD, because it is closed.
Note that for the POD bases, e0 and e2 have the same signs in the streaks but opposite signs in the rolls. That's in line with Wally's contention that the rolls and streaks are dynamically independent. Linear superposition of e0 and e2 can produce roll-streak combinations with different relative strengths of rolls and streaks. That such a superposition is present in the very lowest POD says that such variations in relative roll-streak strength happen frequently in the periodic orbit and turbulent flow.
Next thing is to compute state-space portraits of the orbits and turbulence for the periodic orbit paper. Also, it would be good to compare the e2 POD modes to the linear instability of the streaks a la self-sustaining process.
Addition 2009-01-06: SSP basis I added a column showing an SSP basis that I computed last spring following Waleffe Phys Fluids vol 9 no 4, 1997 closely (same cell size, frozen roll strength V=0.02). Main thing to note here is that the complex instability modes of the SSP basis are very similar to two of the POD modes. I.e., the POD modes arise from the complex instability from Wally's SSP.
e0 and e1 are given by
where “roll” is a [v,w](y,z) eigenfunction of the Stokes operator and “streak” is the asymptotic forced response u(y,z) to a frozen of strength V=0.02 (Wally's terminology, means maximum amplitude of v(y,z) = 0.02). c00 and c11 are determined by the choosing relative strengths of streaks and rolls to match means of DNS data and then normalizing to 1 (i.e. to match e0 of the POD basis sets). The coefficients for e1 are set by orthogonality to e0 and normalization.
e2 and e3 are computed this way: Compute the complex instability of frozen streak+roll in strengths comparable to DNS data. Orthogonalize to get e2, e3. There is an arbitrary choice of phase
Chose to maximize i.e. to best align e2' and e3' with the e1 and e3 basis elements of the turbulent POD, respectively.
I know this probably sounds like an arbitrary concoction, but it does establish something new and important, I think: that those periodic orbits and the turbulent flow really are strongly associated with the complex instability of frozen streak+roll structures.
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 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 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.
In the pics:
And some properties. Will add eigenvalues when they're done cooking.
|T||σ||<D>|||Λ|>1||max |Λ|||max Re λ|||Λ|S >1||max |Λ|S||max Re λS|
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