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Gibson research blog, spring 2009

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.

State space of HKW cell

Basis sets

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:

  • All results here are confined to the S-invariant subspace as defined in our JFM08 paper.
  • Unlike the narrower γ=1.5 cell, in the HKW γ=1.67 cell, the streamwise phase of the roll/streak system never changes, at least in my observation. If the high-speed (red) streak starts in the middle of the box, it stays in the middle of the box, for thousands of time units.
  • Thus, unlike the narrow cell, a translational basis need not include spanwise (z) half-cell shifts in order to (roughly) span the region of state space that a turbulent trajectory explores. That gives just a two-d basis:  e_{\pm} = c_{\pm} u \pm \tau_x u .
  • Since we don't have a good equilibrium to form a translational basis, I used the mean of the P87p89 orbit plus its half-cell shift in x (pictured below). In the figures these basis elements are labeled e0 P87meantrans and e1 P87meantrans.
  • The turbulent POD basis was constructed from about two hundred samples over a trajectory of several thousand time units. The basis clearly reflects the fixed z-phase of the roll-streak structures.
  • The POD basis sets were calculated without removing the mean first, and the mean is far from the origin, so the first (zeroth) POD mode corresponds to the mean of the data. This facilitated comparison with the translational basis.
  • The POD calculations also explicitly enforced symmetry under half-cell x translations. E.g. the averages were computed over both P87p9 and  \tau_x P87p89, and similarly for the samples of the the turbulent trajectory.

The mean of the P87p89 orbit and its half-cell shift in x,  \langle P87.89 + \tau_x P87.89 \rangle , with and without the laminar flow.

Four basis sets

  • en POD : POD of a turbulent trajectory and its  \tau_x symmetric partner.
  • en SSP : ON basis constructed from streak, roll and the Re and Im parts of the complex instability of roll+streak
  • en P87p89 POD : POD of P87p89 and its  \tau_x symmetric partner.
  • en P87p89meantrans:   \langle P87.89 \pm  \tau_x P87.89 \rangle

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  \tau_x 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  \pm 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

 e_0 = c_{00} \text{streak} + c_{01} \text{roll} \\
 e_1 = c_{10} \text{streak} + c_{11} \text{roll}
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

 e_2' = \cos(\theta) e_2 - \sin(\theta) e_3 \\
 e_3' = \sin(\theta) e_2 + \cos(\theta) e_3
Chose \theta to maximize (e_2', \text{POD} e_1)^2 + (e_3', \text{POD} e_3)^2 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.

State space portraits


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 (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 (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.

In the pics:

  • The black dot is laminar flow
  • The blue (green) dots are Nagata lower (upper) branch eqbs (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 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|S >1 max |Λ|S max Re λS
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 fT(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 symmetry.

2009-2-08 Movies of W03 orbits

(If Flash is installed, you can watch a video inside this web page.)

(If Flash is installed, you can watch a video inside this web page.)

(If Flash is installed, you can watch a video inside this web page.)

(If Flash is installed, you can watch a video inside this web page.)


This area is for testing embedded movies in blogs, particularly the namespace issues.

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gtspring2009/gibson/blog.1234149327.txt.gz · Last modified: 2009/02/08 19:15 by gibson