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Ideas for research projects

Open projects

1. most beautiful: [Predrag 2009-09-08] Talked to Itano about plotting our solutions in their beautiful way. He said he cannot do it, has 15 min/week for research, but that it will take us 3 days to learn how to do it ourselves. It is done with IBM's openDX.org. More info here.
2. most profound: [John 2009-07-08, 2009-10-09] Try to find near-wall solutions. Basic idea is to add a pressure gradient to plane Couette flow to break the sigma_x symmetry and make the mean flow profile look more like the mean flow profile in the near-wall region of a turbulent channel flow. That would give you pseudo boundary layer solutions (i.e. solutions that roughly fit the mean-flow profile of a boundary layer, but have a Dirichlet boundary condition on the upper surface). Then you rachet up the Reynolds number and pressure gradient in order to make the total domain larger and larger, relative to the near wall region. Eventually you have solutions living in the high-gradient near-wall region of a boundary layer, with the Dirichlet boundary condition far away.
3. straightforward, most PACE CPU time: Track existing HKW periodic orbits and their bifurcations as functions of Re.
4. most fun: Join/continue J. Elton Lagrangian chaos investigation of individual equilibria. Relative equilibria and periodic orbit solutions have not been touched yet. Read Elton svn repository first.
5. Search for new relative equilibria (traveling waves). In particular, currently we have no relative equilibrium that travels both streamwise and spanwise.
6. Repeat the 67-fold path group theory decomposition of Halcrow and Gibson, Halcrow and Cvitanović for the duct flows (square profile, as opposed to the circular profile of the pipe). Reason: number of groups (Nagata, Botero, Kawahara, etc - see Marburg turbulence conference ETC12 proccedings) are repeating plane Couette kind of investigations for the duct problem. None of them have worked detailed group theory, or the state space visualizations. For us it is simpler than the pipe, as there is only one, streamwise continuous symmetry, and I believe it is pure SO(2) , not O(2). — Predrag Cvitanovic 2009-09-26 13:44
7. Search for new heteroclinic connections, particularly connections to solutions with greater than 1d unstable manifolds. Get a start on this by expanding the existing heteroclinic connections. The EQ4 to EQ1 connection can be extended to a 2d sheet by perturbing along a third real-unstable S-symmetric eigenfunction. Could also explore the possibility of hitting the different translational phases of the EQ1 from EQ4 by perturbing the existing heteroclinic connection with S-antisymmetric eigenfunctions.
8. hard, but important: Construct a control system using linearizations about a set of periodic orbits with blowing/sucking boundary conditions for control. This has a lot of parts, should probably start with just one part e.g. linearizing about a periodic orbit
9. numerical/programming, important: implement an automated continuation system for solutions as a function of parameter (Re, Lx, Lz). This would involve doing a little literature review on continuation algorithms and then implementing an algorithm and hooking it into channelflow's solution-finding code.
10. Try to hit various lower-branch equilibria using small perturbations of the laminar state as initial conditions, or make a close pass and then shoot towards turbulence along the lower branch's unstable manifold. Initial condition would be a Stokes-mode [v,w](y,z) roll plus a perturbation with streamwise (x) variation.
11. Compute subharmonic instabilities of existing solutions
12. Try to construct new solutions as subharmonic perturbations of spatially doubled (tripled,…) solutions
13. Predrag's junk bonds rating: (if you cannot think, compute) Do numerical studies of lifetime of transients
14. Look at state-space structure as a function of Reynolds number, starting from our current state-space portraits and heteroclinic connections. Can we understand transient lifetimes as a function of Reynolds from changes in state space structure?
15. Related to above, try to understand via state-space structure why turbulent lifetime increases dramatically when Lz goes from to

Assigned research projects

Poincaré sections and return maps

most important: Compute an W03 cell unstable manifold Poincaré section and return map for the upper branch solution. Place Gibson's new periodic orbit P47.18 in the W03 cell on it.

I will take the W03 unstable manifold Poincaré sections and return maps for the upper branch project. I have a feeling that I could readily jump into this one given the tools recently provided in channelflow-1.3.4. — Dustin Spieker 2009-03-02 14:39

Continuations of existing solutions

I took the straightforward, lots of PACE CPU time project: Track existing W03 and GHC solutions and their bifurcations as functions of streamwise , spanwise . — Dustin Spieker 2009-07-02

Equilibria of plane Couette with one of the five isotropy groups

most straightforward: Search for equilibria of plane Couette with one of the five isotropy groups

that are currently unexplored, in particular the or isotropies described in “Halcrow et al. on symmetries of plane Couette.” Can be executed with channelflow.org “as is”. If successful, a new contribution.

11/04/09 DWS found the first R-isotropy equilibrium, see spieker_blog:daily_blog.

11/12/09 DWS found the first Rz-isotropy invariant equilibrium, see spieker_blog:daily_blog.

Hiccups

(latest posts at the top)

Let's agree on all graphics being *.png - that works for both the dokuwiki and pdflaTeX. See How to format figures. — Predrag Cvitanovic 2009-02-27 13:29

Let people experiment with the dividing line. The overhead of blogging via latex and subversion is large enough to slow down my research, deter me from blogging things that I should blog, and commenting on others' blogs, and keeping a current PDF on the web of my blog was such a hassle I never did it (should have used a cron job). This motivated me to start a latex-enabled wiki. On the downside, blogging via wiki would take us further away from publication readiness. The web and the wiki prefer pixel-oriented graphics like PNG and JPEG rather than postscript or PDF, for example. But in my experience blogging figures good enough for publication is a waste of time (getting the fonts, axes, labels, etc just right is very time-consuming –I spend at least a couple hours for every published figure), and the text of a paper is very different from what you write when you're trying to make sense of new ideas. That said, I'm not sure that blogging via wiki is right, so I support letting people experiment.

I lean toward putting most text into individual blogs (easiest to migrate to publications), and putting updates as what the study group might want to read into study group main page.

Need to ponder a rational division between what goes into the communal blog, and what stays in individual blogs.