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====== Research projects ====== ===== Assigned projects ===== ==== Poincaré sections and return maps ==== {{gtspring2009:pc.jpg }} **most important:** Compute an W03 cell unstable manifold [[chaosbook:maps#sectionpoincare_sections|Poincaré section and return map]] for the upper branch solution. Place Gibson's new [[gtspring2009:gibson:w03|periodic orbit P47.18 in the W03 cell]] on it. :-) I will take the [[gtspring2009:spieker_blog#projectpoincare_sections_and_return_maps|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@gatech.edu|Dustin Spieker]] 2009-03-02 14:39// ===== Ideas for research projects ===== Please pick a project (or add your own project, if not in the list), write your name next to it with a link to your blog, where you describe your goals in detail. ==== Open projects that can be executed with channelflow.org "as is" ==== - **most straightforward:** Search for equilibria of plane Couette with one of the five isotropy groups \\ {{:gtspring2009:research_projects:gtspring2009:research_projects:pcfisotrpgrps.png?600|}} \\ that are currently unexplored, in particular the <latex>R_x</latex> or <latex>R</latex> isotropies described in "[[http://www.scribd.com/full/4929920?access_key=key-28bjtf1f3w5j7pq0ef7h|Halcrow et al. on symmetries of plane Couette]]." If successful, a new contribution. - **straightforward, lots of PACE CPU time:** Track existing W03 cell states and their bifurcations as functions of streamwise <latex>L_x</latex> - **straightforward, most PACE CPU time:** Track existing HKW periodic orbits and their bifurcations as functions of Re. - **most fun:** Join/continue [[:gtspring2009:research_projects:elton:blog|J. Elton Lagrangian chaos]] investigation of individual equilibria. Relative equilibria and periodic orbit solutions have not been touched yet. Read Elton svn repository first. - Search for **new relative equilibria** (traveling waves). In particular, currently we have no relative equilibrium that travels both streamwise and spanwise. - 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. ==== Other open projects ==== - **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 - **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. - 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. - Compute subharmonic instabilities of existing solutions - Try to construct new solutions as subharmonic perturbations of spatially doubled (tripled,...) solutions - **Predrag's junk bonds rating:** (if you cannot think, compute) Do numerical studies of lifetime of transients - 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? - Related to above, try to understand via state-space structure why turbulent lifetime increases dramatically when Lz goes from <latex> 1.2 \pi </latex> to <latex> 1.75 \pi </latex> ===== Hiccups ===== (latest posts at the top) {{gtspring2009:pc.jpg }} Let's agree on all graphics being *.png - that works for both the dokuwiki and pdflaTeX. See [[gtspring2009:howto:format#figures|How to format figures]]. --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-27 13:29// {{gtspring2009:gibson.png?24}} 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. {{gtspring2009:pc.jpg }} 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 [[:gtspring2009|study group main page]]. {{gtspring2009:gibson.png?24}} Need to ponder a rational division between what goes into the communal blog, and what stays in individual blogs.
