gtspring2009:research_projects
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gtspring2009:research_projects [2009/02/27 14:27] predrag intermediate commit (my browser is not letting me import figures, uurgh) |
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| ====== Research projects ====== | ====== Research projects ====== | ||
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| + | ===== Assigned projects ===== | ||
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| + | ==== Poincaré sections and return maps ==== | ||
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| + | {{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. | ||
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| + | :-) 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// | ||
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| Please pick a project (or add your own project, if not in the list), write your name next to it with a | 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. | link to your blog, where you describe your goals in detail. | ||
| + | ==== Open projects that can be executed with channelflow.org "as is" ==== | ||
| - | ==== 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 |
| - | + | \\ | |
| - | - Search for equilibria of plane Couette with isotropy group | + | {{:gtspring2009:research_projects:gtspring2009:research_projects:pcfisotrpgrps.png?600|}} |
| - | that is currently unexplored, in particular the ?? isotropy described in "[[http://www.scribd.com/full/4929920?access_key=key-28bjtf1f3w5j7pq0ef7h|Halcrow et al. on symmetries of plane Couette]]." | + | \\ |
| - | - Look for equilibria, traveling waves, and periodic orbits of plane Couette in isotropy groups that are currently unexplored. | + | 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. |
| - | - Systemize the computation of periodic orbits by considering how unstable manifolds of different solutions (nearly) map onto each other. | + | - **straightforward, lots of PACE CPU time:** Track existing W03 cell states and their bifurcations as functions of streamwise <latex>L_x</latex> |
| - | - Search for more heteroclinic connections, particularly connections to solutions with greater than 1d unstable manifolds. | + | - **straightforward, most PACE CPU time:** Track existing HKW periodic orbits and their bifurcations as functions of Re. |
| - | - Do a more thorough bifurcation analysis of existing solutions. | + | - **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 ==== | ==== Other open projects ==== | ||
| - | - Look for equilibria, traveling waves, and periodic orbits of plane Couette in isotropy groups that are currently unexplored. | + | - **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 |
| - | - Do numerical studies of lifetime of transients. | + | - **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. |
| - | - 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> | + | |
| - | - 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 | + | |
| - 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. | - 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 | - Compute subharmonic instabilities of existing solutions | ||
| - Try to construct new solutions as subharmonic perturbations of spatially doubled (tripled,...) 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> | ||
| + | - 2009-07-08 **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. I will write this up with plots soon. //John Gibson 2009-07-08 11:27 EST// | ||
| ===== Hiccups ===== | ===== Hiccups ===== | ||
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| {{gtspring2009:gibson.png?24}} Need to ponder a rational division between what goes into the communal blog, and | {{gtspring2009:gibson.png?24}} Need to ponder a rational division between what goes into the communal blog, and | ||
| what stays in individual blogs. | what stays in individual blogs. | ||
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gtspring2009/research_projects.txt · Last modified: 2010/02/02 07:55 (external edit)
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