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--- gtspring2009:research_projects [2009/07/08 08:29]
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+++ gtspring2009:research_projects [2010/02/02 07:55] (current)
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-====== Research projects ======
+====== Ideas for 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
+==== Open projects ====
-\\
- {{:gtspring2009:research_projects:gtspring2009:research_projects:pcfisotrpgrps.png?600|}}
+  - **most beautiful:** [Predrag 2009-09-08] Talked to [[gtspring2009:gibson:eq8|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 [[http://www.opendx.org/|openDX.org]]. More info [[:gtspring2009:spieker_blog:itano|here]].
-\\
+  - **most profound:** [John 2009-07-08, 2009-10-09] [[research:gibson:wallsolns|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.
-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.
+  - Repeat the //67-fold path// **group theory decomposition** of [[http://chaosbook.org/projects/Halcrow/thesis.pdf|Halcrow]] and [[http://www.cns.gatech.edu/~predrag/papers/preprints.html#n00bs|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 [[http://chaosbook.org/library/index.html#etc12_proceed|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@physics.gatech.edu|Predrag Cvitanovic]] 2009-09-26 13:44//
   - 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.
@@ Line 36: / Line 21: @@
   - 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//
+====== Assigned research 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//
+===== Continuations of existing solutions =====
+:-) I took the straightforward, lots of PACE CPU time project: [[gtspring2009:spieker_blog:continuations:lx|Track existing W03 and GHC solutions and their bifurcations]] as functions of streamwise <latex>L_x</latex>, spanwise <latex>L_y</latex>.   --- //[[dustin.spieker@gatech.edu|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
+\\
+ {{: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]]." Can be executed with channelflow.org "as is". If successful, a new contribution.
+**11/04/09 DWS** found the first //R//-isotropy equilibrium, see [[gtspring2009:spieker_blog:daily_blog|spieker_blog:daily_blog]].
+**11/12/09 DWS**  found the first //R<sub>z</sub>//-isotropy invariant equilibrium, see [[gtspring2009:spieker_blog:daily_blog|spieker_blog:daily_blog]].
 ===== Hiccups =====

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