Table of Contents

on which I post my miscellaneous daily activities

to do

  1. Apply for funding for Channelflow. PC 2009-11-30
  2. Get a job a year hence. 2009-11-03
  3. Write periodic orbit paper. 2009-08-25
  4. Write 2d navier-stokes integrator. 2009-08-25
  5. Cylinder wake. 2010-02-15
  6. Localized plane Couette solutions, JFM paper. 2010-02-15

am doing

2009-11-03

2009-11-01 PC: John, you might want to enter Channelflow.org into dynamical systems software list. November already. Mein Gott.

2009-10-20 PC: would be nice to do a man-to-man skype about weltanschanung in general. Click on skype if I show up green

2009-10-16 Localized EQB/TW PRL largely in place. Figures are done, the text is in place but needs to be shrunk. References are not done. I am headed to California for a week and two weekends with Uma, to visit my sister and attend a wedding. Will be back 2009-10-27. Will be in email contact but only sporadic nighttime skype.

Periodic orbit paper:

  1. Thirtyish S-symmetric orbits: an IvD plot of a few of them at HKW Re=400, table of properties, movies.
  2. Methods: computation of Poincare intersections, generation of initial guesses from close returns on I-D=0 Poincare section, restriction of time integration to symmetric subspace
  3. Dynamics: one of
    • an initial stab at linearization about P-section FPs, see how large the region of accurate linearization is,
    • possibly extend unstable manifold of an orbit or two and look for heteroclinic connection between two orbits.
    • symbolic dynamics via Voronoy triangulation or spheres around FPs, based on the estimate of linear accuracy.

2009-10-10 Wrote out idea for near-wall solutions.

2009-10-06 Have 31 HKW orbits at I-D=0 and at 48x33x48 (32x33x32 after dealiasing) dt~0.03, most confirmed to be 1e-04 or 1e-05 accurate at 64x49x64 and dt~0.02 and to converge quickly at that resolution. A few of the longer orbits are perhaps spurious at the low resolution, with initial error 1e-02 or 1e-03 and slower initial convergence. Will have to compare IvD plots at the two resolutions. Next step: try linearization about a specific orbit, or extend unstable manifold within Poincare section.

2009-10-05 Organizing orbits for plotting, posting, writing up: scping from PACE, getting to same gridsize, choosing best Poincare intersection, computing trajectories, plotting I vs D, etc.

2009-10-05 Found ten new HKW orbits since last Thursday using close recurrences on I-D=0 Poincare section for initial guesses. It is so much easier this way. I think a JFM paper describing this, presenting a table of the thirtyish orbits, showing a few in detail, and perhaps a section showing how unstable manifolds of the return map can be continued in the Poincare section, would be a good paper. Also, some interesting new results on continuation in streamwise length for the localized solutions.

2009-09-25 Localized PCF PRL is underway. Continuing existing orbits in aspect ratio, diagonal length, and Reynolds using I-D=0 Poincare section.

have done

“have done” is just “am doing” suitably aged

2009-11-30 Laurette, via PC: I was very nice to you guys? I'd say it was the other way around!

  1. I used one of the channelflow figures to illustrate perfectly the minimal flow unit
  2. I used the Gibson laptop to give my presentation with movie (which my own computer was incapable of doing)

2009-11-30 PC to Laurette: John reports you were very nice to us humbldt plumbers in your talk - mercy buckets!

2009-11-25 We got a lot of love at the APS meeting: mentions in many talks, including a prominent placement in Laurette's invited session (she placed us along with Wedin, Kerswell, Eckhardt, Faisst, Hof and others in a line of progression starting with Reynolds) plus a enthusiastic recommendation of channelflow.org. A number of people came up to me and thanked me for channelflow.org and/or spoke highly of our first JFM paper. And the localized solution work made a big impact. We really got a lot of excitement over that.

2009-09-15 Um, a couple weeks went by there. Have started localized eqb paper w Tobias and John Burke (for PRL). Set up using subversion hosting on svn.channelflow.org to avoid ssh tunneling problems and need for GT accounts. Long discussion w Wally and Divakar about whether transition to turbulence in large (8pi x 2 x 8pi) domains results from close passes to Nagata-Busse-Clever-Waleffe LB EQBs. Found no evidence supporting this idea. Resumed continuations in Re of HKW orbits, using I-D=0 Poincare section to eliminate temporal phase shifts. This improves the quadratic extrapolation in the continuation parameter and reduces the amount of manual oversight needed in continuations. Am just focusing on continuing a few orbits for the PO paper to demonstrate parametric robustness of orbits and a few connections via bifurcation.

2009-08-25 Set up skeleton for localized PCF PRL paper on svn.channelflow.org; thankless professional activity.

2009-08-24 Nailed down saddle-node bifurcation of localized EQB in 4pi,16pi box. Eigenfunction has expected shape (concentrated near edges of solution); another eigenvalue crossing the imaginary axis shortly after bifurcation could give rise to EQB-TW connections.

2009-08-20 Figuring out the symmetries of the EQB and TW bifurcations. It is amazing how much time stuff like this takes, getting all the solutions in compatible spatial phases, figuring out the symmetry groups on the larger domains, calculating eigenfunctions and their symmetries…

2009-08-19 The calculations of I-D=0 intersection of all the HKW orbits should be done today. While those are going on I'm going to see if I can figure out the symmetries of bifurcations of the localized EQB and TW off the globally periodic solution, in prep for a Friday meeting with Tobias and John.

2009-08-18 Calculating intersections of HKW orbits with I-D=0, with some further tinkering on that code to just calculate the crossings you want (within a time range and in desired direction).

2009-08-17 Updated Poincare-section GMRES-hookstep code for fixed points of Poincare map to higher accuracy intersections and general defns of sections. Have tested on coarse-discretized P19p02, gets f(u)-u==0 and I-D==0 to 1e-13.

2009-08-17 Revised interpolation trajectories crossing Poincare section for higher accuracy. After quadratic interpolation of t that gives to f(u(t)) == 0, added a Newton search on t. This increases accuracy of interpolation from O(dt^3) to floating point. Though this accuracy is artificial, I think it will help remove spurious marginal eigenvalues in Arnoldi iteration of the Poincare map. I am going to compute the I==D poincare sections of all the HKW periodic orbits now.

2009-08-14 PACE nodes are about 1/4 as fast as my new home home machine with a AMD Phenom II X4 955 CPU. It'll be faster to run at home and shut down the machine for sleep.

2009-08-14 Generalized the Poincare-section code so that user can supply arbitrary f(u) to define section. Uses a functor class so that f can retain internal data for things like f(u) = (u - u*, e*) (internal data for f is u* and e*). Am using this to compute intersection of HKW orbits with f(u) = I(u) - D(u) = 0. I think this will be a good section for partitioning and Markov model, since all orbits intersect this section.

2009-08-13 Thankless professional activity.

2009-08-10 Moved continuation calculations for localized solutions from home machine to PACE. Slept better last night without the computer humming.

2009-08-04 Have refactored the adhoc Poincare-section / fundamental domain integration routines in such a way that they can be reused in different command-line routines. Imperfect but acceptable. Imperfect in that Poincare section is defined in a particular way, (u(t) - u*, e*) == 0, same for fundamental domain (u(t), e[n]) > 0, with symmetry sigma[n] mapping crossings back into fundamental domain via sigma[n] e[n] = -e[n]. Also imperfect in that function signatures are not identical to time-T integrations so swapping into Arnoldi and Newton_krylov code will require some tinkering. I am currently recomputing EQ2, its eigenfunctions, and P47 orbit in lower discretization (32 x 33 x 32) so that I can test this code with faster integrations.

2009-08-04 Continuations of localized solutions continue running in background. Understanding how the localized EQB and TW bifurcate from the globally periodic EQB will take some thought and numerical effort. Thought is about expanding our Couette symmetry group classification to deal with quarter-shifts, eighth-shifts, etc., or better, to rephrase it in terms of finite shifts lx, lz in infinite domain. Numerical effort is to determine the symmetries/antisymmetries of the periodic EQB's eigenfunctions w.r.t the larger symmetry group and then center the EQBs and TWs already computed in proper phases so that they bifurcate from the same EQB, coordinated with the symmetries of the EQB eigenfunctions.

2009-08-07 APS-DFD abstract submitted.

2009-08-03 Finished paper review.

2009-07-31 Sent off n00bs proofs.

2009-07-30 Also figured out how to scp -pr from PACE to my home machine through an ssh tunnel on ssh.physics.gatech.edu, so I am copying my measly-but-overflowing 11 GB PACE account to my spacious home rig to make room for more PACE calculations. Watch out, pace-cns queue!

2009-07-28 Found the typo that broke all traveling-wave searches so far in my career: The FieldSymmetry *= Real operator definition had ax_ = c*az_ instead of az_ = c*az_, where ax_, az_ are finite translations and c is the multiplicative constant. When used in the Newton-Krylov-hookstep algorithm, this obliterated the small perturbations in finite-time shifts and made it impossible to converge onto solutions of the traveling-wave equations. Now am happily continuing localized traveling waves for 'even' half of odd/even snaking ladder. 2009-07-30 Refactoring DNS algorithm advance functions to uniting end-on-section with end-at-time-T algorithms for unified findsoln, arnoldi, etc. is not so easy. At times like this I wish high-level channelflow were in Python. Might lower the goal and produce kludgier special-purpose command-line utils for poincare sections.

2009-07-24 Significant progress yesterday and today on incorporating integration to Poincare sections into a DNS algorithm, so that section-to-section calculations is more self-contained and can be used within Arnoldi iteration, periodic orbit calculations, a command-line utility, etc. Today I figured out how to factor the previous command-line code into a DNS::advance function that works very similarly to the given-time-interval advance function. Have the code up and running and am testing it for proper behavior around EQ2.

2009-07-23 Am working on the poincare sections. I believe I recently properly readjusted my understanding of what Predrag wants here and am revising my Poincare-section computing code to accomplish it. Namely, I'm adding moving what was previously a post-processing calculation into a DNS algorithm, so that simulations can be specified to end at a certain time or a certain geometric value (still using polynomial interpolation on u(t-2dt), u(t-dt), u(t) rather than replacing time with a space variable, which would require recalculating and refactoring implicit integration matrices at every step –difficult and inefficient). This will simplify the process of continuing and comparing the unstable manifolds of the equilibrium and the periodic orbit.

2009-07-24 Respond to queries about forcing functions in channelflow howto blog. Posted 2009-07-16.

2009-07-22 Move channelflow SVN to channelflow.org. Will allow better access to source code for those outside Georgia Tech. Done right away because the svndump Roman made for me was getting stale. Posted 2007-07-20

2009-07-18 Center the snaking-bifurcation solutions and compute their eigenspectra.

2009-07-17 Write an ad-hoc program to translate localized solutions so they're s_xyz-symmetric about the center of the box. Took a whole day, longer than I thought.

2009-07-16. Tobias is inclined to keep this to ourselves until publishing. I'll follow his lead since he let me in on the project.

2009-06-11. PC: You might want to get in touch with Yohann Duguet, KTH Stockholm. He has some interesting spatially localized equilibria for plane Couette.

2009-06-11. I have started examining spatially localized equilibria in collaboration with Tobias Schneider, who found them with Daniel Marinc and Bruno in Marinc's 2008 Master's thesis. Tobias has some interesting plans to continue this work; I am reproducing some continuations from Marinc's thesis, will attempt to continue around bifurcations and produce more solutions with different numbers of roll-streak pairs, upper branch solutions, etc.

2009-06-09. Computer upgrade I ended the semester with my PACE account (10 GB) and home computer (180 GB) crammed with thousands of binary flow fields from time integrations and parametric continuations. The time integrations can be recomputed pretty easily, but parametric continuation is not cheap. So I decided to upgrade to larger disks at home, and a faster CPU while at it. After some research I settled on the AMD Phenom II 955 ($245). The Intel Core i7 920 ($280) is a faster chip, at least for single threaded applications (it boosts the clock speed when running just a single core), but according to the few scientific benchmarks I could find the i7 multicore scaling seems to be poorer than the Phenom II. Also, X58 motherboards for Core i7 are all > $200 high-end dual-graphics monsters, whereas for AMD I found a good $85 just-the-basics Gigabyte motherboard for AM3. I also trust AMD to keep putting out chips with good price/performance ratios, compared to Intel. So, here's my new rig 

1 x ($245.00) CPU AMD|PH II X4 955 3.2G AM3  R  $245.00
1 x ($64.99) MEM 2Gx2|GSK F3-12800CL9D-4GBNQ R   $64.99
2 x ($94.99) HD 1T|WD 7K 32M SATA2 WD1001FALS   $189.98
1 x ($84.99) MB GIGABYTE | GA-MA770T-UD3P AMD770 $84.99
1 x ($29.99) A64 COOLER |ARCTIC FREEZER 64 PRO R $29.99

totalling some $600. I used my current Lian-Li aluminum case, DVD burner, power supply, graphics card, etc. The new computer is up and running (opensuse-11.1). Each core is about twice as fast as the cores in my old Athlon X2 4200+ and my Core 2 Duo laptop, and it has 4 cores instead of 2. I have the disks on RAID-1 (mirroring) and 870 GB is devoted to my home directory. Backups are onto a 500 GB external USB drive, automated daily/weekly/monthly snapshots with rsnapshot. I will soon have to modify my backup scripts so that data-* directories (time integrations) are left out.

My old ASUS motherboard, memory, and Athlon X2 cpu are going to my buddy David Johnston in Chicago.