Just so y'all didn't think that I was doing nothing productive recently. I have been working, unfortunately just not blogging. This plot will be updated as updates are made available by my computer.

On the aggregate plot, from top to bottom, the intersections go EQ11, EQ2, EQ10, EQ12, EQ5, EQ8, EQ4, EQ1, EQ9, EQ3, EQ7. The jury is still out on EQ6 as I have had trouble generating reliable data for continuations in Lx of EQ6. — Dustin Spieker 2009-07-28 09:08

Ok, now I see what you mentioned on skype this morning, about the lower branch of EQ10 (magenta) recrossing alpha = 1.14 and possibly being a new solution (in our naming scheme that gives a new label to any distinct equilibrium at alpha,gamma = 1.14, 2.5 and Re=400). That then does appear to be a distinct solution but not a new one, as we have already found it (EQ12) by similar continuation in gamma. See fig 11b of our EQB & TW paper. On the other hand, it looks like you successfully continued EQ6 back to (α,γ,R) = (1.14, 2.5, 400), which is something we haven't been able to achieve in Reynolds or gamma continuations (look in that same fig 11b, the green curve stops mid-air). It'd be nice to close that green loop. John Gibson 2009-07-07 15:23 EST

I continued EQ6 at a Reynolds number of 330, as per the database's description of the equilibrium. I will see if the lower branch of EQ6 can be continued to Re 400. — Dustin Spieker 2009-07-07 12:51

I found this solution by continuing EQ7 to an upper branch. It has a dissipation of 1.76985 which is different than any of those found in the the EQB & TW paper. Below is a picture of the solution. If this solution is worth pursuing further, let me know what calculations I need to do on it. — Dustin Spieker 2009-07-23 10:09

Actually, it looks to me like you have the same dissipation for both EQ2 and EQ8 in your EQB & TW paper. Perhaps the solution above is actually EQ8. But the database has EQ8 at a Re of 270, and the solution I found has an Re of 400. Either way, I think the paper and the database need to be updated/corrected. — Dustin Spieker 2009-07-23 10:48

I think that's the same EQ8 that we have in fig 4 of the EQB & TW paper. You're right that the dissipation value in table 1 is incorrectly just a repeat of EQ2. I checked the EQ8 that I have on disk at Re=400 and it is D=1.76977, i.e. the same as yours, within the eror levels of our numerics at this discretization. I will update the paper, double-check the other values, and update the database. You caught this right in time: I'm reviewing the proofs and will submit the very final changes tomorrow. You've done a great job in tracking solutions. Maybe , if you want to find a new solution, you could either try more exhaustive searches from random samples of turbulent trajectories, or move to the virgin territory of other isotropy groups. John Gibson 2009-07-23 21:03 EST

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Excellent! Looks like the upper branch reconnects to the lower in streamwise length continuation as well as spanwise. Compare above plots to these D vs γ=2π/Lz plots. Lx changes over a factor of three in these plots. Have you also changed the Nx spatial discretization to keep pace? If not, I would recompute the solutions at the largest Lx value with triple the Nx gridpoints and see if the results are the same.

BTW, I eventually came around to Fabian Waleffe's point of view that the fundamental wavenumbers (α, γ) = (2π/Lx, 2π/Lz) are better parameters for describing the spatial periodicity of solutions than Lx and Lx, because a solution with a given α can fit in boxes of size Lx, 2Lx, 3Lx, etc, or a box of infinite length. The fundmental wavenumber α is defined as the largest value of α for which the periodic function f(x) can be expanded in the form

Thinking of solutions in terms of (α, γ) defined this way lets you think of them as uniquely defined functions, independent of a choice of box size. John Gibson 2009-06-08 13:53 EST

Gah! I should have been more careful, but could we please put check marks or something next to projects that have already been worked on in the projects page. Never mind, you were doing continuations in γ while I am doing continuations in α. I feel better now.— Dustin Spieker 2009-06-09 10:13

So I recomputed the solutions at the largest Lx value with triple the Nx gridpoints, and the dissipation, as near as I could tell, did not change. Initially, without a change in Nx, the Dissipation was found to be 1.71802726542802. After I increased Nx to 144 (3*48 the initial value) I used fieldprops to determine that the dissipation was 1.71803, and that is all the significant figures I could get out of fieldprops. So, I think calculations as I originally did them are accurate. I will post more as the day winds on. — Dustin Spieker 2009-06-10 08:00

I updated the continuation plot (above) with all of the data that I have generated. The red points are solutions I generated without the continuesoln utility, while the blue points are solutions I generated with the continuesoln utility. They seem to match up nicely, which is good. I'm going to see if I connect the loop. — Dustin Spieker 2009-06-10 08:51

According to our n00bs referee

   the stability calculations by Clever and Busse\rf{CB97} 
   indicate that the Nagata solutions prefer a 2:1 streamwise 
   to spanwise aspect ratio. Hence a study of changes in solutions 
   under variation in both streamwise and spanwise periodicities 
   might shed further light on the physical nature of these solutions.

I expect EQ1/EQ2 to survive for increasing Re as “outer solution”, i.e. solution whose scale is set by the wall-to-wall separation. So you might want to check Busse's claim by continuing EQ2 with increasing Re along fixed Lx/Lz=2/1 ratio. — Predrag Cvitanovic 2009-06-09 12:46

Initial results for continuation in Re of the Lx/Lz = 2/1 Lower Branch Solution…

It looks like solutions in this geometry exist for much higher Reynolds numbers than I am used to channelflow working with. — Dustin Spieker 2009-06-12 08:54

My intuition is this; no matter what Reynolds number, there is always 'outer' scale of wall-to-wall distance 2, and accompanying vortex whose width is roughly 2, and 3-dimensionality (streamwise constant solutions all decay) apparently requires a sinusoidal wiggle of period approx. 4. So you might be able to track upper branch to Re = 10,000, just as John did with the lower branch.

Embedded within that is 'turbulence', ie close-to-wall structures measured in wall units which get smaller and smaller as Re increases. These are the guys we want to pin down as exact invariant structures, present for any Re and any large spanwise/streamwise ratio.

You can see these structures in Daniel's Taylor-Couette flow on the 3rd floor. — Predrag Cvitanovic 2009-06-12 12:21

Yes! It would be very exciting to pin down near-wall structures! For some time I have been sitting on an idea for exploring these that would maybe make an excellent long-term research project for Dustin. I will write out a brief description of this immediatement. John Gibson 2009-07-08 11:15 EST

Continuing Lower Branch is not so exciting, maybe not so surprising as John did it for Lx alone to a high Re. The exciting thing would be if the Upper Branch persisted and thus was shown to belong to the outer scale. Could be that it is actually the near-wall structure, in which case it exists only for the finite range of wall-to-wall Re, but stays a solution in wall units, repeated appropriatelly in spanwise, streamwise extent as those lengths go large (in wall units).

2009-08-05: Dustin, I have been following daily your continuations of solutions streamwise, with the aggregate plot as the centerpiece - a useful piece of the puzzle we had not done. Do you think you have enough new material to write it up as a summer project, in the classical form (John used to teach that too) - prologue, parados, the five dramatic scenes, exodus, the climax and conclusion - you know, the usual?

I am kind of cramming for my qualifying exams which happen next week. I might be able to write a little bit in the time leading up to the exams, but I think that if I do have enough to write up, I am going to have to work on that after the exams. — Dustin Spieker 2009-08-06 09:34

2009-08-06: Dustin, quals are the absolute priority. Forget this project until after the qualifiers.

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