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{{:gtspring2009:spieker_blog:continuations:dvsa_update2.png|}}

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.

{{:gtspring2009:spieker_blog:continuations:dvsaeq1-size.png|}}
{{:gtspring2009:spieker_blog:continuations:dvsaeq3-size.png|}}
{{:gtspring2009:spieker_blog:continuations:dvsaeq5.png|Bottom branch is eq5}}
{{:gtspring2009:spieker_blog:continuations:dvsaeq6.png|}}
{{:gtspring2009:spieker_blog:continuations:dvsaeq7-size.png|}}
{{:gtspring2009:spieker_blog:continuations:dvsaeq9-size.png|}}
{{:gtspring2009:spieker_blog:continuations:dvsaeq10.png|Top branch is eq10}}
{{:gtspring2009:spieker_blog:continuations:dvsaeq11.png|}}
{{:gtspring2009:spieker_blog:continuations:dvsa_all.png|}}

Eq1 and Eq2 are connected via continuation in Lx. --- //[[dustin.spieker@gatech.edu|Dustin Spieker]] 2009-07-07 09:57//

Intersections of the vertical (alpha = 1.14) line and continuation curves, as near as I can tell, represent the following solutions in order from top to bottom: eq11, eq6, eq10, unknown, unknown, eq5, eq4, eq1, eq9, eq3, eq7. --- //[[dustin.spieker@gatech.edu|Dustin Spieker]] 2009-07-07 10:57//

{{gtspring2009:gibson.png?24}} 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 [[http://www.cns.gatech.edu/~gibson/publications/GibsonJFM09resubmit.pdf|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//

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{{gtspring2009:gibson.png?24}} Excellent! Looks like the upper branch reconnects to the lower in streamwise length continuation as well as spanwise. Compare above plots to these [[gtspring2009:gibson:continuation#section| 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

<latex>
f(x) = \sum_{n=-\infty}^{\infty} \hat{f}_n e^{i \alpha n x}
</latex>

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@gatech.edu|Dustin Spieker]] 2009-06-09 10:13//

{{gtspring2009:pc.jpg}} According to our [[http://www.cns.gatech.edu/~predrag/papers/preprints.html#n00bs|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@physics.gatech.edu|Predrag Cvitanovic]] 2009-06-09 12:46//

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@gatech.edu|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@gatech.edu|Dustin Spieker]] 2009-06-10 08:51//

===== Busse Check =====

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

{{:gtspring2009:spieker_blog:continuations:busse_1.png|}}

It looks like solutions in this geometry exist for much higher Reynolds numbers than I am used to channelflow working with. --- //[[dustin.spieker@gatech.edu|Dustin Spieker]] 2009-06-12 08:54//

{{gtspring2009:pc.jpg}} 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@physics.gatech.edu|Predrag Cvitanovic]] 2009-06-12 12:21//
 
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