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====== EQ7,8 AKA HVS (hairpin vortex state) ======

<html><span style="color:red;font-size:80%;text-align:right;">[the latest posts at the top of this page]</span></html>

{{gtspring2009:gibson.png?24}}  Their visualization technique is very beautiful, though I think it must involve some arbitrary choices on where to start the vortex lines. I would like to try the same for the other solutions. It would be nice to get their scripts. I agree with you on "tertiary" etc. I think "asymptoting to laminar" is probably wrong -that was one of the main points of the Wang, Gibson, Waleffe PRL, at least for the Nagata lower branch. //John Gibson 2009-06-09 //

{{gtspring2009:pc.jpg}} Notes on [[http://chaosbook.org/library/ItGe09.pdf|Itano and Generalis paper]]. <html>
<span style="color:red;font-size:90%;text-align:right;">[Nothing earthshaking - there are a few unanswered questions among the comments marked red]. Their visualization is very pretty. It is interesting how hard (or impossible) is to recognize the same equilibrium from different visualizations alone]</span></html> 
  * At higher Reynolds numbers, another predominant structure is the hairpin vortex (horse-shoe, \Lambda- or \Omega-shaped, attached vortex loops), see [Adrian 2007]. This structure was first envisaged by Theodorsen as a ‘‘wake structure’’ in the boundary layer. The equilibrium state that corresponds to such a hairpin has never been identified so far. There is, therefore, still a controversy about the distinction between streamwise and hairpin vortices. We isolate the steady hairpin vortex state in PCF by tracing a homotopy parameter. We find that the streamwise vortex bifurcates from the hairpin vortex via a symmetry breaking in the homotopy parameter space.  <html>
<span style="color:red;font-size:90%;text-align:right;">[PC: here is my problem with botanical apporaches to shmurbulence. How do we agree on what constitutes a "hairpin vortex?"]</span></html> 
  * the mean shear rate at the boundary is an order parameter that characterize the states in PCF. <html><span style="color:red;font-size:90%;text-align:right;">[we call this "The rate of energy input is I", perhaps "mean shear rate" is more standard and better term?]</span></html>
  * The tertiary branch yields "new state termed HVS (hairpin vortex state) in PCF that has never been identified before." <html><span style="color:red;font-size:90%;text-align:right;">[our EQ8, see John Gibson 2009-04-08 posting below]</span></html>
  * HVS has a turning point at //Re = 174//, and the lower branch is likely to asymptote to the laminar state. <html>
<span style="color:red;font-size:90%;text-align:right;">[PC: not a useful concept - all lower branch states probably "asymptote to laminar." Perhaps asymptoting to the "edge state" would be a more precise statement.]</span></html>
  * we also plot ‘‘NBW’’ (Nagata, Busse, and Waleffe) state in PCF. <html>
<span style="color:red;font-size:90%;text-align:right;">[PC: NBW stands for both EQ1 and EQ2. Bit of a slippery slope, as by varying spanwise, stremwise wavelengths, Re or other homotopy parameters many of these equilibria could be related to each other in this way...]</span></html>
  * The vortex lines are integrated from the equivalent points located at //|z| = 0.8// for both EQ8 and EQ2. <html>
<span style="color:red;font-size:90%;text-align:right;">[PC: Not sure about integrated how? One evaluates the 3D vorticity vector at the starting point, then continues it as a curve to which vorticity vector is tangent?]</span></html>
  * NBW is the quaternary branch, while our state is the tertiary branch in the bifurcation sequence for the laminar state.<html>
<span style="color:red;font-size:90%;text-align:right;">[PC: I do not appreciate significance of this. If they stayed within PCF, there would be no bifurcation off the laminar state to begin with]</span></html>
  * The other structure of a low-speed region is visualized by u_x = 0.1 (cyan) isosurface, which exists across the midplane of the channel, is knotted rather than streaky, and shows a staggered pattern of intertwined knots of low-speed regions in the x-y plane <html>
<span style="color:red;font-size:90%;text-align:right;">[PC: I do not really see the 'knots.']</span></html>
  * The knots of the hairpin structure are responsible for the higher stress value and would account for the predominance of the HVS at higher Reynolds numbers. <html>
<span style="color:red;font-size:90%;text-align:right;">[PC: what is "stress value?" How do they know the E8 is predominant at higher Reynolds numbers?]</span></html>

{{gtspring2009:pc.jpg}} They do have a point - they submitted their paper prior to our arXiv submission, and the fault for that is ours. Took them 7 months of refereeing, got told to compare with us at the very end. Would we ever know John discovered a "Hairpin Vortex"  were it not for them? Let us invite Itano and Generalis to join the wiki.  Getting a collaboration started with them would be good for all of us. --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-04-14 05:55//

:-D Whoops, an embarrassing error in my last post.  B is τ^(-1/4)_{xz} u = τ^(1/4)_{xz}σ_{xz} u, which is equivalent to τ_{xz}σ_{xz}.
But, EQ7/8 satisfy this symmetry as well.  A response from Itano: \\

"Unfortunately for both my collaborator and me,
 the current period is overloaded with duties that prevent us
 from launching a serious effort in comparing our data with yours.
We are suspecting though that once time becomes available
 we should concentrate our efforts.
\\
Please bear with us for a little bit, Tomoaki Itano"

--- //[[jonathan.halcrow@gmail.com|Jonathan Halcrow]] 2009-04-13 06:46//

{{gtspring2009:gibson.png?24}} I was able to continue EQ8 past Re=365 using the automated quadratic-extrapolation
continuation code. It undergoes a sharp turn near Re=365 and returns to much lower dissipation values. The turning
point is smooth, as seen in the right-hand figure. So this gives us EQ8 at Re=400 at D=1.77. I will update the 
n00bs text, figs, and tables once the eigenvalue calculation is finished. Should examine the turning point in more detail to make sure it's not switching branches at a bifurcation. EQ6 is making very slow progress around Re=335. It could be turning a similar corner, but it's yet not clear if it'll succeed. // John Gibson 2009-04-11 21:19 EST//

{{gtspring2009:gibson:w03eqbs_dre.png?300}} {{gtspring2009:gibson:w03eqbs_dre_2.png?300}} 

P.S. I agree that the symmetries of our EQ8 and Itano's solution are the same.

:-D Posting from the strange world of Ohio, home of Joe the actual Plumber. Those plots seem to cinch it that the IGs are EQ7/8, but for fun I looked at the symmetries.  Itano identifies three symmetries of their equilibria:  A = τ_x σ_z, B:τ^(1/4)_{xz} u = τ^(1/4)_{xz} σ_{xz} u, and C = σ_{xz}.   It seems they might have made an error here because B is identical to C.  Note also that A = s_1.  The symmetries of EQ7/8 are {τ_x σ_z,τ_xz σ_x, τ_z σ_xz}  x {e,τ_{xz}}.  If we shift EQ7/8 1/4 length streamwise and apply C, then we have σ_{xz} τ^{1/4)_x u  =  τ^{-1/4)_x σ_{xz} u = τ^{-1/4)_x σ_{xz} (τ_x σ_xz u) = τ^{1/4)_x σ_{xz}σ_{xz} u = τ^{1/4)_x u.  So, C is satisfied.  Also, A commutes with streamwise shifts, so the quarter shifted eq7/8 remain invariant with respect to it. So the symmetries are the same. ---//[[http://www.cns.gatech.edu/~jhalcrow/|The Prodigal Son]] 2009-04-11 //

{{gtspring2009:gibson.png?24}} Comparison of Itano & Generalis solution (left fig) to EQ7,8 (right fig) via D vs Re continuation. His τ is out D. I continued our EQ7 to IG's α,γ = 1,2 cell size and continued it in Re. Graphs are identical to eye (sorry the graphs don't line up exactly, it's really hard to do that.) IG give the bifurcation point as roughly Re=174 at this α,γ, and that's what I get, too. The continuation is still running. I'll fill out the upper branch for Re>210 tomorrow when it's done. //John Gibson 2009-04-09 19:24 EST // Update: done, slight discrepancy near D=5, probably just discretization. //2009-04-09 23:29 EST //


{{gtspring2009:gibson:itano_prl_red.png?300}} {{gtspring2009:gibson:eq8a1g2_red.png?320}} 



:-D  to	Tomoaki Itano //<itano@ipcku.kansai-u.ac.jp>//:
\\
Dr Itano, I've received a copy of your paper through one of my colleagues.  (I'm
currently without a university affiliation).  It is very interesting.
Your new equilibria seem similar to those that we refer to as EQ7 and
EQ8, and what Schmiegel referred to as σ in his 1999 thesis.
But, without a copy of your data we cannot say for sure.  Have you
compared your solution to the ones that we have posted at
www.channelflow.org? ---  // [[http://www.cns.gatech.edu/~jhalcrow/|The Prodigal Son]] 2009-04-08//

{{gtspring2009:gibson.png?24}}  Professors these days --don't they read anything? The database also
shows the first posting: 2008-05-09. We should note in the paper and in a note to
Itano that EQ8 was likely first discovered by Schmiegel in 1999. I'll
do what I can to verify that the Schmiegel, our EQ8 and Itano solutions are the same. ---  // John Gibson 2009-04-08//

:-D  My earliest record is the svn  commit 552 to n00bs on May 6 by John,  in
which he says "2 new EQB pairs for narrow box, one NB/NB2-like, one EQ5/EQ6-like.
Found from exhaustive search on a long-lived NB unstable manifold trajectory.
Eigenvalues to follow." ---  // [[http://www.cns.gatech.edu/~jhalcrow/|The Prodigal Son]] 2009-04-08//

{{gtspring2009:pc.jpg}} I'm writing this to a referee: "... one of our solutions (EQ8 found by John Gibson
on May 6, 2008, posted on [[http://arxiv.org/abs/0808.3375|arXiv.org:0808.3375]] Aug 25, 2008) was discovered independently and a letter submitted to PR Lett. on Aug 20, 2008 [[http://link.aps.org/doi/10.1103/PhysRevLett.102.114501|[Itano and Generalis 2009]]]. This single solution was [...] published as a cover article of Physics Review Letters."

:-D  to	Tomoaki Itano //<itano@ipcku.kansai-u.ac.jp>//:
\\
Dr Itano, Congratulations on making the cover of PRL.  I would be happy to
compare your solution to the ones we've found.  Along with the data,
could you please also send me a copy of your paper?  We have our
solutions posted at
[[http://www.channelflow.org/dokuwiki/doku.php/database|Channelflow.org/dokuwiki/doku.php/database]] if you'd like to
see them. ---  // [[http://www.cns.gatech.edu/~jhalcrow/|The Prodigal Son]] 2009-03-24//

8-) Tomoaki Itano to Harclow:
\\
My name is Tomoaki Itano and I am the corresponding author
of the recent article [[http://link.aps.org/doi/10.1103/PhysRevLett.102.114501|Phys Rev Lett 102, 114501 (2009)]]
that has actually made the cover of the Journal for its current issue.
We had a lengthy refereeing process that lasted for many months.
In fact, we, my collaborator and I, had the results ready for publication
for sometime now, but there was some discomfort from the referees to
publish our __new Couette__ solution.

At the final stage of this refereeing process,
one of the referees asked us to contact your group
for the purpose of comparing our results, pointing out to us
your paper that you have submitted on the preprint archive.
We take this opportunity therefore to make contact with you and
perhaps compare data and/or opinions about the
data presented in our independent works.
\\
Yours sincerely,
Tomoaki Itano, PhD. ---  //<itano@ipcku.kansai-u.ac.jp> Mar 24, 2009//
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