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====== EQ7,8 ======

{{gtspring2009:pc.jpg}} Notes on [[http://chaosbook.org/library/ItGe09.pdf|Itano and Generalis paper]].
  * In PCF, a ‘‘wall structure’’ was first identified in [Nagata1990] (see also Refs. [Clever and 1997, Waleffe1998, 2003]) as the nontrivial equilibrium state containing the streamwise vortices with spanwise wavy modulation. [PC: we agree] 
  * 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 standing wave solutions that are periodic in the streamwise and spanwise directions with wavelengths Lx and Ly, respectively, are of particular interest in the present study. Increase of Re produces three-dimensional instabilities, the  tertiary state that evolves naturally consists of weakly oblique (in the x-y plane) spanwise vortical knots. 
  * This state satisfies the three independent symmetries <span style="color:red;font-size:90%;text-align:right;">[PC: //z// is wall-normal]</span></html>: 
    * (A) Streamwise translational and spanwise reflectional symmetry
    * (B) Parity symmetry with respect to ...
    * (C) Parity symmetry with respect to the origin
  * The bifurcation curve as \epsilon decreases does not reach the PCF limit (with \epsilon = 0). This is consistent with a conclusion from recent quests for two-dimensional states in PCF [Mehta and Healey 2005, F. Rincon (2007].
  * 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>
  * 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 three-dimensionality of the low-speed region is relatively weak in EQ2, so that this region has been often referred to as wall streak (streaky pattern aligned in x direction) [Robinson 1991].
  * 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: too much poetry for me - I do not see the 'knots.']</span></html>
  * the mean shear rate of the upper branch of the new state is larger than that of NBW


{{gtspring2009:pc.jpg}} I'm in favor of opening this page to them, life of a cutting-edge plumber is a lonely one, getting some company is more fun than going at it alone. Surely they do not need to recheck, now that John has
shown that they have identical continuations? 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. So they feel it is theirs. Would we ever know John discovered a "Hairpin Vortex"  were it not for them? In n00bs we can set the record straight. Itano is very capable, getting a collaboration started with him would be more profitable than getting annoyed about who did what first. If we do open this page to them, maybe we prune it first a bit for possible lack of politeness here and there? 

Should probably also check what [[http://www.ceac.aston.ac.uk/research/staff/sg/|Sotos Generalis]] does, do not remember meeting him or reading anything by him. 

Not a big deal for me, if you prefer to exchange emails, go right ahead. I think it would be good citizenship either way. --- //[[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.  Also, I got a response from Itano, reproduced below: \\

Dear Dr Halcrow

Thank you for your email.
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, PhD. 

Any thoughts on a response?  --- //[[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. I would prefer to keep our communication with Itano limited to email for now. So far the information flow has been from us to them, and they have not been very generous with attribution. I don't even see a footnote in their PRl about our paper, or a mention that our 2008 JFM paper published a new plane Couette solution. 

:-D Posting from the strange world of Ohio, home of Joe the actual Plumber.  Still no word from Itano.  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 ot it. So the symmetries are the same. ---//[[http://www.cns.gatech.edu/~jhalcrow/|The Prodigal Son]] 2009-04-11 //
 

{{gtspring2009:pc.jpg}} Sure the same pair of equilibria, discovered independently at what might have been the same time (unless they are real pros and know how to write their results up quickly). Should we invite Itano to join the wiki (apparently he has not responded to Jonathan)? Or post the graphs and a brief summary on some public homepage?  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-04-10 09:42//

{{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." Did Itano send you guys his data?  I haven't heard from him
since the email below.
---  // [[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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