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====== EQ7,8 ====== :-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//
