User Tools

Site Tools


EQ7,8 AKA HVS (hairpin vortex state)

[the latest posts at the top of this page]

posts after 2009-07-01 are here

2009-07-01 12:09 EST Got an email from Soto Generalis

Dear Prof Gibson,

We are both so sorry for not contacting you much earlier than we are doing now. It is a very busy period here in the UK. This is not really a good excuse for not replying earlier, but the administrative duties that are coupled with our appointments are temporally demanding although albeit mundane. As we mentioned before it was the suggestion of one of the referees to make contact with your group and your work and compare our solutions with the solutions of your team. We understand from your email below that the manuscript that you submitted to the JFM is now in its final stages prior to its publication. We are both grateful that you have provided the link for the revised version of your manuscript and we both wish you success in publishing your manuscript. We also agree with you that there are other solutions as you describe below. It would be interesting to compare solutions and as a starting point would you agree that the value for Re_{min} \approx 174? Would you be able to confirm to us that the lowest value of the Reynolds number of EQ8 is the above mentioned one when the stream- and span- wise wavenumbers are fixed at the values of 1 and 2 respectively? We could provide you with a slightly more accurate value that we have obtained with our methods, but the value 174 was the one that was reported in our PRL paper.

We will be in contact soon, Tomoaki Itano and Sotos Generalis.

My response:

Dear Profs. Itano and Generalis:

I am glad to hear from you. I certainly understand how busy academic life is! Our EQ7/EQ8 solution has Re_{min} \approx 174 with streamwise,spanwise wavenumbers fixed at alpha. gamma = 1,2, the same as your HVS solution. The lowest value of Re we found was 174.07, but we did not try to determine that precisely. The dissipation vs Reynolds number plot appears to be the same as well. I've attached a figure and some Re vs D data. We have normalized dissipation D == wall shear I so that the laminar solution has D = I = 1. We have also compared the symmetries of our solution to yours, and they are the same (up to phase shifts). So I'm pretty sure the solutions are the same.

I'm not sure that we need further comparisons, but if you would like to examine our solutions, the data is available at in tar-gzip files of ascii data for velocity at Fourier-Chebyshev gridpoints. The data format is explained in detail towards the bottom of the page. The data posted is for alpha=1.14 gamma=2.5; if you would like alpha=1, gamma=2 I can send that to you directly.

We like your visualization scheme very much. Would you consider sharing your visualization scripts with us, so that we could look at some of our other solutions this way?

Our paper has been accepted by JFM and should appear in print soon.

best regards,


Their visualization technique is very beautiful. I think you're right (Predrag) about choosing some initial points arbitrarily and then integrating lines tangent to the vorticity field. 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. “Mean shear rate” is vague, implies (to me) the temporal mean rather than the time-varying spatial mean. Perhaps just “shear rate”. John Gibson 2009-06-09

Notes on Itano and Generalis paper. [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]

  • 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. [PC: here is my problem with botanical apporaches to shmurbulence. How do we agree on what constitutes a "hairpin vortex?"]
  • the mean shear rate at the boundary is an order parameter that characterize the states in PCF. [we call this "The rate of energy input is I", perhaps "mean shear rate" is more standard and better term?]
  • The tertiary branch yields “new state termed HVS (hairpin vortex state) in PCF that has never been identified before.” [our EQ8, see John Gibson 2009-04-08 posting below]
  • HVS has a turning point at Re = 174, and the lower branch is likely to asymptote to the laminar state. [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.]
  • we also plot ‘‘NBW’’ (Nagata, Busse, and Waleffe) state in PCF. [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...]
  • The vortex lines are integrated from the equivalent points located at |z| = 0.8 for both EQ8 and EQ2. [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?]
  • NBW is the quaternary branch, while our state is the tertiary branch in the bifurcation sequence for the laminar state. [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]
  • 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 [PC: I do not really see the 'knots.']
  • 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. [PC: what is "stress value?" How do they know the E8 is predominant at higher Reynolds numbers?]

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 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 2009-04-13 06:46

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

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. —The Prodigal Son 2009-04-11

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

:-D to Tomoaki Itano
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 — The Prodigal Son 2009-04-08

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.” — The Prodigal Son 2009-04-08

I'm writing this to a referee: “… one of our solutions (EQ8 found by John Gibson on May 6, 2008, posted on Aug 25, 2008) was discovered independently and a letter submitted to PR Lett. on Aug 20, 2008 [Itano and Generalis 2009]. This single solution was […] published as a cover article of Physics Review Letters.”

:-D to Tomoaki Itano
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 if you'd like to see them. — 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 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. — Mar 24, 2009

gtspring2009/gibson/eq8.txt · Last modified: 2010/02/02 07:55 (external edit)