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====== Chapter: Turbulence? ======

(ChaosBook.org blog, chapter [[http://chaosbook.org/paper.shtml#PDEs|Turbulence?]])  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-03-08//

(the latest posts within a section at the top, blog-style)

{{gtspring2009:pc.jpg}} 

===== Blog =====

**2011-07-05 Predrag** [[http://arxiv.org/abs/1107.0580|arXiv.org:1107.0580]], "Turbulent transition in a truncated one-dimensional model for shear flow" by J.H.P. Dawes and W.J. Giles merits a closer look.

**2010-06-10 Predrag** Run into a bunch of papers by  Magnitskii about "Feigenbaum-Sharkovskii-Magnitskii scenario of transition to turbulence in
the Rayleigh-Benard convection." Feigenbaum and Sharkovskii can sleep restfully (it is about sequences of bifurcations from the laminar attractor into tori and such, with increasing Rayleigh and Prandtl numbers) but visualizations onto {{:chaosbook:uvw.gif|}} like this one

{{:chaosbook:2torus.jpg|}}

might be useful to us. Of course, they do not explain the flow that generated this plot; presumably they have fixed a point in //(x,y,z)// and then trace //(U,V,W)(x,y,z;t)// as a function of time. Fixing //U=0// also enables them to visualize a section across this torus. I put one of the articles, N.M. Evstigneev, N.A. Magnitskii and S.V. Sidorov, "Nonlinear dynamics of laminar-turbulent transition in three dimensional Rayleigh-Benard convection", //Communications in Nonlinear Science and Numerical Simulation// **15**, 1007-5704 (2010), into [[https://www.zotero.org/groups/cns/items/collection/1931456|https://www.zotero.org/groups/cns library]].

**2009-11-25 Humbledt Plumber to Benny Lautrup:** methinks,

"Pipes and planes" glides down your tongue like poetry

"Plates and pipes"  (chapter title in Lautrup's book) is strictly engineering - no physicists thinks of a "plate"; infinite in all directions is called a "plane".

**2009-07-27 Balu /BALASUBRAMANYA Nadiga/** 
Your comments/suggestions on [[http://ChaosBook.org/library/NaLu01.pdf|Global Bifurcation of Shilnikov Type in a Double-Gyre Ocean Model]] by Nadiga and Luce, a paper on homoclinic chaos
on the way to turbulence in the simpler setting of the barotropic
vorticity eqn. in the context of ocean flows would be very much
appreciated.


===== Plumbers unite! A manifesto =====

[[researchProgram|click here]]



===== A plane Couette blog =====

[[pCf|click here]]

===== A pipe blog =====

[[pipes|click here]]

===== A Taylor-Couette blog =====

[[Taylor-Couette|click here]]

===== A baroclinic blog =====

[[baroclynic|click here]]

===== Rėsumė =====

 Mark here when summary completed.

===== Commentary =====

Here all references to external literature (avoid them in the text proper, it is meant to be self-contained).

==== Nomenclature ====

a **streak** is a region of streamwise velocity deviation from the mean streamwise velocity

**rolls**, **vortices**  are defined in terms of spanwise normal section in-plane velocity (streammwise vorticity) 

**bifurcations**:

F.H. Busse\cite[BC96a]: Many fluid flows exhibit sequences of bifurcations in which new dynamical mechanisms are introduced. Symmetries may be broken at each bifurcation.
  * __primary state__ exhibits full symmetries of the system (example: laminar solution)
  * __secondary states__ take - for planar flows - form of //2d// rolls
  * __tertiary states__ are full //3d// structures (examples: equilibria of plane Couette)
  * __quaternary states__ a time-periodic //3d// invariant solutions  (examples: traveling waves?)
In plane Couette experiments a transition to turbulent from laminar occurs at ''Re'' of order of ''Re=1600/4'', with //3d// structures realized\cite{DD95} as early as ''Re=640/4''


===== Exercises =====

===== References =====

Mark here when correctly ordered. OK to have references not cited, as long as they are relevant to this chapter.


  @InProceedings{BC96a,
  title = {Bifurcation Sequences in Fluid Flows and Coherent Structures in
        the Turbulent State},
  author = {Busse, F. H. and Clever, R. M.},
  year = {1996},
  booktitle = {Advances in Turbulence VI},
  editor = {Gavrilakis, S. and Machiels, L. and Monkewitz, P. A.},
  series = {Proceedings of the Sixth European Turbulence Conference},
  pages = {309-312}
  }

  @Article{ DD95,
    author = "O. Dauchot and F. Daviaud",
    title = "Finite-amplitude perturbation and spots growth mechanism in plane {C}ouette flow",
    journal = "Phys. Fluids",
    volume = "7",
    year = "1995",
    pages = "335--343"
  }

  @Article{KaUhVe12,
  AUTHOR = {van Veen, L. and Kawahara, G.},
  TITLE = {The significance of simple invariant solutions
         in turbulent flows},
  YEAR = {2012},
  JOURNAL = {Ann. Rev. Fluid Mech.},
  VOLUME = {44},
  PAGES = {203–25}
  }

  @Article{VeKa11,
  AUTHOR = {van Veen, L. and Kawahara, G.},
  TITLE = {Homoclinic tangle at the edge of shear turbulence},
  YEAR = {2011},
  JOURNAL = {Phys. Rev. Lett.},
  VOLUME = {107},
  PAGES = {114501}
  }

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