Chapter: Turbulence?

(ChaosBook.org blog, chapter Turbulence?) — Predrag Cvitanovic 2009-03-08

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

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 like this one

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 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 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.

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Mark here when summary completed.

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

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

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"
}