Notebooks
http://bactra.org/notebooks
Cosma's NotebooksenTurbulence, and Fluid Mechanics in General
http://bactra.org/notebooks/2023/12/30#turbulence
<blockquote>(Most of the text below goes back to some point in the late
1990s --- probably 1997, judging by the references. Ordinarily I'd
fix this entry at that date, but I lost it long ago.)</blockquote>
<P>The story is told of many giants of modern physics, but most plausibly of
Heisenberg, that, on his death-bed, he remarked that the two great unsolved
problems were reconciling quantum mechanics and general relativity, and
turbulence. "Now, I'm optimistic about gravity..."
<P>Fluid flow, it should be said, is in one sense very well understood; since
the early 1800s there's been a fine, non-linear, Newtonian equation for the
velocity field that seems to work, the Navier-Stokes equation. (Like Newton's
law of gravitation, it should be branded on to anyone who babbles that <a
href="chaos.html">non-linear</a> physics is <a href="new-physics.html">"new" or
"non-Newtonian"</a>.) One of its properties is that it's invariant so long as
the Reynolds number --- density*(length scale)*(velocity scale)/viscosity ---
stays the same. This is why wind-tunnels work: the model in the tunnel is
shorter than the original, but the mean speed is higher, so the flows are
equivalent. When the Reynolds number is small, the equation is mathematically
nice, the non-linearities are small, and we can solve the equation. The
stream-lines --- the paths followed by small tracer particles dropped into the
fluid --- form nice layers around the boundaries of the flow, which is why the
flow is called <em>laminar</em>, and these laminæ are stable.
<P>As you turn up the Reynolds number, the non-linearities become important,
and the flow gets uglier --- it is no longer steady, but erratic (probably <a
href="chaos.html">chaotic</a> in the strict sense), and the nice regular
stream-lines and their laminæ get snarled and then completely confused;
eddies and vortices form and spin and dissolve without much obvious pattern,
and the develop their own eddies in turn; odd structures with names like "von
Kármán streets" appear. (Pictures make this a lot clearer; van
Dyke's <cite>Album of Fluid Motion</cite> is full of handsome ones, but short
on explanation.) Turbulence --- yea, "fully developed turbulence", even --- is
when this decay into confusion is complete, when there are eddies and motions
on all length scales, from the largest possible in the fluid on down to the
so-called "dissipation scale," which is (roughly!) the minimum eddy size, as
set by the mechanical properties of the fluid (its viscosity and the like).
When faced with this confusion, if not well before, we give up and turn to
statistics; we begin to ask questions about the statistical properties of the
flow --- if you will, about all possible flows we could see under given
conditions. Here we can make some nice observations, and even come up with two
well-confirmed empirical laws about these statistics, and endless graphs.
<P>So what, you may ask, is the fabled "problem of turbulence"? In essence,
this: what on Earth do our statistics and our equation have to do with each
other? A solution to the problem of turbulence would be, more or less, a valid
derivation from the Navier-Stokes equation (and statements about the
appropriate conditions) of our measured statistics. Physicists are very far
from this at present. Our current closest approach stems from the work of
Kolmogorov, who, by means of some statistical hypotheses about small-scale
motion, was able to account for the empirical laws I mentioned. Unfortunately,
no one has managed to coax the hypotheses from the Navier-Stokes equation
(sound familiar?) and the hypotheses hold exactly only in the limit of infinite
Reynolds number, i.e. they are not true of any actual fluid.
<P>So what's to do? Well, all sorts of things, including more or less direct
simulations of flows by cousins of <a href="cellular-automata.html">cellular
automata</a> called "lattice gasses" (which is how I connect to the subject,
though very vaguely). One approach uses the vorticity (the curl of the
velocity field, which tells us about how the fluid swirls), since it turns out
to be possible to identify some (more or less) simple objects in the flow,
called vortex lines or vortex tubes, work out how they interact (there's a
Hamiltonian), and then use <a href="stat-mech.html">statistical mechanics</a>
to calculate various <a href="emergent-properties.html">emergent properties</a>
--- which, if you use just the right approximations, and tolerate negative
temperatures (which are not impossible, and actually hotter than infinity)
gives you the Kolmogorov laws. This could've been custom-tailored for my
philosophical and methodological biases, which makes me suspicious, as do all
the leaps in the approximation scheme used. (For the pro-vorticity case, see
Chorin; reasons for caution are discussed by Frisch, pp. 189f.)
<P>If people <em>must</em> find analogies for society, ecosystems, etc., from
physics and engineering, turbulence is probably a better one than <a
href="cybernetics.html">feedback</a>.
<P>See also:
<a href="geophysical-fluid-dynamics.html">Geophysical Fluid Dynamics</a>
<ul>Recommended, big picture:
<li>George Batchelor, <cite>The Life and Legacy of G. I. Taylor</cite>
[A nice scientific biography of one of the founders of modern mechanics, and of
the statistical theory of turbulence; <a
href="../reviews/batchelor-on-taylor/">review</a>]
<li>Pierre Berge <em>et al.,</em> <cite>Order within Chaos</cite>
<li>Alexandre J. Chorin, <cite>Vorticity and Turbulence</cite> ["This
book provides an introduction to turbulence in vortex systems, and to
turbulence theory for incompressible flow described in terms of the vorticity
field. It is the author's hope that by the end of the book the reader will
believe these subjects are identical, and constitute a special case of fairly
standard statistical mechanics, with both equilibrium and non-equilibrium
aspects." Despite being a re-write of a famously incomprehensible set of
lecture notes, this book is surprisingly well-written, covers a huge amount of
material in about 150 pages (I read it in a night), and makes a very strong
case for the vorticity approach.]
<li><a href="http://www.cns.gatech.edu/~predrag/">Predrag
Cvitanovic</a>,
"<a href="http://www.cns.gatech.edu/~predrag/talks/essay.html">Turbulence, and
what to do about it?</a>" [2002 "essay" on Cvitanovic's approach, based on
identify recurrent patterns and expressing things in terms of them; I'm
sympathetic.]
<li>Uriel Frisch, <cite>Turbulence: The Legacy of A. N.
Kolmogorov</cite> [An excellent introduction, very strong on defending
Kolmogorov's work from mis-understandings and invalid criticisms.]
<li>Victor L'vov and Itamar Procaccia, <a
href="http://lvov.weizmann.ac.il/physword/physword.html">"Hydrodynamic
Turbulence: a 19th Century Problem with a Challenge for the 21st Century"</a>
(=<a href="http://arxiv.org/abs/chao-dyn/9606015">chao-dyn 9606015</a>) [The
first few sections are a very good description of the problem of turbulence
from the physicist's perspective; their version of the "Well, I'm optimistic
about the non-turbulence problem" story attributes it to the great
hydrodynamicist Horace Lamb. L'vov and Procaccia then go on to describe and
extol their particular strategy, which is to try to make <a
href="field-theory.html">field theory</a> work.]
<li>Terence Tao, <a
href="http://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/">"Why
global regularity for Navier-Stokes is hard"</a> [A really good exposition of
some of the difficult mathematical issues connected to proving (or disproving)
the existence of regular solutions to the Navier-Stokes equations. Assumes
little by way of physical or hydrodynamical knowledge, but a good bit of
mathematical maturity.]
<li>H. Tennekes and J. L. Lumley, <cite>A First Course in
Turbulence</cite> [Older --- from the 1970s --- but still good]
<li>van Dyke, <cite>An Album of Fluid Motion</cite> [Pretty pictures,
with Reynolds numbers]
</ul>
<ul>Recommended, close ups:
<li>Ole E. Barndorff-Nielsen, Fred Espen Benth, and Almut E. D. Veraart,
<cite><a href="https://doi.org/10.1007/978-3-319-94129-5">Ambit Stochastics</a></cite>
<li>O. E. Barndorff-Nielsen, J. L. Jensen and M. Sorensen, "Parametric
Modelling of Turbulence," <cite>Philosophical Transactions of the Royal
Society: Physical Science and Engineering,</cite> <strong>332</strong> (1990):
439--455 [<a href="time-series.html">ARMA</a> models work for turbulence! Who
knew? <a href="http://www.jstor.org/pss/76800">JSTOR</a>]
<li>Garrett Birkhoff, <cite><a href="http://bactra.org/weblog/algae-2018-08.html#birkhoff">Hydrodynamics: A Study in Logic, Fact and Similitude</a></cite> [<a href="https://www.jstor.org/stable/j.ctt183prct">JSTOR</a>]
<li>S. C. Chapman, G. Rowlands and Nick W. Watkins, "The Origin of
Universal Fluctuations in Correlated Systems: Explicit Calculation for an
Intermittent Turbulent Cascade," <a
href="http://arxiv.org/abs/cond-mat/0302624">cond-mat/0302624</a>
<li>Alice M. Crawford, Nicolas Mordant, Andy M. Reynolds, Eberhard
Bodenschatz, "Comment on 'Dynamical Foundations of Nonextensive Statistical
Mechanics'", <a href="http://arxiv.org/abs/physics/0212080">physics/0212080</a>
<li>Jordgi Delgado and Ricard V. Solé, "Characterizing
Turbulence in Globally Coupled Maps with Stochastic Finite Automata,"
<cite>Physics Letters A</cite> <strong>314</strong> (2000): 314--319
<li>Gary Doolen, Uriel Frisch, Brosl Hasslacher, Steven Orszag and
Stephen Wolfram (eds.), <cite>Lattice Gas Methods for Partial Differential
Equations: A Volume of Lattice Gas Reprints and Articles, Including Selected
Papers from the Workshop on Large Nonlinear Systems, Held August, 1987, in Los
Alamos, New Mexico</cite> SFI Proceedings Vol. IV
<li>Gregory L. Eyink and Katepalli R. Sreenivasan, "Onsager and the
theory of hydrodynamic
turbulence", <a href="http://dx.doi.org/10.1103/RevModPhys.78.87"><cite>Reviews
of Modern Physics</cite> <strong>78</strong> (2006): 87--135</a> [<a href="http://bactra.org/weblog/433.html">Things like
this aren't supposed to happen in real life.</a>]
<li>Nicolas Mordant, Alice M. Crawford and Eberhard Bodenschatz
<ul>
<li>"Three-Dimensional Structure of the Lagrangian Acceleration in Turbulent
Flows", <a
href="http://dx.doi.org/10.1103/PhysRevLett.93.214501"><cite>Physical Review
Letters</cite> <strong>93</strong> (2004): 214501</a> [It's lognormal! Well,
they don't do any formal tests of goodness of fit, but by eye it's pretty
good.]
<li>"Experimental Lagrangian Acceleration Probability Density Function
Measurement,"
<a href="http://arxiv.org/abs/physics/0303003">physics/0303003</a>
</ul>
<li>M. G. Shats, H. Xia and H. Punzmann, "Self-organization in
turbulence as a route to order in plasma and fluids", <a
href="http://arxiv.org/abs/physics/0409074">physics/0409074</a>
<li>P. Tabeling and O. Cardoso (eds.) <cite>Turbulence: A Tentative
Dictionary</cite>
</ul>
<ul>Recommended, historical:
<li>G. I. Taylor, "Diffusion by Continuous
Movements", <a href="http://dx.doi.org/10.1112/plms/s2-20.1.196"><cite>Proceedings
of the London Mathematical Society</cite>, series 2, volume 20 (1922),
pp. 196--212</a>
[Comments under <a href="ergodic-theory.html">ergodic theory</a>]
</ul>
<ul>To read:
<li>Edsel A. Ammons
<ul>
<li>"The Stationary Statistics of a Turbulent
Environment as an Attractor,"
<a href="http://arxiv.org/abs/physics/0202028">physics/0202028</a>
<li>"An Approach to the Statistics of
Turbulence", <a href="http://arxiv.org/abs/physics/0306068">physics/0306068</a>
</ul>
<li>Claudia Angelini, Daniela Cavab, Gabriel Katul, and Brani
Vidakovic, "Resampling hierarchical processes in the wavelet domain: A case
study using atmospheric turbulence", <a
href="http://dx.doi.org/10.1016/j.physd.2005.05.015"><cite>Physica
D</cite> <strong>207</strong> (2005): 24--40</a>
<li>Alex Arenas, Alexandre Chorin, "On the existence and scaling of
structure functions in turbulence according to the
data", <a href="http://arxiv.org/abs/cond-mat/060161">cond-mat/060161</a>
<li>A. K. Aringazin and M. I. Mazhitov
<ul>
<li>"One-dimensional Langevin models of fluid particle
acceleration in developed
turbulence", <a href="http://dx.doi.org/10.1103/PhysRevE.69.026305"><cite>Physical Review
E</cite> <strong>69</strong> (2004): 026305</a>, <a href="http://arxiv.org/abs/cond-mat/0305186">cond-mat/0305186</a>
<li>"Stochastic models of
Lagrangian acceleration of fluid particle in developed turbulence",
<a
href="http://dx.doi.org/10%2E1142/S0217979204026433">International Journal of
Modern Physics B</cite> <strong>18</strong> (2004): 3095--3168</a>,
<a
href="http://arxiv.org/abs/cond-mat/0408018">cond-mat/0408018</a>
</ul>
<li>V. I. Arnol'd, <cite>Topological Methods in Hydrodynamics</cite>
<li>A. Bershadskii, J.J. Niemela, A. Praskovsky and K.R. Sreenivasan,
"`Clusterization' and intermittency of temperature fluctuations in turbulent
convection", <a href="http://arxiv.org/abs/nlin.CD/0401044">nlin.CD/0401044</a>
[Their "telegraph approximation" is a discretization, which means one can use
all kinds of <a href="symbolic-dynamics.html">symbolic-dynamical</a> tricks can
be used]
<li>Eugene Balkovsky and Boris I. Shraiman, "Olfactory Search at High
Reynolds Number,"
<a href="http://arXiv.org/abs/nlin/0109019">nlin.CD/0109019</a>
<li>M. M. Bandi, J. R. Cressman Jr., W. I. Goldburg, "Test of the
Fluctuation Relation in compressible turbulence on a free
surface", <a href="http://arxiv.org/abs/nlin.CD/0607037">nlin.CD/0607037</a>
<li>M. M. Bandi, W. I. Goldburg, J. R. Cressman Jr, "Measurement of
entropy production rate in compressible
turbulence", <a href="http://arxiv.org/abs/nlin.CD/0607036">nlin.CD/0607036</a>
<li>G. I. Barenblatt, <cite>Scaling Phenomena in Fluid Mechanics</cite>
<li>G. I. Barenblatt and Alexandre J. Chorin, "A New Formulation of
the Near-Equilibrium Theory of Turbulence,"
<a href="http://arxiv.org/abs/math.DS/9909060">math.DS/9909060</a>
<li>George Batchelor [Old, but classics]
<ul>
<li><cite>An Introduction to Fluid Dynamics</cite>
<li><cite>The Theory of Homogeneous Turbulence</cite>
</ul>
<li>G. K. Batchelor et al (eds.), <cite>Perspectives in Fluid Dynamics:
A Collective Introduction to Current Research</cite> ["eleven chapters that
introduce and review different branches of the subject for graduate-level
courses, or for specialists seeking introductions to other areas"]
<li>Christian Beck
<ul>
<li>"Nonextensive methods in turbulence and particle
physics," <a href="http://arxiv.org/abs/cond-mat/0110071">cond-mat/0110071</a>
<li>"Generalized statistical mechanics and fully developed
turbulence,"
<a href="http://arxiv.org/abs/cond-mat/0110073">cond-mat/0110073</a>
<li>"Superstatistical turbulence models", <a
href="http://arxiv.org/abs/physics/0506123">physics/0506123</a>
</ul>
<li>Roberto Benzi, Luca Biferale and Federico Toschi, "Intermittency
in Turbulence: Multiplicative Random Processes in Space and Time",
<cite>Journal of Statistical Physics</cite> <strong>113</strong>
(2003): 783--798
<li>Jacob Berg, "Lagrangian one-particle velocity statistics in a
turbulent flow", <a
href="http://arxiv.org/abs/physics/0610155">physics/0610155</a>
<li>L. Biferale, G. Boffetta, A. Celani, A. Lanotte and F. Toschi
<ul>
<li>"Lagrangian statistics in fully developed turbulence", <a
href="http://arxiv.org/abs/nlin.CD/0402032">nlin.CD/0402032</a>
<li>"Multifractal statistics of Lagrangian velocity and
acceleration in turbulence", <a
href="http://arxiv.org/abs/nlin.CD/0403020">nlin.CD/0403020</a>
</ul>
<li>Dieter Biskamp, <cite><a href="http://cambridge.org/0521810116">Magnetohydrodynamic Turbulence</a></cite>
<li>Tomas Bohr, Morgan Jensen, Giovanni Paladin and Angelo
Vulpiani, <cite><a href="http://cambridge.org/9780521017947">Dynamical Systems Approach to Turbulence</a></cite> [I'm told this
Bohr is a grandson of <em>the</em> Bohr... who is, come to think of it, the
only founder of <a href="quantum-mechanics.html">quantum mechanics</a> who has
never been the hero of a version of my opening anecdote, at least not that I've
run across.]
<li>Bradshaw, <cite>An Introduction to Turbulence and Its
Measurement</cite>
<li>Jean-Michel Caillol, Oksana Patsahan, and Ihor Mryglod,
"Statistical field theory for simple fluids: the collective variables
representation", <a
href="http://arxiv.org/abs/cond-mat/0503213">cond-mat/0503213</a>
<li>Haris J. Catrakis and Paul E. Dimotakis, "Shape Complexity in
Turbulence," <cite>Physical Review Letters</cite> <strong>80</strong> (1998):
968--971
<li>Pierre-Henri Chavanis, "Statistical mechanics of geophysical
turbulence: application to jovian flows and Jupiter's great red spot", <a
href="http://dx.doi.org/10.1016/j.physd.2004.11.004"><cite>Physica
D</cite> <strong>200</strong> (2005): 257--272</a>
<li>L. Chevillard, N. Mazellier, C. Poulain, Y. Gagne, and C. Baudet,
"Statistics of Fourier Modes of Velocity and Vorticity in Turbulent Flows:
Intermittency and Long-Range Correlations", <a
href="http://dx.doi.org/10.1103/PhysRevLett.95.200203"><cite>Physical Review
Letters</cite> <strong>95</strong> (2005): 200203</a>
<li>L. Chevillard and C. Meneveau, "Lagrangian Dynamics and Statistical
Geometric Structure of Turbulence", <a
href="http://dx.doi.org/10.1103/PhysRevLett.97.174501"><cite>Physical Review
Letters</cite> <strong>97</strong> (2006): 174501</a>,
<a href="http://arxiv.org/abs/cond-mat/0606267">cond-mat/0606267</a>
<li>L. Chevillard, S. G. Roux, E. Lévêque, N. Mordant,
J.-F. Pinton, and A. Arnéodo, "Intermittency of Velocity Time Increments
in Turbulence", <a
href="http://dx.doi.org/10.1103/PhysRevLett.95.064501"><cite>Physical Review
Letters</cite> <strong>95</strong> (2005): 064501</a>
<li>Rene Chevray and Jean Mathieu, <cite>Topics in Fluid
Mechanics</cite>
<li>Chorin and Marsden, <cite>A Mathematical Introduction to Fluid
Mechanics</cite>
<li>Igor Chueshov and Annie Millet, "Stochastic 2D hydrodynamical type systems: Well posedness and large deviations", <a href="http://arxiv.org/abs/0807.1810">arxiv:0807.1810</a>
<li>R. Collina, R. Livi and A. Mazzino, "Large Deviation Approach to
the Randomly Forced Navier-Stokes Equation", <a
href="http://arxiv.org/abs/physics/0410148">physics/0410148</a>
<li>Peter Constantin and Ciprian Foias, <cite><a href="http://www.press.uchicago.edu/cgi-bin/hfs.cgi/00/2981.ctl">Navier-Stokes
Equations</a></cite>
<li>Alice M. Crawford, Nicolas Mordant and Eberhard Bodenschatz, "Joint
Statistics of the Lagrangian Acceleration and Velocity in Fully Developed
Turbulence", <a
href="http://link.aps.org/abstract/PRL/v94/e024501"><cite>Physical Review
Letters</cite> <strong>94</strong> (2005): 024501</a>
<li>Olivier Darrigol, <cite><a href="http://www.oup.co.uk/isbn/0-19-856843-6">Worlds of Flow: A history of hydrodynamics
from the Bernoullis to Prandtl</a></cite>
<li>P. A. Davidson and B. R. Pearson, "Identifying Turbulent Energy
Distributions in Real, Rather than Fourier, Space", <a
href="http://dx.doi.org/10.1103/PhysRevLett.95.214501"><cite>Physical Review
Letters</cite> <strong>95</strong> (2005): 214501</a>
<li>Peter D. Ditlevsen, <cite><a href="http://cambridge.org/9780521190367">Turbulence and Shell Models</a></cite>
<li>Bruno Eckhardt, Tobias M. Schneider, "How does flow in a pipe become turbulent?", <a href="http://arxiv.org/abs/0709.3230">arxiv:0709.3230</a>
<li>Richard S. Ellis, Kyle Haven and Bruce Turkington, "The Large
Deviation Principle for Coarse-Grained Processes,"
<a href="http://arxiv.org/abs/math-ph/0012023">math-ph/0012023</a>
<li>Gregory L. Eyink
<ul>
<li>"Locality of turbulent cascades", <a
href="http://dx.doi.org/10.1016/j.physd.2005.05.018"><cite>Physica
D</cite> <strong>207</strong> (2005): 91--116</a>
<li>"Turbulent Diffusion of Lines and Circulations",
<a href="http://arxiv.org/abs/0704.0263">arxiv:0704.0263</a>
</ul>
<li>Gregory L. Eyink, Shiyi Chen and Qiaoning Chen, "Gibbsian
Hypothesis in Turbulence,"
<a href="http://arxiv.org/abs/cond-mat/0205286">cond-mat/0205286</a>
<li>G. Falkovich, K. Gawedzki and M. Vergassola, "Particles and fields
in fluid turbulence," <a
href="http://arxiv.org/abs/cond-mat/0105199">cond-mat/0105199</a> [RMP review]
<li>F. Flandoli and M. Romito, "Markov selections for the 3D stochastic
Navier-Stokes
equations", <a href="http://arxiv.org/abs/math.PR/0602612">math.PR/0602612</a>
<li>C. Foias, O. Manley, R. Rosa and R. Temam
<cite><a href="http://cambridge.org/9780521064606">Navier-Stokes Equations and Turbulence</a></cite>
<li>J. Fontbona, "A probabilistic interpretation and stochastic
particle approximations of the 3-dimensional Navier-Stokes
equations", <a
href="http://dx.doi.org/10.1007/s00440-005-0477-9"><cite>Probability Theory and
Related Fields</cite> <strong>136</strong> (2006): 102--156</a>
<li>Uriel Frisch, Marco Martins Afonso, Andrea Mazzino and Victor
Yakhot, "Does multifractal theory of turbulence have logarithms in the scaling
relations?", <a href="http://arxiv.org/abs/nlin.CD/0506003">nlin.CD/0506003</a>
["The multifractal theory of turbulence uses a saddle-point evaluation in
determining the power-law behaviour of structure functions. Without suitable
precautions, this could lead to the presence of logarithmic corrections,
thereby violating known exact relations such as the four-fifths law. Using the
theory of large deviations applied to the random multiplicative model of
turbulence and calculating subdominant terms, we explain here why such
corrections cannot be present." The LD argument sounds more interesting to me
than the actual result!]
<li>T. Funaki, D. Surgailis and W. A. Woyczynski, "Gibbs-Cox
Random Fields and Burgers Turbulence", <cite>Annals of Applied
Probability</cite> <strong>5</strong> (1995): 461--492
<li>J. D. Gibbon and Charles R. Doering, "Intermittency and regularity
issues in 3D Navier-Stokes turbulence", <a
href="http://arxiv.org/abs/math.DS/0406146">math.DS/0406146</a>
<li>Claude Godrèche and Paul Manneville (eds.),
<cite><a
href="http://www.cup.org/Titles/45/0521455030.html">Hydrodynamics and Nonlinear Instabilities</a></cite>
<li>Nigel Goldenfeld, "Roughness-induced critical phenomena in a
turbulent flow", <a
href="http://arxiv.org/abs/cond-mat/0509439">cond-mat/0509439</a>
<li>Toshiyuki Gotoh, Robert H. Kraichnan, "Turbulence and Tsallis
Statistics", <a href="http://arxiv.org/abs/nlin.CD/0305040">nlin.CD/0305040</a>
<li>Vincent Grenard, Nicolas Garnier, Antoine Naert, "Effective temperature of a stationary dissipative system: fully-developed turbulence",
<a href="http://arxiv.org/abs/0704.0325">arxiv:0704.0325</a>
<li>Gustafson and Sethian (eds.), <cite>Vortex Methods and Vortex
Motion</cite>
<li>Gregory W. Hammett and John C. Bowman, "Non-white noise and a
multiple-rate Markovian closure theory for turbulence,"
<a href="http://arxiv.org/abs/physics/0203031">physics/0203031</a>
<li>Geoff Hewitt and Christos Vassillicos (eds.), <cite><a href="http://cambridge.org/9780521838993">Prediction of
Turbulent Flows</a></cite>
<li>Philip Holmes, John Lumley and Gal Berkooz, <cite>Turbulence,
Coherent Structures, Dynamical Systems, and Symmetry</cite> [Surely they
could've crammed more keywords into the title if they'd really tried]
<li>Sunghwan Jung, P. J. Morrison, and Harry L. Swinney, "Statistical
mechanics of two-dimensional turbulence", <a
href="http://arxiv.org/abs/cond-mat/0503305">cond-mat/0503305</a> [Sounds cool,
from the abstract]
<li>Holger Kantz, Detlef Holstein, Mario Ragwitz and Nikolay K.
Vitanov, "Markov chain model for turbulent wind speed data", <a
href="http://dx.doi.org/10.1016/j.physa.2004.01.070"><cite>Physica
A</cite> <strong>342</strong> (2004): 315--321</a>
<li>Dan Kushnir, Jörg Schumacher, and Achi Brandt,
"Geometry of Intensive Scalar Dissipation Events in Turbulence",
<a href="http://dx.doi.org/10.1103/PhysRevLett.97.124502"><cite>Physical Review
Letters</cite> <strong>97</strong> (2006): 124502</a> [Mostly for
the <a href="http://scitation.aip.org/prl/covers/97_12.jsp">pictures</a>]
<li>Marten Landahl and E. Mollo-Christensen, <cite>Turbulence and
Random Processes in Fluid Mechanics</cite>
<li>Marcel Lesieur, <cite>Turbulence in Fluids</cite> [Lesieur has also
written a much less technical book, <cite>La turbulence,</cite>, which is
supposed to be a visual treat, but I can't seem to find a copy, and my French
would be woefully inadequate.]
<li>Y. Charles Li, "Chaos in Partial Differential Equations, Navier-Stokes Equations and Turbulence", <a href="http://arxiv.org/abs/0712.4026">arxiv:0712.4026</a>
<li>Yi Li and Charles Meneveau, "Origin of Non-Gaussian Statistics in
Hydrodynamic Turbulence", <a
href="http://dx.doi.org/10.1103/PhysRevLett.95.164502"><cite>Physical Review
Letters</cite> <strong>95</strong> (2005): 164502</a>
<li>Victor S. L'vov, Evgenii Podivilov, Anna Pomyalov, Itamar Procaccia
and Damien Vandembroucq, "An Optimal Shell Model of Turbulence,"
<a href="http://arxiv.org/abs/chao-dyn/9803025">chao-dyn/9803025</a>
<li>Paul Manneville, <cite>Instabilities, Chaos and Turbulence</cite>
<li>Mathieu and Scott, <cite>An Introduction to Turbulent Flow</citE>
<li>Manikandan Mathur, George Haller, Thomas Peacock, Jori
E. Ruppert-Felsot, and Harry L. Swinney, "Uncovering the Lagrangian Skeleton of
Turbulence",
<a href="http://dx.doi.org/10.1103/PhysRevLett.98.144502"><citE>Physical Review
Letters</cite> <strong>98</strong> (2007): 144502</a>
<li>Jonathan C. Mattingly, "On Recent Progress for the Stochastic
Navier Stokes Equations", <a
href="http://arxiv.org/abs/math.PR/0409194">math.PR/0409194</a>
<li>James C. McWilliams, <citE><a href="http://cambridge.org/9780521856379">Fundamentals of Geophysical Fluid
Dynamics</a></cite>
<li>C. Meneveau, Y. Li, "On the origin of non-Gaussian statistics in
hydrodynamic
turbulence", <a href="http://arxiv.org/abs/physics/0508211">physics/0508211</a>
["we derive, from the Navier-Stokes equations, a simple nonlinear dynamical
system for the Lagrangian evolution of longitudinal and transverse velocity
increments. ... the ubiquitous non-Gaussian tails in turbulence have their
origin in the inherent self-amplification of longitudinal velocity increments,
and cross amplification of the transverse velocity increments."]
<li>Patrick Milan, Matthias Wächter, and Joachim Peinke, "Turbulent Character of Wind Energy", <a href="http://dx.doi.org/10.1103/PhysRevLett.110.138701"><cite>Physical
Review Letters</cite> <strong>110</strong> (2013): 138701</a>
<li>P. D. Mininni and A. Pouquet, "Persistent cyclonic structures in self-similar turbulent flows", <a href="http://arxiv.org/abs/0903.2294">arxiv:0903.2294</a>
<li>N. Mordant, J. Delour, E. Leveque, A. Arneodo, Jean-Francois
Pinton, "Long time correlations in Lagrangian dynamics: a key to
intermittency in turbulence,"
<a href="http://dx.doi.org/10.1103/PhysRevLett.89.254502"><cite>Physical
Review Letters</cite> <strong>89</strong> (2002): 254502</a>,
<a href="http://arxiv.org/abs/physics/0206013">physics/0206013</a>
<li>H. Mouri, A. Hori, M. Takaoka, "Large-scale lognormal fluctuations in turbulence velocity fields", <a href="http://arxiv.org/abs/0810.2166">arxiv:0810.2166</a>
<li>H. Mouri, M. Takaoka, A. Hori, Y. Kawashima, "On Landau's
prediction for large-scale fluctuation of turbulence energy dissipation", <a
href="http://arxiv.org/abs/physics/0505203">physics/0505203</a>
<li>Mark Nelkin, "Does Kolmogorov mean field theory become exact for
turbulence above some critical dimension?"
<a href="http://arxiv.org/abs/nlin.CD/0103046">nlin.CD/0103046</a> [An
important question when designing five-dimensional hydraulics]
<li>M. Ossiander, "A probabilistic representation of solutions of the
incompressible Navier-Stokes equations in $\mathbb{R}^3$",
<a
href="http://dx.doi.org/10.1007/s00440-004-0418-z"><cite>Probability Theory and
Related Fields</cite> <strong>133</strong> (2005): 267--298</a>,
<a href="http://arxiv.org/abs/math.PR/0412034">math.PR/0412034</a>
<li>Pope, <cite>Turbulent Flows</cite>
<li>Olivier Poujade, "Rayleigh-Taylor turbulence is nothing like
Kolmogorov's in the self similar
regime", <a href="http://arxiv.org/abs/physics/0606136">physics/0606136</a>
<li>Itamar Procaccia, K.R. Sreenivasan, "The State of the Art in Hydrodynamic Turbulence: Past Successes and Future Challenges", <cite>Physica D</cite> <strong>237</strong> (2008): 2167--2183, <a href="http://arxiv.org/abs/0710.5446">arxiv:0710.5446</a>
<li>S. G. Rajeev, "Fuzzy Fluid Mechanics in Three
Dimensions", <a href="http://arxiv.org/abs/0705.2139">arxiv:0705.2139</a>
[<a href="http://sgrajeev.com/fuzzy-fluids/">Background post by
Prof. Rajeev</a>]
<li>Raoul Robert and Vincent Vargas, "Hydrodynamic
Turbulence and Intermittent Random Fields", <a href="http://dx.doi.org/10.1007/s00220-008-0642-y"><cite>Communications in Mathematical Physics</cite> <strong>284</strong> (2008): 649--673</a>
<li>Marco Romito, "Analysis of equilibrium states of Markov solutions
to the 3D Navier-Stokes equations driven by additive noise",
<a href="http://arxiv.org/abs/0709.3267">arxiv:0709.3267</a>
<li>David Ruelle, "Hydrodynamic turbulence as a problem in nonequilibrium statistical mechanics", <a href="http://dx.doi.org//10.1073/pnas.1218747109"><cite>Proceedings of the National Academy of Sciences</cite> (USA) <strong>109</strong> (2012): 20344--20346</a>
<li>Jori E. Ruppert-Felsot, Olivier Praud, Eran Sharon, Harry
L. Swinney, "Extraction of coherent structures in a rotating turbulent flow
experiment", <a href="http://arxiv.org/abs/physics/0410161">physics/0410161</a>
<li>W. Sakikawa and O. Narikiyo, "Kolmogorov scaling for the
epsilon-entropy in a forced turbulence simulation,"
<a href="http://arxiv.org/abs/cond-mat/0208094">cond-mat/0208094</a>
<li>Francois G. Schmitt and Yongxiang Huang, <cite><a href="http://cambridge.org/9781107067615">Stochastic Analysis of Scaling Time Series:
From Turbulence Theory to Applications</a></cite>
<li>S. F. Shandarin and Ya. B. Zeldovich, "The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium",
<a href="http://dx.doi.org/10.1103/RevModPhys.61.185"><cite>Reviews of Modern Physics</cite> <strong>61</strong> (1989): 185--220</a>
<li>Troy R. Smith, Jeff Moehlis and Philip Holmes, "Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial",
<a href="http://dx.doi.org/10.1007/s11071-005-2823-y"><cite>Nonlinear
Dynamics</cite>
<strong>41</strong> (2005): 275--307</a>
<li>F. Spineanu and M. Vlad, "Statistical properties of an ensemble of
vortices interacting with a turbulent field", <a
href="http://arxiv.org/abs/physics/0506099">physics/0506099</a>
<li>R. Stresing, J. Peinke, R. E. Seoud and J. C. Vassilicos, "Defining
a New Class of Turbulent
Flows", <a href="http://dx.doi.org/10.1103/PhysRevLett.104.194501"><cite>Physical
Review Letters</cite> <strong>104</strong> (2010): 194501</a>
<li>Henk Tennekes, <cite>The Simple Science of Flight</cite>
<li>S. A. Thorpe
<ul>
<li><cite><a href="http://cambridge.org/9780521676809">An Introduction to Ocean Turbulence</a></cite>
<li><cite><a href="http://cambridge.org/9780521835435">The Turbulent Ocean</a></cite>
</ul>
<li>F. Toschi, L. Biferale, G. Boffetta, A. Celani, B. J. Devenish, and
A. Lanotte, "Acceleration and vortex filaments in turbulence", <a
href="http://arxiv.org/abs/nlin.CD/0501041">nlin.CD/0501041</a>
<li>Antonio Turiel, Germain Mato, Néstor Parga and Jean-Pierre
Nadal, "Self-Similarity Properties of Natural Images Resemble Those of
Turbulent Flows," <a
href="http://dx.doi.org/10.1103/PhysRevLett.80.1098"><cite>Physical Review
Letters</cite> <strong>80</strong> (1998): 1098--1101</a>
<li>Antonio Turiel, Jordi Isern-Fontanet, Emilio Garcia-Ladona, and
Jordi Font, "Multifractal Method for the Instantaneous Evaluation of the Stream
Function in Geophysical Flows", <a
href="http://dx.doi.org/10.1103/PhysRevLett.95.104502"><cite>Physical Review
Letters</cite> <strong>95</strong> (2005): 104502</a>
<li>S. I. Vainshtein, "Transverse velocities, intermittency and
asymmetry in fully developed turbulence", <a
href="http://arxiv.org/abs/nlin.CD/0505049">nlin.CD/0505049</a>
<li>Mahendra K. Verma, "Introduction to Statstical Theory of Fluid
Turbulence",<a href="http://arxiv.org/abs/nlin.CD/0510069">nlin.CD/0510069</a>
<li>Z. Warhaft, "Turbulence in nature and in the laboratory",
<a href="http://dx.doi.org/10.1073/pnas.012580299"><cite>Proceedings of the National
Academy of Sciences</cite> (USA) <strong>99</strong> (2002): 2481--2486</a>
<li>Herwig Wendt, Patrice Abry and Stephane Jaffard, "Bootstrap for
Empirical Multifractal Analysis", <citE>IEEE Signal Processing Magazine</cite>
July 2007, pp. 38--48 [+ technical papers by these authors]
<li>Henricus H. Wensink, Jörn Dunkel, Sebastian Heidenreich, Knut Drescher, Raymond E. Goldstein, Hartmut L&oum;wen, and Julia M. Yeomans, "Meso-scale turbulence in living fluids", <a href="http://dx.doi.org/10.1073/pnas.1202032109"><cite>Proceedings of the National Academy of Sciences</cite> (USA) <strong>109</strong> (2012): 14308--14313</a>
<li>Haitao Xu, Nicholas T. Ouellette, Eberhard Bodenschatz, "Evolution
of geometric structures in intense
turbulence", <a href="http://arxiv.org/abs/0708.3955">arxiv:0708.3955</a>
<li>Huidan Yu, Sharath S. Girimaji and Li-Shi Luo, "Lattice Boltzmann
simulations of decaying homogeneous isotropic turbulence", <a
href="http://dx.doi.org/10.1103/PhysRevE.71.016708"><cite>Physical Review
E</cite> <strong>71</strong> (2005): 016708</a>
</ul>
<ul>To do:
<li>Reconstruct causal states from turbulence data; calculate
correlation functions therefrom
</ul>