## Turbulence, and Fluid Mechanics in General

*12 Mar 2018 13:36*

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..."

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 non-linear physics is "new" or
"non-Newtonian".) 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 *laminar*, and these laminæ are stable.

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 chaotic 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 Album of Fluid Motion 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.

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.

So what's to do? Well, all sorts of things, including more or less direct simulations of flows by cousins of cellular automata 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 statistical mechanics to calculate various emergent properties --- 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.)

If people *must* find analogies for society, ecosystems, etc., from
physics and engineering, turbulence is probably a better one than feedback.

- Recommended, big picture:
- George Batchelor, The Life and Legacy of G. I. Taylor [A nice scientific biography of one of the founders of modern mechanics, and of the statistical theory of turbulence; review]
- Pierre Berge
*et al.,*Order within Chaos - Alexandre J. Chorin, Vorticity and Turbulence ["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.]
- Predrag Cvitanovic, "Turbulence, and what to do about it?" [2002 "essay" on Cvitanovic's approach, based on identify recurrent patterns and expressing things in terms of them; I'm sympathetic.]
- Uriel Frisch, Turbulence: The Legacy of A. N. Kolmogorov [An excellent introduction, very strong on defending Kolmogorov's work from mis-understandings and invalid criticisms.]
- Victor L'vov and Itamar Procaccia, "Hydrodynamic Turbulence: a 19th Century Problem with a Challenge for the 21st Century" (=chao-dyn 9606015) [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 field theory work.]
- Terence Tao, "Why global regularity for Navier-Stokes is hard" [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.]
- H. Tennekes and J. L. Lumley, A First Course in Turbulence [Older --- from the 1970s --- but still good]
- van Dyke, An Album of Fluid Motion [Pretty pictures, with Reynolds numbers]

- Recommended, close ups:
- O. E. Barndorff-Nielsen, J. L. Jensen and M. Sorensen, "Parametric
Modelling of Turbulence," Philosophical Transactions of the Royal
Society: Physical Science and Engineering,
**332**(1990): 439--455 [ARMA models work for turbulence! Who knew? JSTOR] - S. C. Chapman, G. Rowlands and Nick W. Watkins, "The Origin of Universal Fluctuations in Correlated Systems: Explicit Calculation for an Intermittent Turbulent Cascade," cond-mat/0302624
- Alice M. Crawford, Nicolas Mordant, Andy M. Reynolds, Eberhard Bodenschatz, "Comment on 'Dynamical Foundations of Nonextensive Statistical Mechanics'", physics/0212080
- Jordgi Delgado and Ricard V. Solé, "Characterizing
Turbulence in Globally Coupled Maps with Stochastic Finite Automata,"
Physics Letters A
**314**(2000): 314--319 - Gary Doolen, Uriel Frisch, Brosl Hasslacher, Steven Orszag and Stephen Wolfram (eds.), 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 SFI Proceedings Vol. IV
- Gregory L. Eyink and Katepalli R. Sreenivasan, "Onsager and the
theory of hydrodynamic
turbulence", Reviews
of Modern Physics
**78**(2006): 87--135 [Things like this aren't supposed to happen in real life.] - Nicolas Mordant, Alice M. Crawford and Eberhard Bodenschatz
- "Three-Dimensional Structure of the Lagrangian Acceleration in Turbulent
Flows", Physical Review
Letters
**93**(2004): 214501 [It's lognormal! Well, they don't do any formal tests of goodness of fit, but by eye it's pretty good.] - "Experimental Lagrangian Acceleration Probability Density Function Measurement," physics/0303003

`<li>M. G. Shats, H. Xia and H. Punzmann, "Self-organization in`

turbulence as a route to order in plasma and fluids", physics/0409074

- "Three-Dimensional Structure of the Lagrangian Acceleration in Turbulent
Flows", Physical Review
Letters
- P. Tabeling and O. Cardoso (eds.) Turbulence: A Tentative Dictionary

- To read:
- Edsel A. Ammons
- "The Stationary Statistics of a Turbulent Environment as an Attractor," physics/0202028
- "An Approach to the Statistics of Turbulence", physics/0306068

`<li>Claudia Angelini, Daniela Cavab, Gabriel Katul, and Brani`

Vidakovic, "Resampling hierarchical processes in the wavelet domain: A case study using atmospheric turbulence", Physica D

**207**(2005): 24--40 - Alex Arenas, Alexandre Chorin, "On the existence and scaling of structure functions in turbulence according to the data", cond-mat/060161
- A. K. Aringazin and M. I. Mazhitov
- "One-dimensional Langevin models of fluid particle
acceleration in developed
turbulence", Physical Review
E
**69**(2004): 026305, cond-mat/0305186 - "Stochastic models of
Lagrangian acceleration of fluid particle in developed turbulence",
International Journal of
Modern Physics B
**18**(2004): 3095--3168, cond-mat/0408018

- "One-dimensional Langevin models of fluid particle
acceleration in developed
turbulence", Physical Review
E
- V. I. Arnol'd, Topological Methods in Hydrodynamics
- A. Bershadskii, J.J. Niemela, A. Praskovsky and K.R. Sreenivasan, "`Clusterization' and intermittency of temperature fluctuations in turbulent convection", nlin.CD/0401044 [Their "telegraph approximation" is a discretization, which means one can use all kinds of symbolic-dynamical tricks can be used]
- Eugene Balkovsky and Boris I. Shraiman, "Olfactory Search at High Reynolds Number," nlin.CD/0109019
- M. M. Bandi, J. R. Cressman Jr., W. I. Goldburg, "Test of the Fluctuation Relation in compressible turbulence on a free surface", nlin.CD/0607037
- M. M. Bandi, W. I. Goldburg, J. R. Cressman Jr, "Measurement of entropy production rate in compressible turbulence", nlin.CD/0607036
- G. I. Barenblatt, Scaling Phenomena in Fluid Mechanics
- G. I. Barenblatt and Alexandre J. Chorin, "A New Formulation of the Near-Equilibrium Theory of Turbulence," math.DS/9909060
- George Batchelor [Old, but classics]
- An Introduction to Fluid Dynamics
- The Theory of Homogeneous Turbulence

- G. K. Batchelor et al (eds.), Perspectives in Fluid Dynamics: A Collective Introduction to Current Research ["eleven chapters that introduce and review different branches of the subject for graduate-level courses, or for specialists seeking introductions to other areas"]
- Christian Beck
- "Nonextensive methods in turbulence and particle physics," cond-mat/0110071
- "Generalized statistical mechanics and fully developed turbulence," cond-mat/0110073
- "Superstatistical turbulence models", physics/0506123

- Roberto Benzi, Luca Biferale and Federico Toschi, "Intermittency
in Turbulence: Multiplicative Random Processes in Space and Time",
Journal of Statistical Physics
**113**(2003): 783--798 - Jacob Berg, "Lagrangian one-particle velocity statistics in a turbulent flow", physics/0610155
- L. Biferale, G. Boffetta, A. Celani, A. Lanotte and F. Toschi
- "Lagrangian statistics in fully developed turbulence", nlin.CD/0402032
- "Multifractal statistics of Lagrangian velocity and acceleration in turbulence", nlin.CD/0403020

- Birkhoff, Hydrodynamics: A Study in Logic, Fact and Similitude
- Dieter Biskamp, Magnetohydrodynamic Turbulence
- Tomas Bohr, Morgan Jensen, Giovanni Paladin and Angelo
Vulpiani, Dynamical Systems Approach to Turbulence [I'm told this
Bohr is a grandson of
*the*Bohr... who is, come to think of it, the only founder of quantum mechanics who has never been the hero of a version of my opening anecdote, at least not that I've run across.] - Bradshaw, An Introduction to Turbulence and Its Measurement
- Jean-Michel Caillol, Oksana Patsahan, and Ihor Mryglod, "Statistical field theory for simple fluids: the collective variables representation", cond-mat/0503213
- Haris J. Catrakis and Paul E. Dimotakis, "Shape Complexity in
Turbulence," Physical Review Letters
**80**(1998): 968--971 - Pierre-Henri Chavanis, "Statistical mechanics of geophysical
turbulence: application to jovian flows and Jupiter's great red spot", Physica
D
**200**(2005): 257--272 - 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", Physical Review
Letters
**95**(2005): 200203 - L. Chevillard and C. Meneveau, "Lagrangian Dynamics and Statistical
Geometric Structure of Turbulence", Physical Review
Letters
**97**(2006): 174501, cond-mat/0606267 - L. Chevillard, S. G. Roux, E. Lévêque, N. Mordant,
J.-F. Pinton, and A. Arnéodo, "Intermittency of Velocity Time Increments
in Turbulence", Physical Review
Letters
**95**(2005): 064501 - Rene Chevray and Jean Mathieu, Topics in Fluid Mechanics
- Chorin and Marsden, A Mathematical Introduction to Fluid Mechanics
- Igor Chueshov and Annie Millet, "Stochastic 2D hydrodynamical type systems: Well posedness and large deviations", arxiv:0807.1810
- R. Collina, R. Livi and A. Mazzino, "Large Deviation Approach to the Randomly Forced Navier-Stokes Equation", physics/0410148
- Peter Constantin and Ciprian Foias, Navier-Stokes Equations
- Alice M. Crawford, Nicolas Mordant and Eberhard Bodenschatz, "Joint
Statistics of the Lagrangian Acceleration and Velocity in Fully Developed
Turbulence", Physical Review
Letters
**94**(2005): 024501 - Olivier Darrigol, Worlds of Flow: A history of hydrodynamics from the Bernoullis to Prandtl
- P. A. Davidson and B. R. Pearson, "Identifying Turbulent Energy
Distributions in Real, Rather than Fourier, Space", Physical Review
Letters
**95**(2005): 214501 - Peter D. Ditlevsen, Turbulence and Shell Models
- Bruno Eckhardt, Tobias M. Schneider, "How does flow in a pipe become turbulent?", arxiv:0709.3230
- Richard S. Ellis, Kyle Haven and Bruce Turkington, "The Large Deviation Principle for Coarse-Grained Processes," math-ph/0012023
- Gregory L. Eyink
- "Locality of turbulent cascades", Physica
D
**207**(2005): 91--116 - "Turbulent Diffusion of Lines and Circulations", arxiv:0704.0263

- "Locality of turbulent cascades", Physica
D
- Gregory L. Eyink, Shiyi Chen and Qiaoning Chen, "Gibbsian Hypothesis in Turbulence," cond-mat/0205286
- G. Falkovich, K. Gawedzki and M. Vergassola, "Particles and fields in fluid turbulence," cond-mat/0105199 [RMP review]
- F. Flandoli and M. Romito, "Markov selections for the 3D stochastic Navier-Stokes equations", math.PR/0602612
- C. Foias, O. Manley, R. Rosa and R. Temam Navier-Stokes Equations and Turbulence
- J. Fontbona, "A probabilistic interpretation and stochastic
particle approximations of the 3-dimensional Navier-Stokes
equations", Probability Theory and
Related Fields
**136**(2006): 102--156 - Uriel Frisch, Marco Martins Afonso, Andrea Mazzino and Victor Yakhot, "Does multifractal theory of turbulence have logarithms in the scaling relations?", nlin.CD/0506003 ["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!]
- T. Funaki, D. Surgailis and W. A. Woyczynski, "Gibbs-Cox
Random Fields and Burgers Turbulence", Annals of Applied
Probability
**5**(1995): 461--492 - J. D. Gibbon and Charles R. Doering, "Intermittency and regularity issues in 3D Navier-Stokes turbulence", math.DS/0406146
- Claude Godrèche and Paul Manneville (eds.), Hydrodynamics and Nonlinear Instabilities
- Nigel Goldenfeld, "Roughness-induced critical phenomena in a turbulent flow", cond-mat/0509439
- Toshiyuki Gotoh, Robert H. Kraichnan, "Turbulence and Tsallis Statistics", nlin.CD/0305040
- Vincent Grenard, Nicolas Garnier, Antoine Naert, "Effective temperature of a stationary dissipative system: fully-developed turbulence", arxiv:0704.0325
- Gustafson and Sethian (eds.), Vortex Methods and Vortex Motion
- Gregory W. Hammett and John C. Bowman, "Non-white noise and a multiple-rate Markovian closure theory for turbulence," physics/0203031
- Geoff Hewitt and Christos Vassillicos (eds.), Prediction of Turbulent Flows
- Philip Holmes, John Lumley and Gal Berkooz, Turbulence, Coherent Structures, Dynamical Systems, and Symmetry [Surely they could've crammed more keywords into the title if they'd really tried]
- Sunghwan Jung, P. J. Morrison, and Harry L. Swinney, "Statistical mechanics of two-dimensional turbulence", cond-mat/0503305 [Sounds cool, from the abstract]
- Holger Kantz, Detlef Holstein, Mario Ragwitz and Nikolay K.
Vitanov, "Markov chain model for turbulent wind speed data", Physica
A
**342**(2004): 315--321 - Dan Kushnir, Jörg Schumacher, and Achi Brandt,
"Geometry of Intensive Scalar Dissipation Events in Turbulence",
Physical Review
Letters
**97**(2006): 124502 [Mostly for the pictures] - Marten Landahl and E. Mollo-Christensen, Turbulence and Random Processes in Fluid Mechanics
- Marcel Lesieur, Turbulence in Fluids [Lesieur has also written a much less technical book, La turbulence,, which is supposed to be a visual treat, but I can't seem to find a copy, and my French would be woefully inadequate.]
- Y. Charles Li, "Chaos in Partial Differential Equations, Navier-Stokes Equations and Turbulence", arxiv:0712.4026
- Yi Li and Charles Meneveau, "Origin of Non-Gaussian Statistics in
Hydrodynamic Turbulence", Physical Review
Letters
**95**(2005): 164502 - Victor S. L'vov, Evgenii Podivilov, Anna Pomyalov, Itamar Procaccia and Damien Vandembroucq, "An Optimal Shell Model of Turbulence," chao-dyn/9803025
- Paul Manneville, Instabilities, Chaos and Turbulence
- Mathieu and Scott, An Introduction to Turbulent Flow
- Manikandan Mathur, George Haller, Thomas Peacock, Jori
E. Ruppert-Felsot, and Harry L. Swinney, "Uncovering the Lagrangian Skeleton of
Turbulence",
Physical Review
Letters
**98**(2007): 144502 - Jonathan C. Mattingly, "On Recent Progress for the Stochastic Navier Stokes Equations", math.PR/0409194 ["We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations."]
- James C. McWilliams, Fundamentals of Geophysical Fluid Dynamics
- C. Meneveau, Y. Li, "On the origin of non-Gaussian statistics in hydrodynamic turbulence", physics/0508211 ["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."]
- P. D. Mininni and A. Pouquet, "Persistent cyclonic structures in self-similar turbulent flows", arxiv:0903.2294
- N. Mordant, J. Delour, E. Leveque, A. Arneodo, Jean-Francois
Pinton, "Long time correlations in Lagrangian dynamics: a key to
intermittency in turbulence,"
Physical
Review Letters
**89**(2002): 254502, physics/0206013 - H. Mouri, A. Hori, M. Takaoka, "Large-scale lognormal fluctuations in turbulence velocity fields", arxiv:0810.2166
- H. Mouri, M. Takaoka, A. Hori, Y. Kawashima, "On Landau's prediction for large-scale fluctuation of turbulence energy dissipation", physics/0505203
- Mark Nelkin, "Does Kolmogorov mean field theory become exact for turbulence above some critical dimension?" nlin.CD/0103046 [An important question when designing five-dimensional hydraulics]
- M. Ossiander, "A probabilistic representation of solutions of the
incompressible Navier-Stokes equations in $\mathbb{R}^3$",
Probability Theory and
Related Fields
**133**(2005): 267--298, math.PR/0412034 ["A new probabilistic representation is presented for solutions of the incompressible Navier-Stokes equations in R3 with given forcing and initial velocity. This representation expresses solutions as scaled conditional expectations of functionals of a Markov process indexed by the nodes of a binary tree."] - Pope, Turbulent Flows
- Olivier Poujade, "Rayleigh-Taylor turbulence is nothing like Kolmogorov's in the self similar regime", physics/0606136
- Itamar Procaccia, K.R. Sreenivasan, "The State of the Art in Hydrodynamic Turbulence: Past Successes and Future Challenges", Physica D
**237**(2008): 2167--2183, arxiv:0710.5446 - S. G. Rajeev, "Fuzzy Fluid Mechanics in Three Dimensions", arxiv:0705.2139 [Background post by Prof. Rajeev]
- Raoul Robert and Vincent Vargas, "Hydrodynamic
Turbulence and Intermittent Random Fields", Communications in Mathematical Physics
**284**(2008): 649--673 ["construct two families of multifractal random vector fields with non-symmetrical increments ... to model the velocity field of turbulent flows."] - Marco Romito, "Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise", arxiv:0709.3267
- Jori E. Ruppert-Felsot, Olivier Praud, Eran Sharon, Harry L. Swinney, "Extraction of coherent structures in a rotating turbulent flow experiment", physics/0410161
- W. Sakikawa and O. Narikiyo, "Kolmogorov scaling for the epsilon-entropy in a forced turbulence simulation," cond-mat/0208094
- Francois G. Schmitt and Yongxiang Huang, Stochastic Analysis of Scaling Time Series: From Turbulence Theory to Applications
- Troy R. Smith, Jeff Moehlis and Philip Holmes, "Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial",
Nonlinear
Dynamics
**41**(2005): 275--307 - F. Spineanu and M. Vlad, "Statistical properties of an ensemble of vortices interacting with a turbulent field", physics/0506099
- R. Stresing, J. Peinke, R. E. Seoud and J. C. Vassilicos, "Defining
a New Class of Turbulent
Flows", Physical
Review Letters
**104**(2010): 194501 - Henk Tennekes, The Simple Science of Flight
- S. A. Thorpe, An Introduction to Ocean Turbulence
- F. Toschi, L. Biferale, G. Boffetta, A. Celani, B. J. Devenish, and A. Lanotte, "Acceleration and vortex filaments in turbulence", nlin.CD/0501041
- Antonio Turiel, Germain Mato, Néstor Parga and Jean-Pierre
Nadal, "Self-Similarity Properties of Natural Images Resemble Those of
Turbulent Flows," Physical Review
Letters
**80**(1998): 1098--1101 - Antonio Turiel, Jordi Isern-Fontanet, Emilio Garcia-Ladona, and
Jordi Font, "Multifractal Method for the Instantaneous Evaluation of the Stream
Function in Geophysical Flows", Physical Review
Letters
**95**(2005): 104502 - S. I. Vainshtein, "Transverse velocities, intermittency and asymmetry in fully developed turbulence", nlin.CD/0505049
- Mahendra K. Verma, "Introduction to Statstical Theory of Fluid Turbulence",nlin.CD/0510069
- Z. Warhaft, "Turbulence in nature and in the laboratory",
Proceedings of the National
Academy of Sciences (USA)
**99**(2002): 2481--2486 - Herwig Wendt, Patrice Abry and Stephane Jaffard, "Bootstrap for Empirical Multifractal Analysis", IEEE Signal Processing Magazine July 2007, pp. 38--48 [+ technical papers by these authors]
- 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", Proceedings of the National Academy of Sciences (USA)
**109**(2012): 14308--14313 - Haitao Xu, Nicholas T. Ouellette, Eberhard Bodenschatz, "Evolution of geometric structures in intense turbulence", arxiv:0708.3955
- Huidan Yu, Sharath S. Girimaji and Li-Shi Luo, "Lattice Boltzmann
simulations of decaying homogeneous isotropic turbulence", Physical Review
E
**71**(2005): 016708

- To do:
- Reconstruct causal states from turbulence data; calculate correlation functions therefrom