Turbulence, and Fluid Mechanics in General

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.

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]

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. Further discussion here.]
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.]
M. G. Shats, H. Xia and H. Punzmann, "Self-organization in turbulence as a route to order in plasma and fluids", physics/0409074
P. Tabeling and O. Cardoso (eds.) Turbulence: A Tentative Dictionary

Edsel A. Ammons
- "The Stationary Statistics of a Turbulent Environment as an Attractor," physics/0202028
- "An Approach to the Statistics of Turbulence", physics/0306068
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", cond-mat/0305186 = Physical Review E 69 (2004): 026305
- "Stochastic models of Lagrangian acceleration of fluid particle in developed turbulence", cond-mat/0408018 = International Journal of Modern Physics B 18 (2004): 3095--3168 [A review of Langevin-type stochastic models]
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 [Blurb]
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. Blurb]
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 [Blurb]
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 [Blurb]
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 [Blurb]
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
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 [blurb]
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 [Blurb]
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 [blurb]
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 [blurb]
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
Nicolas Mordant, Alice M. Crawford and Eberhard Bodenschatz, "Experimental Lagrangian Acceleration Probability Density Function Measurement," physics/0303003
N. Mordant, J. Delour, E. Leveque, A. Arneodo, Jean-Francois Pinton, "Long time correlations in Lagrangian dynamics: a key to intermittency in turbulence," physics/0206013 = Physical Review Letters 89 (2002): 254502
H. Mouri, M. Takaoka, A. Hori, Y. Kawashima, "On Landau's prediction for large-scale fluctuation of turbulence energy dissipation", physics/0505203 ["Kolmogorov's theory for turbulence in 1941 is based on a hypothesis that small-scale statistics are uniquely determined by the kinematic viscosity and the mean rate of energy dissipation. Landau remarked that the local rate of energy dissipation should fluctuate in space over scales of large eddies and hence should affect small-scale statistics. Experimentally, we confirm the significance of this fluctuation, which is comparable to the mean rate of energy dissipation at the typical scale of large eddies. The significance is independent of the Reynolds number and the configuration for turbulence production. With an increase of scale r above the scale of largest eddies, the fluctuation comes to have the scaling r^{-1/2} and tends Gaussian. We also confirm that the fluctuation affects small-scale statistics."]
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 R³", math.PR/0412034 = Probability Theory and Related Fields 133 (2005): 267--298 ["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
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 [blurb]
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]
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

Reconstruct causal states from turbulence data; calculate correlation functions therefrom

Notebooks

Turbulence, and Fluid Mechanics in General