Turbulence
| Roberto Benzi and Uriel Frisch (2010), Scholarpedia, 5(3):3439. | doi:10.4249/scholarpedia.3439 | revision #137205 [link to/cite this article] |
Turbulence, a scientific term to describe certain complex and unpredictable motions of a fluid, is part of our daily experience and has been for a long time. No telescope or microscope is needed to contemplate the volutes of smoke from a cigarette, the elegant arabesques of cream poured into coffee and the vigorous eddies of a mountain stream. In an airplane we sometimes experience bursts of "clear air turbulence". Ultrasonography can reveal turbulent blood flow in our arteries; satellite pictures may show turbulent meteorological perturbations; computer simulations reveal turbulent fluctuations of mass in the Universe on scales of tens of megaparsecs. Without turbulence, urban pollution would linger around for centuries, the heat produced by nuclear reactions in the interior of stars would not be able to escape on an acceptable time scale and meteorological phenomena would be predictable almost for ever.
Actually the word "turbulence" (Latin: turbulentia) originally refers to the disorderly motion of a crowd (turba). In the Middle Ages it was frequently used to mean just "trouble", a word which derives from it. Even today "turbulent" may refer to social or personal behaviour. Its scientific usage refers to irregular and seemingly random motion of a fluid. This definition, which is far from exhaustive, tries to express in a synthetic way one of the most complex and fascinating phenomenon of natural science, from Antiquity to present days.
The subject has indeed a very long history. More than two thousand years ago Lucretius described eddy motion in his De rerum natura. Over five centuries ago Leonardo was probably the first to use the word turbulence (in Italian turbolenza) with its modern meaning and to observe the slow decay of eddies formed behind the pillars of a bridge. Just over a quarter of a millennium back, Euler wrote the equations of incompressible ideal or inviscid (zero-viscosity) flow in both two and three dimensions and realized the importance of vorticity. Seventy years later Navier generalized these equation to include viscosity. Because of further work by Stokes, the equations are known as the Navier–Stokes equation. They constitute a set of nonlinear and nonlocal evolution equations for the three-dimensional velocity field. In modern notation, the first equation which expresses Newton's law as applied to arbitrary fluid elements, reads \[\tag{1} \partial_t{ v} +{ v}\cdot\nabla { v} = -\nabla p +\nu\nabla ^2 { v}, \]
where \(p\) is the pressure (divided by a the constant density of the fluid) and \(\nu\) is the (kinematic) viscosity. The second equation, due to d'Alembert, expresses incompressibility \[\tag{2} \nabla \cdot { v} =0.\]
Kelvin was the first to propose
studying turbulence using random solution of the Navier–Stokes
equations. Reynolds showed that, for a given geometry of the flow,
the different regimes that can take place (laminar, turbulent, ...)
are controlled by the dimensionless number (now called the Reynolds number)
\[\tag{3}
Re = LV/\nu,\]
where \(L\) and \(V\) are a typical scale and a typical velocity of the flow. For a lot more information on the early history of the subject we refer the reader to Worlds of Flow by Darrigol.
With the advent of aeronautics, the development of meteorology, astrophysics, plasma physics and nuclear weapons the understanding of turbulent flow became a very important issue. Progress remained however quite slow, for reasons we shall come back to. In more recent years a paradigmatic shift took place: as predicted by von Neumann 60 years ago it became possible to simulate turbulent flow on computers, thereby leading to a new kind of experimentation which somewhat blurs the traditional distinction between theory and experiments.
The aim of this review is to focus on some open questions for which significant progress can be expected on the scale of the next decade. We shall of course not try to review the entire field of turbulence: it has become very large and has to some extent developed a babel-tower excessive diversity for lack of a unifying language. We encourage the reader to look up the other Scholarpedia articles on fluid mechanics and particularly those focusing on specific aspects which we could not cover in detail, e.g. direct numerical simulations [1] [2].
After the great breakthrough due to Kolmogorov , dimensional and scaling arguments seemed to provide such a language, at least in the limit of very large Reynolds numbers (fully developed turbulence). Although it turned out that the true story was much more complicated, we feel that it is still useful to examine many topics arising in turbulence from the point of view of scaling and of its shortcomings. The topics addressed here are listed in the Contents.
Contents |
Tools for turbulence
Theory
Since the basic equations are known [link to section on NS], the question is: how much of a theoretical handle do we have on the Navier–Stokes (NS) equations? The short answer is: very little. We cannot, for example, show that the solutions of the NS equations with nice and smooth initial conditions stay nice, smooth and unique for all times, at least not in 3D (but in 2D, yes, we can). There will be more on this in Section 4 on blow up. It has even been speculated by Jean Leray in the thirties that the random character of turbulence originates from non-uniqueness of the solutions to the NS equations. Nowadays, we know enough about how chaos can appear in deterministic dynamical systems that there is no need to resort to non-uniqueness to explain turbulence.
When we try tackling random solutions of the NS equations we have to face a closure problem: because the equations are quadratically nonlinear, the time-rate of change of the correlation functions of the velocity at \(n\) different points involves similar correlation functions, but with \(n+1\) arguments. An infinite hierarchy of equations is then obtained. The simplest form of closure, introduced by Kolmogorov's student Milionshchikov, is to arbitrarily close this hierarchy by relating fourth-order correlation functions to second-order ones as if the velocity had Gaussian statistics. This is of course unjustified and leads to problems such as negative values for energy-like quantities, which are by definition non-negative. Cures for such diseases can be found but they are frequently ad hoc with no possibility to control the errors made with respect to the correct solutions. The main difficulty is the absence of a small parameter which would permit to start a suitable perturbative approach. The smallness of the viscosity - or equivalently a large value for the Reynolds number - is of no use so far because very little is understood about the Euler equations, that is the Navier–Stokes equations with the viscosity set to zero (see Euler 250 and references therein).
The most fruitful theoretical approaches have been based on scaling arguments, that is essentially on dimensional analysis. This is presented in Section 2. Another rather fruitful kind of approach is through the use of toy models: after having identified certain properties of the basic equations which are believed to play a key role in the behavior of turbulence (for example invariance and conservations properties) one tries to find simpler models sharing those properties and which either can be solved analytically or at least for which numerical solutions are much simpler than for the full 3D NS equations. Precise definitions of such models would take up too much space, but we can give an idea of what has been achieved with the use of some of these toy models.
The 1D Burgers equation (Frisch and Bec, Bec and Khanin) gives a concrete example of how energy dissipation can have a finite non-vanishing limit when the viscosity tends to zero in spite of the fact that the inviscid equation formally conserves energy. As pointed out by Saffman, it also shows what can go wrong with naive application of dimensional arguments.
The random coupling model of Kraichnan starts with N independent replicas of the random NS equations and then couples them artificially by random Gaussian coefficients chosen in such a way as to preserve most invariance and conservation laws. In the limit \( N \to \infty \) closed equations, called the Direct Interaction (DIA) equations, are obtained for suitable statistical quantities (including the two-time and two-point velocity correlation functions). The solutions are not compatible with the Kolmogorov 1941 theory to be discussed in Section 2; this happens not because scale invariance is broken but because the model does not preserve a certain form of Galilean invariance. Fortunately, this shortcoming can be repaired by resorting to a kind of Lagrangian description (Kraichnan ) or by making the coupling coefficients scale dependent to obtain the Eddy Damped Quasi-Normal Markovian (EDQNM) model (Orszag).
Shell models start from the NS equations written in spatial Fourier space and replace all the Fourier modes in the shell having wavenumbers between \(2^n\) and \(2^{n+1}\) by just a few degrees of freedom, typically one complex number. The interactions between these "shell amplitudes" are of course chosen again to preserve as many features as possible of the original equations. Some shell models, such as GOY or SABRA (Gledzer, Ohkitani and Yamada, Biferale), are known to display the same anomalous scaling as the full equations, see Section 2. Unfortunately little theoretical progress has been achieved and the shell models main advantage remains their ability to run very high Reynolds numbers using just a workstation.
Much more drastic simplifications of the true dynamics are involved in the multiplicative random model (Novikov and Stewart, Yaglom, Benzi et al.) in which the amplitudes in the shell \(n+1\) are just obtained by multiplying the amplitude in the shell \(n\) by a random variable with a suitable distribution (independent identically distributed random variables are assumed for different \(n\)). Correlation functions can then be calculated explicitly; they display anomalous scaling and multifractality through a mechanism involving large deviations (Varadhan).
Finally purely qualitative models such as the Swift–Richardson flea-eddy model (Section 2) can be made which are stimulating for the imagination.
Experiments
As stated before, turbulent flows abound everywhere around us. However high-Reynolds-number flows displaying good scaling properties of the kind discussed in Section 2 require very large scales, as is the case in the natural atmospheric and astrophysical environments. The former keeps changing (with the weather) and the latter is not so easily accessed. Large-scale facilities, such as major wind tunnels, are very expensive and it is difficult to have a fundamental experiment running there for a duration of several days to several weeks (needed, e.g. to accumulate good statistical data on rare violent events) when they are competing with important industrial applications such as the testing of new designs for cars and airplanes.
During the last ten years a new technique has been developed involving low temperature Helium (above the lambda point where it becomes superfluid) (Chavanne et al., Niimela and Sreenivasan). It takes advantage of the fact that Helium can have a kinematic viscosity two orders of magnitude lower than air to achieve fairly high Reynolds numbers with facilities which still fit on a table. Special measurement techniques had to be developed, since the usual hot-wire techniques cannot cope with such experiments. It is now planned to use much larger Helium facilities using know-how developed by major high-energy centers such as CERN.
Finally, very promising results have been recently obtained in understanding the dynamics of Lagrangian (tracer) particles in turbulent flows. Rapid technological advances in optical particle tracking allow scientists to measure accurately the positions, velocities and accelerations of such tracer particles (for a review see Toschi and Bodenschatz). Detailed comparison between experiments and numerical simulations on Lagrangian properties of turbulent flows has opened interesting new directions of investigation with many applications where transport and/or aggregation of particles is important.
Numerical simulations
In 1949 von Neumann predicted that the advent of digital computers would revolutionize the study of turbulence since it would become possible to simulate the NS equations in 3D in turbulent regimes. Actually the first genuine 3D simulations of turbulence had to wait about 20 years with the advent of supercomputers and the development of (pseudo)-spectral numerical technique taking advantage of fast Fourier transform algorithms (Orszag and Patterson). For homogeneous turbulence without boundaries (or more precisely, using periodic boundary conditions) the achievable spatial resolutions, which at first was only about \(32^3\) has now reached \(4096^3\) (Kaneda et al.). It can be safely predicted that within a decade or less the Reynolds numbers achievable by such simulations will be comparable to those of the best (present-day) experimental facilities. Numerical simulations have of course the advantage that one has access to the whole spatial structure of the velocity, of the vorticity, of the local energy dissipation, etc. However a single simulation at very high Reynold numbers may take from weeks to months of CPU time which can be prohibitive when exploring parameter space. Simulations are of course rather demanding for those problems in which boundaries are essential. Furthermore, when complex boundaries or physics are involved setting up suitable simulations can become difficult. One interesting way to handle such problems is through Lattice Boltzmann simulations which combine the hydrodynamical aspect with the microscopic physics of the problem while being not more demanding than, say, finite difference techniques (Benzi et al. , Succi).
For engineering applications, one usually needs to suitably model turbulence
while using as much as possible of our basic scientific
understanding. This can be done in a variety of ways, including
\(K-\epsilon\) theory (Mohammadi and Pironneau), Reynolds Averaged Navier–Stokes modelling
(Pope )
and renormalization group methods( Orszag et al. ). Such topics are beyond the present
review which focusses on the physics of turbulence.
K41 and intermittency
Scaling ideas have a long history in fluid mechanics, starting with Newton's derivation of the quadratic dependence of the drag on the relative velocity of motion between a body and the ambient fluid. Scaling arguments and dimensional analysis play a key role in the first serious attempt to understand the statistical properties of turbulent flow.
In 1941 Kolmogorov defined a conceptual framework for turbulence, now referred to as K41 theory ( Kolmogorov 1941a, Kolmogorov 1941b, Frisch) which applies to homogeneous, isotropic turbulence, that is turbulence statistically invariant under translations and rotations of the sort frequently obtained at very high Reynolds numbers when there is no large-scale shear.
A few years earlier, Richardson had proposed a qualitative vision of the energy cascade for the way energy flows from larger to smaller eddies. He proposed that it would be similar to the way blood flows from larger to smaller fleas in a famous poem of Swift.
In K41, this is made quantitative by two postulates regarding the
large Reynolds number limit. On the one hand
Kolmogorov assumes that the energy dissipation rate \(\varepsilon\) has
a finite non-vanishing limit as the viscosity tends to zero while
keeping the scale and velocity characteristic of the production
of the turbulence fixed (for a recent experimental investigation on this point see Sreenivasan 1984). On the other hand he assumes that a statistical scale
invariance of the cascade is achieved in the limit of very large Reynolds
numbers. The former assumption, which is now generally called the existence
of a dissipative anomaly (in a laminar fluid, the dissipation goes
to zero with the viscosity), is well supported by experimental
and numerical results. The latter assumption holds only in an approximate
way (see below).
In its mathematical formulation, the invariance postulated by Kolmogorov resembles that of the Brownian motion process, in the development of which Kolmogorov was strongly involved. If \(x(t)\) is the position at time \(t\) of a Brownian particle, then for any \(t>0\ ,\) \(h>0\) and \(\lambda >0\ ,\) the statistical distribution of \(x(t+\lambda h) - x(t)\) is the same as that of \(\lambda ^{1/2}[x(t+h) -x(t)]\ .\) In plain language, the increments of the position of the Brownian particle scale as the square root of the time increments. In K41, temporal position increments become spatial velocity increments and the square root becomes a cubic root. The latter is dictated by a dimensional argument: let \(\varepsilon\) denote the energy dissipation per unit mass of the fluid and \(l\) the separation between two points. If we try relating velocity increments to these quantities by a formula of the form \[\tag{4} \hbox{velocity increment} = C \varepsilon ^\alpha l ^ \beta,\]
where \(C\) is a dimensionless constant, we immediately find that \[\tag{5} \alpha = \beta = 1/3.\]
Kolmogorov was using the Richardson cascade idea to equate the energy dissipation rate to the rate of energy transfer from scale to scale and assuming that viscosity should become irrelevant. K41 scaling immediately implies scaling laws for structure functions, that is moments of velocity increments. The simplest instances are the longitudinal structure functions for homogeneous isotropic turbulence, defined as \[\tag{6} S_p (l)\equiv \left \langle\left\{\left[{\vec v}( {\vec r} +{\vec \ell}) - {\vec v}({\vec r})\right ]\cdot\frac{\vec \ell}{l}\right\}^p\right \rangle,\]
where \(p\) is a positive integer, \({\vec \ell}\) denotes a spatial vector increment and \(l\) its modulus. According to K41 one should have
\[\tag{7} S_p (l) =C_p \varepsilon ^{p/3} l ^{p/3} \]
for any positive integer \(p\ .\) The case \(p=3\) deserves special mention
because Kolmogorov showed that the four-fifths relation
\[\tag{8}
S_3 (l) = -\frac{4}{5} \varepsilon l \]
holds without any need to assume self-similarity (Kolmogorov 1941c).
All these scaling relations are meant to apply within the inertial range, that is the range of scales much smaller than the scales at which turbulence is produced and much larger than the Kolmogorov dissipation scale \(\eta\) at which direct energy dissipation into heat becomes important. K41 gives \(\eta = C_{\rm diss} (\nu ^3/\varepsilon)^{1/4}\ .\)
Eq. (7) and one of its consequences, namely that the energy spectrum of turbulence should follow a \(k^{-5/3}\) law (where \(k\) is the wavenumber) are reasonably well supported by experimental and numerical data. However, a careful examination of the scaling laws, using for example the Extended Self Similarity method, reveals small but measurable discrepancies from K41. Indeed, structure functions at inertial-range separations do display power-law behavior \( S_p(l) \propto l ^{\zeta_p}\ ,\) but the graph of the scaling exponents \(\zeta_p\) is not exactly the straight line \( \zeta_p= p/3\) predicted by K41: it displays curvature as shown in (Figure 2). Hence the self-similarity assumed in K41 may actually be broken. Presently the scaling exponents \(\zeta_p\) are known with an accuracy of a few percent and could well be universal, that is independent of the mechanism by which the turbulence is driven. Obtaining better evidence for or against universality is important.
This is in fact not surprising if we go back to the Swift–Richardson picture of turbulence. Real fleas may not want to cluster in the Swiftian fashion (he actually had in mind poets, not fleas) but there are many natural instances of hierarchical clustering, for example in plants as shown in (Figure 3)
Thanks to the work of Mandelbrot we know that such objects are fractals. Mandelbrot was also the first to conjecture that at infinite Reynold numbers the energy dissipation of turbulence concentrates in a fractal set of Hausdorff dimension less than three ( Mandelbrot 1968).
The fact that small-scale activity in high-Reynold number turbulence becomes increasingly clumpy and that self-similarity is broken is generally referred to as intermittency. It can be quantified by measuring for example the flatness of velocity increments \[\tag{9} F(l) \equiv S_4(l) /S_2^2(l),\]
a quantity which should be independent of \(l\) in K41 and which actually grows as \(l\) decreases.
Once it was realized that K41 is probably not exactly correct –
although it had already found many practical applications – a flurry
of activity started to understand intermittency. At first
phenomenological models were developed, such as the multiplicative
random model, which displays not only fractal dissipation but the
phenomenon of multifractality, a concept with applications far beyond
turbulence (Benzi et al., Jaffard).
In the mid-nineties Kraichnan
conjectured that intermittency and anomalous scaling are already
present in a simple passive scalar model in which a scalar field
(say, a temperature field) is being advected by a prescribed Gaussian
random field with K41-type scaling and a very short correlation time.
The conjecture was proven and – for the first time – a result about intermittency was derived ab initio, that is from the basic equations without recourse to ad hoc steps. This breakthrough made use of techniques borrowed from quantum field theory. Very roughly, to calculate the correlation function \(\psi_n\) of order \(n\) of the passive scalar, one has to solve a multi-dimensional linear partial differential equation of the form: \[\tag{10} L_n \psi_n = f_n,\]
where \(f_n\) is a prescribed function and \(L_n\) a prescribed linear operator, both scale-invariant with known scaling properties. This seems to determine the scaling properties of the solution, the exception being those functions which sit in the null space of \(L_n\ .\) These are the zero modes which allow the breaking of scale invariance (Gawedzki and Kupianen , Chertkov , Shraiman and Siggia, Falkovich et al. ). Such zero modes, which can be interpreted as quantities statistically conserved under transport by the flow, are hard to calculate but can be shown not to depend on initial and boundary condition and on the way the scalar is fed into the flow, thus ensuring universality of the (anomalous) scaling properties. It is not known if such conservation laws are associated to any symmetries, such as the invariance properties of the passive scalar transport in the absence of molecular diffusion. It is generally believed that a somewhat similar mechanism will guarantee the universality of scaling for the full nonlinear problem of turbulence but suitable techniques to derive such results ab initio have yet to be found.
Bulk quantities, drag and its reduction
The K41 theory makes remarkable predictions on the statistical properties of homogeneous and isotropic turbulence. Although the effects of intermittency and breaking of scale invariance tell us that K41 theory is not exactly true, Kolmogorov was able to build a conceptual framework which enables us to understand in a quantitative way what we mean by turbulence. One obvious question is whether one can use the same framework to make predictions on bulk (global) quantities for turbulent flows.
As a simple but not trivial example, consider the question of how much fluid can be carried by a pipe of radius \(H\) for a given pressure gradient \(\nabla p\ .\) In a laminar flow, the velocity profile is parabolic with a maximum at the centre of the channel equal to \(U_0 = v_*^2 H/\nu\ ,\) where \(v_*^2 \equiv H\nabla p /(2\rho)\) and \(H\) is the pipe radius and \( \rho\) the density. Consequently, the Reynolds number of the flow is given by \(Re =U_0 H / \nu = v_*^2 H^2 / \nu^2\ .\) For large enough \(Re\) the flow becomes turbulent and the mass throughput becomes smaller with respect to the laminar case: a substantial power input is spent for maintaining turbulent fluctuations and the maximum average velocity at the centre of the channel \(U_M\) becomes substantially smaller than \(U_0\ .\) For a pipe of length \(L\ ,\) the total work done by the pressure gradient is proportional to \(H^2 \nabla p L / \rho\) while the total average kinetic energy proportional to \(\rho U_M^2 L\ .\) The ratio between these two quantities is named drag coefficient \(C_D\ ,\) namely: \[ C_D \equiv \frac{\nabla p H^2 }{\rho U_M^2} = \frac{v_*^2}{U_M^2}. \] The interpretation of the drag coefficient \(C_D\) is rather intuitive: it is a measure of how much kinetic energy is acquired by the mean flow with respect to the work done by the external forces. For laminar flow in a pipe, \(U_M = U_0\) and \(C_D \sim 1/Re\ .\) At the onset of turbulence \(C_D\) increases and eventually decreases very slowly for increasing \(Re\ .\) There exists no systematic theory capable of predicting the Reynolds-number dependence of \(C_D\ .\)
A beautiful although phenomenological approach has been developed by
von Kármán in the thirties. Using ideas from the slightly
later K41 theory, von Kármán's
approach can be recast as follows. For homogenous pipe flow,
the velocity profile \(U\) depends only on the distance \(y\) from the
boundary and the description of the different statistical quantities becomes
easier by introducing Prandtl's dimensionless variables
\[\tag{11}
U^+(y^+) \equiv \frac{U(y)}{v_*} :\,\,\, \ y^+ \equiv \frac{y}{\delta} :\,\,\, \ \delta \equiv \frac{\nu}{v_*}.
\]
In all turbulent flows, one can always introduce the average velocity field and the turbulent fluctuations. In homogenous wall bounded turbulent flows, the average velocity field has a non zero component only in the streamwise direction \(x\ .\) Denoting by \(\langle \ldots \rangle\) the average along the \(x\) direction, the momentum equation reads: \[\tag{12} \nu \frac { \partial U}{ \partial y} + W= v_*^2(1-\frac { y^+ }{ H^+ }), \]
where \(W\) is the momentum flux towards the wall. Turbulent fluctuations take energy from the mean flow at a rate \(W\partial_y U(y)\ .\) On the average the energy source of turbulent fluctuations must balance \(\varepsilon_t\) which is the rate of energy dissipation due to turbulence. Both energy production and energy dissipation are mostly concentrated near the boundaries. The conceptual advance made by von Kármán is to stress that in the range \(1 \ll y^+ \ll H^+\ ,\) one can expect that the effect of viscosity should be negligible, i.e. the mean flow should not depend on \(\delta\ .\) Then using (12) one can obtain \(W=v_*^2\ .\) Next, using the Kolmogorov theory, one can assume that energy dissipation is independent of the viscosity. By dimensional analysis, the most general expression for \(\varepsilon_t\) can be written as \(v_*^3F(y^+)/\delta\ .\) Finally, by balancing energy source and energy, we obtain \[\tag{13} W \frac{\partial U}{\partial y } = \varepsilon(y) = \frac{v_*^3}{\delta} F(y^+) \ \rightarrow \frac{\partial U}{\partial y} = \frac{v_*}{\delta} F(y^+/\delta), \]
where \(F\) is a universal function. As it is written, the r.h.s of the above equation depends explicitly on \(\delta\ .\) For \(Re \rightarrow \infty\ ,\) according to the Kolmogorov theory, \(\partial_y U\) should become independent of the viscosity which is parameterized by the scale \(\delta\ .\) Therefore we reach the conclusion that either \(F=0\) or \(F \sim 1/y^+\ ,\) and the most general form of \(U^+\) must be \[\tag{14} U^ +(y^+) = A + B\log \left(y^+\right), \]
where \(A\) and \(B\) are universal constant. Equation (14) is the prediction originally made by von Kármán and it allows us to compute the value of \(C_D\) as a function of \(Re\) and the two constants \(A\) and \(B\ .\) Note that for finite Reynolds number, we may expect deviation (of the order of \(1/\log (Re)\)) from (14) which cannot be computed by using dimensional analysis. Existing data from laboratory experiments and numerical simulations show a remarkable good agreement with (14) (Procaccia and Sreenivasan).
Let us remark that we did use the K41 theory although the turbulent flow is certainly neither isotropic nor homogeneous (in the \(y\) direction). There is, thus, a hidden assumption that the basic relation of the K41 theory can be extended beyond the framework where the theory has been proposed. This idea can be worked out by assuming that turbulent kinetic energy and energy dissipation are always related according to the K41 dimensional analysis, which is the basis of the so-called \(K - \varepsilon \) model of turbulent flow.
Within the obvious limitations of the argument so far discussed, we want only to stress that the fundamental assumption in the K41 theory, namely that energy dissipation is independent of the viscosity for large Reynolds number, is the basic guideline in understanding the dynamics of high-Reynolds number turbulent flows, and this beyond the case of homogeneous and isotropic turbulence. It follows that the effect of boundary conditions does not necessarily show a different world with respect to the K41 theory.
According to our discussion in the previous section, we should expect
intermittency acting in wall-bounded turbulent flows as in any other
case of turbulence. Does the intermittency change the von Kármán
theory? As far as bulk quantities are concerned, it seems that the
answer is no. However, if one looks at high-order moment of pressure
fluctuations at the wall (pressure is force per unit area from which
we compute the drag coefficient \(C_D\)), then one discovers strong
intermittency and breakdown of the predictions from dimensional analysis. A quantitative
explanation of intermittent fluctuations in wall bounded turbulence
flows is beyond the von Kármán as well as the K41 theory and it is
a research problem under investigation (Casciola et al., Toschi et al.).
Although successful, the von Kármán theory of boundary layers is not "closed" as far as one does not predict the universal constants \(A\) and \(B\ .\) Clearly, any scaling argument, such as those obtained by dimensional analysis, cannot be helpful in predicting the two constants. Numerical simulations give estimates of \(A\) and \(B\) which are in very good agreement with experimental data. An open question is whether one can go on theoretically and make progress to predict pure constants beyond dimensional analysis.
There are many physical problems where dimensional analysis does not provide an answer. One example is the problem of drag reduction in a turbulent pipe flow when a small amount of flexible polymers is added (Lumley, Virk). This phenomenon, known also as Tom's effect, was discovered in the early forties and it cannot be explained by using arguments similar to the von Kármán or the Kolmogorov theory (for recent reviews see (Christopher et al. , Procaccia et al. ). Since polymers can be stretched by turbulent fluctuations, a certain amount of turbulent kinetic energy is transferred to the polymers. This mechanism decreases turbulent fluctuations especially near the wall. Thus the momentum flux towards the wall decreases as well, which may eventually lead to decrease in the drag coefficient \(C_D\ ,\) i.e. more mass throughput for the same power input. It is a challenging question to obtain a quantitative theory on the amount of drag reduction as a function of the polymer concentration \(C\) and the physical properties of the polymer (notably its maximum extension length \(L_p\) and characteristic time \(\tau_p\) for the stretched polymer to relax its extension from \(L_p\) ) . de Gennes and Tabor suggested that the presence of polymers in high-Reynolds-number turbulence modifies the Kolmogorov dissipation scale, leading to a different energy balance. However, this amounts to a change in the effective Reynolds number that can hardly affect the balance in the energy-containing range and thus the drag. Actually, it has been argued (Procaccia et al.) that drag reduction by addition of polymers is equivalent to introducing a space-dependent viscosity which increases linearly from the wall boundary: the amount of drag reduction depends on the slope of the effective viscosity which is a rather complex function of both \( C\) and \(\tau_p\ .\)
Polymer addition is not the only way to reduce drag. Experiments and numerical
simulations show that addition of surfactants or air bubbles in water can lead
to drag reductions. It is unclear whether one will eventually discover a universal
mechanism for drag reduction. The whole subject, of considerable importance for many
engineering applications, is under active investigation.
Blow up
A fundamental milestone in three-dimensional turbulence is the idea that energy dissipation becomes Reynolds-number independent for large Reynolds numbers. It is present both in K41 and in subsequent theories of intermittency and in good agreement with experiments and numerical simulations (see Section 5.2 of Frisch and references therein). Since dissipation is proportional to the viscosity, it is somewhat paradoxical that it can tend to a non-zero value when viscosity vanishes, hence the name viscous anomaly. For the passive scalar problem discussed in Section 2. a result which is the counterpart of the viscous anomaly can be derived rigorously. The mathematical problem remains open for the three-dimensional Navier–Stokes equations and is connected to some deep issues of regularity for both the Euler and the Navier–Stokes equations.
Onsager was the first to observe that solutions to the 3D incompressible Euler equations need not conserve energy if the solutions lack regularity (more precisely if they are not Hölder continuous with an exponent larger than \(1/3\)); Duchon and Robert gave an expression of the dissipative anomaly (the local amount of dissipation) for such solutions of the Euler equations. For such questions see the review paper by Eyink and Sreenivasan.
The key issue, of course, is: how regular are the solutions to the Navier–Stokes and Euler equations when the initial conditions are sufficiently smooth? Such issues are reviewed in Majda and Bertozzi and Temam and, in more elementary way, in Rose and Sulem . In two dimensions in a bounded domain it has been known since the thirties that Euler flow preserves (sufficient) initial smoothness forever. In three dimensions only finite-time regularity is guaranteed. This can be understood somewhat naively by the following argument (part of which actually survives in much more complex functional analysis estimates).
As was shown by Cauchy, if one takes the curl of the 3D Euler equation one obtains the following equation for the vorticity \({ \omega} \equiv \nabla \wedge { u}\ :\) \[D_t { \omega} \equiv \partial_t{ \omega} +{ u}\cdot \nabla { \omega} = { \omega}\cdot \nabla { u}.\]
Now, let us be sloppy: we identify the velocity gradient \(\nabla { u}\) and the vorticity (which is actually its antisymmetric part) and furthermore blur the distinction between scalars, vectors and tensors to rewrite the preceding equation as \[D_t { \omega} \simeq { \omega}^2,\] which we consider as an ordinary differential equation (written in Lagrangian coordinates). It is obvious that if initially \({ \omega}_0 \equiv { \omega}(0) >0\ ,\) then \({ \omega}(t)\) will blow up (become infinite) at \(t_\star = 1/{ \omega}_0 \ .\)
An identical equation giving blow up can actually be derived
from the inviscid (zero viscosity) Burgers equation
\[\partial_t u +u\partial_x u =0.\]
Setting \(w\equiv -\partial_xu\ ,\) we obtain upon space differentiation
\[D_t w \equiv \partial_t w +u\partial_x w = w^2.\]
There is however a huge difference between (infinitely)
compressible Burgers dynamics and incompressible hydrodynamics.
In the former case singularities (mostly shocks) appear because
faster elements of fluid catch up with slower ones, a scenario
which hardly carries over to incompressible flow.
However our understanding of the mathematics of the Euler equations is so limited – even a quarter of a millennium after their introduction– that we cannot do much better than obtaining for various function-space norms an upper bound which behaves as for the Burgers equation. The numerical evidence about 3D Euler flow is not clearly for or against finite-time blow up. There are actually mechanisms which could make 3D incompressible flow much tamer than compressible flow. One of them, depletion is discussed in Section 4.
One might think that the presence of viscous dissipation makes the problem much easier and that blow up would then be ruled out. This is indeed conjectured by most specialists but has never been proven. Actually the Clay Foundation has decided to devote one of its Millennium million-dollar prizes to precisely this issue. Finally there is a tricky issue at the interface of the Euler and Navier-Stokes blow up problems. In the absence of boundaries one can show that hypothetical all-time regularity for the Euler equations implies the same for Navier–Stokes but not so with boundaries ( Kato )
Increase of predictability and depletion of non linearity
Turbulence in two space dimensions shows some interesting new features with respect to the three dimensional case so far discussed. Vorticity conservation prevents energy dissipation from staying constant when the viscosity is decreased. Thus the basic assumption of K41 theory becomes inapplicable. However enstrophy, i.e. the integral of the vorticity squared, can be dissipated and the energy cascade of the K41 picture can by replaced by an enstrophy cascade along with an inverse cascade of energy towards large scales (Kraichnan and Montgomery ). As soon as high resolution numerical simulations of two dimensional flows became available to the scientific community, one striking feature emerged from flow visualization and, in particular, from visualization of the vorticity field (Mc Williams, Benzi et al., Legras et al. ): the emergence of long lived coherent structure mostly in the form of circular vortex. A careful analysis of the flow field shows that these structures are stable non linear solutions of the Euler equations. As such, coherent structures represent, locally, a depletion of non-linearity in a turbulent flow: within a coherent structure the enstrophy cascade is inhibited and the structure can survive for extremely long times. Whenever the vorticity field is dominated by few coherent structures, the predictability time of large scale dynamics increases, being no longer constrained by the small scale fluctuations. The above picture is quite generic and is not necessarily limited to two dimensional turbulence, although strong evidence of depletion of non linearity has been observed mostly in the 2D case.
The increase of predictability time by depletion of nonlinearity is certainly relevant for geophysical applications. In particular, for atmospheric and oceanic dynamics, one can show the relevance of some form of 2D dynamics. For example in the case of atmospheric flows, one can prove that, in absence of dissipation and of mechanical and thermodynamical forcing, there exists a new form of vorticity, named Ertel potential vorticity, which is conserved. The Ertel potential vorticity is defined as \(\vec{\Omega}\cdot \vec{\nabla} \theta/\rho\ ,\) where \(\vec{\Omega}\) is the absolute vorticity (the sum of the flow vorticity and the Coriolis force on a rotating sphere) and \(\theta\) is the potential temperature (surface at \(\theta=const\) are isoentropic). A particular case of Ertel potential vorticity can be obtained at midlatitudes where the Coriolis force is almost balanced by the pressure gradients (geostrophic balance). Quasigeostrophic flows can be described using the idea of potential enstrophy (square of potential vorticity) cascade from large to small scales (where the flow is assumed to be dissipated as three dimensional turbulence). One can then expect that quasigeostrophic flows can form stable coherent structure similar to those observed in ordinary 2D turbulence, and we can again expect that coherent structures must will be more predictable than predicted by dimensional arguments.
We want to emphasize that the above scenario, still
rather speculative at this stage, implies that there
is an enhancement of the predictability time, i.e. an enhancement of
our ability to forecast weather and/or ocean circulation, because of coherent structures.
This enhancement in the predictability time could be an intermittent process, i.e. it could happen at random times, depending on whether or not
coherent structures dominate the flow.
Also, the effect of coherent structures can change the whole energy budget in
the flow and in particular momentum and heat fluxes may be changed in
a non trivial way, with relevant consequences on the long-time
behavior of the general circulation, i.e. the climate.
Overture of turbulence flows in different scientific fields
There are many important and fascinating problems in physics where turbulence plays an important role. In many cases, one can develop arguments similar to those of the K41 theory. but in other cases, new concepts are needed.
Here is an example of a long-standing problem not easily handled by a simplistic argument: when the density is a slowly varying function of temperature, one can observe rather strong turbulence whenever a temperature gradient is acting in the direction opposite to that of the gravitational field (natural convection). In this case, one is usually interested in predicting the heat flux in terms of the temperature difference \(\Delta T\ ,\) the buoyancy force \(g\beta\ ,\) the temperature gradient \(\Delta T/H\ ,\) the viscosity \(\nu\) and the thermal diffusivity \(\kappa\ .\) Here, \(\beta\) designates the volume expansivity and \(H\) the thickness of the layer. One can easily show that the dimensional turbulent heat flux \( Nu \) must be a function of the Rayleigh number \(Ra = g\beta \Delta T H^3/\nu\kappa\) and the Prandtl number \(Pr= \nu/\kappa\ .\) From the equations of motion, one is able to compute the energy dissipation of the turbulent flow which must be proportional to \(Nu Ra\ .\) According to the K41 approach, energy dissipation should also be of the order of \(U^3/H\) where, as usual, \(U\) is characteristic flow velocity in turbulent flow. Since the source of energy in natural convection is the potential energy, \(g \beta \Delta T H\ ,\) we can estimate \(U\sim (g \beta \Delta TH)^{1/2}\ ,\) i.e. \(U \sim Ra^{1/2}\ .\) It follows that the straightforward prediction based on K41 assumption is that \(Nu \sim Ra^{1/2}\) (Kraichnan). It so happens that the above prediction, with a suitable estimate of constants, is also an upper bound of the turbulent heat flux for any Prandtl number. However, for infinite Pr number, a rigorous upper bound gives at most \(Nu \sim Ra^{1/3}\ .\) When \(Pr \rightarrow \infty\ ,\) the boundary layers are laminar for any \(Ra\) and, consequently, the turbulent heat flux can change dramatically. Thus, a rather complex behavior of the relation Nu versus Ra must be expected, depending on the many parameters entering in the systems: one definitively must go beyond a simple dimensional analysis (for recent reviews, Ahlers et al., Procaccia and Sreenivasan and references therein).
Thermal convection, with its complex and fascinating properties, is just one example of the many important systems which are still in need of a full systematic scientific understanding from the theoretical, experimental and computational point of view. Here is a short and incomplete list of some of several interesting fields where understanding the nature of turbulence can potentially lead to a significant scientific breakthrough:
- MHD (magnetohydrodynamic) turbulence plays an important role in many astrophysical and geophysical problems. such as the generation of magnetic fields in heavenly bodies, in planets and (sometime) in large-scale industrial facilities;
- Compressible turbulence which is crucial in many engineering applications for example in aeronautics and combustion;
- Geophysical fluid Dynamics and Climate theory. We already briefly mention some interesting links between two dimensional turbulence and enhancement in the predictability time. This is just one aspect of a major scientific problem concerning the large-scale atmospheric and oceanic circulations and climate.
- Superfluid turbulence.
The above list is neither exhaustive nor indicates any scientific priority. It just tells us that the understanding of turbulent flows remains a fundamental issue in modern physics.
References
- Lucretius, P (1947). De Rerum Natura (3 vols. Latin text books) Comprehensive commentary by Cyril Bailey. Oxford Univ. Press, Oxford.
- Darrigol, O (2005). World of Flow: A History of Hydrodynamics from Bernoullis to Prandt. Oxford Univ. Press, Oxford ISBN 0198568436, 9780198568438.
- Glimm, J; Impagliazzo, J and Singer, I.M (1990). The Legacy of John Von Neumann. American Mathematical Society., New York.
- Kolmogorov, AN (1941a). The local structure of turbulence in incompressible viscous fluid for very large Reynolds number Dokl. Akad. Nauk. SSSR 30: 9-13. doi:10.1007/978-94-011-3030-1_45.
- Millionshchikov, MD (1941). Theory of homogeneous isotropic turbulence Izv. Akad. Nauk. SSSR Ser. Geogr. Geofiz. 5: 433-446.
- Eyink, G; Frisch, U; Moreau, R and Sobolevsky, A (2008). Euler Equations: 250 Years, Proceedings of an International Conference held in Aussois, June18–23, 2007. Physica D : 450.
- Burgers, JM (1948). A mathematical model illustrating the theory of turbulence. Adv. in Appl. Mech. 1: 171-199. doi:10.1016/s0065-2156(08)70100-5.
- Frisch, U and Bec, J (2001). Burgulence. Les Houches 2000: New Trends in Turbulence, Springer EDP Science.
- Bec(2007). Phys. Reports. 447: 1.
- Saffman, PG (1968). Lectures on homogeneous turbulence, Topics in Nonlinear Physics. Springer, .
- Kraichnan, RH (1961). Dynamics of nonlinear stochastic systems. Math. Phys. 2: 124-148. doi:10.1063/1.1724206.
- Kraichnan, RH (1965). Lagrangian history closure approximation for turbulence. Phys. Fluids 8: 575-598. doi:10.1063/1.1761271.
- Orszag, S (1973). Statistical Theory of Turbulence, in Fluid Dynamics, Les Houches 1973. R. Balian, J.L.Peube eds, Gordon and Breach, NY : 237.
- Gledzer, EB (18). System of hydrodynamic type admitting two quadratic integrals of motion. Sov. Phys. Dokl. 1973: 216-217.
- Ohkitani(1989). Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model turbulence. Prog. Theoret. Phys. 89: 329-341. doi:10.1143/ptp.81.329.
- Biferale, label (2003). Annual Rev. Fluid Mech. 35: 441.
- Novikov(1964). The intermittency of turbulence and the spectrum of energy dissipation. Izv. Akad. Nauk. SSSR : 408-413.
- Yaglom, AM (1966). Effect of fluctuations in energy dissipation rate on the form of turbulence characteristics in the inertial subrange. Dokl. Akad. Nauk SSSR 166: 49-52.
- Benzi, R; Biferale, L and Sbragaglia, M (2004). A Gibbs-Like Measure for Single-Time, Multi-Scale Energy Transfer in Stochastic Signals and Shell Model of Turbulence J. Stat. Phys. 114: 137.
- Varadhan, SRS (1985). Large Deviations and Applications. Springer, Philadelphia.
- Chavanne, X; Chilla, F; Chabaud, B; Castaing, B and Hebral, B (2001). Phys. Fluids. 13: 1300.
- Niemela(2006). J. Low Temp. Phys. 143: 163-212.
- Toschi(2009). Annual Review of Fluid Mechanics. 41: 375-404.
- Orszag(1972). Numerical simulation of turbulence Statistical Models and Turbulence, Led. Notes in Phys., Springer 12: 127-147. doi:10.1007/3-540-05716-1_8.
- Kaneda, Y; Ishihara, T and Yokokawa, M (2003). Phys. Fluids : 15.
- Benzi, R; Succi, S and Vergassola, M (1992). Lattice Boltzmann equation: theory and applications Phys. Rep 222: 145. doi:10.1016/0370-1573(92)90090-m.
- Succi, S (2001). Lattice Boltzmann Equation for Fluid Dynamics and beyond. Oxford Univ Press, Oxford.
- Mohammadi, B and Pironneau, O (1994). Analysis of the k-epsilon turbulence model. Masson, Paris.
- Pope, SB (2000). Turbulent flows. Cambridge Univ. Press, Cambridge.
- Orszag, S et al. (1993). Renormalization Group Modeling and Turbulence Simulations Proc. Int. Conf. on Near Wall Turbulent Flows, eds RMC So, CG Speziale, BE Launder, Elsevier, Amsterdam : 1031-1046.
- Newton, I (1687). Mathematical Principles of Natural Philosophy. Book II, Proposition XXXII.
- Kolmogorov, AN (1941b). On degeneration (decay) of isotropic turbulence in an incompressible viscous liquid Dokl. Akad. Nauk. SSSR 31: 538-540. doi:10.1007/978-94-011-3030-1_46.
- Frisch, U (1995). Turbulence, the Legacy of A.N. Kolmogorov. Cambridge Univ. Press, Cambridge.
- Richardson, LF (1922). Weather Prediction by Numerical Process. Cambridge Univ. Press, Cambridge.
- Sreenivasan, KR (1984). On the scaling of the turbulent energy dissipation rate Phys Fluids 27: 1048-1051. doi:10.1063/1.864731.
- Kolmogorov, AN (1941c). Dissipation of energy in locally isotropic turbulence Dokl. Akad. Nauk. SSSR 32: 16-18. doi:10.1098/rspa.1991.0076.
- Benzi, R et al. (1993). Extended Self-Similarity in Turbulent Flows Phys. Rev. E 48: R29-R32. doi:10.1103/physreve.48.r29.
- Benzi, R et al. (2008). Phys. Rev. Lett. 100: 234503.
- Gotoh, T; Fukayama, D and Nakano, T (2002). Phys. Fluids 14: 1065.
- Mandelbrot, B (1977). Fractals: Form, Chance and Dimension. Freeman and Co., San Francisco, San Francisco.
- Mandelbrot, B (1968). On intermittent free turbulence. Turbulence of Fluids and Plasmas NY, Brooklyn Polytechnic Int, New York.
- Chen, SY; Dhruva, B; Kurien, S; Sreenivasan, KR and Taylor, MA (2005). Anomalous scaling of low-order structure functions of turbulent velocity J Fluid Mech 533: 183-192. doi:10.1017/s002211200500443x.
- Benzi, R; Paladin, G; Parisi, G and Vulpiani, A (1984). On the Multi Fractal Natur of Fully Developed Turbulence and Chaotic System J. Phys. A:Math, Gen. 17: 3521. doi:10.1088/0305-4470/17/18/021.
- Jaffard, S (2001). Wavelet expansions, function spaces and multifractal analysis. Twentieth Century Harmonic Analysis, a Celebration, Eds JS Byrnes, NATO, Springer.
- Kraichnan, RH (1994). Phys. Rev. Lett. 72: 1016.
- Celani, A; Lanotte, A; Mazzino, A and Vergassola, M (2001). Fronts in a passive scalar turbulence Phys. Fluids 13: 1768-1783. doi:10.1063/1.1367325.
- Gawedski(1995). Phys. Rev. Lett 75: 3834.
- Chertkov, M; Falkovich, G; Kolokolov, I and Lebedev, V (1995). Phys. Rev.E 52: 4924.
- Shraiman(1995). C.R. Acad. Sci. 321: 279.
- Falkovich, G; Gawedzki, K and Vergassola, M (2001). Rev. Mod. Phys. 73: 913.
- von Kårmån, Th (1930). Mechanische Ahnlichkeit und Turbulenz. Nachrichten der Akademie der Wissenschaften Gottingen Math. Phys. Klasse 58: 1.
- Prandtl, L (1932). Zur turbulenten Stromung in Rohren und langs Platten Ergebn. Aerodyn. Versuchsanst 4: 18-29. doi:10.1007/978-3-662-11836-8_50.
- Procaccia(2008). The State of the art in Hydrodynamic Turbulence: Past Successes and Future Challanges Physica D 237: 1825-2250. doi:10.1016/j.physd.2008.01.025.
- Casciola, CM; Gualtieri, P; Benzi, R and Piva, R (2003). J. Fluid Mech. 476: 105-114.
- Toschi, F; Amati, G; Succi, S; Benzi, R and Piva, R (1999). Phys. Rev. Lett 82: 5044-5047.
- Lumley, JL (1969). Ann. Rev. Fluid. Mech. 1: 367.
- Virk, PS (1969). AlChe J. 21: 625.
- Christopher, M; White, M and Mungal, G (2008). Ann. Rev. Fluid Mech. 40: 235-256.
- Procaccia, I; L'vov, V and Benzi, R (2008). Theory of drag reduction by polymers in wall bounded turbulence Rev. Mod. Phys. 80: 225. doi:10.1103/revmodphys.80.225.
- Tabor(1986). A cascade theory of drag reduction Europhys. Lett 2: 519-522. doi:10.1209/0295-5075/2/7/005.
- Eyink(2006). Onsager and the theory of hydrodynamical turbulence Rev. Mod. Phys. 78: 87. doi:10.1103/revmodphys.78.87.
- Majda, AJ and Bertozzi, AL (2001). Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics, Cambridge.
- Temam, R (2001). Navier-Stokes equations. Theory and numerical analysis. AMS Bookstore, .
- Rose(1978). Fully Developed Turbulence and Statistical Mechanics J. Phys. France : 441-484.
- Fefferman, C (2000). Existence and smoothness of the Navier–Stokes equation. Millennium problems of the Clay Mathematics Institute (2000), www.claymath.org.
- Kato, T ( 1983). Remarks on the zero viscosity limit for non stationary Navier-Stokes flows with boundary, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983). S.S. Chern, Springer, Berkeley.
- Kraichnan(1980). Two dimensional turbulence Rep. Prog. Phys. 43: 547-619. doi:10.1088/0034-4885/43/5/001.
- Williams, J (1984). The emergence of isolated coherent vortices in turbulent flows J. Fluid Mech 146: 21-43. doi:10.1017/s0022112084001750.
- Benzi, R; Patarnello, S and Santangelo, P (1986). On the statistical properties of two dimensional decaying turbulence Europhys. Lett. 3: 811. doi:10.1209/0295-5075/3/7/007.
- Legras, B; Santangelo, P and Benzi, R (1988). High-resolution numerical experiments for forced two-dimensional turbulence Europhys. Lett 5: 37-42. doi:10.1209/0295-5075/5/1/007.
- McIntyre, ME (2003). Potential Vorticity. Encycl. of Atmospheric Sciences, JR Holton, JA Pyle, JA Curry Eds, Academic Press.
- Kraichnan, RH (1962). Phys. Fluids 5: 1374.
- Ahlers, G; Grossmann, S and Lohse, D (2009). Heat transfer and large-scale dynamics in turbulent Rayleigh-Benard convection. Rev. Mod. Phys 58: 503. doi:10.1103/revmodphys.81.503.
Internal references
- Paul M.B. Vitanyi (2007) Andrey Nikolaevich Kolmogorov. Scholarpedia, 2(2):2798.
- Edward Ott (2008) Attractor dimensions. Scholarpedia, 3(3):2110.
- Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Giovanni Gallavotti (2008) Fluctuations. Scholarpedia, 3(6):5893.
- Søren Bertil F. Dorch (2007) Magnetohydrodynamics. Scholarpedia, 2(4):2295.
- Andrei D. Polyanin, William E. Schiesser, Alexei I. Zhurov (2008) Partial differential equation. Scholarpedia, 3(10):4605.
- Benjamin Widom (2009) Scaling laws. Scholarpedia, 4(10):9054.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
