Turbulence

Curator:Dr.Roberto Benzi, Dipartimento di Fisica and INFN, Università di Roma"Tor Vergata", Via della Ricerca Scientifica 1, I-00133, Roma,Italy
Curator:Dr.Uriel Frisch, UNS, CNRS, Laboratoire Cassiopée, OCA, BP 4229, 06304Nice cedex 4, France

Turbulence, a scientific term to describe certaincomplex and unpredictable motions of a fluid, is part of our dailyexperience and has been for a long time. No telescope or microscopeis needed to contemplate the volutes of smoke from a cigarette, theelegant arabesques of cream poured into coffee and the vigorouseddies of a mountain stream. In an airplane we sometimesexperienceburstsof"clear air turbulence". Ultrasonography can reveal turbulent bloodflow in our arteries; satellite pictures may show turbulentmeteorological perturbations; computer simulations revealturbulentfluctuationsofmass in the Universe on scales of tens of megaparsecs. Withoutturbulence, urban pollution would linger around for centuries, theheat produced by nuclear reactions in the interior of stars wouldnot be able to escape on an acceptable time scale andmeteorological phenomena would be predictable almost for ever.

Actually the word "turbulence"(Latin:turbulentia) originally refers tothe disorderly motion of a crowd (turba). In the Middle Agesit was frequently used to mean just "trouble", a word which derivesfrom it. Even today "turbulent" may refer to social or personalbehaviour. Its scientific usage refers to irregular and seeminglyrandom motion of a fluid. This definition, which is far fromexhaustive, tries to express in a synthetic way one of the mostcomplex and fascinating phenomenon of natural science, fromAntiquity to present days.

The subject has indeed a very long history. More than two thousandyears agoLucretiusdescribededdy motion in hisDe rerum natura. Overfive centuries ago Leonardo was probably the first to use the wordturbulence (in Italianturbolenza) with itsmodern meaning and to observe the slow decay of eddies formedbehind the pillars of a bridge. Just over a quarter of a millenniumback, Euler wrote the equations of incompressible ideal or inviscid(zero-viscosity) flow in both two and three dimensions and realizedthe importance of vorticity. Seventy years later Navier generalizedthese equation to include viscosity. Because of further work byStokes, the equations are known as the Navier–Stokes equation. Theyconstitute a set of nonlinear and nonlocal evolution equations forthe three-dimensional velocity field. In modern notation, the firstequation which expresses Newton's law as applied to arbitrary fluidelements, reads

(1)

whereisthe pressure (divided by a the constant density of the fluid)andisthe (kinematic) viscosity. The second equation, due to d'Alembert,expresses incompressibility

(2)

Kelvin was the first to propose studying turbulence using randomsolution of the Navier–Stokes equations. Reynolds showed that, fora given geometry of the flow, the different regimes that can takeplace (laminar, turbulent, ...) are controlled by the dimensionlessnumber (now called the Reynolds number)

(3)

whereandarea typical scale and a typical velocity of the flow. For a lot moreinformation on the early history of the subject we refer the readertoWorlds ofFlowbyDarrigol.

With the advent of aeronautics, the development of meteorology,astrophysics, plasma physics and nuclear weapons the understandingof turbulent flow became a very important issue. Progress remainedhowever quite slow, for reasons we shall come back to. In morerecent years a paradigmatic shift took place: as predictedbyvonNeumann60 years ago it became possible tosimulate turbulent flow on computers, thereby leading to a new kindof experimentation which somewhat blurs the traditional distinctionbetween theory and experiments.

The aim of this review is to focus on some open questions for whichsignificant progress can be expected on the scale of the nextdecade. We shall of course not try to review the entire field ofturbulence: it has become very large and has to some extentdeveloped a babel-tower excessive diversity for lack of aunifyinglanguage.We encourage the reader to look up theotherScholarpediaarticleson fluid mechanics and particularly those focusing on specificaspects which we could not cover in detail,e.g.directnumerical simulations[1][2].

After the great breakthrough due toKolmogorov,dimensional and scaling arguments seemed to provide such alanguage, at least in the limit of very large Reynolds numbers(fully developed turbulence). Although it turned out that the truestory was much more complicated, we feel that it is still useful toexamine many topics arising in turbulence from the point of view ofscaling and of its shortcomings. The topics addressed here arelisted in the Contents.

Tools forturbulence

Theory

Since the basic equations are known [link to section on NS], thequestion is: how much of a theoretical handle do we have on theNavier–Stokes (NS) equations? The short answer is: very little. Wecannot, for example, show that the solutions of the NS equationswith nice and smooth initial conditions stay nice, smooth andunique for all times, at least not in 3D (but in 2D, yes, we can).There will be more on this in Section4 on blowup. It has even been speculated by Jean Leray in the thirties thatthe random character of turbulence originates from non-uniquenessof the solutions to the NS equations. Nowadays, we know enoughabout howchaoscanappear in deterministicdynamicalsystemsthat there is no need to resort tonon-uniqueness to explain turbulence.

When we try tackling random solutions of the NS equations we haveto face a closure problem: because the equations are quadraticallynonlinear, the time-rate of change of the correlation functions ofthe velocity atdifferentpoints involves similar correlation functions, butwitharguments.An infinite hierarchy of equations is then obtained. The simplestform of closure, introduced byKolmogorov'sstudentMilionshchikov,is to arbitrarily close this hierarchy by relating fourth-ordercorrelation functions to second-order ones as if the velocity hadGaussian statistics. This is of course unjustified and leads toproblems such as negative values for energy-like quantities, whichare by definition non-negative. Cures for such diseases can befound but they are frequently ad hoc with no possibility to controlthe errors made with respect to the correct solutions. The maindifficulty is the absence of a small parameter which would permitto start a suitable perturbative approach. The smallness of theviscosity - or equivalently a large value for the Reynolds number -is of no use so far because very little is understood abouttheEuler equations, that is theNavier–Stokes equations with the viscosity set to zero(seeEuler250and references therein).

The most fruitful theoretical approaches have been based on scalingarguments, that is essentially on dimensional analysis. This ispresented in Section2. Another rather fruitfulkind of approach is through the use of toy models: after havingidentified certain properties of the basic equations which arebelieved to play a key role in the behavior of turbulence (forexample invariance and conservations properties) one tries to findsimpler models sharing those properties and which either can besolved analytically or at least for which numerical solutions aremuch simpler than for the full 3D NS equations. Precise definitionsof such models would take up too much space, but we can give anidea of what has been achieved with the use of some of these toymodels.

The 1DBurgersequation(Frischand Bec,Becand Khanin) gives a concrete example of how energy dissipationcan have a finite non-vanishing limit when the viscosity tends tozero in spite of the fact that the inviscid equation formallyconserves energy. As pointed out bySaffman,it also shows what can go wrong with naive application ofdimensional arguments.

The random coupling model ofKraichnanstartswith N independent replicas of the random NS equations and thencouples them artificially by random Gaussian coefficients chosen insuch a way as to preserve most invariance and conservation laws. Inthe limitclosedequations, called the Direct Interaction (DIA) equations, areobtained for suitable statistical quantities (including thetwo-time and two-point velocity correlation functions). Thesolutions are not compatible with the Kolmogorov 1941 theory to bediscussed in Section2; this happens not becausescale invariance is broken but because the model does not preservea certain form of Galilean invariance. Fortunately, thisshortcoming can be repaired by resorting to a kindofLagrangiandescription(Kraichnan)or by making the coupling coefficients scale dependent to obtainthe Eddy Damped Quasi-Normal Markovian (EDQNM) model (Orszag).

Shell models start from the NS equations written in spatial Fourierspace and replace all the Fourier modes in the shell havingwavenumbers betweenandbyjust a few degrees of freedom, typically one complex number. Theinteractions between these "shell amplitudes" are of course chosenagain to preserve as many features as possible of the originalequations. Some shell models, such as GOY or SABRA (Gledzer,Ohkitaniand Yamada,Biferale),are known to display the same anomalous scaling as the fullequations, see Section2. Unfortunately littletheoretical progress has been achieved and the shell models mainadvantage remains their ability to run very high Reynolds numbersusing just a workstation.

Much more drastic simplifications of thetruedynamicsareinvolved in the multiplicative random model (Novikovand Stewart,Yaglom,Benziet. al.) in which the amplitudes in theshellarejust obtained by multiplying the amplitude in theshellbya random variable with a suitable distribution (independentidentically distributed random variables are assumed fordifferent).Correlation functions can then be calculated explicitly; theydisplay anomalous scaling andmultifractalitythrougha mechanism involving large deviations (Varadhan).

Finally purely qualitative models such as the Swift–Richardsonflea-eddy model (Section2) can be made which arestimulating for the imagination.

Experiments

As stated before, turbulent flows abound everywhere around us.However high-Reynolds-number flows displaying good scalingproperties of the kind discussed in Section2require very large scales, as is the case in the naturalatmospheric and astrophysical environments. The former keepschanging (with the weather) and the latter is not so easilyaccessed. Large-scale facilities, such as major wind tunnels, arevery expensive and it is difficult to have a fundamental experimentrunning there for a duration of several days to several weeks(needed, e.g. to accumulate good statistical data on rare violentevents) when they are competing with important industrialapplications such as the testing of new designs for cars andairplanes.

During the last ten years a new technique has been developedinvolving low temperature Helium (above the lambda point where itbecomes superfluid) (Chavanneet al.,Niimelaand Sreenivasan). It takes advantage of the fact that Heliumcan have a kinematic viscosity two orders of magnitude lower thanair to achieve fairly high Reynolds numbers with facilities whichstill fit on a table. Special measurement techniques had to bedeveloped, since the usual hot-wire techniques cannot cope withsuch experiments. It is now planned to use much larger Heliumfacilities using know-how developed by major high-energy centerssuch as CERN.

Finally, very promising results have been recently obtained inunderstanding the dynamics of Lagrangian (tracer) particles inturbulent flows. Rapid technological advances in optical particletracking allow scientists to measure accurately the positions,velocities and accelerations of such tracer particles (for a reviewseeToschiand Bodenschatz). Detailed comparison between experiments andnumerical simulations on Lagrangian properties of turbulent flowshas opened interesting new directions of investigation with manyapplications where transport and/or aggregation of particles isimportant.

Numericalsimulations

In 1949vonNeumannpredicted that the advent of digitalcomputers would revolutionize the study of turbulence since itwould become possible to simulate the NS equations in 3D inturbulent regimes. Actually the first genuine 3D simulations ofturbulence had to wait about 20 years with the advent ofsupercomputers and the development of (pseudo)-spectral numericaltechnique taking advantage of fast Fouriertransformalgorithms(Orszagand Patterson). For homogeneous turbulence without boundaries(or more precisely, using periodic boundary conditions) theachievable spatial resolutions, which at first was onlyabouthasnow reached(Kanedaet al.). It can be safely predicted that within a decade orless the Reynolds numbers achievable by such simulations will becomparable to those of the best (present-day) experimentalfacilities. Numerical simulations have of course the advantage thatone has access to the whole spatial structure of the velocity, ofthe vorticity, of the local energy dissipation, etc. However asingle simulation at very high Reynold numbers may take from weeksto months of CPU time which can be prohibitive when exploringparameter space. Simulations are of course rather demanding forthose problems in which boundaries are essential. Furthermore, whencomplex boundaries or physics are involved setting up suitablesimulations can become difficult. One interesting way to handlesuch problems is through Lattice Boltzmann simulations whichcombine the hydrodynamical aspect with the microscopic physics ofthe problem while being not more demanding than, say, finitedifference techniques (Benziet al.,Succi).

For engineering applications, one usually needs to suitably modelturbulence while using as much as possible of our basic scientificunderstanding. This can be done in a variety of ways,includingtheory(Mohammadiand Pironneau), Reynolds Averaged Navier–Stokes modelling(Pope)andrenormalizationgroupmethods(Orszaget al.). Such topics are beyond the presentreview which focusses on the physics of turbulence.

K41 andintermittency

Scaling ideas have a long history in fluid mechanics, startingwithNewton'sderivationof the quadratic dependence of the drag on the relative velocity ofmotion between a body and the ambient fluid. Scaling arguments anddimensional analysis play a key role in the first serious attemptto understand the statistical properties of turbulent flow.

In 1941 Kolmogorov defined a conceptual framework for turbulence,now referred to as K41 theory (Kolmogorov1941a,Kolmogorov1941b,Frisch)which applies tohomogeneous,isotropicturbulence, that is turbulencestatistically invariant under translations and rotations of thesort frequently obtained at very high Reynolds numbers when thereis no large-scale shear.

A few years earlier,Richardsonhadproposed a qualitativevisionofthe energy cascade for the way energy flows from larger to smallereddies. He proposed that it would be similar to the way blood flowsfrom larger to smaller fleas in a famous poem of Swift.

In K41, this is made quantitative by two postulates regarding thelarge Reynolds number limit. On the one hand Kolmogorov assumesthat the energy dissipation ratehasa finite non-vanishing limit as the viscosity tends to zero whilekeeping the scale and velocity characteristic of the production ofthe turbulence fixed (for a recent experimental investigation onthis point seeSreenivasan1984). On the other hand he assumes that a statistical scaleinvariance of the cascade is achieved in the limit of very largeReynolds numbers. The former assumption, which is now generallycalled the existence of a dissipative anomaly (in a laminar fluid,the dissipation goes to zero with the viscosity), is well supportedby experimental and numerical results. The latter assumption holdsonly in an approximate way (see below).

In its mathematical formulation, the invariance postulated byKolmogorov resembles that of the Brownian motion process, in thedevelopment of which Kolmogorov was strongly involved.Ifisthe position at timeofa Brownian particle, then for any,and,the statistical distribution ofisthe same as that of.In plain language, the increments of the position of the Brownianparticle scale as the square root of the time increments. In K41,temporal position increments become spatial velocity increments andthe square root becomes a cubic root. The latter is dictated by adimensional argument: letdenotethe energy dissipation per unit mass of the fluidandtheseparation between two points. If we try relating velocityincrements to these quantities by a formula of the form

(4)

whereisa dimensionless constant, we immediately find that

Kolmogorov was using the Richardson cascade idea to equate theenergy dissipation rate to the rate of energy transfer from scaleto scale and assuming that viscosity should become irrelevant. K41scaling immediately impliesscalinglawsforstructurefunctions, that is moments of velocity increments. The simplestinstances are the longitudinal structure functions for homogeneousisotropic turbulence, defined as

(5)

whereisa positive integer,denotesa spatial vector increment anditsmodulus. According to K41 one should have

(6)

for any positive integer.The casedeservesspecial mention because Kolmogorov showed thatthefour-fifthsrelation

(7)

holds without any need to assume self-similarity (Kolmogorov1941c).

All these scaling relations are meant to apply withintheinertial range, that is the range ofscales much smaller than the scales at which turbulence is producedand much larger than the Kolmogorov dissipationscaleatwhich direct energy dissipation into heat becomes important. K41gives.

Eq. (6)and one of its consequences, namely that the energy spectrum ofturbulence should follow alaw(whereisthe wavenumber) are reasonably well supported by experimental andnumerical data. However, a careful examination of the scaling laws,using for example theExtendedSelf Similaritymethod, reveals small butmeasurable discrepancies from K41. Indeed, structure functions atinertial-range separations do displaypower-lawbehavior,but thegraphofthescalingexponentsisnot exactly the straight linepredictedby K41: it displays curvature as shown in Fig. (1).Hence the self-similarity assumed in K41 may actually be broken.Presently the scaling exponentsareknown with an accuracy of a few percent and could wellbeuniversal, that is independent of themechanism by which the turbulence is driven. Obtaining betterevidence for or against universality is important.

This is in fact not surprising if we go back to theSwift–Richardson picture of turbulence. Real fleas may not want tocluster in the Swiftian fashion (he actually had in mind poets, notfleas) but there are many natural instances of hierarchicalclustering, for example in plants as shown in Fig. (2)

Thanks to the work ofMandelbrotweknow that such objects arefractals.Mandelbrot was also the first to conjecture that at infiniteReynold numbers the energy dissipation of turbulence concentratesin a fractal set ofHausdorffdimensionless than three(Mandelbrot1968).

The fact that small-scale activity in high-Reynold numberturbulence becomes increasingly clumpy and that self-similarity isbroken is generally referred to asintermittency. It can bequantified by measuring for example the flatness of velocityincrements

(8)

a quantity which should be independentofinK41 and which actually grows asdecreases.

Once it was realized that K41 is probably not exactly correct –although it had already found many practical applications – aflurry of activity started to understand intermittency. At firstphenomenological models were developed, such as the multiplicativerandom model, which displays not only fractal dissipation but thephenomenon of multifractality, a concept with applications farbeyond turbulence (Benziet. al.,Jaffard).In the mid-ninetiesKraichnanconjecturedthat intermittency and anomalous scaling are already present in asimplepassivescalarmodel in which a a scalar field (say, atemperature field) is being advected by a prescribed Gaussianrandom field with K41-type scaling and a very short correlationtime.

The conjecture was proven and – for the first time – a result aboutintermittency was derivedab initio, thatis from the basic equations without recourse to ad hoc steps. Thisbreakthrough made use of techniques borrowed from quantum fieldtheory. Very roughly, to calculate the correlationfunctionoforderofthe passive scalar, one has to solve a multi-dimensionallinearpartialdifferential equationof the form:

(9)

whereisa prescribed function andaprescribed linear operator, both scale-invariant with known scalingproperties. This seems to determine the scaling properties of thesolution, the exception being those functions which sit in the nullspace of.These are thezeromodeswhich allow the breaking of scaleinvariance (Gawedzkiand Kupianen,Chertkov,Shraimanand Siggia,Falkovichet al.). Such zero modes, which can beinterpreted as quantities statistically conserved under transportby the flow, are hard to calculate but can be shown not to dependon initial and boundary condition and on the way the scalar is fedinto the flow, thusensuringuniversalityofthe (anomalous) scaling properties. It is not known if suchconservation laws are associated to any symmetries, such as theinvariance properties of the passive scalar transport in theabsence of molecular diffusion. It is generally believed that asomewhat similar mechanism will guarantee the universality ofscaling for the full nonlinear problem of turbulence but suitabletechniques to derive such resultsabinitiohave yet to be found.

Bulk quantities, dragand its reduction

The K41 theory makes remarkable predictions on the statisticalproperties of homogeneous and isotropic turbulence. Although theeffects of intermittency and breaking of scale invariance tell usthat K41 theory is not exactly true, Kolmogorov was able to build aconceptual framework which enables us to understand in aquantitative way what we mean by turbulence. One obvious questionis whether one can use the same framework to make predictions onbulk (global) quantities for turbulent flows.

As a simple but not trivial example, consider the question of howmuch fluid can be carried by a pipe ofradiusfora given pressure gradient.In a laminar flow, the velocity profile is parabolic with a maximumat the centre of the channel equal to,whereandisthe pipe radius andthedensity. Consequently, the Reynolds number of the flow is givenby.For large enoughtheflow becomes turbulent and the mass throughput becomes smaller withrespect to the laminar case: a substantial power input is spent formaintaining turbulent fluctuations and the maximum average velocityat the centre of the channelbecomessubstantially smaller than.For a pipe of length,the total work done by the pressure gradient is proportionaltowhilethe total average kinetic energy proportionalto.The ratio between these two quantities is named dragcoefficient,namely:

The interpretation of the dragcoefficientisrather intuitive: it is a measure of how much kinetic energy isacquiredby the meanflowwith respect to the work done by theexternal forces. For laminar flow in apipe,and.At the onset of turbulenceincreasesand eventually decreases very slowly forincreasing.There exists no systematic theory capable of predicting theReynolds-number dependence of.

A beautiful although phenomenological approach has been developedbyvonKármánin the thirties. Using ideas from theslightly later K41 theory, von Kármán's approach can be recast asfollows. For homogenous pipe flow, the velocityprofiledependsonly on the distancefromthe boundary and the description of the different statisticalquantities becomes easier by introducingPrandtl'sdimensionlessvariables

(10)

In all turbulent flows, one can always introduce the averagevelocity field and the turbulent fluctuations. In homogenous wallbounded turbulent flows, the average velocity field has a non zerocomponent only in the streamwisedirection.Denoting bytheaverage along thedirection,the momentum equation reads:

(11)

whereisthe momentum flux towards the wall. Turbulent fluctuations takeenergy from the mean flow at a rate.On the average the energy source of turbulent fluctuations mustbalancewhichis the rate of energy dissipation due to turbulence. Both energyproduction and energy dissipation are mostly concentrated near theboundaries. The conceptual advance made by von Kármán is to stressthat in the range,one can expect that the effect of viscosity should be negligible,i.e. the mean flow should not dependon.Then using (11)one can obtain.Next, using the Kolmogorov theory, one can assume that energydissipation is independent of the viscosity. By dimensionalanalysis, the most general expressionforcanbe written as.Finally, by balancing energy source and energy, we obtain

(12)

whereisa universal function. As it is written, the r.h.s of the aboveequation depends explicitly on.For,according to the Kolmogorov theory,shouldbecome independent of the viscosity which is parameterized by thescale.Therefore we reach the conclusion thateitheror,and the most general form ofmustbe

(13)

whereandareuniversal constant. Equation (13)is the prediction originally made by von Kármán and it allows us tocompute the value ofasa function ofandthe two constantsand.Note that for finite Reynolds number, we may expect deviation (ofthe order of)from (13)which cannot be computed by using dimensional analysis. Existingdata from laboratory experiments and numerical simulations show aremarkable good agreement with (13)(Procacciaand Sreenivasan).

Let us remark that we did use the K41 theory although the turbulentflow is certainly neither isotropic nor homogeneous (inthedirection).There is, thus, a hidden assumption that the basic relation of theK41 theory can be extended beyond the framework where the theoryhas been proposed. This idea can be worked out by assuming thatturbulent kinetic energy and energy dissipation are always relatedaccording to the K41 dimensional analysis, which is the basis ofthe so calledmodelof turbulent flow.

Within the obvious limitations of the argument so far discussed, wewant only to stress that the fundamental assumption in the K41theory, namely that energy dissipation is independent of theviscosity for large Reynolds number, is the basic guideline inunderstanding 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 notnecessarily show a different world with respect to the K41theory.

According to our discussion in the previous section, we shouldexpect intermittency acting in wall-bounded turbulent flows as inany other case of turbulence. Does the intermittency change the vonKármán theory? As far as bulk quantities are concerned, it seemsthat the answer is no. However, if one looks at high-order momentof pressure fluctuations at the wall (pressure is force per unitarea from which we compute the dragcoefficient),then one discovers strong intermittency and breakdown of thepredictions from dimensional analysis. A quantitative explanationof intermittent fluctuations in wall bounded turbulence flows isbeyond the von Kármán as well as the K41 theory and it is aresearch problem under investigation (Casciolaet al.,Toschiet al.).

Although successful, the von Kármán theory of boundary layers isnot "closed" as far as one does not predict the universalconstantsand.Clearly, any scaling argument, such as those obtained bydimensional analysis, cannot be helpful in predicting the twoconstants. Numerical simulations give estimatesofandwhichare in very good agreement with experimental data. An open questionis whether one can go on theoretically and make progress to predictpure constants beyond dimensional analysis.

There are many physical problems where dimensional analysis doesnot provide an answer. One example is the problem of drag reductionin a turbulent pipe flow when a small amount of flexible polymersis added (Lumley,Virk).This phenomenon, known also as Tom's effect, was discovered in theearly forties and it cannot be explained by using arguments similarto the von Kármán or the Kolmogorov theory (for recent reviews see(Christopheret al.,Procacciaet al.). Since polymers can be stretched byturbulent fluctuations, a certain amount of turbulent kineticenergy is transferred to the polymers. This mechanism decreasesturbulent fluctuations especially near the wall. Thus the momentumflux towards the wall decreases as well, which may eventually leadto decrease in the drag coefficient,i.e. more mass throughput for the same power input. It is achallenging question to obtain a quantitative theory on the amountof drag reduction as a function of the polymerconcentrationandthe physical properties of the polymer (notably its maximumextension lengthandcharacteristic timeforthe stretched polymer to relax its extensionfrom).deGennes and Taborsuggested that the presenceof polymers in high-Reynolds-number turbulence modifies theKolmogorov dissipation scale, leading to a different energybalance. However, this amounts to a change in the effectiveReynolds number that can hardly affect the balance in theenergy-containing range and thus the drag. Actually, it has beenargued (Procacciaet al.) that drag reduction by addition of polymers isequivalent to introducing a space-dependent viscosity whichincreases linearly from the wall boundary: the amount of dragreduction depends on the slope of the effective viscosity which isa rather complex function of bothand.

Polymer addition is not the only way to reduce drag. Experimentsand numerical simulations show that addition of surfactants or airbubbles in water can lead to drag reductions. It is unclear whetherone will eventually discover a universal mechanism for dragreduction. The whole subject, of considerable importance for manyengineering applications, is under active investigation.

Blow up

A fundamental milestone in three-dimensional turbulence is the ideathat energy dissipation becomes Reynolds-number independent forlarge Reynolds numbers. It is present both in K41 and in subsequenttheories of intermittency and in good agreement with experimentsand numerical simulations (see Section5.2ofFrischandreferences therein). Since dissipation is proportional to theviscosity, it is somewhat paradoxical that it can tend to anon-zero value when viscosity vanishes, hence thenameviscous anomaly.Forthe passive scalar problem discussed in Section2.a result which is the counterpart of the viscous anomaly can bederived rigorously. The mathematical problem remains open for thethree-dimensional Navier–Stokes equations and is connected to somedeep issues of regularity for both the Euler and the Navier–Stokesequations.

Onsager was the first to observe that solutions to the 3Dincompressible Euler equations need not conserve energy if thesolutions lack regularity (more precisely if they are not Höldercontinuous with an exponent largerthan);Duchon and Robert gave an expression of the dissipative anomaly(the local amount of dissipation) for such solutions of the Eulerequations. For such questions see the review paperbyEyinkand Sreenivasan.

The key issue, of course, is: how regular are the solutions to theNavier–Stokes and Euler equations when the initial conditions aresufficiently smooth? Such issues are reviewedinMajdaand BertozziandTemamand,in more elementary way, inRoseand Sulem. In two dimensions in a boundeddomain it has been known since the thirties that Euler flowpreserves (sufficient) initial smoothness forever. In threedimensions only finite-time regularity is guaranteed. This can beunderstood somewhat naively by the following argument (part ofwhich actually survives in much more complex functional analysisestimates).

As was shown by Cauchy, if one takes the curl of the 3D Eulerequation one obtains the following equation for thevorticity:

Now, let us be sloppy: we identify the velocitygradientandthe vorticity (which is actually its antisymmetric part) andfurthermore blur the distinction between scalars, vectors andtensors to rewrite the preceding equation as

which we consider as anordinarydifferential equation(writen in Lagrangiancoordinates). It is obvious that ifinitially,thenwillblow up (become infinite) at.

An identical equation giving blow up can actually be derived fromthe inviscid (zero viscosity) Burgers equation

Setting,we obtain upon space differentiation

There is however a huge difference between (infinitely)compressible Burgers dynamics and incompressible hydrodynamics. Inthe former case singularities (mostly shocks) appear because fasterelements of fluid catch up with slower ones, a scenario whichhardly carries over to incompressible flow.

However our understanding of the mathematics of the Euler equationsis so limited – even a quarter of a millennium after theirintroduction– that we cannot do much better than obtaining forvarious function-space norms an upper bound which behaves as forthe Burgers equation. The numerical evidence about 3D Euler flow isnot clearly for or against finite-time blow up. There are actuallymechanisms which could make 3D incompressible flow much tamer thancompressible flow. One of them, depletion is discussed in Section4.

One might think that the presence of viscous dissipation makes theproblem much easier and that blow up would then be ruled out. Thisis indeed conjectured by most specialists but has never beenproven. Actually theClayFoundationhas decided to devote one of itsMillennium million-dollar prizes to precisely this issue. Finallythere is a tricky issue at the interface of the Euler andNavier-Stokes blow up problems. In the absence of boundaries onecan show that hypothetical all-time regularity for the Eulerequations implies the same for Navier–Stokes but not so withboundaries (Kato)

Increaseof predictability and depletion of non linearity

Turbulence in two space dimensions shows some interesting newfeatures with respect to the three dimensional case so fardiscussed. Vorticity conservation prevents energy dissipation fromstaying constant when the viscosity is decreased. Thus the basicassumption of K41 theory becomes inapplicable. However enstrophy,i.e. the integral of the vorticity squared, can be dissipated andthe energy cascade of the K41 picture can by replaced by anenstrophy cascade along with an inverse cascade of energy towardslarge scales (Kraichnanand Montgomery). As soon as high resolutionnumerical simulations of two dimensional flows became available tothe scientific community, one striking feature emerged from flowvisualization and, in particular, from visualization of thevorticity field (McWilliams,Benziet al.,Legraset al.): the emergence of long lived coherentstructure mostly in the form of circular vortex. A careful analysisof the flow field shows that these structuresarestablenonlinear solutions of the Euler equations. As such, coherentstructures represent, locally, a depletion of non-linearity in aturbulent flow: within a coherent structure the enstrophy cascadeis inhibited and the structure can survive for extremely longtimes. Whenever the vorticity field is dominated by few coherentstructures, the predictability time of large scale dynamicsincreases, being no longer constrained by the small scalefluctuations. The above picture is quite generic and is notnecessarily limited to two dimensional turbulence, although strongevidence of depletion of non linearity has been observed mostly inthe 2D case.

The increase of predictability time by depletion of nonlinearity iscertainly relevant for geophysical applications. In particular, foratmospheric and oceanic dynamics, one can show the relevance ofsome form of 2D dynamics. For example in the case of atmosphericflows, one can prove that, in absence of dissipation and ofmechanical and thermodynamical forcing, there exists a new form ofvorticity, namedErtelpotentialvorticity, which is conserved. The Ertel potential vorticity isdefined as,whereisthe absolute vorticity (the sum of the flow vorticity and theCoriolis force on a rotating sphere)andisthe potential temperature (surface atareisoentropic). A particular case of Ertel potential vorticity can beobtained at midlatitudes where the Coriolis force is almostbalanced by the pressure gradients (geostrophic balance).Quasigeostrophic flows can be described using the idea of potentialenstrophy (square of potential vorticity) cascade from large tosmall scales (where the flow is assumed to be dissipated as threedimensional turbulence). One can then expect that quasigeostrophicflows can form stable coherent structure similar to those observedin ordinary 2D turbulence, and we can again expect that coherentstructures must will be more predictable than predicted bydimensional arguments.

We want to emphasize that the above scenario, still ratherspeculative at this stage, implies that there is an enhancement ofthe predictability time, i.e. an enhancement of our ability toforecast weather and/or ocean circulation, because of coherentstructures. This enhancement in the predictability time could be anintermittent 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 energybudget in the flow and in particular momentum and heat fluxes maybe changed in a non trivial way, with relevant consequences on thelong-time behavior of the general circulation, i.e. theclimate.

Overtureof turbulence flows in different scientific fields

There are many important and fascinating problems in physics whereturbulence plays an important role. In many cases, one can developarguments 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 bya simplisitc argument: when the density is a slowly varyingfunction of temperature, one can observe rather strong turbulencewhenever a temperature gradient is acting in the direction oppositeto that of the gravitational field (natural convection). In thiscase, one is usually interested in predicting the heat flux interms of the temperature difference,the buoyancy force,the temperature gradient,the viscosityandthe thermal diffusivity.Here,designatesthe volume expansivity andthethickness of the layer. One can easily show that the dimensionalturbulent heat fluxmustbe a function of the Rayleigh numberandthe Prandtl number.From the equations of motion, one is able to compute the energydissipation of the turbulent flow which must be proportionalto.According to the K41 approach, energy dissipation should also be ofthe order ofwhere,as usual,ischaracteristic flow velocity in turbulent flow. Since the source ofenergy in natural convection is the potentialenergy,,we can estimate,i.e..It follows that the straightforward prediction based on K41assumption is that(Kraichnan).It so happens that the above prediction, with a suitable estimateof constants, is also an upper bound of the turbulent heat flux forany Prandtl number. However,for infinite Prnumber, a rigorous upper bound gives atmost.When,the boundary layers are laminar foranyand,consequently, the turbulent heat flux can change dramatically.Thus, a rather complex behavior of the relation Nu versus Ra mustbe expected, depending on the many parameters entering in thesystems: one definitively must go beyond a simple dimensionalanalysis (for recent reviews,Ahlerset al.,Procacciaand Sreenivasanand references therein).

Thermal convection, with its complex and fascinating properties, isjust one example of the many important systems which are still inneed of a full systematic scientific understanding from thetheoretical, experimental and computational point of view. Here isa short and incomplete list of some of several interesting fieldswhere understanding the nature of turbulence can potentially leadto a significant scientific breakthrough:

The above list is neither exhaustive nor indicates any scientificpriority. It just tells us that the understanding of turbulentflows remains a fundamental issue in modern physics.

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See also

Chaos

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