Turbulence Review

124 Los Alamos Science Number 29 2005

The Turbulence Problem An Experimentalist’s Perspective

Turbulent fluid flow is a complex, nonlinear multiscale phenomenon, which poses some of the most difficult and fundamental problems in classical physics. It is also of tremendouspractical importance in making predictions—for example, about heat transfer in nuclearreactors, drag in oil pipelines, the weather, and the circulation of the atmosphere and theoceans. But what is turbulence? Why is it so difficult to understand, to model, or even toapproximate with confidence? And what kinds of solutions can we expect to obtain? Thisbrief survey starts with a short history and then introduces both the modern search for uni-versal statistical properties and the new engineering models for computing turbulent flows.It highlights the application of modern supercomputers in simulating the multiscale velocityfield of turbulence and the use of computerized data acquisition systems to follow the trajectories of individual fluid parcels in a turbulent flow. Finally, it suggests that these tools, combined with a resurgence in theoretical research, may lead to a “solution” of theturbulence problem.

Robert Ecke

Leonardo da Vinci’s illustration of the swirlingflow of turbulence.(The Royal Collection 2004,Her Majesty Queen Elizabeth II)

Many generations of scientistshave struggled valiantly to understandboth the physical essence and themathematical structure of turbulentfluid motion (McComb 1990, Frisch1995, Lesieur 1997). Leonardo daVinci (refer to Richter 1970), who in1507 named the phenomenon heobserved in swirling flow “la tur-bolenza” (see the drawing on theopening page), described the follow-ing picture: “Observe the motion ofthe surface of the water, which resem-bles that of hair, which has twomotions, of which one is caused bythe weight of the hair, the other by thedirection of the curls; thus the waterhas eddying motions, one part ofwhich is due to the principal current,the other to the random and reversemotion.”

Two aspects of da Vinci’s observa-tions remain with us today. First, hisseparation of the flow into a mean anda fluctuating part anticipates byalmost 400 years the approach takenby Osborne Reynolds (1894). The“Reynolds decomposition” of thefluid velocity into mean and fluctuat-ing parts underpins many engineeringmodels of turbulence in use today.1

Second, da Vinci’s identification of“eddies” as intrinsic elements in tur-bulent motion has a modern counter-part: Scientists today are activelyinvestigating the role of such struc-tures as the possible “sinews” of tur-bulent dynamics.

Long after da Vinci’s insightfulobservations, a major step in thedescription of fluid flows was thedevelopment of the basic dynamical

equation of fluid motion. The Eulerequation of motion (written down inthe 18th century) describes the con-servation of momentum for a fluidwithout viscosity, whereas the Navier-Stokes equation (19th century)describes the rate of change ofmomentum at each point in a viscousfluid. The Navier-Stokes equation fora fluid with constant density ρ andconstant kinematic viscosity ν is

(1) ,

with ∇ ⋅ u = 0, which is a statementof fluid incompressibility and withsuitable conditions imposed at theboundaries of the flow. The variableu(x,t) is the (incompressible) fluidvelocity field, and P(x,t) is the pres-sure field determined by the preserva-tion of incompressibility. Thisequation (when multiplied by ρ to getforce per unit volume) is simplyNewton’s law for a fluid: Force equalsmass times acceleration. The left sideof Equation (1) is the acceleration ofthe fluid,2 and the right side is thesum of the forces per unit mass on aunit volume of the fluid:3 the pressureforce and the viscous force arising

from momentum diffusion throughmolecular collisions. Remarkably, asimple equation representing a simplephysical concept describes enor-mously complex phenomena.

The Navier-Stokes equations aredeterministic in the sense that, oncethe initial flow and the boundary con-ditions are specified, the evolution ofthe state is completely determined, atleast in principle. The nonlinear termin Equation (1), u · ∇u, describes theadvective transport of fluid momen-tum. Solutions of the nonlinearNavier-Stokes equations may dependsensitively on the initial conditions sothat, after a short time, two realiza-tions of the flow with infinitesimallydifferent initial conditions may becompletely uncorrelated with eachother. Changes in the external forcingor variations in the boundary condi-tions can produce flows that varyfrom smooth laminar flow to morecomplicated motions with an identifi-able length or time scale, and fromthere to the most complicated flow ofthem all, namely, fully developed tur-bulence with its spectrum of motionsover many length scales. Dependingon the specific system (for example,flow in a pipe or behind a grid), thetransition from smooth laminar flowto fully developed turbulence mayoccur abruptly, or by successivelymore complex states, as the forcing isincreased.

The difficulties of finding solutionsto the Navier-Stokes equations thataccurately predict and/or describe thetransition to turbulence and the turbu-lent state itself are legendary, prompt-ing the British physicist Sir HoraceLamb to remark, “I am an old man

Number 29 2005 Los Alamos Science 125

The Turbulence Problem

1 Reynolds rewrote the Navier-Stokesfluid equation as two equations—one forthe mean velocity, which includes a quad-ratic term in the fluctuating velocity calledthe Reynolds stress, and one for the fluc-tuations, which is usually modeled bysome suitable approximation. Thisapproach underpins commonly used engi-neering models of turbulent fluid motionknown as Reynolds-Averaged Navier-Stokes (RANS)—refer to Taylor (1938).

2 The acceleration term looks compli-cated because of the advection term u ⋅ ∇u, which arises from the coordinatetransformation from a frame moving withthe fluid parcels (the “Lagrangian” frame,in which Newton's law has the usualform) to a frame of reference fixed inspace (the “Eulerian” frame, in whichother aspects of the mathematics are sim-pler). Specifically, acceleration of thefluid is, by definition, the second timederivative of the Lagrangian fluid trajec-tory x(t), which describes the motion ofthe fluid element that was initially atposition x(0). The first time derivative isthe Lagrangian fluid velocity, dx(t)/dt,which is related to the Eulerian fluidvelocity by dx(t)/dt = u(t,x(t)). Because uis a function of time t and position x(t),which itself is a function of time, theEulerian expression for the Lagrangiansecond time derivative (the fluid accelera-tion) is obtained through the chain ruleand equals du/dt = ∂u/∂t + u ⋅ ∇u.

3 Often, there is an additional term addedto the right side of the equation that repre-sents an external forcing of the flow perunit volume such as by gravity.Alternatively, the forcing can arise fromthe imposition of boundary conditions,whereby energy is injected by stresses atthose boundaries.

now, and when I die and go to heaventhere are two matters on which I hopefor enlightenment. One is quantumelectrodynamics, and the other is theturbulent motion of fluids. And aboutthe former I am rather optimistic”—1932 (in Tannehill et al. 1984). One of

the most influential turbulence theo-rists in the last 40 years, RobertKraichnan, started studying turbulencewhile working with Albert Einstein atPrinceton, when he noticed the simi-larity between problems in gravita-tional field theory and classical fluid

dynamics. His contributions includefield-theoretic approaches to turbu-lence that have had recent stunningsuccess when applied to the turbulenttransport of passive scalar concentra-tion (see the article “Field Theory and Statistical Hydrodynamics” onpage 181).

What Is Turbulence?

So, what is turbulence and why isit so difficult to describe theoretically?In this article, we shall ignore thetransition to turbulence and focusinstead on fully developed turbulence.One of the most challenging aspects isthat, in fully developed turbulence,the velocity fluctuates over a largerange of coupled spatial and temporalscales. Examples of turbulence(Figure 1) are everywhere: the flow ofwater from a common faucet, waterfrom a garden hose, the flow past acurved wall, and noisy rapids result-ing from flow past rocks in an ener-getically flowing river. Anotherexample is the dramatic pyroclasticflow in a volcanic eruption. InFigure 2, the explosive eruption ofMount St. Helens is illustrated at suc-cessively higher magnification, show-ing structure at many length scales. Inall these examples, large velocity dif-ferences (as opposed to large veloci-ties) resulting from shear forcesapplied to the fluid (or from intrinsicfluid instability) produce strong fluidturbulence, a state that can be definedas a solution of the Navier-Stokesequations whose statistics exhibit spa-tial and temporal fluctuations.

Historically, investigations of tur-bulence have progressed throughalternating advances in experimentalmeasurements, theoretical descrip-tions, and most recently, the introduc-tion of numerical simulation ofturbulence on high-speed computers.Similarly, there has been a rich inter-play between fundamental understand-



Figure 1. Common Examples of Fluid Turbulence Turbulence is commonly apparent in everyday life, as revealed by the collage of pic-tures above: (a) water flow from a faucet, (b) water from a garden hose, (c) flow pasta curved wall, and (d) and (e) whitewater rapids whose turbulent fluctuations are sointense that air is entrained by the flow and produces small bubbles that diffuselyreflect light and cause the water to appear white.

(a) (b)

(c)

(d) (e)

ing and applications. For example, tur-bulence researchers in the early to mid20th century were motivated by twoimportant practical problems: predict-ing the weather and building evermore sophisticated aircraft. Aircraftdevelopment led to the construction oflarge wind tunnels, where measure-ments of the drag and lift on scaledmodel aircraft were used in the designof airplanes. On the other hand,weather prediction was severely ham-pered by the difficulty in doing numer-ical computation and was only madepractical after the development, manydecades later, of digital computers; inthe early days, the calculation of theweather change in a day requiredweeks of calculation by hand! In addi-tion to these two large problems, manyother aspects of turbulent flow wereinvestigated and attempts were madeto factor in the effects of turbulence onthe design and operation of realmachines and devices.

To understand what turbulence isand why it makes a big difference inpractical situations, we consider flowthrough a long cylindrical pipe ofdiameter D, a problem consideredover a century ago by OsborneReynolds (1894). Reynolds measuredmean quantities such as the averageflow rate and the mean pressure dropin the pipe. A practical concern is todetermine the flow rate, or meanvelocity U, as a function of theapplied pressure, and its profile, as afunction of distance from the wall.Because the fluid is incompressible,the volume of fluid entering any crosssection of the pipe is the same as thevolume flowing out of the pipe. Thus,the volume flux is constant along theflow direction. We can useEquation (1) to get a naive estimate ofthe mean velocity U for flow in a hor-izontal pipe. Consider as a concreteexample the flow of water in a rigidpipe hooked up to the backyard waterfaucet. Taking a 3-meter length of a2.5-centimeter diameter pipe and esti-

mating the water pressure at30 pounds per square inch (psi), the imposed pressure gradient ∇P is0.1 psi/cm or 7000 dynes/cm3. Weassume the simplest case, namely, thatthe flow is smooth, or “laminar,” sothat the nonlinear term in Equation (1)can be neglected, and that the flowhas reached its limiting velocity with∂U/∂t = 0. In that case, the density-normalized pressure gradient ∇P/ρ would be balanced by the vis-cous acceleration (or drag), ν∇2u.Using dimensional arguments andtaking into account that u = 0 at thepipe wall, we estimate thatν∇2u ≈ νU/D2, which yields the esti-

mate for the mean flow velocity ofU ~ ∇PD2/ρν. Thus, for water withviscosity ν = 0.01 cm2/s flowing in apipe with diameter D = 2.5 centime-ters, the laminar flow velocity wouldreach U ~ 40,000 m/s or almost30 times the speed of sound in water!Clearly, something is wrong with thisargument. It turns out that the flow insuch a pipe is turbulent (it has highlyirregular spatial and temporal velocityfluctuations) and the measured meanflow velocity U is smaller by a factorof about 4000, or only about 10 m/s!

How can we improve our estimate?For the turbulent case, we mightassume, as Reynolds did, that the non-linear term dominates over the vis-

cous term and then equate the nonlin-ear term (u · ∇u ~ U2/D) to the pres-sure gradient, thereby obtaining themuch more realistic estimate of U ~ (∇PD/ρ)1/2 ~ 1.5 m/s. This esti-mate actually overestimates theeffects of the nonlinear term.

As illustrated in Figure 3, the solu-tion for the laminar-flow velocity pro-file is quite gradual, whereas theturbulent velocity profile is muchsteeper at the walls and levels off inthe center of the pipe. Evidently, theeffect of turbulence is to greatlyincrease the momentum exchange inthe central regions of the pipe, aslarge-scale eddies effectively ‘lockup’ the flow and thereby shift thevelocity gradient (or velocity shear)closer to the wall. Because the flowresistance in the pipe is proportionalto the steepness of the velocity profilenear the wall, the practical conse-quence of the turbulence is a largeincrease in the flow resistance of thepipe—that is, less flow for a givenapplied pressure. The real-worldimplications of this increase in flowresistance are enormous: A large frac-tion of the world’s energy consump-tion is devoted to compensating forturbulent energy loss! Nevertheless,the detailed understanding and predic-tion from first principles still eludeturbulence theory.



Figure 2. Scale-Independence in Turbulent FlowsThe turbulent structure of the pyroclastic volcanic eruption of Mt. St. Helens shownin (a) is expanded by a factor of 2 in (b) and by another factor of 2 in (c). The char-acteristic scale of the plume is approximately 5 km. Note that the expanded imagesreveal the increasingly finer scale structure of the turbulent flow. The feature ofscale independence, namely, that spatial images or temporal signals look the same(statistically) under increasing magnification is called self-similarity.

(a) (b) (c)

The example of pipe flow illus-trates an important feature of turbu-lence—the ratio of the nonlinear termto the viscous dissipation term pro-vides a good measure of the strengthof turbulence. In fact, this ratio,Re = UD/ν, where D is the size of thelarge-scale forcing (typically shear), isknown as the Reynolds number afterReynolds’ seminal work on pipe flow(1894). For a small Reynolds number,Re << 1, the nonlinearity can be neg-lected, and analytic solutions to theNavier-Stokes equation correspondingto laminar flow can often be found.When Re >> 1, however, there are nostable stationary solutions,4 and thefluid flow is highly fluctuating inspace and time, corresponding to tur-bulent flow. In particular, the flow is

fully developed turbulence when Re islarge compared with the Re for transi-tion to turbulence for a particular setof forcing and boundary conditions.For example, in the problem above,where D = 2.5 cm and U = 10 m/s,the Reynolds number is Re ~ 3 × 105

compared with a typical Re ~ 2000 forthe onset of turbulence in pipe flow.

The Search for UniversalProperties and the

Kolmogorov Scaling Laws

In early laboratory experiments onturbulence, Reynolds and others sup-plemented their measurements ofapplied pressure and average velocityby observing the rapidly fluctuatingcharacter of the flow when they useddyes and other qualitative flow-visuali-zation tools. In the atmosphere, how-ever, one could measure muchlonger-term fluctuations, at a fixedlocation, and such Eulerian measure-ments intrigued the young theoreticalphysicist G. I. Taylor (1938).Turbulence is difficult to measure

because the turbulent state changesrapidly in space and time. Taylor pro-posed a probabilistic/statisticalapproach based on averaging overensembles of individual realizations,although he soon replaced ensembleaverages by time averages at a fixedpoint in space. Taylor also used theidealized concept (originally intro-duced by Lord Kelvin in 1887) of sta-tistically homogeneous, isotropicturbulence. Homogeneity and isotropyimply that spatial translations and rota-tions, respectively, do not change theaverage values of physical variables.5

Lewis F. Richardson was anotherinfluential fluid dynamicist of the early20th century. Richardson performedthe first numerical computation forpredicting the weather (on a hand cal-culator)! He also proposed (1926) apictorial description of turbulencecalled a cascade, in which nonlinearitytransforms large-scale velocity circula-tions (or eddies, or whorls) into circu-lations at successively smaller scales(sizes) until they reach such a smallscale that the circulation of the eddiesis efficiently dissipated into heat byviscosity. Richardson captured thisenergy cascade in a poetic take-off onJonathan Swift’s famous description offleas.6 In Richardson’s words, “Bigwhorls have little whorls that feed ontheir velocity, and little whorls havelesser whorls and so on to viscosity”(circa 1922). A schematic illustrationof the energy cascade picture is shownin Figure 4, where the mean energy



Figure 3. Mean Velocity Profiles for Laminar and Turbulent Pipe FlowThe velocity profile across the diameter (D = 2R where R is the radius) of a pipe forlaminar-flow conditions (black curve) shows a gradual velocity gradient comparedwith the very steep gradients near the walls resulting from turbulent flow conditions(red curve). Those steep gradients are proportional to the flow resistance in thepipe. Thus turbulence results in significantly less flow for a given applied pressure.

4 How large Re must be to get a turbulentstate depends on the particular source offorcing and on the boundary conditions.For example, the transition to turbulencein pipe flow can occur anywhere in therange 1000 < Re < 50,000 depending oninlet boundary conditions and the rough-ness of the pipe wall. For most com-monly encountered conditions, thetransition is near Re = 2000.

5 Many theoretical descriptions use theseassumptions, but typical turbulenceencountered in the real world often obeysneither condition at large scales. A keyquestion in real-world situations iswhether the assumptions of homogeneityand isotropy are satisfied at small scales,thus justifying application of a generalframework for those smaller scales.

6 “So, the naturalists observe, the flea,/hath smaller fleas that on him prey;/ Andthese have smaller still to bite ‘em;/ Andso proceed, ad infinitum.”—JonathanSwift, Poetry, a Rhapsody

injection rate ε at large scales is bal-anced by the mean energy dissipationrate at small scales. Richardson andTaylor also appreciated that genericproperties of turbulence may be dis-covered in the statistics of velocity dif-ferences between two locationsseparated by a distance r, denoted asδu(x, x+r) = u(x) – u(x+r). The statis-tics of velocity differences at two loca-tions are an improvement over thestatistics of velocity fluctuations at asingle location for a number of techni-cal reasons, which we do not discusshere. Longitudinal projections ofvelocity differences

are often measured in modern experi-ments and are one of the main quanti-ties of interest in the analysis of fluidturbulence.

Measuring velocity differences onfast time scales and with high preci-sion was a difficult proposition in theearly 20th century and required thedevelopment of the hot-wireanemometer, in which fluid flowingpast a thin heated wire carries heataway at a rate, proportional to thefluid velocity. Hot-wire anemometrymade possible the measurement, onlaboratory time scales, of the fluctuat-ing turbulent velocity field at a singlepoint in space (see Figure 5). For aflow whose mean velocity is large,velocity differences were inferredfrom those single-point measurementsby Taylor’s “frozen-turbulence”hypothesis.7



Figure 4. The Energy Cascade Picture of Turbulence This figure represents a one-dimensional simplification of the cascade process withββ representing the scale factor (usually taken to be 1/2 because of the quadraticnonlinearity in the Navier-Stokes equation). The eddies are purposely shown to be“space filling” in a lateral sense as they decrease in size.(This figure was modified with permission from Uriel Frisch. 1995. Turbulence: The Legacy of A. N. Kolmogorov.

Cambridge, UK: Cambridge University Press.)

Figure 5. Time Series of Velocities in a Turbulent Boundary LayerThis time series of velocities for an atmospheric turbulent boundary layer withReynolds number Re ~ 2 ×× 107 was measured at a single location with a hot-wireanemometer. The velocity fluctuations are apparently random. (This figure is courtesy of

Susan Kurien of Los Alamos, who has used data recorded in 1997 by Brindesh Dhruva of Yale University.)

7 If the mean velocity is large comparedwith velocity fluctuations, the turbulencecan be considered “frozen” in the sensethat velocity fluctuations are swept past asingle point faster than they would changebecause of turbulent dynamics. In thatcase, the spatial separation ∆r is related tothe time increment ∆t by ∆r = – U∆t,where U is the mean velocity. See also thearticle “Taylor’s Hypothesis, Hamilton’sPrinciple, and the LANS-α Model forComputing Turbulence” on page 152 .

Single-point measurements of tur-bulent velocity fluctuations have beenperformed for many systems and havecontributed both to a fundamentalunderstanding of turbulence and toengineering modeling of the effects ofturbulence at small scales on the flowat larger scales. (See the section onengineering models.) Single-pointmeasurements of velocity fluctuationshave been the primary tool for investi-gating fluid turbulence.They remain incommon use because of their largedynamic range and high signal-to-noiseratio relative to more modern develop-ments such as particle image velocime-try, in which the goal is to measure

whole velocity fields. For now, we con-sider results that were motivated ormeasured with the limitations of single-point experiments in mind.

The Kolmogorov Scaling Laws.In 1938, von Kármán and Howarthderived an exact theoretical relationfor the dynamics of turbulence statis-tics. Starting from the Navier-Stokesequation and assuming homogeneityand isotropy, the two scientistsderived an equation for the dynamicsof the lowest-order two-point velocitycorrelation function. (This function is⟨u(x) · u(x+r)⟩, where the anglebrackets denote an ensemble average,

that is, an average over many statisti-cally independent realizations of theflow. The two-point velocity correla-tion functions cannot describe univer-sal features of turbulence becausethey are scale dependent (the large-scale flow dominates their behavior)and they lack Galilean invariance.Nevertheless, their derivation inspireda real breakthrough. In 1941, AndreiKolmogorov recast the Kármán-Howarth equation in terms of themoments of δu(r), the velocity differ-ences across scales, thereby producinga relationship between the secondmoment ⟨[δu(r)]2⟩ and the thirdmoment ⟨[δu(r)]3⟩. These statisticalobjects, which retain Galilean invari-ance and hence hold the promise ofuniversality, are now known as struc-ture functions.

Kolmogorov then proposed thenotion of an “inertial range” of scalesbased on Richardson’s picture of theenergy cascade: Kinetic energy isinjected at the largest scales L of theflow at an average rate ε and gener-ates large-scale fluctuations. Theinjected energy cascades to smallerscales via nonlinear inertial (energy-conserving) processes until it reachesa scale of order ld, where viscous dis-sipation becomes dominant and thekinetic energy is converted into heat.In other words, the intermediate spa-tial scales r, in the interval ld << r <<L, define an inertial range in whichlarge-scale forcing and viscous forceshave negligible effects. With theseassumptions and the Kármán-Howarthequation recast for structure functions,Kolmogorov derived the famous“four-fifths law.” The equation defin-ing this law describes an exact rela-tionship for the third-order structurefunction within the inertial range:

where ε is assumed to be the finiteenergy-dissipation rate (per unit mass)of the turbulent state. This relation-



Figure 6. Kolmogorov-like Energy Spectrum for a Turbulent JetThe graph shows experimental data for the energy spectrum (computed from veloc-ity times series like that in Figure 5) as a function of wave number k, or E(k) vs k,for a turbulent jet with Reynolds number Re ~ 24,000. Note that the measured spec-trum goes as E(k) ∝∝ k–5/3. (Champagne 1978. Redrawn with the permission of Cambridge University Press.)

ship is a statement of conservation ofenergy in the inertial range of scalesof a turbulent fluid; the third moment,which arises from the nonlinear termin Equation (1), is thus an indirectmeasure of the flux of energy throughspatial scales of size r. (HighReynold’s-number numerical simula-tions are compared with the four-fifthslaw and with other statistical charac-terizations of turbulence in the article“Direct Numerical Simulations ofTurbulence” on page 142.)

Kolmogorov further assumed thatthe cascade process occurs in a self-similar way. That is, eddies of a givensize behave statistically the same aseddies of a different size. Thisassumption, along with the four-fifthslaw, gave rise to the general scalingprediction of Kolmogorov, whichstates that the nth order structure func-tion (referred to in the article “DirectNumerical Simulations of Turbulence”as Sn(r)) must scale as rn/3. During thedecades that have passed sinceKolmogorov’s seminal papers (1941),empirical departures from his scalingprediction have been measured for ndifferent from 3, leading to our pres-ent understanding that turbulent scalesare not self-similar, but that theybecome increasingly intermittent asthe scale size decreases. The charac-terization and understanding of thesedeviations, known as the “anomalous”scaling feature of turbulence, havebeen of sustained and current interest(see the box “Intermittency andAnomalous Scaling in Turbulence” onpage 136).

Empirical observations show that aflow becomes fully turbulent onlywhen a large range of scales separatesthe injection scale L and the dissipa-tion scale ld. A convenient measure ofthis range of spatial scales for fluidturbulence, which also characterizesthe number of degrees of freedom ofthe turbulent state, is the large-scaleReynolds number, Re. The Reynoldsnumber is also the ratio of nonlinear

to viscous forces introduced earlier inthe context of pipe flow. Most theo-ries of turbulence deal with asymptot-ically large Re, that is, Re → ∞, sothat an arbitrarily large range of scalesseparates the injection scales from thedissipation scales.

Because energy cascading downthrough spatial scales is a central fea-ture of fluid turbulence, it is natural toconsider the distribution of energyamong spatial scales in wave number(or Fourier) space, as suggested byTaylor (1938). The energy distributionE(k) = 1/2|u~(k)|2, where u~(k) is theFourier transform of the velocity fieldand the wave number k is related tothe spatial scale l by k = 2π/l.8 Wave-number space is very useful for therepresentation of fluid turbulencebecause differential operators in realspace transform to multiplicativeoperators in k-space. For example, thediffusion operator in the term ν∇2ubecomes νk2u~ in the Fourier repre-sentation. Another appealing featureof the wave number representation isthe nonlocal property of the Fouriertransform, which causes each Fouriermode represented by wave number kto represent cleanly the correspondingscale l. On the other hand, the k-spacerepresentation is difficult from theperspective of understanding how spa-tial structures, such as intense eddies,affect the transfer of energy betweenscales, that is between eddies of dif-ferent sizes.

The consequence of energy conser-vation on the form of E(k) was inde-pendently discovered by Obukov(1941), Heisenberg (1948), andOnsager (1949), all of whom obtainedthe scaling relationship for the energyspectrum E(k) ~ k–5/3 for the inertialscales in fully developed homoge-neous isotropic turbulence. This result

is not independent of the picture pre-sented above in terms of real space-velocity differences but is anotherway of looking at the consequences ofenergy conservation. Many subse-quent experiments and numerical sim-ulations have observed thisrelationship to within experimental/numerical uncertainty, thereby lendingcredence to the energy cascade pic-ture. Figure 6 shows the energy spec-trum obtained from time seriesmeasurements at a single point in aturbulent jet, where the spatial scale isrelated to time by Taylor’s hypothesisthat the large mean velocity sweepsthe “frozen-in” turbulent field past themeasurement point.

Vorticity. Another importantquantity in the characterization andunderstanding of fluid turbulence isthe vorticity field, ωω(x,t) = ∇∇ × u(x,t),which roughly measures the localswirl of the flow as picturesquelydrawn by da Vinci in the openingillustration. The notion of an “eddy”or “whorl” is naturally associatedwith one’s idea of a vortex—a com-pact swirling object such as a dustdevil, a tornado, or a hurricane—butthis association is schematic at best.In three-dimensional (3-D) turbu-lence, vorticity plays a quantitativerole in that the average rate of energydissipation ε is related to the mean-square vorticity by the relation ε = –u⟨ωω2⟩. Vorticity plays a differentrole in two-dimensional (2-D) turbu-lence. Vortex stretching has long beenrecognized as an important ingredientin fluid turbulence (Taylor 1938); if avortex tube is stretched so that itscross section is reduced, the mean-square vorticity in that cross sectionwill increase, thereby causing strongspatially localized vortex filamentsthat dissipate energy. The notion ofvortex stretching and energy dissipa-tion is discussed in the article “TheLANS-α model for ComputingTurbulence” on page 152.



8If isotropy is assumed, the energy distri-bution E(k), where k is a vector quantity,depends only on the magnitude k = |k| andone can denote the energy as E(k) withoutloss of generality.

Engineering Models of Turbulence

It is worth stressing again that tur-bulence is both fundamentally inter-esting and of tremendous practicalimportance. As mentioned above,modeling complex practical problemsrequires a perspective different fromthat needed for studying fundamentalissues. Foremost is the ability to getfairly accurate results with minimalcomputational effort. That goal canoften be accomplished by makingsimple models for the effects of turbu-lence and adjusting coefficients in themodel by fitting computational resultsto experimental data. Provided thatthe parameter range used in the modelis well covered by experimental data,this approach is very efficient.Examples that have benefited fromcopious amounts of reliable data areaircraft design—aerodynamics ofbody and wing design have been atthe heart of a huge internationalindustry—and aerodynamic dragreduction for automobiles to achievebetter fuel efficiency. Global climatemodeling and the design of nuclearweapons, on the other hand, areexamples for which data are eitherimpossible or quite difficult to obtain.In such situations, the utmost careneeds to be taken when one attemptsto extrapolate models to circum-stances outside the validation regime.

The main goal of many engineeringmodels is to estimate transport proper-ties—not just the net transport ofenergy and momentum by a single fluidbut the transport of matter such as pol-lutants in the atmosphere or the mixingof one material with another. Mixing isa critical process in inertial confine-ment fusion and in weapons physicsapplications. It is crucial for certifica-tion of the nuclear weapons stockpilethat scientists know how well engineer-ing models are performing and use thatknowledge to predict outcomes with aknown degree of certainty.

The Closure Problem forEngineering Models. Engineeringmodels are constructed for computa-tional efficiency rather than perfectrepresentation of turbulence. The classof engineering models known asReynolds-Averaged Navier-Stokes(RANS) provides a good example ofhow the problem of “closure” arisesand the parameters that need to bedetermined experimentally to makethose models work. Consider againthe flow of a fluid with viscosity ν ina pipe with a pressure gradient alongthe pipe. When the pressure applied atthe pipe inlet and the pipe diameter Dare small, the fluid flow is laminar,and the velocity profile in the pipe isquadratic with a peak velocity U thatis proportional to the applied pressure(refer to Figure 3). When the forcingpressure gets large enough to producea high flow velocity and therefore alarge Reynolds number, typically Re =UD/ν ≥ 2000, the flow becomes tur-bulent, large velocity fluctuations arepresent in the flow, and the velocityprofile changes substantially (referagain to Figure 3). An engineeringchallenge is to compute the spatial dis-tribution of the mean velocity of theturbulent flow. Following the proce-dure first written down by Reynolds,the velocity and pressure fields areseparated into mean (the overbardenotes a time average) and fluctuat-ing (denoted by the prime) parts:

where i denotes one of the compo-nents of the vector field u(x,t) and theaverage of the fluctuating part of thevelocity is zero by definition. Also,

Substituting these expressions intoEquation (1), using the constant-den-sity continuity condition

and averaging term by term yields anequation for the mean velocity:

(2)

Note that the equation for the meanflow looks the same as the Navier-Stokes equation for the full velocity u,Equation (1), with the addition of aterm involving the time average of aproduct of the fluctuating parts of thevelocity, namely, the Reynolds stresstensor,

That additional term, which repre-sents the transport of momentumcaused by turbulent fluctuations, actslike an effective stress on the flow andcannot, at this time, be determinedcompletely from first principles. As aresult, many schemes have beendeveloped to approximate theReynolds stress.

The simplest formulation for theReynolds stress tensor is

where νT(x) is called the turbulenteddy viscosity because the additionalterm looks like a viscous diffusionterm. A more sophisticated approachis to solve for the Reynolds stress bywriting an equation for the time evo-lution of u′iu′j (Johnson et al. 1986).This equation has multiple undeter-mined coefficients and depends on thethird moment u′iu′ju′k. . Again, the third-order moment is unknown and needsto be approximated or written in termsof fourth-order moments. In principle,an infinite set of equations for higher-order moments is required, so oneneeds to “close” the set at a smallnumber to achieve computational effi-



ciency. At any stage of approximation,undetermined coefficients are set bycomparison with experimental ordirect numerical simulation data. Thisapproach is often very effective,although it does depend on the qualityof the data and on the operatingparameter regime covered by the data.

Modern Developments

By the end of the 1940s, greatprogress had been made in the studyof turbulence. The statistical approachto turbulence, the importance of theenergy and its wave number represen-tation, the notion of measuring veloc-ity differences, and the dynamics ofvortex structures as an explanation ofthe mechanism of fluid turbulence hadall been articulated by Taylor. Thecascade picture of Richardson hadbeen made quantitative byKolmogorov and others. The conceptsof universal scaling and self-similaritywere key ingredients in that descrip-tion. On the practical side, tremen-dous advances had been made inaeronautics, with newly emerging jetaircraft flying at speeds approachingthe speed of sound. Empirical modelsbased on the engineering approachdescribed above were being used todescribe these practical turbulenceproblems.

The next 50 years were marked bysteady progress in theory and model-ing, increasingly sophisticated experi-ments, and the introduction andwidespread use of the digital com-puter as a tool for the study of turbu-lence. In the remainder of this review,we touch on some of the advances ofthe post-Kolmogorov era, paying par-ticular attention to the ever-increasingimpact of the digital computer onthree aspects of turbulence research:direct numerical simulations of ideal-ized turbulence, increasingly sophisti-cated engineering models ofturbulence, and the extraordinary

enhancement in the quality and quan-tity of experimental data achieved bycomputer data acquisition. As farback as the Manhattan Project, thecomputer (more exactly, numericalschemes implemented on a roomfulof Marchand calculators) began toplay a major role in the calculationsof fluid problems. A leading figure inthat project, John von Neumann(1963), noted in a 1949 review ofturbulence that “… a considerablemathematical effort towards adetailed understanding of the mecha-nism of turbulence is called for” butthat, given the analytic difficultiespresented by the turbulence problem,“… there might be some hope to‘break the deadlock’ by extensive,but well-planned, computationalefforts.” Von Neumann’s foresight inunderstanding the important role ofcomputers for the study of turbulencepredated the first direct numericalsimulation of the turbulent state bymore than 20 years.

From a fundamental perspective,the direct numerical simulation ofidealized isotropic, homogeneous tur-bulence has been revolutionary in itsimpact on turbulence researchbecause of the ability to simulate anddisplay the full 3-D velocity field atincreasingly large Reynolds number.Similarly, experimentation on turbu-lence has advanced tremendously byusing computer data acquisition;20 years ago it was possible to meas-ure and analyze time series data fromsingle-point probes that totaled nomore than 10 megabytes of informa-tion, whereas today statistical ensem-bles of thousands of spatially andtemporally resolved velocity fields,taking 10 terabytes of storage spacecan be obtained and processed. Thismillionfold increase in experimentalcapability has opened the door togreat new possibilities in turbulenceresearch that will be enhanced evenfurther by expected future increasesin computational power.

Below, we briefly addressadvances in numerical simulation, inturbulence modeling, and theoreticalunderstanding of passive scalar trans-port, topics dealt with more exten-sively in the articles immediatelyfollowing this one. We then describeseveral exciting new experimentaladvances in fluid turbulence researchand close this introduction with aview toward “solving” at least someaspect of the turbulence problem.

Direct Numerical Simulation of Turbulence

Recent advances in large-scale sci-entific computing have made possibledirect numerical simulations of theNavier-Stokes equation under turbu-lent conditions. In other words, forsimulations performed on the world’slargest supercomputers, no closure orsubgrid approximations are used tosimplify the flow, but rather the simu-lated flow follows all the twisting-turning and stretching-folding motionsof the full-blown Navier-Stokes equa-tions at effective large-scale Reynoldsnumbers of about 105. These simula-tions render available for analysis theentire 3-D velocity field down to thedissipation scale. With these numeri-cally generated data, one can studythe structures of the flow and corre-late them with turbulent transferprocesses, the nonlinear processes thatcarry energy from large to smallscales.

An especially efficient techniquefor studying isotropic, homogeneousturbulence is to consider the flow ina box of length L with periodicboundary conditions and use thespectral method, an orthogonaldecomposition into Fourier modes, tosimulate the Navier-Stokes equation.Forcing is typically generated bymaintaining constant energy in thesmallest k mode (or a small numberof the longest-wavelength modes).



The first direct numerical simulationof fluid turbulence (Orszag andPatterson 1972) had a resolution of323, corresponding to Re ~ 100. Bythe early 1990s, Reynolds numbersof about 6000 for a 5123 simulationcould be obtained (She et al. 1993);the separation between the box sizeand the dissipation scale was justshort of a decade. Recent calcula-tions on the Los Alamos Q machine,using 20483 spatial resolution, andon the Japanese Earth Simulator with40963 modes (Kaneda et al. 2003)achieved a Reynolds number ofabout 105, corresponding to about1.5 decades of turbulent scales,which is approaching fully developedturbulence in modestly sized windtunnels. It is important to appreciatethat the Re of direct numerical turbu-lence simulations grows only as Re ∝N4/9, where N is the number ofdegrees of freedom that are com-puted: A factor of 2 increase in thelinear dimension of the box meanscomputing 23 more modes for a cor-responding increase in Re of about2.5. Nevertheless, for isotropic,homogeneous fully developed turbulence, direct numerical simula-tion has become the tool of choicefor detailed characterization of fundamental flow properties andcomparison with Kolmogorov-typetheories. More details regardingnumerical simulation of turbulencecan be found in the article “DirectNumerical Simulations ofTurbulence” on page 142.

Modern Turbulence Models

Although the RANS modelsdescribed above maintain a dominantrole in turbulence modeling, otherapproaches have become tractablebecause of increases in computationalpower. A more recent approach tomodeling turbulent processes is todecompose spatial scales of the flow

into Fourier modes and then to trun-cate the expansion at some intermedi-ate scale (usually with a smoothGaussian filter) and model the smallscales with a subgrid model. One thencomputes the large scales explicitlyand approximates the effect of thesmall scales with the subgrid model.This class of methods (Meneveauand Katz 2000) is called large eddysimulation (LES) and has become analternative to RANS modeling whenmore-accurate spatial information isrequired. Because of the spatial fil-tering, LES modeling has problemswith boundaries and is less computa-tionally efficient than RANS tech-niques. Nevertheless, LES modelsmay be more universal than RANSmodels and therefore rely less on adhoc coefficients determined fromexperimental data.

Another variant of the subgrid-model approach recently invented atLos Alamos is the Lagrangian-aver-aged Navier-Stokes alpha (LANS-α)model. Although not obtainable by filtering the Navier-Stokes equations,the LANS-α model has a spatial cut-off determined by the coefficient α.For spatial scales larger than α, thedynamics are computed exactly (ineffect, the Navier-Stokes equationsare used) and yield the energy spectrum E(k) ∝ k–5/3, whereas forspatial scales less than α, the energyfalls more rapidly, E(k) ∝ k–3. TheLANS-α model can be derived from aLagrangian-averaging procedure start-ing from Hamilton’s principle of leastaction. It is the first-closure (or sub-grid) scheme to modify the nonlinearterm rather than the dissipation termand, as a result, has some uniqueadvantages relative to more traditionalLES schemes (see the articles “TheLANS-α Model for ComputingTurbulence” and “Taylor’sHypothesis, Hamilton’s Principle, andthe LANS-α Model for ComputingTurbulence” on pages 152 and 172,respectively).

Beyond the Kolmogorov Theory

The 50 years that have passed,from about 1950 until the new millen-nium, were notable for increasinglysophisticated theoretical descriptionsof fluid turbulence, including the sem-inal contributions of RobertKraichnan (1965, 1967, 1968, 1975),who pioneered the foundations of themodern statistical field-theoryapproach to hydrodynamics, particu-larly by predicting the inverse energycascade from small scales to largescales in 2-D turbulence, and GeorgeBatchelor (1952, 1953, 1959, 1969).Those developments are generallybeyond the scope of the presentreview, and we have already referredthe reader to recent books in whichthey are surveyed in detail (McComb1990, Frisch 1995, Lesieur 1997). Wetouch briefly, however, on a fewrecent developments that grew out ofthose efforts and on the influence ofthe Lagrangian perspective of fluidturbulence.

The physical picture that emergesfrom the Kolmogorov phenomenologyis that the turbulent scales of motionare self-similar; that is, the statisticalfeatures of the system are independentof spatial scale. One measure of thisself-similarity is the nondimensionalratio of the fourth moment to thesquare of the second moment, F = ⟨δu4⟩/⟨δu2⟩2, as a function ofscale separation. If the velocity distri-bution is self-similar, then F shouldbe constant, or flat, as a function oflength scale. Indeed, the nondimen-sional ratio of any combination ofvelocity-increment moments shouldbe scale independent. If, however, Fbehaves as a power law in the separa-tion r, then the system is not self-sim-ilar, but instead it is characterized byintermittency: short bursts (in time) orisolated regions (in space) of high-amplitude fluctuations separated byrelatively quiescent periods or



regions. From the 1960s to the 1980s,experimentalists reported departuresfrom the Kolmogorov scaling. Themeasured fourth-order and highermoments of velocity differences didnot scale as rn/3, but rather as a lowerpower of the separation r,⟨δun⟩ ~ rξn , with ξn < n/3 for n > 3.To preserve the correct dimensions ofthe nth-order velocity differencemoments, the deviations fromKolmogorov scaling, or from self-similarity, can be written as a correc-tion factor given by the ratio of thelarge scale L to the separation r tosome power ∆n, or (L/ r)∆n (see thebox “Intermittency and AnomalousScaling in Turbulence” on page 136).Some recent analytic progress towardunderstanding the origin of theobserved anomalous scaling has beenmade in the context of passive scalarturbulence and involves the applica-tion of nonperturbative field-theorytechniques to that problem (see thearticle “Field Theory and StatisticalHydrodynamics” on page 181).

The passive scalar problemdescribes the transport and effectivediffusion of material by a turbulent

velocity field. This stirring processcharacterizes fluid mixing, which hasmany important scientific and techni-cal applications. Whereas intermit-tency is rather weak in turbulentvelocity statistics, the distribution of apassive scalar concentration carriedby a turbulent flow is very intermit-tent. In other words, there is a muchlarger probability (compared withwhat one would expect for a random,or Gaussian, distribution) of findinglocal concentrations that differ greatlyfrom the mean value. For characteriz-ing fluid mixing, the Lagrangianframe of reference (which moves withthe fluid element as opposed to theEulerian frame, which is fixed inspace) is very useful theoreticallybecause a passive scalar is carried byfluid elements. Figure 7 shows thedistribution of a virtual drop of yellowdye carried by a 2-D turbulent flow ina stratified layer experiment.9 Thestructured distribution of the dye illus-

trates how the velocity field stretchesand folds fluid elements to producemixing. Adopting the Lagrangianframe of reference is rapidly emergingas a powerful new approach for mod-eling turbulent mechanisms of energytransfer. This approach has led toLagrangian tetrad methods, a phenom-enological model arising from thenonperturbative field-theoreticalapproach to turbulence, and to theLANS-α model mentioned above.

Recent ExperimentalDevelopments

Quantitative single-point measure-ments of velocity combined withqualitative flow visualization (vanDyke 1982) have characterized almostall experimental measurements offluid turbulence for most of the 20thcentury. Recently, however, newexperimental techniques enabled bydigital data acquisition, powerfulpulsed-laser technology, and fast digi-tal imaging systems have emergedand are causing great excitement inthe field of turbulence.



Figure 7. Passive Scalar Turbulence in a Stratified LayerAn effective blob of yellow dye is carried by a forced 2-D turbulent flow in a stratified layer. The images show (a) the initial dyeconcentration and (b) the concentration after about one rotation of a large-scale eddy. The sharp gradients in the concentrationlead to the very strong anomalous scaling in the transport of the passive scalar field.

9The “virtual” drop consists of more than10,000 fluid elements, whose evolution iscomputed by solving the Lagrangianequation dx(t)/dt = u(t,x(t)) from experi-mental velocity fields.

(a) (b)



Intermittency is associated with the violent, atypi-cal discontinuous nature of turbulence. When asignal from turbulent flow (for example, the veloc-ity along a particular direction) is measured at asingle spatial point and a sequence of times (an Eulerian measurement), the fluctuations in thevalues appear to be random. Since any randomsequence is most naturally explained in terms of itsstatistical distribution, one typically determines thestatistics by constructing a histogram of the signal.The violent nature of turbulence shows itself in thevery special shape of the histogram, or the proba-bility distribution function (PDF), of the turbulentvelocity signal. It is typically wider than theGaussian distributions emerging in the context ofequilibrium statistical physics—for example, theGaussian distribution that describes the velocity of(or the distance traveled by) a molecule undergo-ing Brownian motion.

The PDF of the energy dissipation rate, P(ε), illus-trates how far from Gaussian a turbulent distribu-tion can be. At values far above the average, ε >>⟨ε⟩, where ε ≡ ν(∇u)2, the probability distributionP(ε) has a stretched exponential tail, ln P(ε) ∝ –εa

(La Porta 2001). The extended tail of the turbulentPDF illustrates the important role played by theatypical, violent, and rare events in turbulence.

Intermittency has many faces. In the context oftwo-point measurements, intermittency is associ-ated with the notion of anomalous scaling.Statistics of the longitudinal velocity increments,δu(r) (the difference in the velocity componentsparallel to the line separating the two points) indeveloped turbulence becomes extremely non-Gaussian as the scale decreases. In particular, if thescale r separating the two points is deep inside theinertial interval, L >>r >> ld, then the nth momentof the longitudinal velocity increment is given by

(1)

where L is the integral (pumping, energy-contain-ing) scale of turbulence. The first thing to mentionabout Equation (1) is that the viscous, Kolmogorovscale ld does not enter the relation in the devel-oped turbulence regime. This fact is simply relatedto the direction of the energy cascade: On average,energy flows from the large scale, where it ispumped into the system, toward the smaller scale,ld, where it is dissipated; it does not flow from thesmall scale. Secondly, ∆n on the right side ofEquation (1) is the anomalous scaling exponent. Inthe phenomenology proposed by Kolmogorov in1941, the flow is assumed to be self-similar in theinertial range of scales, which implies that anom-alous scaling is absent, ∆n = 0, for all values of n.The self-similar scaling phenomenology is anextension of the four-fifths law proven byKolmogorov in 1941 for the third moment

(See discussion of the four-fifths law in “DirectNumerical Simulations of Turbulence” on page142). This law is a statement of conservation ofenergy from scale to scale in the inertial regime ofhomogeneous isotropic turbulence. Modern experimental and numerical tests (Frisch 1995) unequivocally dismiss the self-similarity assump-tion, ∆n = 0, as invalid. But so far, theory is stillincapable of adding any other exact relation to thecelebrated four-fifths law.

On the other hand, even though a comprehensivetheoretical analysis of developed isotropic turbu-lence remains elusive, there has been an importantbreakthrough in understanding anomalous scalingin the simpler problem of passive scalar turbulence(see the article “Field Theory and StatisticalHydrodynamics” on page 181).

Intermittency and Anomalous Scaling in Turbulence

Misha Chertkov

One innovative example of a newturbulence experiment (La Porta2001) is the use of silicon strip detec-tors from high-energy physics to tracka single small particle in a turbulentflow with high Reynolds number (seeFigure 8). This is an example of thedirect measurement of Lagrangian(moving with the fluid) properties ofthe fluid flow. Because the particletrajectories are time resolved, theacceleration statistics can be obtaineddirectly from experiment, and theoret-ical predictions for those statistics canbe tested.

Another application of new tech-nology has made possible the localtime-resolved determination of thefull 3-D velocity gradient tensor10 at apoint in space (Zeff 2003). Knowingthe local velocity gradients allows oneto calculate the energy dissipation rateε and mean-square vorticity, Ω =⟨ω2⟩/2, and thereby provide an experi-mental measure of intense and inter-mittent dissipation events (seeFigure 9).

Both these techniques can be usedto obtain large data samples that arestatistically converged and have onthe order of 106 data sets per parame-ter value. At present, however, thephysical length scales accessible tothese two techniques are constrainedto lie within or close to the dissipationscale ld. By using holographic meth-ods, one can obtain highly resolved,fully 3-D velocity fields (seeFigure 10), which allow the full turbu-lent inertial range of scales to beinvestigated (Zhang et al. 1997). Forholographic measurements, however,one is limited to a small number ofsuch realizations, and time-resolvedmeasurements are not currentlyachievable. Finally, for physical realizations of

2-D flows, full-velocity fields can bemeasured with high resolution in bothspace and time (Rivera et al. 2003),and the 2-D velocity gradient tensorcan be used to identify topologicalstructures in the flow and correlate

them with turbulent cascade mecha-nisms. The technique used to makethese measurements and those repre-sented in Figure 10 is particle-imagevelocimetry (PIV) or its improvedversion, particle-tracking velocimetry(PTV). Two digital images, taken



Figure 8. Three-Dimensional Particle Trajectory in a Turbulent Fluid A high-speed silicon-strip detector was used to record this trajectory of a particlein a turbulent fluid with Re = 63,000 (La Porta 2001). The magnitude of the instanta-neous acceleration is color-coded. Averaging over many such trajectories allowscomparison with the theory of Lagrangian acceleration statistics.(Modified with permission from Nature. This research was performed at Cornell University.)

Figure 9. Intermittency of Energy Dissipation and Enstrophy at Re = 48,000Time traces of the local energy dissipation εε (crosses) and enstrophy ΩΩ = ⟨ωω2⟩/2(solid curve) illustrate the very intermittent behavior of these dissipation quantitiesfor turbulent flow with Re = 48,000 (Zeff 2003). (Modified with permission from Nature. This

research was performed at the University of Maryland.)

10 The velocity gradient tensor is a 3 × 3matrix consisting of the spatial derivativesof three components of velocity. Forexample, for the velocity component ui,the derivatives are ∂ui/∂x1, ∂ui/∂x2, and∂ui/∂x3.

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closely spaced in time, track themotion of small particles that seed theflow and move with the fluid. Thebasic notion is illustrated in Figure 11,where two superimposed digitalimages of particle fields, separated by∆t = 0.03 second, are shown. Within

small subregions of the domain, pat-terns of red particles in image 1 canbe matched with very similar patternsof blue particles in image 2 by maxi-mizing the pattern correlation. Anaverage velocity vector for the match-ing patterns is then calculated over the

entire box from which a velocity fieldis obtained, as shown in Figure 12(a).Notice that there are some anomalousvectors caused by bad matching thatneed to be fixed by some interpolationscheme. PTV, on the other hand, usesa particle-matching algorithm to trackindividual particles between frames.The resulting vector field is shown inFigure 12(b). The PTV method hashigher spatial resolution than PIV butalso greater computational-processingdemands and more stringent image-quality constraints. From the PTVvelocity field, the full vorticity field ωω can be computed as shown inFigure 12(c).

An additional advantage of thePTV approach is that, for high enoughtemporal resolution, individual parti-cle tracks can be measured over manycontiguous frames, and informationabout Lagrangian particle trajectoriescan be obtained. Some 2-D particletracks are shown in Figure 13.

This capability can be combinedwith new analysis methods for turbu-lence to produce remarkable newvisualization tools for turbulence.Figure 14 shows the full backwardand forward time evolution of amarked region of fluid within an iden-tified stable coherent structure (a vor-tex). These fully resolvedmeasurements in two dimensions willhelp build intuition for the eventualdevelopment of similar capabilities inthree dimensions. Further, the physi-cal mechanisms of 2-D turbulence arefascinating in their own right and maybe highly relevant to atmospheric oroceanic turbulence.

The Prospects for“Solving” Turbulence

Until recently, the study of turbu-lence has been hampered by limitedexperimental and numerical data on theone hand and the extremely intractableform of the Navier-Stokes equation on



Figure 10. High-Resolution 3-D Turbulent Velocity Fields These images were obtained using digital holographic particle-imaging velocimetry.(Zhang et al. 1997. Modified with permission from Experiments in Fluids.)

Figure 11. Superimposed Digital Images of a Particle FieldTwo digital images of suspended particles taken 0.003 s apart are superimposed.The first exposure is in red; the second, in blue. In PIV, the pattern of particlesover a small subregion is correlated between exposures, and an average velocityis computed by the mean displacement δδx of the pattern. In PTV, each particle ismatched between exposures; as a result, spatial resolution is higher, and there isno spatial averaging. These data allow one to infer the velocity field connectingthe two images.



Figure 14. Time Evolution of aCompact Distribution of 104

Points in a Coherent Vortex atTime t2Vortex merging and splitting happenat times t1 and t3, respectively. Thecolor-coding of the surface repre-sents the spatially local energy fluxto larger (red) and smaller (blue) spatial scales.(M. K. Rivera, W. B. Daniel, and R. E. Ecke, to be

published in Gallery of Fluid Images, Chaos, 2004.)

Figure 13. Individual Lagrangian Particle Tracks for Forced 2-D Turbulence(a) Approximately 104 particles are tracked for short periods.(b) Several individual trajectories are shown for several injection-scale turnover times.

Figure 12. Two-Dimensional VectorVelocity and Vorticity FieldsThe velocities in (a) and (b) wereobtained from data similar to those inFigure 11 using PIV and PTV techniques,respectively. The vorticity field in (c) iscalculated from the velocity field in (b).

(a)

(c)

(b)

(a)

(b)

the other. Today, the advent of large-scale scientific computation, combinedwith new capabilities in data acquisi-tion and analysis, enables us to simu-late and measure whole velocity fieldsat high spatial and temporal resolu-tions. Those data promise to revolu-tionize the study of fluid turbulence.Further, new emerging ideas in statisti-cal hydrodynamics derived from fieldtheory methods and concepts are pro-viding new theoretical insights into thestructure of turbulence (Falkovitch etal. 2001). We will soon have many ofthe necessary tools to attack the turbu-lence problem with some hope of solv-ing it from the physics perspective ifnot with the mathematical rigor or theextremely precise prediction of proper-ties obtained in, say, quantum electro-dynamics. Let me explain then what Imean by that solution. In condensedmatter physics, for example, the mys-tery of ordinary superconductivity wassolved by the theory of Bardeen,Cooper, and Schrieffer (1957), whichdescribed how electron pairing medi-ated by phonons led to a Bose-Einsteincondensation and gave rise to thesuperconducting state. Despite thissolution, there has been no accuratecalculation of a superconducting transi-tion temperature for any superconduct-ing material because of complicationsemerging from material properties. Ithink that there is hope for understand-ing the mechanisms of turbulentenergy, vorticity, and mass transferbetween scales and between points inspace. This advance may turn out to beelegant enough and profound enoughto be considered a solution to the mys-tery of turbulence. Nevertheless,because turbulence is probably a wholeset of problems rather than a singleone, many aspects of turbulence willlikely require different approaches. Itwill certainly be interesting to see howour improved understanding of turbu-lence contributes to new predictabilityof one of the oldest and richest areas inphysics. n

Acknowledgments

Three important books on fluidturbulence, McComb (1990), Frisch(1995), and Lesieur (1997), publishedin the last 20 years have helped pro-vide the basis for this review. I wouldlike to thank Misha Chertkov, GregEyink, Susan Kurien, Beth Wingate,Darryl Holm, and Greg Swift forvaluable suggestions. I am indebtedto my scientific collaborators—ShiyiChen, Brent Daniel, Greg Eyink,Michael Rivera, and PeterVorobieff—for many excitingmoments in learning about fluid tur-bulence, and I look forward to morein the future. I would also like toacknowledge the LDRD program atLos Alamos for its strong support offundamental research in turbulenceduring the last 5 years

Further Reading

Bardeen, J., L. N. Cooper, and J. R. Schrieffer.1957. Theory of Superconductivity. Phys.Rev. 108 (5): 1175.

Batchelor, G. K. 1952. The Effect ofHomogeneous Turbulence on MaterialLines and Surfaces. Proc. R. Soc. London,Ser. A 213: 349.

———. 1953. The Theory of HomogeneousTurbulence. Cambridge, UK: CambridgeUniversity Press.

———. 1959. Small-Scale Variation ofConvected Quantities Like Temperature inTurbulent Fluid. Part 1. General Discussionand the Case of Small Conductivity. J.Fluid Mech. 5: 113.

———. 1969. Computation of the EnergySpectrum in Homogeneous Two-Dimensional Turbulence. Phys. Fluids 12,Supplement II: 233.

Besnard, D., F. H. Harlow, N. L. Johnson, R.Rauenzahn, and J. Wolfe. 1987. Instabilitiesand Turbulence. Los Alamos Science 15:145.

Champagne, F. H. 1978. The Fine-ScaleStructure of the Turbulent Velocity Field.J. Fluid Mech. 86: 67.

Falkovitch, G., K. Gawedzki, and M.Vergassola. 2001. Particles and Fields inFluid Turbulence. Rev. Mod. Phys. 73: 913.

Frisch, U. 1995. Turbulence: The Legacy ofA. N. Kolmogorov. Cambridge, UK:Cambridge University Press.

Heisenberg, W. 1948. Zur Statistischen Theorieder Turbulenz. Z. Phys. 124: 628.

Kaneda, Y., T. Ishihara, M. Yokokawa, K.Itakura, and A. Uno. 2003. EnergyDissipation Rate and Energy Spectrum inHigh Resolution Direct NumericalSimulations of Turbulence in a PeriodicBox. Phys. Fluids 15: L21.

Kolmogorov, A. N. 1941a. The Local Structureof Turbulence in Incompressible ViscousFluid for Very Large Reynolds Number.Dok. Akad. Nauk. SSSR 30: 9

———. 1941b. Decay of Isotropic Turbulencein an Incompressible Viscous Fluid. Dok.Akad. Nauk. SSSR 31: 538.

———. 1941c. Energy Dissipation in LocallyIsotropic Turbulence. Dok. Akad. Nauk.SSSR 32: 19.

Kraichnan, R. H. 1959. The Structure ofIsotropic Turbulence at Very High ReynoldsNumbers. J. Fluid Mech. 5: 497.

———. 1965. Lagrangian-History ClosureApproximation for Turbulence. Phys. Fluids8: 575.

———. 1967. Inertial Ranges in Two-Dimensional Turbulence. Phys. Fluids 10: 1417.

———. 1968. Small-Scale Structure of a ScalarField Convected by Turbulence. Phys.Fluids 11: 945.

———. 1974. On Kolmogorov’s Inertial-RangeTheories. J. Fluid Mech. 62: 305.



Lamb, C. 1932. Address to the BritishAssociation for the Advancement of Science.Quoted in Computational Fluid Mechanicsand Heat Transfer by J. C. Tannehill, D. A.Anderson, and R. H. Pletcher. 1984. NewYork: Hemisphere Publishers.

La Porta, A., G. A. Voth, A. M. Crawford, J.Alexander, and E. Bodenschatz. 2001. FluidParticle Accelerations in Fully DevelopedTurbulence. Nature 409: 1017.

Lesieur, M. 1997. Turbulence in Fluids.Dordrecht, The Netherlands: KluwerAcademic Publishers.

McComb, W. D. 1990. The Physics of FluidTurbulence. Oxford, UK: Oxford UniversityPress.

Meneveau, C., and J. Katz. 2000. Scale-Invariance and Turbulence Models forLarge-Eddy Simulation. Annu. Rev. FluidMech. 32: 1.

Obukhov, A. M. 1941. Spectral EnergyDistribution in a Turbulent Flow. Dok.Akad. Nauk. SSSR 32: 22.

Onsager, L. 1945. The Distribution of Energy inTurbulence. Phys. Rev. 68: 286.

———. 1949. Statistical Hydrodynamics.Nuovo Cimento 6: 279.

Orszag, S. A., and G. S. Patterson. 1972. InStatistical Models and Turbulence, LectureNotes in Physics, Vol. 12, p. 127. Edited byM. Rosenblatt and C. M. Van Atta. Berlin:Springer-Verlag.

Reynolds, O. 1895. On the Dynamical Theoryof Incompressible Viscous Fluids and theDetermination of the Criterion. Philos.Trans. R. Soc. London, Ser. A 186: 123.

Richter, J. P. 1970. Plate 20 and Note 389. InThe Notebooks of Leonardo Da Vinci. NewYork: Dover Publications.

Richardson, L. F. 1922. Weather Prediction byNumerical Process. London: CambridgeUniversity Press.

———. 1926. Atmospheric Diffusion Shownon a Distance-Neighbour Graph. Proc. R.Soc. London, Ser. A 110: 709.

Rivera, M. K., W. B. Daniel, S. Y. Chen, and R.E. Ecke. 2003. Energy and EnstrophyTransfer in Decaying Two-DimensionalTurbulence. Phys. Rev. Lett. 90: 104502.

She, Z. S., S. Y. Chen, G. Doolen, R. H.Kraichnan, and S. A. Orszag. 1993.Reynolds-Number Dependence of IsotropicNavier-Stokes Turbulence. Phys. Rev. Lett.70: 3251.

Taylor, G. I. 1935. Statistical Theory ofTurbulence. Proc. R. Soc. London, Ser. A151: 421.

———. 1938. The Spectrum of Turbulence.Proc. R. Soc. London, Ser. A 164: 476.

Van Dyke, M. 1982. An Album of Fluid Motion.Stanford, CA: Parabolic Press.

von Kármán, T., and L. Howarth. 1938. On theStatistical Theory of Isotropic Turbulence.Proc. R. Soc. London, Ser. A 164: 192.

von Neumann, J. 1963. Recent Theories ofTurbulence. In Collected Works(1949–1963), Vol. 6, p. 437. Edited by A.H. Taub. Oxford, UK: Pergamon Press.

Zeff, B. W., D. D. Lanterman, R. McAllister, R.Roy, E. J. Kostelich, and D. P. Lathrop.2003. Measuring Intense Rotation andDissipation in Turbulent Flows. Nature 421:146.

Zhang, J., B. Tao, and J. Katz. 1997. TurbulentFlow Measurement in a Square Duct withHybrid Holographic PIV. Exp. Fluids 23:373.



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