The Turbulence Problem an Experimentalist’S Perspective

Home , Turbulence

Leonardo da Vinci’s illustration of the swirling flow of turbulence. (The Royal Collection  2004, Her Majesty Queen Elizabeth II) The Turbulence Problem An Experimentalist’s Perspective

Robert Ecke

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 tremendous practical importance in making predictions—for example, about heat transfer in nuclear reactors, drag in oil pipelines, the weather, and the circulation of the atmosphere and the oceans. But what is turbulence? Why is it so difficult to understand, to model, or even to approximate with confidence? And what kinds of solutions can we expect to obtain? This brief survey starts with a short history and then introduces both the modern search for universal statistical properties and the new engineering models for computing turbulent flows. It highlights the application of modern supercomputers in simulating the multiscale velocity field 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 the turbulence problem.

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Many generations of scientists equation of fluid motion. The Euler from momentum diffusion through have struggled valiantly to understand equation of motion (written down in molecular collisions. Remarkably, a both the physical essence and the the 18th century) describes the con- simple equation representing a simple mathematical structure of turbulent servation of momentum for a fluid physical concept describes enor- fluid motion (McComb 1990, Frisch without viscosity, whereas the Navier- mously complex phenomena. 1995, Lesieur 1997). Leonardo da Stokes equation (19th century) The Navier-Stokes equations are Vinci (refer to Richter 1970), who in describes the rate of change of deterministic in the sense that, once 1507 named the phenomenon he momentum at each point in a viscous the initial flow and the boundary con- observed in swirling flow “la tur- fluid. The Navier-Stokes equation for ditions are specified, the evolution of bolenza” (see the drawing on the a fluid with constant density ρ and the state is completely determined, at opening page), described the follow- constant kinematic viscosity ν is least in principle. The nonlinear term ing picture: “Observe the motion of in Equation (1), u · ∇u, describes the the surface of the water, which resem- advective transport of fluid momen- (1) bles that of hair, which has two , tum. Solutions of the nonlinear motions, of which one is caused by Navier-Stokes equations may depend the weight of the hair, the other by the with ∇ ⋅ u = 0, which is a statement sensitively on the initial conditions so direction of the curls; thus the water of fluid incompressibility and with that, after a short time, two realiza- has eddying motions, one part of suitable conditions imposed at the tions of the flow with infinitesimally which is due to the principal current, boundaries of the flow. The variable different initial conditions may be the other to the random and reverse u(x,t) is the (incompressible) fluid completely uncorrelated with each motion.” velocity field, and P(x,t) is the pres- other. Changes in the external forcing Two aspects of da Vinci’s observa- sure field determined by the preserva- or variations in the boundary conditions remain with us today. First, his tion of incompressibility. This tions can produce flows that vary separation of the flow into a mean and equation (when multiplied by ρ to get from smooth laminar flow to more a fluctuating part anticipates by force per unit volume) is simply complicated motions with an identifi- almost 400 years the approach taken Newton’s law for a fluid: Force equals able length or time scale, and from by Osborne Reynolds (1894). The mass times acceleration. The left side there to the most complicated flow of “Reynolds decomposition” of the of Equation (1) is the acceleration of them all, namely, fully developed tur- fluid velocity into mean and fluctuat- the fluid,2 and the right side is the bulence with its spectrum of motions ing parts underpins many engineering sum of the forces per unit mass on a over many length scales. Depending models of turbulence in use today.1 unit volume of the fluid:3 the pressure on the specific system (for example, Second, da Vinci’s identification of force and the viscous force arising flow in a pipe or behind a grid), the “eddies” as intrinsic elements in tur- transition from smooth laminar flow bulent motion has a modern counter- to fully developed turbulence may part: Scientists today are actively 2 The acceleration term looks compli- occur abruptly, or by successively investigating the role of such struc- cated because of the advection term more complex states, as the forcing is u ⋅∇u, which arises from the coordinate tures as the possible “sinews” of tur- transformation from a frame moving with increased. bulent dynamics. the fluid parcels (the “Lagrangian” frame, The difficulties of finding solutions Long after da Vinci’s insightful in which Newton's law has the usual to the Navier-Stokes equations that form) to a frame of reference fixed in observations, a major step in the space (the “Eulerian” frame, in which accurately predict and/or describe the description of fluid flows was the other aspects of the mathematics are sim- transition to turbulence and the turbu- development of the basic dynamical pler). Specifically, acceleration of the lent state itself are legendary, prompt- fluid is, by definition, the second time derivative of the Lagrangian fluid trajec- ing the British physicist Sir Horace tory x(t), which describes the motion of Lamb to remark, “I am an old man 1 Reynolds rewrote the Navier-Stokes the fluid element that was initially at fluid equation as two equations—one for position x(0). The first time derivative is the mean velocity, which includes a quad- the Lagrangian fluid velocity, dx(t)/dt, ratic term in the fluctuating velocity called which is related to the Eulerian fluid 3 Often, there is an additional term added the Reynolds stress, and one for the fluc- velocity by dx(t)/dt = u(t,x(t)). Because u to the right side of the equation that repre- tuations, which is usually modeled by is a function of time t and position x(t), sents an external forcing of the flow per some suitable approximation. This which itself is a function of time, the unit volume such as by gravity. approach underpins commonly used engi- Eulerian expression for the Lagrangian Alternatively, the forcing can arise from neering models of turbulent fluid motion second time derivative (the fluid accelera- the imposition of boundary conditions, known as Reynolds-Averaged Navier- tion) is obtained through the chain rule whereby energy is injected by stresses at Stokes (RANS)—refer to Taylor (1938). and equals du/dt = ∂u/∂t + u ⋅ ∇u. those boundaries.

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dynamics. His contributions include (a) (b) field-theoretic approaches to turbulence that have had recent stunning success when applied to the turbulent transport of passive scalar concentration (see the article “Field Theory and Statistical Hydrodynamics” on page 181).

What Is Turbulence?

So, what is turbulence and why is it so difficult to describe theoretically? (c) In this article, we shall ignore the transition to turbulence and focus instead on fully developed turbulence. One of the most challenging aspects is that, in fully developed turbulence, the velocity fluctuates over a large range of coupled spatial and temporal scales. Examples of turbulence (Figure 1) are everywhere: the flow of water from a common faucet, water from a garden hose, the flow past a (d) (e) curved wall, and noisy rapids resulting from flow past rocks in an ener- getically flowing river. Another example is the dramatic pyroclastic flow in a volcanic eruption. In Figure 2, the explosive eruption of Mount St. Helens is illustrated at successively higher magnification, show- ing structure at many length scales. In all these examples, large velocity differences (as opposed to large velocities) resulting from shear forces applied to the fluid (or from intrinsic Figure 1. Common Examples of Fluid Turbulence fluid instability) produce strong fluid Turbulence is commonly apparent in everyday life, as revealed by the collage of pic- turbulence, a state that can be defined tures above: (a) water flow from a faucet, (b) water from a garden hose, (c) flow past as a solution of the Navier-Stokes a curved wall, and (d) and (e) whitewater rapids whose turbulent fluctuations are so equations whose statistics exhibit spa- intense that air is entrained by the flow and produces small bubbles that diffusely tial and temporal fluctuations. reflect light and cause the water to appear white. Historically, investigations of turbulence have progressed through now, and when I die and go to heaven the most influential turbulence theo- alternating advances in experimental there are two matters on which I hope rists in the last 40 years, Robert measurements, theoretical descrip- for enlightenment. One is quantum Kraichnan, started studying turbulence tions, and most recently, the introduc- electrodynamics, and the other is the while working with Albert Einstein at tion of numerical simulation of turbulent motion of fluids. And about Princeton, when he noticed the simi- turbulence on high-speed computers. the former I am rather optimistic”— larity between problems in gravita- Similarly, there has been a rich inter- 1932 (in Tannehill et al. 1984). One of tional field theory and classical fluid play between fundamental understand-

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ing and applications. For example, tur- (a) (b) (c) bulence researchers in the early to mid 20th century were motivated by two important practical problems: predicting the weather and building ever more sophisticated aircraft. Aircraft development led to the construction of large wind tunnels, where measurements of the drag and lift on scaled model aircraft were used in the design Figure 2. Scale-Independence in Turbulent Flows of airplanes. On the other hand, The turbulent structure of the pyroclastic volcanic eruption of Mt. St. Helens shown weather prediction was severely ham- in (a) is expanded by a factor of 2 in (b) and by another factor of 2 in (c). The char- pered by the difficulty in doing numer- acteristic scale of the plume is approximately 5 km. Note that the expanded images ical computation and was only made reveal the increasingly finer scale structure of the turbulent flow. The feature of practical after the development, many scale independence, namely, that spatial images or temporal signals look the same decades later, of digital computers; in (statistically) under increasing magnification is called self-similarity. the early days, the calculation of the weather change in a day required mating the water pressure at cous term and then equate the nonlin- weeks of calculation by hand! In addi- 30 pounds per square inch (psi), ear term (u · ∇u ~ U2/D) to the pres- tion to these two large problems, many the imposed pressure gradient ∇P is sure gradient, thereby obtaining the other aspects of turbulent flow were 0.1 psi/cm or 7000 dynes/cm3. We much more realistic estimate of investigated and attempts were made assume the simplest case, namely, that U ~ (∇PD/ρ)1/2 ~ 1.5 m/s. This esti- to factor in the effects of turbulence on the flow is smooth, or “laminar,” so mate actually overestimates the the design and operation of real that the nonlinear term in Equation (1) effects of the nonlinear term. machines and devices. can be neglected, and that the flow As illustrated in Figure 3, the solu- To understand what turbulence is has reached its limiting velocity with tion for the laminar-flow velocity pro- and why it makes a big difference in ∂U/∂t = 0. In that case, the density- file is quite gradual, whereas the practical situations, we consider flow normalized pressure gradient turbulent velocity profile is much through a long cylindrical pipe of ∇P/ρ would be balanced by the vis- steeper at the walls and levels off in diameter D, a problem considered cous acceleration (or drag), ν∇2u. the center of the pipe. Evidently, the over a century ago by Osborne Using dimensional arguments and effect of turbulence is to greatly Reynolds (1894). Reynolds measured taking into account that u = 0 at the increase the momentum exchange in mean quantities such as the average pipe wall, we estimate that the central regions of the pipe, as flow rate and the mean pressure drop ν∇2u ≈ νU/D2, which yields the esti- large-scale eddies effectively ‘lock in the pipe. A practical concern is to mate for the mean flow velocity of up’ the flow and thereby shift the determine the flow rate, or mean U ~ ∇PD2/ρν. Thus, for water with velocity gradient (or velocity shear) velocity U, as a function of the viscosity ν = 0.01 cm2/s flowing in a closer to the wall. Because the flow applied pressure, and its profile, as a pipe with diameter D = 2.5 centime- resistance in the pipe is proportional function of distance from the wall. ters, the laminar flow velocity would to the steepness of the velocity profile Because the fluid is incompressible, reach U ~ 40,000 m/s or almost near the wall, the practical conse- the volume of fluid entering any cross 30 times the speed of sound in water! quence of the turbulence is a large section of the pipe is the same as the Clearly, something is wrong with this increase in the flow resistance of the volume flowing out of the pipe. Thus, argument. It turns out that the flow in pipe—that is, less flow for a given the volume flux is constant along the such a pipe is turbulent (it has highly applied pressure. The real-world flow direction. We can use irregular spatial and temporal velocity implications of this increase in flow Equation (1) to get a naive estimate of fluctuations) and the measured mean resistance are enormous: A large frac- the mean velocity U for flow in a hor- flow velocity U is smaller by a factor tion of the world’s energy consump- izontal pipe. Consider as a concrete of about 4000, or only about 10 m/s! tion is devoted to compensating for example the flow of water in a rigid How can we improve our estimate? turbulent energy loss! Nevertheless, pipe hooked up to the backyard water For the turbulent case, we might the detailed understanding and predic- faucet. Taking a 3-meter length of a assume, as Reynolds did, that the non- tion from first principles still elude 2.5-centimeter diameter pipe and esti- linear term dominates over the vis- turbulence theory.

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because the turbulent state changes rapidly in space and time. Taylor proposed a probabilistic/statistical approach based on averaging over ensembles of individual realizations, although he soon replaced ensemble averages by time averages at a fixed point in space. Taylor also used the idealized concept (originally introduced by Lord Kelvin in 1887) of statistically homogeneous, isotropic turbulence. Homogeneity and isotropy imply that spatial translations and rota- tions, respectively, do not change the average values of physical variables.5 Lewis F. Richardson was another Figure 3. Mean Velocity Profiles for Laminar and Turbulent Pipe Flow influential fluid dynamicist of the early The velocity profile across the diameter (D = 2R where R is the radius) of a pipe for 20th century. Richardson performed laminar-flow conditions (black curve) shows a gradual velocity gradient compared the first numerical computation for with the very steep gradients near the walls resulting from turbulent flow conditions (red curve). Those steep gradients are proportional to the flow resistance in the predicting the weather (on a hand cal- pipe. Thus turbulence results in significantly less flow for a given applied pressure. culator)! He also proposed (1926) a pictorial description of turbulence called a cascade, in which nonlinearity The example of pipe flow illus- fully developed turbulence when Re is transforms large-scale velocity circula- trates an important feature of turbu- large compared with the Re for transi- tions (or eddies, or whorls) into circu- lence—the ratio of the nonlinear term tion to turbulence for a particular set lations at successively smaller scales to the viscous dissipation term pro- of forcing and boundary conditions. (sizes) until they reach such a small vides a good measure of the strength For example, in the problem above, scale that the circulation of the eddies of turbulence. In fact, this ratio, where D = 2.5 cm and U = 10 m/s, is efficiently dissipated into heat by Re = UD/ν, where D is the size of the the Reynolds number is Re ~ 3 × 105 viscosity. Richardson captured this large-scale forcing (typically shear), is compared with a typical Re ~ 2000 for energy cascade in a poetic take-off on known as the Reynolds number after the onset of turbulence in pipe flow. Jonathan Swift’s famous description of Reynolds’ seminal work on pipe flow fleas.6 In Richardson’s words, “Big (1894). For a small Reynolds number, whorls have little whorls that feed on Re << 1, the nonlinearity can be neg- The Search for Universal their velocity, and little whorls have lected, and analytic solutions to the Properties and the lesser whorls and so on to viscosity” Navier-Stokes equation corresponding Kolmogorov Scaling Laws (circa 1922). A schematic illustration to laminar flow can often be found. of the energy cascade picture is shown When Re >> 1, however, there are no In early laboratory experiments on in Figure 4, where the mean energy stable stationary solutions,4 and the turbulence, Reynolds and others sup- fluid flow is highly fluctuating in plemented their measurements of 5 space and time, corresponding to tur- applied pressure and average velocity Many theoretical descriptions use these assumptions, but typical turbulence bulent flow. In particular, the flow is by observing the rapidly fluctuating encountered in the real world often obeys character of the flow when they used neither condition at large scales. A key dyes and other qualitative flow-visuali- question in real-world situations is 4 How large Re must be to get a turbulent whether the assumptions of homogeneity state depends on the particular source of zation tools. In the atmosphere, how- and isotropy are satisfied at small scales, forcing and on the boundary conditions. ever, one could measure much thus justifying application of a general For example, the transition to turbulence framework for those smaller scales. in pipe flow can occur anywhere in the longer-term fluctuations, at a fixed range 1000 < Re < 50,000 depending on location, and such Eulerian measure- 6 “So, the naturalists observe, the flea,/ inlet boundary conditions and the rough- ments intrigued the young theoretical hath smaller fleas that on him prey;/ And ness of the pipe wall. For most com- these have smaller still to bite ‘em;/ And monly encountered conditions, the physicist G. I. Taylor (1938). so proceed, ad infinitum.”—Jonathan transition is near Re = 2000. Turbulence is difficult to measure Swift, Poetry, a Rhapsody

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injection rate ε at large scales is balanced by the mean energy dissipation rate at small scales. Richardson and Taylor also appreciated that generic properties of turbulence may be discovered in the statistics of velocity differences between two locations separated by a distance r, denoted as δu(x, x+r) = u(x) – u(x+r). The statistics of velocity differences at two locations are an improvement over the statistics of velocity fluctuations at a single location for a number of techni- cal reasons, which we do not discuss here. Longitudinal projections of velocity differences

Figure 4. The Energy Cascade Picture of Turbulence are often measured in modern experi- This figure represents a one-dimensional simplification of the cascade process with ments and are one of the main quanti- β representing the scale factor (usually taken to be 1/2 because of the quadratic ties of interest in the analysis of fluid nonlinearity in the Navier-Stokes equation). The eddies are purposely shown to be turbulence. “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. Measuring velocity differences on Cambridge, UK: Cambridge University Press.) fast time scales and with high preci- sion was a difficult proposition in the early 20th century and required the development of the hot-wire anemometer, in which fluid flowing past a thin heated wire carries heat away at a rate, proportional to the fluid velocity. Hot-wire anemometry made possible the measurement, on laboratory time scales, of the fluctuating turbulent velocity field at a single point in space (see Figure 5). For a flow whose mean velocity is large, velocity differences were inferred from those single-point measurements by Taylor’s “frozen-turbulence” hypothesis.7

7 If the mean velocity is large compared with velocity fluctuations, the turbulence can be considered “frozen” in the sense that velocity fluctuations are swept past a single point faster than they would change because of turbulent dynamics. In that case, the spatial separation ∆r is related to Figure 5. Time Series of Velocities in a Turbulent Boundary Layer the time increment ∆t by ∆r = – U∆t, This time series of velocities for an atmospheric turbulent boundary layer with where U is the mean velocity. See also the 7 article “Taylor’s Hypothesis, Hamilton’s Reynolds number Re ~ 2 × 10 was measured at a single location with a hot-wire Principle, and the LANS-α Model for anemometer. The velocity fluctuations are apparently random. (This figure is courtesy of Computing Turbulence” on page 152 . Susan Kurien of Los Alamos, who has used data recorded in 1997 by Brindesh Dhruva of Yale University.)

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that is, an average over many statistically independent realizations of the flow. The two-point velocity correlation functions cannot describe universal features of turbulence because they are scale dependent (the large- scale flow dominates their behavior) and they lack Galilean invariance. Nevertheless, their derivation inspired a real breakthrough. In 1941, Andrei Kolmogorov recast the Kármán- Howarth equation in terms of the moments of δu(r), the velocity differences across scales, thereby producing a relationship between the second moment 〈[δu(r)]2〉 and the third moment 〈[δu(r)]3〉. These statistical objects, which retain Galilean invariance and hence hold the promise of universality, are now known as structure functions. Kolmogorov then proposed the notion of an “inertial range” of scales based on Richardson’s picture of the energy cascade: Kinetic energy is injected at the largest scales L of the flow at an average rate ε and gener- ates large-scale fluctuations. The Figure 6. Kolmogorov-like Energy Spectrum for a Turbulent Jet injected energy cascades to smaller The graph shows experimental data for the energy spectrum (computed from veloc- scales via nonlinear inertial (energy- ity times series like that in Figure 5) as a function of wave number k, or E(k) vs k, conserving) processes until it reaches for a turbulent jet with Reynolds number Re ~ 24,000. Note that the measured spec- a scale of order ld, where viscous dis- trum goes as E(k) ∝ k–5/3. (Champagne 1978. Redrawn with the permission of Cambridge University Press.) sipation becomes dominant and the kinetic energy is converted into heat. In other words, the intermediate spa- Single-point measurements of tur- whole velocity fields. For now, we con- tial scales r, in the interval ld << r << bulent velocity fluctuations have been sider results that were motivated or L, define an inertial range in which performed for many systems and have measured with the limitations of single- large-scale forcing and viscous forces contributed both to a fundamental point experiments in mind. have negligible effects. With these understanding of turbulence and to assumptions and the Kármán-Howarth engineering modeling of the effects of The Kolmogorov Scaling Laws. equation recast for structure functions, turbulence at small scales on the flow In 1938, von Kármán and Howarth Kolmogorov derived the famous at larger scales. (See the section on derived an exact theoretical relation “four-fifths law.” The equation defin- engineering models.) Single-point for the dynamics of turbulence statis- ing this law describes an exact rela- measurements of velocity fluctuations tics. Starting from the Navier-Stokes tionship for the third-order structure have been the primary tool for investi- equation and assuming homogeneity function within the inertial range: gating fluid turbulence.They remain in and isotropy, the two scientists common use because of their large derived an equation for the dynamics dynamic range and high signal-to-noise of the lowest-order two-point velocity ratio relative to more modern develop- correlation function. (This function is where ε is assumed to be the finite ments such as particle image velocime- 〈u(x) · u(x+r)〉, where the angle energy-dissipation rate (per unit mass) try, in which the goal is to measure brackets denote an ensemble average, of the turbulent state. This relation-

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ship is a statement of conservation of to viscous forces introduced earlier in is not independent of the picture pre- energy in the inertial range of scales the context of pipe flow. Most theo- sented above in terms of real space- of a turbulent fluid; the third moment, ries of turbulence deal with asymptot- velocity differences but is another which arises from the nonlinear term ically large Re, that is, Re → ∞, so way of looking at the consequences of in Equation (1), is thus an indirect that an arbitrarily large range of scales energy conservation. Many subse- measure of the flux of energy through separates the injection scales from the quent experiments and numerical sim- spatial scales of size r. (High dissipation scales. ulations have observed this Reynold’s-number numerical simula- Because energy cascading down relationship to within experimental/ tions are compared with the four-fifths through spatial scales is a central fea- numerical uncertainty, thereby lending law and with other statistical charac- ture of fluid turbulence, it is natural to credence to the energy cascade pic- terizations of turbulence in the article consider the distribution of energy ture. Figure 6 shows the energy spec- “Direct Numerical Simulations of among spatial scales in wave number trum obtained from time series Turbulence” on page 142.) (or Fourier) space, as suggested by measurements at a single point in a Kolmogorov further assumed that Taylor (1938). The energy distribution turbulent jet, where the spatial scale is the cascade process occurs in a self- E(k) = 1/2|u~(k)|2, where u~(k) is the related to time by Taylor’s hypothesis similar way. That is, eddies of a given Fourier transform of the velocity field that the large mean velocity sweeps size behave statistically the same as and the wave number k is related to the “frozen-in” turbulent field past the eddies of a different size. This the spatial scale l by k = 2π/l.8 Wave- measurement point. assumption, along with the four-fifths number space is very useful for the law, gave rise to the general scaling representation of fluid turbulence Vorticity. Another important prediction of Kolmogorov, which because differential operators in real quantity in the characterization and states that the nth order structure func- space transform to multiplicative understanding of fluid turbulence is tion (referred to in the article “Direct operators in k-space. For example, the the vorticity field, ω(x,t) = ∇ × u(x,t), Numerical Simulations of Turbulence” diffusion operator in the term ν∇2u which roughly measures the local n/3 2~ as Sn(r)) must scale as r . During the becomes νk u in the Fourier repre- swirl of the flow as picturesquely decades that have passed since sentation. Another appealing feature drawn by da Vinci in the opening Kolmogorov’s seminal papers (1941), of the wave number representation is illustration. The notion of an “eddy” empirical departures from his scaling the nonlocal property of the Fourier or “whorl” is naturally associated prediction have been measured for n transform, which causes each Fourier with one’s idea of a vortex—a com- different from 3, leading to our pres- mode represented by wave number k pact swirling object such as a dust ent understanding that turbulent scales to represent cleanly the corresponding devil, a tornado, or a hurricane—but are not self-similar, but that they scale l. On the other hand, the k-space this association is schematic at best. become increasingly intermittent as representation is difficult from the In three-dimensional (3-D) turbu- the scale size decreases. The charac- perspective of understanding how spa- lence, vorticity plays a quantitative terization and understanding of these tial structures, such as intense eddies, role in that the average rate of energy deviations, known as the “anomalous” affect the transfer of energy between dissipation ε is related to the mean- scaling feature of turbulence, have scales, that is between eddies of dif- square vorticity by the relation ε = – been of sustained and current interest ferent sizes. u〈ω2〉. Vorticity plays a different (see the box “Intermittency and The consequence of energy conser- role in two-dimensional (2-D) turbu- Anomalous Scaling in Turbulence” on vation on the form of E(k) was inde- lence. Vortex stretching has long been page 136). pendently discovered by Obukov recognized as an important ingredient Empirical observations show that a (1941), Heisenberg (1948), and in fluid turbulence (Taylor 1938); if a flow becomes fully turbulent only Onsager (1949), all of whom obtained vortex tube is stretched so that its when a large range of scales separates the scaling relationship for the energy cross section is reduced, the mean- the injection scale L and the dissipa- spectrum E(k) ~ k–5/3 for the inertial square vorticity in that cross section tion scale ld. A convenient measure of scales in fully developed homoge- will increase, thereby causing strong this range of spatial scales for fluid neous isotropic turbulence. This result spatially localized vortex filaments turbulence, which also characterizes that dissipate energy. The notion of the number of degrees of freedom of 8If isotropy is assumed, the energy distri- vortex stretching and energy dissipa- the turbulent state, is the large-scale bution E(k), where k is a vector quantity, tion is discussed in the article “The depends only on the magnitude k = |k| and Reynolds number, Re. The Reynolds one can denote the energy as E(k) without LANS-α model for Computing number is also the ratio of nonlinear loss of generality. Turbulence” on page 152.

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Engineering Models The Closure Problem for and averaging term by term yields an of Turbulence Engineering Models. Engineering equation for the mean velocity: models are constructed for computa- It is worth stressing again that tur- tional efficiency rather than perfect bulence is both fundamentally inter- representation of turbulence. The class esting and of tremendous practical of engineering models known as (2) importance. As mentioned above, Reynolds-Averaged Navier-Stokes modeling complex practical problems (RANS) provides a good example of requires a perspective different from how the problem of “closure” arises that needed for studying fundamental and the parameters that need to be issues. Foremost is the ability to get determined experimentally to make Note that the equation for the mean fairly accurate results with minimal those models work. Consider again flow looks the same as the Navier- computational effort. That goal can the flow of a fluid with viscosity ν in Stokes equation for the full velocity u, often be accomplished by making a pipe with a pressure gradient along Equation (1), with the addition of a simple models for the effects of turbu- the pipe. When the pressure applied at term involving the time average of a lence and adjusting coefficients in the the pipe inlet and the pipe diameter D product of the fluctuating parts of the model by fitting computational results are small, the fluid flow is laminar, velocity, namely, the Reynolds stress to experimental data. Provided that and the velocity profile in the pipe is tensor, the parameter range used in the model quadratic with a peak velocity U that is well covered by experimental data, is proportional to the applied pressure this approach is very efficient. (refer to Figure 3). When the forcing Examples that have benefited from pressure gets large enough to produce That additional term, which repre- copious amounts of reliable data are a high flow velocity and therefore a sents the transport of momentum aircraft design—aerodynamics of large Reynolds number, typically Re = caused by turbulent fluctuations, acts body and wing design have been at UD/ν ≥ 2000, the flow becomes tur- like an effective stress on the flow and the heart of a huge international bulent, large velocity fluctuations are cannot, at this time, be determined industry—and aerodynamic drag present in the flow, and the velocity completely from first principles. As a reduction for automobiles to achieve profile changes substantially (refer result, many schemes have been better fuel efficiency. Global climate again to Figure 3). An engineering developed to approximate the modeling and the design of nuclear challenge is to compute the spatial dis- Reynolds stress. weapons, on the other hand, are tribution of the mean velocity of the The simplest formulation for the examples for which data are either turbulent flow. Following the proce- Reynolds stress tensor is impossible or quite difficult to obtain. dure first written down by Reynolds, In such situations, the utmost care the velocity and pressure fields are needs to be taken when one attempts separated into mean (the overbar to extrapolate models to circum- denotes a time average) and fluctuat- where νT(x) is called the turbulent stances outside the validation regime. ing (denoted by the prime) parts: eddy viscosity because the additional The main goal of many engineering term looks like a viscous diffusion models is to estimate transport proper- term. A more sophisticated approach ties—not just the net transport of is to solve for the Reynolds stress by energy and momentum by a single fluid where i denotes one of the compo- writing an equation for the time evo- but the transport of matter such as pol- nents of the vector field u(x,t) and the lution of ui′uj′ (Johnson et al. 1986). lutants in the atmosphere or the mixing average of the fluctuating part of the This equation has multiple undeter- of one material with another. Mixing is velocity is zero by definition. Also, mined coefficients and depends on the a critical process in inertial confine- third moment ui′uj′uk.′ . Again, the third- ment fusion and in weapons physics order moment is unknown and needs applications. It is crucial for certifica- to be approximated or written in terms tion of the nuclear weapons stockpile Substituting these expressions into of fourth-order moments. In principle, that scientists know how well engineer- Equation (1), using the constant-den- an infinite set of equations for higher- ing models are performing and use that sity continuity condition order moments is required, so one knowledge to predict outcomes with a needs to “close” the set at a small known degree of certainty. number to achieve computational effi-

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ciency. At any stage of approximation, enhancement in the quality and quan- Below, we briefly address undetermined coefficients are set by tity of experimental data achieved by advances in numerical simulation, in comparison with experimental or computer data acquisition. As far turbulence modeling, and theoretical direct numerical simulation data. This back as the Manhattan Project, the understanding of passive scalar trans- approach is often very effective, computer (more exactly, numerical port, topics dealt with more exten- although it does depend on the quality schemes implemented on a roomful sively in the articles immediately of the data and on the operating of Marchand calculators) began to following this one. We then describe parameter regime covered by the data. play a major role in the calculations several exciting new experimental of fluid problems. A leading figure in advances in fluid turbulence research that project, John von Neumann and close this introduction with a Modern Developments (1963), noted in a 1949 review of view toward “solving” at least some turbulence that “… a considerable aspect of the turbulence problem. By the end of the 1940s, great mathematical effort towards a progress had been made in the study detailed understanding of the mecha- of turbulence. The statistical approach nism of turbulence is called for” but Direct Numerical to turbulence, the importance of the that, given the analytic difficulties Simulation of Turbulence energy and its wave number represen- presented by the turbulence problem, tation, the notion of measuring veloc- “… there might be some hope to Recent advances in large-scale sci- ity differences, and the dynamics of ‘break the deadlock’ by extensive, entific computing have made possible vortex structures as an explanation of but well-planned, computational direct numerical simulations of the the mechanism of fluid turbulence had efforts.” Von Neumann’s foresight in Navier-Stokes equation under turbu- all been articulated by Taylor. The understanding the important role of lent conditions. In other words, for cascade picture of Richardson had computers for the study of turbulence simulations performed on the world’s been made quantitative by predated the first direct numerical largest supercomputers, no closure or Kolmogorov and others. The concepts simulation of the turbulent state by subgrid approximations are used to of universal scaling and self-similarity more than 20 years. simplify the flow, but rather the simu- were key ingredients in that descrip- From a fundamental perspective, lated flow follows all the twisting- tion. On the practical side, tremen- the direct numerical simulation of turning and stretching-folding motions dous advances had been made in idealized isotropic, homogeneous tur- of the full-blown Navier-Stokes equa- aeronautics, with newly emerging jet bulence has been revolutionary in its tions at effective large-scale Reynolds aircraft flying at speeds approaching impact on turbulence research numbers of about 105. These simula- the speed of sound. Empirical models because of the ability to simulate and tions render available for analysis the based on the engineering approach display the full 3-D velocity field at entire 3-D velocity field down to the described above were being used to increasingly large Reynolds number. dissipation scale. With these numeri- describe these practical turbulence Similarly, experimentation on turbu- cally generated data, one can study problems. lence has advanced tremendously by the structures of the flow and corre- The next 50 years were marked by using computer data acquisition; late them with turbulent transfer steady progress in theory and model- 20 years ago it was possible to meas- processes, the nonlinear processes that ing, increasingly sophisticated experi- ure and analyze time series data from carry energy from large to small ments, and the introduction and single-point probes that totaled no scales. widespread use of the digital com- more than 10 megabytes of informa- An especially efficient technique puter as a tool for the study of turbu- tion, whereas today statistical ensem- for studying isotropic, homogeneous lence. In the remainder of this review, bles of thousands of spatially and turbulence is to consider the flow in we touch on some of the advances of temporally resolved velocity fields, a box of length L with periodic the post-Kolmogorov era, paying par- taking 10 terabytes of storage space boundary conditions and use the ticular attention to the ever-increasing can be obtained and processed. This spectral method, an orthogonal impact of the digital computer on millionfold increase in experimental decomposition into Fourier modes, to three aspects of turbulence research: capability has opened the door to simulate the Navier-Stokes equation. direct numerical simulations of ideal- great new possibilities in turbulence Forcing is typically generated by ized turbulence, increasingly sophisti- research that will be enhanced even maintaining constant energy in the cated engineering models of further by expected future increases smallest k mode (or a small number turbulence, and the extraordinary in computational power. of the longest-wavelength modes).

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The first direct numerical simulation into Fourier modes and then to trun- Beyond the of fluid turbulence (Orszag and cate the expansion at some intermedi- Kolmogorov Theory Patterson 1972) had a resolution of ate scale (usually with a smooth 323, corresponding to Re ~ 100. By Gaussian filter) and model the small The 50 years that have passed, the early 1990s, Reynolds numbers scales with a subgrid model. One then from about 1950 until the new millen- of about 6000 for a 5123 simulation computes the large scales explicitly nium, were notable for increasingly could be obtained (She et al. 1993); and approximates the effect of the sophisticated theoretical descriptions the separation between the box size small scales with the subgrid model. of fluid turbulence, including the sem- and the dissipation scale was just This class of methods (Meneveau inal contributions of Robert short of a decade. Recent calcula- and Katz 2000) is called large eddy Kraichnan (1965, 1967, 1968, 1975), tions on the Los Alamos Q machine, simulation (LES) and has become an who pioneered the foundations of the using 20483 spatial resolution, and alternative to RANS modeling when modern statistical field-theory on the Japanese Earth Simulator with more-accurate spatial information is approach to hydrodynamics, particu- 40963 modes (Kaneda et al. 2003) required. Because of the spatial fil- larly by predicting the inverse energy achieved a Reynolds number of tering, LES modeling has problems cascade from small scales to large about 105, corresponding to about with boundaries and is less computa- scales in 2-D turbulence, and George 1.5 decades of turbulent scales, tionally efficient than RANS tech- Batchelor (1952, 1953, 1959, 1969). which is approaching fully developed niques. Nevertheless, LES models Those developments are generally turbulence in modestly sized wind may be more universal than RANS beyond the scope of the present tunnels. It is important to appreciate models and therefore rely less on ad review, and we have already referred that the Re of direct numerical turbu- hoc coefficients determined from the reader to recent books in which lence simulations grows only as Re ∝ experimental data. they are surveyed in detail (McComb N4/9, where N is the number of Another variant of the subgrid- 1990, Frisch 1995, Lesieur 1997). We degrees of freedom that are com- model approach recently invented at touch briefly, however, on a few puted: A factor of 2 increase in the Los Alamos is the Lagrangian-aver- recent developments that grew out of linear dimension of the box means aged Navier-Stokes alpha (LANS-α) those efforts and on the influence of computing 23 more modes for a cor- model. Although not obtainable by the Lagrangian perspective of fluid responding increase in Re of about filtering the Navier-Stokes equations, turbulence. 2.5. Nevertheless, for isotropic, the LANS-α model has a spatial cut- The physical picture that emerges homogeneous fully developed off determined by the coefficient α. from the Kolmogorov phenomenology turbulence, direct numerical simula- For spatial scales larger than α, the is that the turbulent scales of motion tion has become the tool of choice dynamics are computed exactly (in are self-similar; that is, the statistical for detailed characterization of effect, the Navier-Stokes equations features of the system are independent fundamental flow properties and are used) and yield the energy of spatial scale. One measure of this comparison with Kolmogorov-type spectrum E(k) ∝ k–5/3, whereas for self-similarity is the nondimensional theories. More details regarding spatial scales less than α, the energy ratio of the fourth moment to the numerical simulation of turbulence falls more rapidly, E(k) ∝ k–3. The square of the second moment, can be found in the article “Direct LANS-α model can be derived from a F = 〈δu4〉/〈δu2〉2, as a function of Numerical Simulations of Lagrangian-averaging procedure start- scale separation. If the velocity distri- Turbulence” on page 142. ing from Hamilton’s principle of least bution is self-similar, then F should action. It is the first-closure (or sub- be constant, or flat, as a function of grid) scheme to modify the nonlinear length scale. Indeed, the nondimen- Modern Turbulence Models term rather than the dissipation term sional ratio of any combination of and, as a result, has some unique velocity-increment moments should Although the RANS models advantages relative to more traditional be scale independent. If, however, F described above maintain a dominant LES schemes (see the articles “The behaves as a power law in the separa- role in turbulence modeling, other LANS-α Model for Computing tion r, then the system is not self-sim- approaches have become tractable Turbulence” and “Taylor’s ilar, but instead it is characterized by because of increases in computational Hypothesis, Hamilton’s Principle, and intermittency: short bursts (in time) or power. A more recent approach to the LANS-α Model for Computing isolated regions (in space) of high- modeling turbulent processes is to Turbulence” on pages 152 and 172, amplitude fluctuations separated by decompose spatial scales of the flow respectively). relatively quiescent periods or

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(a) (b)

Figure 7. Passive Scalar Turbulence in a Stratified Layer An effective blob of yellow dye is carried by a forced 2-D turbulent flow in a stratified layer. The images show (a) the initial dye concentration and (b) the concentration after about one rotation of a large-scale eddy. The sharp gradients in the concentration lead to the very strong anomalous scaling in the transport of the passive scalar field. regions. From the 1960s to the 1980s, velocity field. This stirring process trates how the velocity field stretches experimentalists reported departures characterizes fluid mixing, which has and folds fluid elements to produce from the Kolmogorov scaling. The many important scientific and techni- mixing. Adopting the Lagrangian measured fourth-order and higher cal applications. Whereas intermit- frame of reference is rapidly emerging moments of velocity differences did tency is rather weak in turbulent as a powerful new approach for mod- not scale as rn/3, but rather as a lower velocity statistics, the distribution of a eling turbulent mechanisms of energy power of the separation r, passive scalar concentration carried transfer. This approach has led to n ξn 〈δu 〉 ~ r , with ξn < n/3 for n > 3. by a turbulent flow is very intermit- Lagrangian tetrad methods, a phenom- To preserve the correct dimensions of tent. In other words, there is a much enological model arising from the the nth-order velocity difference larger probability (compared with nonperturbative field-theoretical moments, the deviations from what one would expect for a random, approach to turbulence, and to the Kolmogorov scaling, or from self- or Gaussian, distribution) of finding LANS-α model mentioned above. similarity, can be written as a correc- local concentrations that differ greatly tion factor given by the ratio of the from the mean value. For characteriz- large scale L to the separation r to ing fluid mixing, the Lagrangian Recent Experimental ∆n some power ∆n, or (L/ r) (see the frame of reference (which moves with Developments box “Intermittency and Anomalous the fluid element as opposed to the Scaling in Turbulence” on page 136). Eulerian frame, which is fixed in Quantitative single-point measure- Some recent analytic progress toward space) is very useful theoretically ments of velocity combined with understanding the origin of the because a passive scalar is carried by qualitative flow visualization (van observed anomalous scaling has been fluid elements. Figure 7 shows the Dyke 1982) have characterized almost made in the context of passive scalar distribution of a virtual drop of yellow all experimental measurements of turbulence and involves the applica- dye carried by a 2-D turbulent flow in fluid turbulence for most of the 20th tion of nonperturbative field-theory a stratified layer experiment.9 The century. Recently, however, new techniques to that problem (see the structured distribution of the dye illus- experimental techniques enabled by article “Field Theory and Statistical digital data acquisition, powerful Hydrodynamics” on page 181). 9The “virtual” drop consists of more than pulsed-laser technology, and fast digi- The passive scalar problem 10,000 fluid elements, whose evolution is tal imaging systems have emerged describes the transport and effective computed by solving the Lagrangian and are causing great excitement in equation dx(t)/dt = u(t,x(t)) from experi- diffusion of material by a turbulent mental velocity fields. the field of turbulence.

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Intermittency and Anomalous Scaling in Turbulence

Misha Chertkov

Intermittency is associated with the violent, atypi- where L is the integral (pumping, energy-contain- cal discontinuous nature of turbulence. When a ing) scale of turbulence. The first thing to mention signal from turbulent flow (for example, the veloc- about Equation (1) is that the viscous, Kolmogorov ity along a particular direction) is measured at a scale ld does not enter the relation in the devel- single spatial point and a sequence of times oped turbulence regime. This fact is simply related (an Eulerian measurement), the fluctuations in the to the direction of the energy cascade: On average, values appear to be random. Since any random energy flows from the large scale, where it is sequence is most naturally explained in terms of its pumped into the system, toward the smaller scale, statistical distribution, one typically determines the ld, where it is dissipated; it does not flow from the statistics by constructing a histogram of the signal. small scale. Secondly, ∆n on the right side of The violent nature of turbulence shows itself in the Equation (1) is the anomalous scaling exponent. In very special shape of the histogram, or the proba- the phenomenology proposed by Kolmogorov in bility distribution function (PDF), of the turbulent 1941, the flow is assumed to be self-similar in the velocity signal. It is typically wider than the inertial range of scales, which implies that anom- Gaussian distributions emerging in the context of alous scaling is absent, ∆n = 0, for all values of n. equilibrium statistical physics—for example, the The self-similar scaling phenomenology is an Gaussian distribution that describes the velocity of extension of the four-fifths law proven by (or the distance traveled by) a molecule undergo- Kolmogorov in 1941 for the third moment ing Brownian motion.

The PDF of the energy dissipation rate, P(ε), illustrates how far from Gaussian a turbulent distribution can be. At values far above the average, ε >> 〈ε〉, where ε ≡ ν(∇u)2, the probability distribution (See discussion of the four-fifths law in “Direct P(ε) has a stretched exponential tail, ln P(ε) ∝ –εa Numerical Simulations of Turbulence” on page (La Porta 2001). The extended tail of the turbulent 142). This law is a statement of conservation of PDF illustrates the important role played by the energy from scale to scale in the inertial regime of atypical, violent, and rare events in turbulence. homogeneous isotropic turbulence. Modern experimental and numerical tests (Frisch 1995) Intermittency has many faces. In the context of unequivocally dismiss the self-similarity assump- two-point measurements, intermittency is associ- tion, ∆n = 0, as invalid. But so far, theory is still ated with the notion of anomalous scaling. incapable of adding any other exact relation to the Statistics of the longitudinal velocity increments, celebrated four-fifths law. δu(r) (the difference in the velocity components parallel to the line separating the two points) in On the other hand, even though a comprehensive developed turbulence becomes extremely non- theoretical analysis of developed isotropic turbu- Gaussian as the scale decreases. In particular, if the lence remains elusive, there has been an important scale r separating the two points is deep inside the breakthrough in understanding anomalous scaling inertial interval, L >>r >> ld, then the nth moment in the simpler problem of passive scalar turbulence of the longitudinal velocity increment is given by (see the article “Field Theory and Statistical Hydrodynamics” on page 181). (1)

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One innovative example of a new turbulence experiment (La Porta 2001) is the use of silicon strip detec- tors from high-energy physics to track a single small particle in a turbulent flow with high Reynolds number (see Figure 8). This is an example of the direct measurement of Lagrangian (moving with the fluid) properties of the fluid flow. Because the particle trajectories are time resolved, the acceleration statistics can be obtained directly from experiment, and theoretical predictions for those statistics can be tested. Another application of new technology has made possible the local time-resolved determination of the full 3-D velocity gradient tensor10 at a Figure 8. Three-Dimensional Particle Trajectory in a Turbulent Fluid point in space (Zeff 2003). Knowing A high-speed silicon-strip detector was used to record this trajectory of a particle the local velocity gradients allows one in a turbulent fluid with Re = 63,000 (La Porta 2001). The magnitude of the instanta- to calculate the energy dissipation rate neous acceleration is color-coded. Averaging over many such trajectories allows ε and mean-square vorticity, Ω = comparison with the theory of Lagrangian acceleration statistics. 〈ω2〉/2, and thereby provide an experi- (Modified with permission from Nature. This research was performed at Cornell University.) mental measure of intense and intermittent dissipation events (see Figure 9). Both these techniques can be used to obtain large data samples that are statistically converged and have on the order of 106 data sets per parameter value. At present, however, the physical length scales accessible to these two techniques are constrained to lie within or close to the dissipation scale ld. By using holographic methods, one can obtain highly resolved, fully 3-D velocity fields (see ) Figure 10 , which allow the full turbu- Figure 9. Intermittency of Energy Dissipation and Enstrophy at lent inertial range of scales to be Re = 48,000 investigated (Zhang et al. 1997). For Time traces of the local energy dissipation ε (crosses) and enstrophy Ω = 〈ω2〉/2 holographic measurements, however, (solid curve) illustrate the very intermittent behavior of these dissipation quantities one is limited to a small number of for turbulent flow with Re = 48,000 (Zeff 2003). (Modified with permission from Nature. This such realizations, and time-resolved research was performed at the University of Maryland.) measurements are not currently achievable. Finally, for physical realizations of them with turbulent cascade mecha- 2-D flows, full-velocity fields can be nisms. The technique used to make measured with high resolution in both these measurements and those repre- 10 The velocity gradient tensor is a 3 × 3 matrix consisting of the spatial derivatives space and time (Rivera et al. 2003), sented in Figure 10 is particle-image of three components of velocity. For and the 2-D velocity gradient tensor velocimetry (PIV) or its improved example, for the velocity component ui, can be used to identify topological version, particle-tracking velocimetry the derivatives are ∂u /∂x , ∂u /∂x , and i 1 i 2 structures in the flow and correlate (PTV). Two digital images, taken ∂ui/∂x3.

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entire box from which a velocity field is obtained, as shown in Figure 12(a). Notice that there are some anomalous vectors caused by bad matching that need to be fixed by some interpolation scheme. PTV, on the other hand, uses a particle-matching algorithm to track individual particles between frames. The resulting vector field is shown in Figure 12(b). The PTV method has higher spatial resolution than PIV but also greater computational-processing demands and more stringent image- quality constraints. From the PTV velocity field, the full vorticity field ω can be computed as shown in Figure 12(c). An additional advantage of the Figure 10. High-Resolution 3-D Turbulent Velocity Fields PTV approach is that, for high enough These images were obtained using digital holographic particle-imaging velocimetry. temporal resolution, individual parti- (Zhang et al. 1997. Modified with permission from Experiments in Fluids.) cle tracks can be measured over many contiguous frames, and information about Lagrangian particle trajectories can be obtained. Some 2-D particle tracks are shown in Figure 13. This capability can be combined with new analysis methods for turbulence to produce remarkable new visualization tools for turbulence. Figure 14 shows the full backward and forward time evolution of a marked region of fluid within an iden- tified stable coherent structure (a vortex). These fully resolved measurements in two dimensions will help build intuition for the eventual Figure 11. Superimposed Digital Images of a Particle Field development of similar capabilities in Two digital images of suspended particles taken 0.003 s apart are superimposed. three dimensions. Further, the physi- The first exposure is in red; the second, in blue. In PIV, the pattern of particles cal mechanisms of 2-D turbulence are over a small subregion is correlated between exposures, and an average velocity fascinating in their own right and may is computed by the mean displacement δx of the pattern. In PTV, each particle is be highly relevant to atmospheric or matched between exposures; as a result, spatial resolution is higher, and there is oceanic turbulence. no spatial averaging. These data allow one to infer the velocity field connecting the two images. The Prospects for closely spaced in time, track the small subregions of the domain, pat- “Solving” Turbulence motion of small particles that seed the terns of red particles in image 1 can flow and move with the fluid. The be matched with very similar patterns Until recently, the study of turbu- basic notion is illustrated in Figure 11, of blue particles in image 2 by maxi- lence has been hampered by limited where two superimposed digital mizing the pattern correlation. An experimental and numerical data on the images of particle fields, separated by average velocity vector for the match- one hand and the extremely intractable ∆t = 0.03 second, are shown. Within ing patterns is then calculated over the form of the Navier-Stokes equation on

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(a) (a)

(b) (b)

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.

(c)

Figure 14. Time Evolution of a Compact Distribution of 104 Points in a Coherent Vortex at Time t2 Vortex merging and splitting happen

at times t1 and t3, respectively. The color-coding of the surface represents the spatially local energy flux to 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 12. Two-Dimensional Vector Velocity and Vorticity Fields The velocities in (a) and (b) were obtained from data similar to those in Figure 11 using PIV and PTV techniques, respectively. The vorticity field in (c) is calculated from the velocity field in (b).

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the other. Today, the advent of large- Acknowledgments Champagne, F. H. 1978. The Fine-Scale scale scientific computation, combined Structure of the Turbulent Velocity Field. with new capabilities in data acquisi- Three important books on fluid J. Fluid Mech. 86: 67. tion and analysis, enables us to simu- turbulence, McComb (1990), Frisch Falkovitch, G., K. Gawedzki, and M. late and measure whole velocity fields (1995), and Lesieur (1997), published Vergassola. 2001. Particles and Fields in at high spatial and temporal resolu- in the last 20 years have helped pro- Fluid Turbulence. Rev. Mod. Phys. 73: 913. tions. Those data promise to revolu- vide the basis for this review. I would Frisch, U. 1995. Turbulence: The Legacy of tionize the study of fluid turbulence. like to thank Misha Chertkov, Greg A. N. Kolmogorov. Cambridge, UK: Further, new emerging ideas in statisti- Eyink, Susan Kurien, Beth Wingate, Cambridge University Press. cal hydrodynamics derived from field Darryl Holm, and Greg Swift for theory methods and concepts are pro- valuable suggestions. I am indebted Heisenberg, W. 1948. Zur Statistischen Theorie viding new theoretical insights into the to my scientific collaborators—Shiyi der Turbulenz. Z. Phys. 124: 628. structure of turbulence (Falkovitch et Chen, Brent Daniel, Greg Eyink, Kaneda, Y., T. Ishihara, M. Yokokawa, K. al. 2001). We will soon have many of Michael Rivera, and Peter Itakura, and A. Uno. 2003. Energy the necessary tools to attack the turbu- Vorobieff—for many exciting Dissipation Rate and Energy Spectrum in lence problem with some hope of solv- moments in learning about fluid tur- High Resolution Direct Numerical Simulations of Turbulence in a Periodic ing it from the physics perspective if bulence, and I look forward to more Box. Phys. Fluids 15: L21. not with the mathematical rigor or the in the future. I would also like to extremely precise prediction of proper- acknowledge the LDRD program at Kolmogorov, A. N. 1941a. The Local Structure ties obtained in, say, quantum electro- Los Alamos for its strong support of of Turbulence in Incompressible Viscous dynamics. Let me explain then what I fundamental research in turbulence Fluid for Very Large Reynolds Number. Dok. Akad. Nauk. SSSR 30: 9 mean by that solution. In condensed during the last 5 years matter physics, for example, the mys- ———. 1941b. Decay of Isotropic Turbulence tery of ordinary superconductivity was in an Incompressible Viscous Fluid. Dok. Akad. Nauk. SSSR 31: 538. solved by the theory of Bardeen, Further Reading Cooper, and Schrieffer (1957), which ———. 1941c. Energy Dissipation in Locally described how electron pairing medi- Isotropic Turbulence. Dok. Akad. Nauk. ated by phonons led to a Bose-Einstein Bardeen, J., L. N. Cooper, and J. R. Schrieffer. SSSR 32: 19. condensation and gave rise to the 1957. Theory of Superconductivity. Phys. Rev. 108 (5): 1175. Kraichnan, R. H. 1959. The Structure of superconducting state. Despite this Isotropic Turbulence at Very High Reynolds solution, there has been no accurate Batchelor, G. K. 1952. The Effect of Numbers. J. Fluid Mech. 5: 497. calculation of a superconducting transi- Homogeneous Turbulence on Material tion temperature for any superconduct- Lines and Surfaces. Proc. R. Soc. London, ———. 1965. Lagrangian-History Closure ing material because of complications Ser. A 213: 349. Approximation for Turbulence. Phys. Fluids 8: 575. emerging from material properties. I think that there is hope for understand- ———. 1953. The Theory of Homogeneous ———. 1967. Inertial Ranges in Two- ing the mechanisms of turbulent Turbulence. Cambridge, UK: Cambridge Dimensional Turbulence. Phys. Fluids University Press. energy, vorticity, and mass transfer 10: 1417. between scales and between points in ———. 1959. Small-Scale Variation of ———. 1968. Small-Scale Structure of a Scalar space. This advance may turn out to be Convected Quantities Like Temperature in Field Convected by Turbulence. Phys. elegant enough and profound enough Turbulent Fluid. Part 1. General Discussion Fluids 11: 945. and the Case of Small Conductivity. J. to be considered a solution to the mys- Fluid Mech. 5: 113. tery of turbulence. Nevertheless, ———. 1974. On Kolmogorov’s Inertial-Range Theories. J. Fluid Mech. 62: 305. because turbulence is probably a whole ———. 1969. Computation of the Energy set of problems rather than a single Spectrum in Homogeneous Two- Phys Fluids 12 one, many aspects of turbulence will Dimensional Turbulence. . , Supplement II: 233. likely require different approaches. It will certainly be interesting to see how Besnard, D., F. H. Harlow, N. L. Johnson, R. our improved understanding of turbu- Rauenzahn, and J. Wolfe. 1987. Instabilities lence contributes to new predictability and Turbulence. Los Alamos Science 15: 145. of one of the oldest and richest areas in physics. n

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Lamb, C. 1932. Address to the British She, Z. S., S. Y. Chen, G. Doolen, R. H. Association for the Advancement of Science. Kraichnan, and S. A. Orszag. 1993. Quoted in Computational Fluid Mechanics Reynolds-Number Dependence of Isotropic and Heat Transfer by J. C. Tannehill, D. A. Navier-Stokes Turbulence. Phys. Rev. Lett. Anderson, and R. H. Pletcher. 1984. New 70: 3251. York: Hemisphere Publishers. Taylor, G. I. 1935. Statistical Theory of La Porta, A., G. A. Voth, A. M. Crawford, J. Turbulence. Proc. R. Soc. London, Ser. A Alexander, and E. Bodenschatz. 2001. Fluid 151: 421. Particle Accelerations in Fully Developed Turbulence. Nature 409: 1017. ———. 1938. The Spectrum of Turbulence. Proc. R. Soc. London, Ser. A 164: 476. Lesieur, M. 1997. Turbulence in Fluids. Dordrecht, The Netherlands: Kluwer Van Dyke, M. 1982. An Album of Fluid Motion. Academic Publishers. Stanford, CA: Parabolic Press.

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Obukhov, A. M. 1941. Spectral Energy Zeff, B. W., D. D. Lanterman, R. McAllister, R. Distribution in a Turbulent Flow. Dok. Roy, E. J. Kostelich, and D. P. Lathrop. Akad. Nauk. SSSR 32: 22. 2003. Measuring Intense Rotation and Dissipation in Turbulent Flows. Nature 421: Onsager, L. 1945. The Distribution of Energy in 146. Turbulence. Phys. Rev. 68: 286. Zhang, J., B. Tao, and J. Katz. 1997. Turbulent ———. 1949. Statistical Hydrodynamics. Flow Measurement in a Square Duct with Nuovo Cimento 6: 279. Hybrid Holographic PIV. Exp. Fluids 23: 373. Orszag, S. A., and G. S. Patterson. 1972. In Statistical Models and Turbulence, Lecture Notes in Physics, Vol. 12, p. 127. Edited by For further information, contact M. Rosenblatt and C. M. Van Atta. Berlin: Robert Ecke (505) 667-6733 Springer-Verlag. ([email protected]). Reynolds, O. 1895. On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion. Philos. Trans. R. Soc. London, Ser. A 186: 123.

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