grupo01-lagrangian investigations of turbulence

29 Oct 2001 19:3 AR AR151-06.tex AR151-06.SGM ARv2(2001/05/10)P1: GSR

Annu. Rev. Fluid Mech. 2002. 34:115–42Copyright c© 2002 by Annual Reviews. All rights reserved

LAGRANGIAN INVESTIGATIONS OF TURBULENCE

P. K. YeungSchool of Aerospace Engineering, Georgia Institute of Technology, Atlanta,Georgia 30332; e-mail: [email protected]

Key Words Lagrangian statistics, scaling, mixing, dispersion, particle tracking

■ Abstract A Lagrangian description of turbulence has unique physical advantagesthat are especially important in studies of mixing and dispersion. We focus on funda-mental aspects, using mainly data from direct numerical simulations capable of greatdetail and precision when specific accuracy requirements are met. Differences betweentime evolution in Eulerian and Lagrangian frames illustrate the dominance of advectivetransport. We examine basic results in Kolmogorov similarity, giving an estimate of aninertial-range universal constant and the grid resolution and Reynolds number neededto attain the requisite scaling range of time lags. The Lagrangian statistics of passivescalars are discussed in view of current efforts in model development, with differentialdiffusion between multiple scalars being characterized by shorter timescales. We alsonote the need for new data in more complex flows and in other applications where aLagrangian viewpoint is especially useful.

1. INTRODUCTION

1.1. Motivation

It is well known that important transport processes in turbulence, including themixing of passive scalars and the dispersion of contaminants, are dominated bythe advective action of disorderly velocity fluctuations in time and space. As aresult, a Lagrangian viewpoint following the motion of infinitesimal material fluidelements (which by definition move with the local instantaneous flow) is concep-tually natural and practically useful for describing turbulent transport. Importantearly contributions in this line of research include those of Taylor (1921) who stud-ied the statistics of displacement of a single fluid particle, Richardson (1926) whostudied the dispersion of particle pairs relative to each other, and Batchelor (1949,1952) who established formal connections between the statistics of fluid particlemotion and the concentration field of a diffusing contaminant. Many classical ideaswere also summarized in Monin & Yaglom (1971, 1975).

It can be seen from two relatively recent reviews in this series (Pope 1994,Sawford 2001) that Lagrangian stochastic models of fluid behavior are increas-ingly important in problems of turbulent mixing and dispersion. Knowledge of theLagrangian history of passive scalar concentration following fluid particle paths

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is, in particular, critical in the probability density function (PDF) approach (Pope1985, Dopazo 1994) for the modeling of turbulent mixing, although it is currentlynot well understood. Similarly, in dispersion problems, both the displacement ofcenter of mass and the increase in size measured in length, area, and volume of a“cloud” of contaminants emerging from a localized source can be captured directlyvia the motion of fluid particles taken singly, in pairs, and in groups of three orfour. Pope (1994) reviewed the use of Lagrangian PDF methods in modeling, withemphasis on the velocity field, whereas Sawford (2001) focused on two-particlerelative dispersion, which is also closely related to the modeling of concentrationvariance (e.g., Thomson 1990) as a measure of the stochastic variability of harmfulpollutants in air-quality applications (Weil et al. 1992).

1.2. The Contents of This Review

In this review we emphasize fundamental aspects and recent progress in investigat-ing turbulence and turbulent mixing from a Lagrangian viewpoint. Traditionally,progress in this area has been hampered by a general lack of Lagrangian experi-mental data, which, despite some recent advances (Voth et al. 1998, Ott & Mann2000), is still very difficult to obtain. However, Lagrangian data can be extractedfrom direct numerical simulations (DNS) with relative ease, the first such effortbeing that of Riley & Patterson (1974). Advances in computational power in thelast 15 years have spurred more of such numerical investigations (e.g., Yeung& Pope 1989; Bernard & Handler 1990; Squires & Eaton 1991; Kontomaris &Hanratty 1992; Yeung 1994, 1997, 2001; Kimura & Herring 1996). In many casesthe DNS data were sufficiently detailed and precise for use in model developmentand evaluation (e.g., Sawford 1991, Heppe 1997, Yeung & Borgas 1998). Althoughbrief attention is given to other methods, here we rely on, in part out of necessity,DNS as the primary source of Lagrangian data.

It is perhaps fair to say that the Lagrangian approach is used only in a rela-tively small fraction of research papers on turbulence and, except for the idea ofthe material derivative, hardly mentioned in general textbooks on fluid mechan-ics. It is thus instructive to highlight important differences between turbulencestatistics in Eulerian versus Lagrangian reference frames. Such a comparison alsoprovides quantitative measures bearing on the validity of the “random sweeping”hypothesis of Tennekes (1975), which can be viewed as an extension of Taylor’sfrozen-turbulence approximation commonly used to obtain spatial derivatives inexperimental work. For similar reasons, we give special attention to extensionsof Kolmogorov’s similarity hypotheses to a Lagrangian frame (Monin & Yaglom1971, 1975), which generally have not been tested adequately in the literature.Reynolds number requirements for quantities important in modeling are also ad-dressed here.

Compared to those for the velocity field, Lagrangian time series of passivescalars are, as noted by Pope (1994), less understood and harder to model. Theyare also virtually unmeasurable in the laboratory, making DNS (Yeung 2001) even

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LAGRANGIAN INVESTIGATIONS OF TURBULENCE 117

more important as a tool for basic understanding. On the other hand, new stochasticmodels (Fox 1997, 1999) have been proposed, which can be improved with theaid of newly available Lagrangian data from DNS. Additional challenges do arisefor multiple scalars of different molecular diffusivities undergoing differentialdiffusion (Bilger & Dibble 1982) relative to each other, and for problems withnonpremixed configurations (Kronenburg & Bilger 1997) where unsteady andnon-Gaussian effects are important.

Whereas most of the information presented in this review is concerned with fluidparticles and passive scalars in homogeneous turbulence without body forces, werecognize that the scope for Lagrangian studies of turbulence is much wider. Oneobvious goal of future research would be to study more general flows where existingdata are very sketchy. In addition, we also discuss briefly other applications such asthe study of local fluid element deformation, the motion and contact rate betweendifferent species of marine micro-organisms in turbulent flow, and the dispersionof foreign species molecules undergoing Brownian motion relative to the fluid.Related topics in multiphase flows with suspended gas bubbles or solid particlesthat may in turn modify the turbulence are covered in other recent reviews (e.g.,Loth 2000).

As a note in passing, despite difficulties in quantitative measurements,Lagrangian trajectories in the form of streaklines and pathlines are, with the aidof smoke or weakly diffusing color dyes, paradoxically easier to visualize thaninstantaneous Eulerian streamlines. Although single-particle paths are typicallysmooth and nonintermittent (Borgas 1993), multiparticle statistics, such as thosein the relative dispersion of particle pairs (Yeung 1994), are characterized by strongintermittency associated with the small scales. Fluid element deformation in turbu-lence can be easily visualized on a macroscopic level, via a dye streak in a turbulentstream and via the tendency of an initially plane material surface (consisting of aset of fluid particles as material points) to become rapidly distorted. The severityof this shape distortion clearly favors the use of statistical (e.g., Pope et al. 1989),rather than deterministic approaches.

The following sections are organized as follows. In Section 2 we introducesome basic notation and discuss the use of different approaches for obtaining La-grangian data in light of requirements for ensuring high-quality results. In Section 3we compare the properties of Eulerian time series recorded at a fixed point inspace with Lagrangian data taken along the paths of marked fluid particles. Anassessment of current evidence for Lagrangian similarity hypotheses and the con-ditions required for them to hold is given in Section 4. In Section 5 we focuson passive scalars in isotropic turbulence. To some extent, the Lagrangian per-spective taken in Sections 4 and 5 may be regarded as complementary to recentreviews on similarity scaling (Sreenivasan & Antonia 1997) and passive scalarmixing (Warhaft 2000), both of which emphasized issues of Eulerian structurein space. Section 6 concerns Lagrangian data in more complex flows and in sev-eral other problems of recent interest. Finally, concluding remarks are given inSection 7.

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2. BACKGROUND AND METHODS

The Lagrangian description of fluid mechanics is based on an observer followingthe trajectories of fluid particles, which are moving mathematical fluid pointsof infinitesimal size but large compared to molecular dimensions as required bycontinuum assumptions. Each particle may be identified by its originating positiony at a reference time, usually taken ast = 0. Flow properties are expressed asfunctions of the initial particle position and current timet: e.g., the instantaneousposition isx+(y, t), which evolves simply as

∂x+(y, t)

∂t= u+(y, t), (1)

where the superscript+ denotes Lagrangian quantities andu+ is an instantaneousvelocity, which may consist of nontrivial mean and fluctuating contributions. Tosimplify the notation it is common to omit the initial positiony = x+(y, 0), whichis statistically immaterial if the turbulence is homogeneous.

2.1. Numerical Simulations

A fundamental relationship between Eulerian and Lagrangian frames of referenceis

u+(y, t) = u(x+(y, t), t) , (2)

i.e., the Lagrangian fluid particle velocity is given by the Eulerian velocity field(u) taken at the instantaneous particle position, which is, in general, randomlydistributed in space. The calculation ofu+(t) thus involves two key ingredients:namely, the availability of the instantaneous Eulerian velocity field, which is usu-ally supplied at a set of grid points in DNS, and an interpolation scheme for calcu-latingu+ from Eulerian values at adjacent grid points. Onceu+ is determined, eachparticle can be tracked forward in time by integrating Equation 1 as an ordinarydifferential equation using a time-differencing scheme consistent with the under-lying Eulerian solution in DNS. In dispersion problems it is sometimes useful (seeThomson 1990, Sawford 2001) to track particles backwards in time instead, e.g.,to determine whether fluid particles in the vicinity of a chosen observation pointoriginated from a contaminated source. However, this “backward dispersion” ismore difficult to simulate in DNS because it would require the laborious storageand recall of the velocity field at every time step archived on mass-storage devices.

Several requirements must be met in order to obtain high-quality Lagrangiandata from numerical simulations. First, a well-resolved instantaneous (not themean) velocity field evolving according to the Navier-Stokes equations (or mi-nor modifications thereof) must be available. In DNS this is little more than thatrequired for accuracy in Eulerian statistics, with the grid spacing being suffi-ciently small to resolve the small scales. However, this requirement is also relevant

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for kinematic simulations (KS) where a modeled velocity field constructed fromrandom Fourier modes is used in place ofu(x, t), above, (Fung et al. 1992,Malik & Vassilicos 1999) and for large-eddy simulations (LES) where the re-solved velocity field computed using a subgrid scale model is employed (Wanget al. 1995, Armenio et al. 1999). It is possible (and argued by the authors inthese references) that some of the Lagrangian statistics computed using KS orLES do not differ substantially from DNS. Nevertheless, for example, becausevelocity fields in KS have a modeled spatial structure and do not evolve naturallyin time, some uncertainties in the Lagrangian timescales can be expected. [Inci-dentally, the same limitation applies to theoretical models (Kraichnan 1966) basedon random velocity fields with no finite-time correlation properties.] Likewise,use of LES carries uncertainties from subgrid scale modeling, which can affectsmall-scale aspects such as Lagrangian spectra at high frequencies or particle-pair dispersion for small initial separations in problems with localized pollutantsources.

A second requirement for accuracy lies in the choice of an interpolation scheme,which also needs to be reasonably efficient when large numbers of particles aretracked. Whereas it is generally agreed that linear interpolation is inadequate,a variety of choices have been used in the literature (e.g., Yeung & Pope 1988,Kontomaris & Hanratty 1992, Rovelstad et al. 1994). In addition to formal order ofaccuracy, an important attribute is the differentiability of the approximation. In par-ticular, in piecewise-continuous approximations numerical noise is generated whenparticles cross grid-line boundaries, which can contribute to poor results when theresulting Lagrangian time series is differentiated, e.g., to obtain acceleration fromthe velocity. A scheme that is fourth-order accurate and twice-continuously differ-entiable is cubic splines, which was proposed by Yeung & Pope (1988) and alsoadopted by others (Kimura & Herring 1996).

A third requirement is statistical—namely, that a sufficient number of parti-cles with a high degree of statistical independence should be tracked and includedin ensemble averages. Although intuitively obvious, this requirement is often notevaluated rigorously. One simple test, applicable for homogeneous turbulence, is tocheck that some overall statistic, like the Lagrangian r.m.s velocity, always remainswithin a small tolerance, say 2%, of the corresponding Eulerian value. More rigor-ously, the entire population of particles can be divided into several subensemblesand the size of confidence intervals evaluated using standard methods of statisticalestimation theory. Because (short of conducting expensive multiple simulations)each particle evolves in the same velocity field, statistical independence can beachieved only approximately: by requiring that the average initial interparticlespacing be at least several grid intervals apart, so that the particle velocities arenot initially strongly correlated with each other. Because the particles should alsobe (roughly) evenly spaced throughout the solution domain, the number of par-ticles needed for a given level of statistical accuracy (as found in P.K. Yeung &M.S. Borgas 2001, unpublished) generally increases with the range of scales inspace or, equivalently, with the Reynolds number. Sampling requirements are also

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more demanding for higher-order moments sensitive to the occurrence of intense,but infrequent, events as well as for inhomogeneous turbulence. In the latter case,for instance in the boundary layer, because the initial position becomes a determin-istic parameter, ensemble averaging can, at most, be taken over particles startingon a plane at fixed distance from the wall. This implies that more particles mustbe tracked overall to ensure adequate sample sizes.

2.2. Measurements

In experiments designed to track fluid particles, the essential task is to obtainrecords of particle positions over time by optical or other advanced particle-detection techniques, whereupon Equation 1 is used in reverse to obtain the particlevelocity. Several major requirements can be identified, including the availabilityof small tracer particles that follow the flow without diffusion or buoyancy effects,the ability to “locate” a particle while maintaining its identity for a significantperiod of time spent inside a small sampling volume in space, and the numericaldifferentiability of particle tracks that clearly demands very rapid sampling rates.As can be seen from a collection of earlier papers (Snyder & Lumley 1971, Shlien& Corrsin 1974, Sato & Yamamoto 1987), none of these requirements is easyto satisfy. Because of these difficulties, indirect measurements have been made(e.g., Karnik & Tavoularis 1990) where the velocity autocorrelation is estimatedby double differentiation of mean-squared dispersion inferred from data on pas-sive temperature fluctuations. Attempts have also been made to relate Eulerianand Lagrangian quantities in a semiempirical manner (see McComb 1990), e.g.,by assuming proportionality between the Lagrangian integral timescale (TL ) andthe Eulerian eddy-turnover timeTE = L1/u′, whereL1 is the longitudinal integrallength scale andu′ is an r.m.s velocity. Some support for this type of reasoning wasprovided by Corrsin (1963), who used truncated inertial range approximations forEulerian and Lagrangian spectra to argue thatTL andTE are of the same order.However, because (especially for inhomogeneous flows) a fluid particle is expectedto encounter the effects of eddies of varying sizes as it moves, these estimates are,at best, qualitative.

Given the challenges cited in the preceding paragraph, it is encouraging tonote a recent resurgence in Lagrangian experimental measurements (Virant &Dracos 1997, Voth et al. 1998, Ott & Mann 2000, La Porta et al. 2001). Of specialpromise is a new method developed by Bodenschatz and coworkers (La Portaet al. 2001) using silicon strip detectors as optical imaging elements, which is incontrast to the particle-tracking velocimetry techniques used by other authors inexperiments at lower Reynolds number. This new silicon-detector method is basedon technology taken from high-energy physics and is capable of data rates upto 70,000 measurements per second. This high data rate is very useful at highReynolds numbers when the Kolmogorov frequency of the small scales becomesrelatively large and has made reliable fluid-particle acceleration measurementsnow possible in the laboratory.

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2.3. Discussion

It appears likely that advances in experimental techniques and supercomputertechnology will both be important for obtaining new Lagrangian data in the future.As might be expected, each method has its own limitations: e.g., considerable noiselevels in experiments and Reynolds-number restrictions in numerical simulations.Results from DNS can be made highly accurate, provided only that the usualresolution and statistical sampling requirements are met. It is true, of course, thatDNS is computationally intensive, which has sometimes been used as an argumentin favor of other simulation approaches. However, there is also the importantadditional benefit that Lagrangian data on other fluid dynamic quantities such asscalar fluctuations (see Section 5) and velocity gradients can be extracted by simpleextensions of Equation 2 (withu replaced by these other variables on both sides).These other quantities are essentially impossible to obtain using the other methodsdiscussed above.

3. EULERIAN AND LAGRANGIAN REFERENCE FRAMES

3.1. Time Series and Timescales

To initiate the discussion here, we show in Figure 1 some typical sample timeseries, both Eulerian and Lagrangian, based on DNS (Yeung 2001) in stationaryisotropic turbulence forced at low wavenumbers in the velocity field (Eswaran &Pope 1988). Scalar fluctuations at different Schmidt numbers (Sc) are sustainedby a uniform mean gradient. Selected sample traces are often useful in revealingqualitative features in the data (e.g., Ott & Mann 2000, La Porta et al. 2001) andmay suggest specific approaches for quantitative analysis. In characterizing thedata in Figure 1, one can ask at least two (related) questions: namely, how quicklydoes the “signal” change in time, and how closely are the fluctuating values atdifferent times (sayt andt + τ ) related to each other as a function of the time lag(τ ). The first of these is measured by the statistics of the time derivative, whereasthe second is best expressed by the autocorrelation function. We are interested inhow these characteristics differ between Eulerian and Lagrangian reference framesand between different turbulent flows. It may be noted that (Tennekes & Lumley1972) in incompressible homogeneous turbulence the Eulerian single-point PDFand Lagrangian one-particle PDF at a given time are equivalent. However, therespective two-time PDFs and timescales are always different.

Time derivatives in Eulerian and Lagrangian frames are given, respectively, bythe unsteady (local) time derivative (∂/∂t) taken at a fixed point in space and thematerial derivative (D/Dt) taken along the trajectory of a specified fluid particle.These are related by

D/Dt = ∂/∂t + u · ∇, (3)

with the difference between them being the convective operator. In analogy to the

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Figure 1 Comparison between Eulerian (bottom half) and Lagrangian (top half)time series for velocity component and scalar fluctuations withSc = 1/8 (φ1) andSc= 1(φ2) normalized by respective rms values. Each line with letters represents adifferent sample. The data were taken from Fourier pseudo-spectral (Rogallo 1981)DNS at 2563 grid resolution andRλ 140, with the particle tracking algorithm ofYeung & Pope (1988).

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Taylor length scale (λ) in turbulence, for a general, fluid dynamic variableX, wecan define Eulerian and Lagrangian “Taylor” timescales (also called microscales;see Tennekes 1975, Pope 1990) as

τE,X = {Var(X)/Var(∂X/∂t)}1/2; τL ,X = {Var(X)/Var(DX/Dt)}1/2 , (4)

where Var(. . .) denotes the variance of each respective quantity. (It should benoted that a factor of

√2 is used by some authors.) These timescales, however,

only represent changes over short periods of time.A more complete view of time evolution is given by the autocorrelation func-

tion. If X is statistically stationary and of meanµX and variance Var(X), theautocorrelation is an even function of time lag alone defined by

ρX(τ ) = 〈(X(t)− µX)(X(t + τ )− µX)〉Var(X)

. (5)

(Eulerian and Lagrangian autocorrelations are, of course, different; but for brevityin notation we omit additional subscriptsEandL here.) As a correlation coefficientbetweenX(t) andX(t + τ ), ρX(τ ) is unity atτ = 0 but falls to zero asτ →∞,provided only that the process{X(t)} has a finite memory. Generally, except whenX(t) is itself the time derivative of a stationary random function (e.g., accelerationin relation to velocity), the decay ofρX(τ ) is monotonic, with the integral timescaledefined by

TX =∫ ∞

0ρX(τ ) dτ (6)

giving a measure of the length of memory inX(t). Knowledge of integral timescalesis important because it provides a practical criterion for the the length of the simula-tion period, which should be long compared toTX in order to ensure independenceof samples over time. Furthermore, if the variableX(t) is close to (or modeledas such) a first-order autoregressive process (also called a linear Markov process)(Priestley 1981), the autocorrelation function becomes dependent onTX alone inthe form exp(−τ/TX). This makes integral timescales often the most importantparameters for comparisons between DNS and stochastic model results.

Considerable care must, in general, be taken (Priestley 1981, Yeung 2001)to ensure statistical accuracy in the estimation of autocorrelation functions andintegral timescales from time series of finite length. This is especially true if thetime series are not stationary, such as in decaying isotropic turbulence (Huang &Leonard 1995), homogeneous shear flow (Squires & Eaton 1991, Shen & Yeung1997), or unsteady mixing with multiple scalars evolving from segregated initialconditions. A simple way to allow for nonstationarity effects is to retain explicitdependence on botht andτ in Equation 5. Another is to render the time series ascovariance stationary (with constant first and second moments) by subtracting themean and dividing by the standard deviation. It may also be possible to define anonlinear stretching of the time axis to render the process stationary in normalizedtime.

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3.2. Quantitative results

Returning to Figure 1, it can be seen that in an Eulerian frame changes in scalarfluctuation occur more slowly than the velocity if the Schmidt number is low,but more rapidly at higher Schmidt numbers. More remarkable, however, is thecontrast in a Lagrangian frame where both scalars shown evolve very slowly,with effects of molecular diffusivity being relatively weak. Quantitatively, thisbehavior is reflected in the Taylor timescales, which are shown normalized bythe Kolmogorov timescale (τη) in Figure 2. As the Reynolds number increases,the Lagrangian Taylor scale for velocity becomes large compared to its Euleriancounterpart, meaning that the variance of the material derivative becomes smallcompared to the local time derivative. This observation is qualitatively consis-tent with the random-sweeping hypothesis of Tennekes (1975), which suggestsτE,u/τη∼

√3 butτE,u/τL ,u∝ R1/2

λ in the high-Reynolds-number limit. Other cal-culations (Tsinober et al. 2001) show that this hypothesis is quite general, beingvalid also for Gaussian fields constructed kinematically to have a specified energyspectrum without evolving according to Navier-Stokes dynamics. In addition, be-cause the Taylor timescale also determines the curvature of the autocorrelationfunction at zero time lag (Tennekes & Lumley 1972), the resultτE,u ≤ τL ,u

Figure 2 Taylor timescales (normalized byτη) at different Reynolds numbers (Rλ 38,90, 140, 240) in isotropic turbulence at four grid resolutions (643, 1283, 2563, 5123).Eulerian velocity (4), Lagrangian velocity (e); Eulerian scalar atSc= 1 (N), La-grangian scalar atSc= 1 ( u); Lagrangian difference between scalars atSc= (1/8, 1)(¥, but Sc= [1/4, 1] at Rλ 38).

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impliesρE(τ ) ≤ ρL (τ ) for the respective autocorrelations at smallτ. This resultwas also predicted theoretically by Kaneda & Gotoh (1991) for decaying isotropicturbulence. Random-sweeping properties may, however, be altered by other phys-ical processes: e.g., Rubinstein & Zhou (1999) showed that helicity causes anincrease inτE,u while having no effect onτL ,u.

The data in Figure 2 also indicate that, at a given Reynolds number, the contrastbetween Eulerian and Lagrangian Taylor scales (and rates of change) impliedby the random-sweeping hypothesis is even stronger for the scalars than for thevelocity. Indeed, a large value ofτL ,φ indicates that scalar fluctuations changelittle as they are carried along a fluid particle trajectory. However, the LagrangianTaylor timescale of the difference (z) between two scalars of different moleculardiffusivities appears to be universal when scaled byτη, as essentially a constantmultiple of τη at all Reynolds and Schmidt numbers. As noted in Yeung (2001)this is consistent with other results [e.g., the scalar-gradient cross correlation inYeung (1998) and Fox (1999)] showing that the effects of differential diffusion,although being of molecular origin, remain important at high Reynolds number—specifically at the small scales (or high frequencies) that contribute most to timeand space derivatives.

3.3. Other Properties

In addition to velocity and scalar fluctuations, the Lagrangian characteristics ofother flow quantities are also of interest. For instance, the evolution of velocitygradients and their tensor invariants was studied by Ooi et al. (1999) in isotropic tur-bulence and by Chacin & Cantwell (2000) in a turbulent boundary layer. Stochasticmodels for these quantities were constructed by Girimaji & Pope (1990) and Martinet al. (1998). Pope & Chen (1990) developed a stochastic model for the energydissipation rate that is used to account for intermittency in PDF modeling of thevelocity field (Pope 2000). Properties of the scalar dissipation rate are importantin the Lagrangian Spectral Relaxation (LSR) model (Fox 1997, 1999), which isdescribed further in Section 5. In general (Pope 1990, Yeung 2001), vector magni-tudes and nonnegative quantities (such as energy dissipation) have longer integraltimescales than Cartesian vector components. This implies that fluctuating vectorsin turbulence tend to undergo changes in direction quite rapidly, which can bedemonstrated by computing the statistics of polar angles in specified coordinateplanes (Yeung 1997).

Lagrangian concepts have also been usefully employed in several situations,including subgrid scale modeling (Meneveau et al. 1996), which are traditionallyconsidered in an Eulerian frame. Recently, new methods have been proposed re-lating the motion of multi-point Lagrangian objects (Gat et al. 1998, Frisch et al.1999) to the scaling exponents of Eulerian structure functions of advected passivescalars and to a new theory of intermittency in turbulence (Falkovich et al. 2001).These methods have been applied to scalars in a stochastic velocity field with pre-scribed power-law spectrum and infinitely fast decorrelation in time (Kraichnan

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1994). Extensions to scalars in a velocity field with true Navier-Stokes dynamicswould be of interest.

4. SCALING OF VELOCITY AND ACCELERATION

4.1. Kolmogorov Similarity

In studies of the Eulerian spatial structure of turbulence (see Sreenivasan & Anto-nia 1997) there is much interest in the scaling properties of the velocity differencebetween two fixed points in space as a function of the spatial separation. Cor-respondingly, in the Lagrangian frame the core issue is in the behavior of theincrement of velocity in time following a single fluid particle. The moments ofthis increment are given by them-th order Lagrangian structure function, definedas

DLm(τ ) = 〈[u+(t + τ )− u+(t)]m〉 , (7)

for different values ofm over nontrivial ranges of the time lag (τ ). The case ofm= 2 is the most important.

The classical account of the application of Kolmogorov’s similarity hypothesesto Lagrangian quantities is given by Monin & Yaglom (1975). At sufficiently highReynolds number the first hypothesis of small-scale universality suggests

DL2 (τ ) = a0〈ε〉3/2ν−1/2τ 2 (for τ ¿ τη) , (8)

where〈ε〉 is the mean-energy dissipation rate anda0 is assumed to be a universalconstant. Furthermore, the second hypothesis suggests inertial range behavior inthe form

DL2 (τ ) = C0〈ε〉τ (for τη ¿ τ ¿ TL ), (9)

with C0 also assumed to be universal at high Reynolds number.

4.2. Evidence and Estimates

The first-hypothesis prediction from Equation 8 implies that the acceleration vari-ance is universal (of valuea0) when scaled by the Kolmogorov variables (〈ε〉, ν).However, this is one of the unresolved issues in turbulence (Nelkin 1994), andrecent evidence is mixed. The latest Lagrangian measurements in a flow drivenby counter-rotating disks (La Porta et al. 2001) suggest that universality holds forRλ beyond 500, witha0 = 5± 1.5 within the bounds of experimental uncertainty.At lower Reynolds numbers the experimental data are consistent with DNS forstationary isotropic turbulence atRλ 38–240 (Vedula & Yeung 1999), which sug-gests thata0 is nonuniversal and increasing asR1/2

λ . Vedula & Yeung related thisresult to the fact that (based on the Navier-Stokes equations) the acceleration isdominated by pressure gradients, which are highly intermittent and concentrated

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at intermediate scales (instead of the small scales) in the turbulence spectrum.Some support for these arguments can also be found in the earlier works of Hill& Wilzcak (1995) and Hill & Thoroddsen (1997). Some new DNS data atRλ478 (Gotoh & Fukayama 2001) have recently been obtained, suggesting a veryweak Reynolds number dependence; but further checks by averaging over a longerperiod of time are desirable, and the underlying physical mechanisms remain tobe understood. Lagrangian data in other laboratory configurations would also behelpful in addressing universality between different types of turbulent flows.

In Equation 9C0 is essentially a Lagrangian Kolmogorov constant, correspond-ing to that (denoted byC2) in the Eulerian structure function, i.e.,⟨

(1ur )2⟩ = C2(〈ε〉r )2/3 , (10)

where1ur is a longitudinal velocity difference between two points at a distanceof r apart. The value ofC2 is relatively well known (approximately 2.1, in Yeung& Zhou 1997) and related directly to Kolmogorov constants in one- and three-dimensional energy spectra in wavenumber space (Sreenivasan 1995). In contrast,the value ofC0 is rather uncertain (Sawford 1991, Du et al. 1995, Sawford & Yeung2001) but important as a key parameter in stochastic modeling. The general picturefrom DNS is that (Yeung & Pope 1989) most Lagrangian statistics have a strongReynolds number dependence, so that if Kolmogorov similarity in the form ofEquation 9 is to apply, then it would require Reynolds numbers considerably higherthan for Eulerian quantities. Although intermittency corrections inherent in themultifractal character of turbulence (Borgas 1993) and manifested in fluctuationsof the energy dissipation rate are possible, this effect is not present in Equation 9because dissipation appears there in a linear form. Instead, the most likely reason isthat the range of timescales relevant to Lagrangian quantities widens with Reynoldsnumber less rapidly than the range of length scales for Eulerian quantities.

We attempt to make some projections here as to what Reynolds numbers andnumerical grid resolutions it would take to settle these questions. For this pur-pose, Figure 3 shows several scale ratios and estimates ofC0 versusRλ, takenfrom isotropic turbulence simulated at four different grid resolutions as recordedin Yeung (2001). Except for some deviations at the low Reynolds number end,Eulerian length- and timescale ratios appear to follow classical scalings, i.e., asL1/η ∼ R3/2

λ (L1 being the longitudinal integral length scale) and (K/〈ε〉)/τη ∼ Rλ(K being the turbulence kinetic energy). The Lagrangian timescale ratioTL/τη isconsiderably less than its Eulerian counterpart. Nevertheless, althoughTL/τη in-creases less rapidly thanRλ in the observed data range, it also appears to followclassical scaling more closely at higherRλ. Consequently, if we estimate that (e.g.,in Jimenez et al. 1993, Yeung & Zhou 1997)Rλ 90 is just sufficient to give a rangeof length scales for the beginnings of an inertial range in the Eulerian structurefunction and energy spectrum, then an extrapolation forTL/τη yields correspond-ingly Rλ ≈ 600–700 for the attainment of a proper inertial range (τη ¿ τ ¿ TL )in the Lagrangian structure function. Furthermore, reference to the number of gridpoints (also plotted in Figure 3) suggests this Reynolds number can be reached

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Figure 3 Scale ratios and dimensionless parameters at different Reynolds numbers:L1/η(4), TL/τη( e), C0(N),C∗0( u), (K/〈ε〉)/τη(¤), TL/(K/〈ε〉)(¥), and number ofgrid points (N) in each direction (unmarked solid line). Dashed lines and filled stars(F) indicate approximate slopes and projected data points for discussion in Section 4.

(or nearly so) using a 20483 grid. A simulation at this resolution (more than 8billion grid points) requires the use of extremely powerful computers at Teraflopspeeds.

It is also possible to use existing data to estimate the asymptotic value ofC0 athigh Reynolds number. We use the fact that for stationary turbulence the structurefunction DL

2 (τ ) is related to the autocorrelation function in a simple way, by

DL2 (τ ) = 2〈u2〉[1− ρL (τ )] , (11)

where, except for small time lags,ρL (τ ) is well approximated by the exponentialform exp(−τ/TL ). For time lags in the rangeτη ¿ τ ¿ TL , we can match theinertial range form ofDL

2 (τ ) on the left with a Taylor-series expansion of theexponential forρL (τ ) on the right. Assuming isotropy (i.e.,〈u2〉 = 2K/3), thisleads to

C0 = 4

3

K/〈ε〉TL

(12)

as an estimate forC0. This result is, incidentally, the same as a consistency conditionfor the Langevin equation used in PDF modeling (Pope 2000).

Figure 3 includes data on a comparison betweenC0 andC∗0, which is the estimateobtained from the peak of the curve obtained by plottingDL

2 (τ )/(〈ε〉τ ) versus time

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lag (Yeung & Pope 1989). It can be seen that, although in generalC0 > C∗0, thesetwo estimates become progressively closer at higherRλ and seem likely to share thesame asymptotic value in the high-Reynolds-number limit. In this limit we obtain

C0 ≈ 4

3

K/〈ε〉TL= 2

TE/TL

〈ε〉L1/u′3, (13)

where the eddy-turnover timeTE is an Eulerian quantity. From DNS results givingTL/TE (whose inverse appears above) about 0.78 nearly independent of Reynoldsnumber (Yeung 2001) and〈ε〉L1/u′3 approaching 0.4 at high Reynolds number(Sreenivasan 1998), we obtainC0 ≈ 6.4. Although both numerator and denomi-nator in Equation 13 are determined by the large scales and may thus be affectedunnaturally by the forcing scheme, it is encouraging to note that this estimate fitsremarkably well in the range 6–7 predicted by Sawford (1991) and Sawford &Yeung (2001).

In addition to the aspects discussed above, several important questions canbe raised about Lagrangian similarity that have not been addressed (or only verybriefly so) in the literature. First, in a way similar to Eulerian paradigms but mostlyfor theoretical interest, one can investigate the scaling of higher-order structurefunctions. In the absence of Lagrangian data with extended inertial range proper-ties, it is not known whether (form> 2 and even)DL

m(τ ) varies in the form [〈ε〉τ ]ζm

and whether the exponentζm shows “anomalous” behavior in deviating fromm/2.Second, for anisotropic flows there are inevitable questions about whether scalingconstants (such asC0 anda0) might vary across different directions of the flow.Although data in both numerical simulations (Sawford & Yeung 2001) and exper-iments (La Porta et al. 2001) show that there are variations, such differences maybecome small at high Reynolds number. Finally, in strongly inhomogeneous flowsthere may be nontrivial variations of the energy dissipation rate along the path inspace traced out by some fluid particle over time. It is not clear, however, if theseeffects can lead to appreciable ambiguities in scaling.

5. MIXING OF PASSIVE SCALARS

5.1. Numerical Simulation

The properties of Lagrangian time series of passive scalars in turbulent flow are veryimportant in the application of PDF methods for which nonlinear terms are closedbut molecular mixing needs to be modeled in a stochastic manner. Compared tothe velocity field, they are both less understood and more difficult to model (Pope1994). Experimentally, because it would be necessary to combine particle trackingwith time-resolved full-field measurements of the scalar field in space, it is notsurprising that no such measurements have been reported. In DNS there is nodifficulty to do so, but yet only very recently have Lagrangian scalar statisticsbeen obtained using this approach (Yeung 2001; and for a more limited parameterrange, see Brethouwer 2000).

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Figure 4 Lagrangian autocorrelations from 5123 DNS atRλ 240: velocity (unmarkedsolid line), energy dissipation (¤), scalarφ1 at Sc= 1/8(4), scalarφ2 at Sc= 1 ( e),dissipation ofφ1(N), and dissipation ofφ2 ( u), and differencez= φ1−φ2 (dotted line).The dashed curve shows exponential approximation to the velocity autocorrelation.The area under each curve is proportional to the integral timescale of the respectivequantity.

The work of Yeung (2001) was directed at the idealized case of stationaryscalar fields with a uniform mean gradient in isotropic turbulence. Figure 4 showsa comparison of Lagrangian autocorrelations of several quantities based on thefluctuating velocity and scalar fields (with the latter taken relative to the meanscalar field at the instantaneous particle position). It can be seen that, espe-cially at higher Schmidt number, the scalar autocorrelation decays more slowlythan the velocity, implying a longer memory time, which is consistent withchanges due to molecular diffusion along particle trajectories that are relativelyweak.

On the other hand, the difference between two scalars has a much shortertimescale, which indicates that the effects of differential diffusion are significantmainly in the small scales at high Reynolds number. The scalar dissipation rates[which are needed in the LSR model of Fox (1997, 1999)] have integral timescalesshorter than that of the energy dissipation and nearly independent of Schmidtnumber. Further results in Yeung (2001) also show that the scalar autocorrelationsdeviate from an exponential form more than the velocity, which implies that amodel of the Langevin type would be less satisfactory.

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The scaling of the Lagrangian frequency spectrum of scalar fluctuations wasalso studied briefly in Yeung (2001), but with inconclusive results. Dimensionalanalysis suggests an inertial range form of〈χ〉ω−2, where〈χ〉 is the mean scalardissipation rate andω is the frequency. However, there are doubts (B.L. Sawfordprivate communication) whether use of Kolmogorov scaling is appropriate. Thiscan be seen by considering the total scalar fieldφ+ = Gx++φ+ for the case of amean gradient of magnitudeG in thex direction. At high Reynolds numbers,φ+changes slowly, so that changes inφ+ are primarily due to particle displacements,which are in turn determined by the large scales in the velocity field. On theother hand, the difference between the fluctuations for two scalars in differentialdiffusion does have a greater degree of local character in time.

In problems of turbulent mixing, the details can depend strongly on the specificproblem configuration, including the form of initial conditions. Another impor-tant, and more complex, case is that of two scalars representing chemical speciesof different molecular diffusivities with nonpremixed (or segregated) initial con-ditions (e.g., Kronenburg & Bilger 1997). In this case, the mixing is unsteady (thevariances decay), the scalar values are bounded (concentrations remain nonneg-ative), and the scalar fluctuations are non-Gaussian in the transient period (withEulerian statistics taking on more complex forms). One possible approach, whichis not exact during the early transient period when the PDFs undergo the greatestchange, is to treat scalar and scalar-dissipation fluctuations normalized by r.m.sand mean values, respectively, as stationary processes.

5.2. Stochastic Modeling

In the context of this review, one of the most important uses of DNS is to helpproduce improvements in stochastic modeling. Ideally, Eulerian data usually in theform of conditional statistics are first used (e.g., Borgas & Yeung 1998, Sawford &Yeung 2000) in an a priori manner to examine the closure assumptions. Lagrangiandata are then used for direct comparisons with stochastic modeling results (e.g.,Yeung & Borgas 1998, Sawford & Yeung 2001). Figure 5 shows single-scalarresults corresponding to Figure 4, from the latest version of the LSR model (Fox1997) incorporating recent changes suggested by Eulerian results in Vedula et al.(2001). Very good agreement is evident.

The Lagrangian modeling of differential diffusion is, in general and especiallyfor the nonpremixed case, a more difficult task. Even for a single scalar (seeDopazo 1994), known model deficiencies include commonly used interaction-by-exchange-with-the-mean (IEM) models (which are simple to implement) pro-ducing correct scalar variance but failing to capture evolution of the shape ofthe PDF, stochastic particle-interaction models producing discontinuous time se-ries, and standard Langevin models (as the sum of a deterministic drift termand a diffusive Wiener process) violating boundedness properties. Vali˜no &Dopazo (1991) proposed a binomial Langevin model that overcomes the

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132 YEUNG

Figure 5 Results from latest version of Lagrangian Spectral Relaxation model(Fox 1997) corresponding to DNS data in Figure 4, excluding the autocorrelationfor z= φ1−φ2. (Special thanks to Rodney Fox for the preparation of this figure.)

limitations noted above but yet depends on the specification of a mixing timescale(ratio of scalar variance to its dissipation rate) for each scalar. Although thistimescale can be used to parameterize Schmidt number dependence in single-scalar statistics, it does not account directly for the effects of differentialdiffusion.

Because differential diffusion has a strong scale dependence, length scale in-formation is essential for modeling. In the LSR model of Fox (1997, 1999), thisinformation is incorporated via a model for fluctuations of the dissipation rate ofeach scalar as well as the joint dissipation rate between them. Although calculationof the difference autocorrelation (not shown in Figure 5) from the model still awaitsan update of the two-scalar version of this model incorporating Eulerian input fromDNS, a degree of success similar to that attained for single-scalar quantities canbe expected. On the other hand, in its most general form, the LSR model requiresknowledge of the conditional dissipation rate and two-scalar joint-dissipation rategiven fluctuations of the velocity, scalar, and energy dissipation rate. Considerablymore work still needs to be done for application of the model to more general situ-ations such as unsteady, nonpremixed problems and mixing in anisotropic and/orinhomogeneous turbulence. In anisotropic turbulence any deviations of scalar gra-dient fluctuations (which constitute the scalar dissipation rate) from local isotropyas a result of couplings between directions of mean velocity and scalar gradientswill have to be addressed.

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6. EXTENSIONS

In Sections 3 through 5 we focused on basic issues in simplified turbulent flows,which hopefully provides a proper foundation for physical understanding. How-ever, it is clear that in practice more complex and often inhomogeneous turbulentflows have to be considered. In addition, there are several types of physical phenom-ena for which extensions of Lagrangian concepts are potentially very fruitful andare being employed in recent research. These aspects are addressed briefly below.

6.1. More Complex Flows

Homogeneous shear flow with mean velocity profile of the form〈U 〉= Sy (forfixed shear rateS) is commonly used as a canonical configuration for studyingthe effects of anisotropy and shear due to production via mean velocity gradients.Classical results from similarity hypotheses giving different rates of dispersionin different coordinate directions were described in Monin & Yaglom (1971).Qualitatively, these predictions are supported by DNS (Squires & Eaton 1991, Shen& Yeung 1997), although for numerical reasons the stationarity in time assumedin classical theory has not been attained exactly. The Lagrangian autocorrelationfor the streamwise velocity persists significantly longer than for the cross-streamand spanwise components. Recent experimental work (Shen & Warhaft 2000) hasled to new questions about the applicability of local isotropy, especially in thehigher-order moments. In the Lagrangian frame, this issue can be checked bycomputing autocorrelations and frequency spectra for small-scale quantities suchas velocity-gradient components.

Another classical anisotropic flow is stratified homogeneous turbulence, whichis important in atmospheric and oceanic contexts where buoyancy effects due toa vertical mean temperature profile play an important role. Kimura & Herring(1996) obtained Lagrangian statistics for this flow using DNS for the case ofstable stratification. Both single-particle displacement and particle-pair separationare found to be significantly reduced in the direction of stratification. In addition,autocorrelations for the velocity component in this direction exhibit systematicoscillations corresponding to the Brunt-V¨aisala frequency, which is a measure ofthe strength of stratification effects.

As Kimura & Herring (1996) pointed out, an important property of stably strat-ified turbulence that is also seen in rotating turbulence is that of reduced energytransfer from the large scales to the small scales (Yeung & Zhou 1998). Dispersionin rotating turbulence is itself of interest in rotating mixers in engineering equip-ment, and in pollution problems over geophysical scales where the rotation of theEarth becomes important. In the limit of strong rotation, one might expect thefluid particles to have a tendency to follow spiraling paths in space (Jacquin et al.1990, Borgas et al. 1997), which would imply that their transverse velocity com-ponents evolve relatively rapidly in time. Less persistent velocity autocorrelationsand larger acceleration variances are thus expected in the transverse plane.

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134 YEUNG

For the important problem of pollutant transport in the atmosphere, the mostrelevant geometry is the classical boundary layer on a flat plate. Lagrangian dis-persion models are especially appropriate for air-quality applications involvinglocalized sources, as well as situations where Eulerian gradient transport closuresfail to represent the physics (Sawford 1985). However, because of the lack of datain the literature, classical theories given in Monin & Yaglom (1971) are still largelyuntested, and relatively simple models of limited physical basis are in commonuse. The Lagrangian characteristics of this flow are expected to be strongly in-fluenced by inhomogeneity, which can be investigated by releasing ensembles ofparticles at different heights from the wall. Choices can be made to correspondto different regimes in the mean-velocity profile, including viscous, buffer, iner-tial, and outer layers. For best physical realism, DNS codes for spatially evolvingflows allowing for the streamwise growth of the boundary layer are preferred overtemporally evolving ones which are artificially periodic in space. Both stable andunstable (convective) conditions in the atmosphere (e.g., Wyngaard 1992, Luharet al. 2000) are of interest. An earlier review of turbulent diffusion in more complexflows with emphasis on inhomogeneity was given by Hunt (1985).

6.2. Other Applications

The Lagrangian approach is useful in many more different ways than can bedescribed within the scope of this review, with some obvious examples beingaerosol science, multiphase flows, and medical tracer imaging. Below, we considerbriefly three topic areas that have attracted recent attention and can be studied (inDNS) using exact equations without resort to other approximations or modeledrelationships.

Velocity gradients and more generally the multipoint structure of the velocityfield carry unique information on the kinematics and topological properties ofa turbulent flow. A useful question to ask in Lagrangian terms concerns the na-ture of the local deformation experienced by a fluid particle in regions of intensestrain rate or vorticity. Ooi et al. (1999) studied in isotropic turbulence the statis-tics of material derivatives of second and third invariants of the velocity-gradienttensor, conditioned on the instantaneous values of these invariants. Their resultswere consistent with those by Ashurst et al. (1987) showing that the intermediateprincipal strain rate tends to be positive in regions of intense dissipation. Chacin& Cantwell (2000) also calculated Lagrangian phase-plane trajectories of theseinvariants using DNS of a turbulent boundary layer. The statistics were appar-ently averaged over all particles without explicit consideration for the effects ofinhomogeneity through initial or instantaneous particle positions. The nature offlow deformation can also be analyzed via the motion of Lagrangian multipointclusters, e.g., “tetrads” of four points, whose shape can be quantified via a momentof inertia (Chertkov et al. 1999). Whereas the volume enclosed in the tetrad gen-erally increases in time, the formation of sheet-like structures would be indicatedby tetrads that are nearly coplanar or have small volume versus their linear size

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(Pumir et al. 2000). Although useful results have been obtained in the papers citedabove, the Reynolds numbers were limited, and there was no strong evidence ofinertial behavior in the energy spectrum or mean-velocity profile. A related topicthat is important but difficult to address precisely is the exploration of connectionsbetween coherent structures and the motion of Lagrangian tracer particles (e.g.,Elhamidi et al. 1993).

In the field of biological oceanography, there is increasing awareness(Rothschild & Osborn 1988, Yamazaki 1993, MacKenzie 2000, Peters & Marrase2000) of the important effects of turbulence on the dynamics and life history ofsmall organisms such as plankton and fish larvae in the ocean. A better understand-ing of the associated issues in marine-population dynamics can have substantialeconomic benefits, such as those via improved techniques of fishery management.The primary question is how small organisms of different species possibly inpredator-prey relationships come into close contact and interact. Because the primeconcern is in the small scales, the hypothesis of local isotropy allows this prob-lem to be studied effectively via numerical simulations of isotropic turbulence.Although these organisms usually have some degree of independent swimmingability (which can be modeled stochastically), to a first approximation they can betreated as fluid particles passively transported in turbulent flow. For a cluster offluid particles with specified initial number density, we can determine the distancebetween each particle and its nearest neighbor and declare that an encounter hasoccurred if this distance falls below a prescribed “contact radius” threshold.

Yamazaki et al. (1991) were perhaps the first to study planktonic dynamicsusing DNS. However, data were provided at only one relatively low Reynoldsnumber. More recently, Lewis & Pedley (2000) provided a more rigorous mathe-matical framework and obtained results using kinematic simulations. Qualitatively,in addition to being dispersive, turbulence also has aggregative effects, which areexpected to cause increased encounter rates at higher Reynolds number. Althoughbiological systems do respond differently, there are conceptual similarities withthe phenomenon of preferential concentration of solid particles that may collidewith each other in turbulent suspensions (Fessler et al. 1994, Sundaram & Collins1997). However, many of these effects, and their scaling with Reynolds number,have yet to be quantified precisely. It is also important to add buoyancy effectsto better simulate oceanic conditions. Oceanic turbulence is in fact rather com-plex; observational and modeling aspects concerning Lagrangian trajectories wereaddressed respectively by Davis (1991) andOzgokem et al. (2000).

Finally, besides solid particles and gas bubbles in suspensions, another exampleof entities primarily, but not exactly, transported by velocity fluctuations in turbu-lence is provided by molecules of foreign chemical species undergoing Brownianmotion relative to the fluid. The theoretical foundation was laid by Saffman (1960).Molecular diffusion is often thought of as only of minor importance in high-Reynolds-number turbulence. However, as noted by Sawford & Hunt (1986), thereare practical situations (small diffusion times, close to small contaminant sources)where these effects are important. [A recent application (Borgas 2000) concerns

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136 YEUNG

the study of odor from accumulations of waste materials.] Similarly, for a bettermatch between results from stochastic modeling and measurements of temperatureor concentration statistics in wind-tunnel experiments at modest Reynolds num-bers, molecular and viscous effects must be considered (Borgas & Sawford 1996).It is convenient in DNS to follow the motion of molecular trajectories and usedata for stochastic modeling (Yeung & Borgas 1997). In addition, because (beingfree of additional Eulerian resolution requirements) this approach is applicable atarbitrary Schmidt numbers, it may be useful for exploring differential diffusionover very wide Schmidt-number parameter ranges.

7. SUMMARY AND CONCLUDING REMARKS

The basic rationale for writing this review is the fact that the Lagrangian approachof following fluid particle trajectories is very useful for studying turbulent trans-port processes, including stochastic modeling and PDF methods for velocity andscalar fields. In part because of measurement difficulties, much of turbulence re-search is conducted in an Eulerian reference frame, and many important aspects ofLagrangian behavior are still unresolved. Although recent advances in experimentscarry considerable promise, most investigations aimed at fundamental understand-ing of Lagrangian flow properties have been undertaken using direct numericalsimulations. We have attempted to summarize both current understanding and theneeds for future research.

Several key requirements for obtaining high-quality Lagrangian data in numer-ical simulations are identified in this article (Section 2). Well-resolved Eulerianinstantaneous velocity and scalar fields evolving according to exact conservationequations must be available. An accurate interpolation scheme, preferably one thatprovides differentiable approximations, is needed to obtain Lagrangian values fromEulerian ones at grid points close to the instantaneous particle position. Care isneeded in assessing and maintaining statistical accuracy and in having sufficientlylarge ensembles of independent particle trajectories. Although computationallyintensive, DNS appears to be the most powerful approach for investigating basicLagrangian behavior, especially if derivative quantities such as velocity and scalargradients sampled along particle paths are required.

Sample time series of the type presented in Figure 1 (based on Yeung 2001)give a qualitative view of the differences between time evolution in Eulerian andLagrangian reference frames. Taylor timescales can be defined to quantify the rateof change, whereas autocorrelations and integral timescales give a measure of thelength of memory in time. Passive scalar fluctuations are found to evolve slowlyin a Lagrangian frame, which is consistent with the dominant role of advectivetransport, especially at higher Reynolds numbers. The DNS data suggests thatthe random-sweeping hypothesis of Tennekes (1975) can be extended to passivescalars in turbulent flow.

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Similarity scaling based on extensions of Kolmogorov’s hypotheses to the La-grangian frame (Monin & Yaglom 1975) is at present a subject of considerableuncertainty. Some of the recent data on acceleration variance from experimentsand DNS in different flows share both similarities and differences, with the finalword awaiting further improvements in both of these lines of research. Because therange of timescales widens with Reynolds numbers more slowly than the range oflength scales, there is great difficulty in attaining an inertial scaling range for timelagsτη¿ τ¿ TL , which is needed for unambiguous deductions ofC0, the assumeduniversal constant in the second-order Lagrangian structure function. Projectionsbased on existing data (Figure 3) suggest that an asymptotic value ofC0 ≈ 6.4 (ingood agreement with at least two other estimates in the literature) can be reachedby 20483 simulations of stationary isotropic turbulence atRλ 600–700.

Data on Lagrangian scalar time series, which are virtually beyond present tech-niques of experimental measurement, have been obtained very recently from DNS(Yeung 2001). Together with Eulerian conditional statistics for the case of station-ary, Gaussian-distributed scalar fluctuations, the new data are currently finding use(Figures 4 and 5) for evaluation and improvements of stochastic modeling. Themodeling of differential diffusion in a Lagrangian frame remains to be addressed insimilar rigor, for scalars in stationary state and in unsteady nonpremixed problems.

Finally, in Section 6 we pointed out the need for systematic studies ofLagrangian behavior in several flows more complex than isotropic turbulence,including rotating flows where many industrial mixing operations take place (as inturbomachinery), and the flat-plate boundary layer as a first conceptual model forthe atmospheric surface layer where pollutant dispersion is of primary concern.In addition, we have noted briefly several phenomena for which the Lagrangianapproach is finding new application, with planktonic dynamics in marine environ-ments as an especially noteworthy example.

We conclude by stating again the original theme that Lagrangian descriptionof turbulent flows is important, though not used very often and in need of furtherphysical understanding. Hopefully, continuing advances in experiments, numer-ical simulations and stochastic modeling will benefit synergistically from eachother.

ACKNOWLEDGMENTS

The author is indebted to Eberhard Bodenschatz, Michael Borgas, Rodney Fox,Stephen Pope, Brian Sawford, Katepalli Sreenivasan, and Zellman Warhaft forvery helpful discussions concerning a draft of this paper as well as different formsof kindness over a period of many years. In addition he also gratefully acknowl-edges current and past support in grants and resources from the National ScienceFoundation, U.S. Environmental Protection Agency, Cornell Theory Center, andthe National Partnership for Advanced Computational Infrastructure at the SanDiego Supercomputer Center for parts of his research that are described here.

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P1: FDS

November 5, 2001 10:30 Annual Reviews AR151-FM

Annual Review of Fluid MechanicsVolume 34, 2002

CONTENTS

FRONTISPIECE xii

MILTON VAN DYKE, THE MAN AND HIS WORK, Leonard W. Schwartz 1

G.K. BATCHELOR AND THE HOMOGENIZATION OF TURBULENCE,H.K. Moffatt 19

DAVID CRIGHTON, 1942–2000: A COMMENTARY ON HIS CAREER ANDHIS INFLUENCE ON AEROACOUSTIC THEORY, John E. Ffowcs Williams 37

SOUND PROPAGATION CLOSE TO THE GROUND, Keith Attenborough 51

ELLIPTICAL INSTABILITY, Richard R. Kerswell 83

LAGRANGIAN INVESTIGATIONS OF TURBULENCE, P.K. Yeung 115

CAVITATION IN VORTICAL FLOWS, Roger E.A. Arndt 143

MICROSTRUCTURAL EVOLUTION IN POLYMER BLENDS, Charles L.Tucker III and Paula Moldenaers 177

CELLULAR FLUID MECHANICS, Roger D. Kamm 211

DYNAMICAL PHENOMENA IN LIQUID-CRYSTALLINE MATERIALS,Alejandro D. Rey and Morton M. Denn 233

NONCOALESCENCE AND NONWETTING BEHAVIOR OF LIQUIDS, G. PaulNeitzel and Pasquale Dell’Aversana 267

BOUNDARY-LAYER RECEPTIVITY TO FREESTREAM DISTURBANCES,William S. Saric, Helen L. Reed, and Edward J. Kerschen 291

ONE-POINT CLOSURE MODELS FOR BUOYANCY-DRIVEN TURBULENTFLOWS, K. Hanjalic 321

WALL-LAYER MODELS FOR LARGE-EDDY SIMULATIONS, Ugo Piomelliand Elias Balaras 349

FILAMENT-STRETCHING RHEOMETRY OF COMPLEX FLUIDS, Gareth H.McKinley and Tamarapu Sridhar 375

MOLECULAR ORIENTATION EFFECTS IN VISCOELASTICITY, Jason K.C.Suen, Yong Lak Joo, and Robert C. Armstrong 417

THE RICHTMYER-MESHKOV INSTABILITY, Martin Brouillette 445

SHIP WAKES AND THEIR RADAR IMAGES, Arthur M. Reed and JeromeH. Milgram 469

vi

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November 5, 2001 10:30 Annual Reviews AR151-FM

CONTENTS vii

SYNTHETIC JETS, Ari Glezer and Michael Amitay 503

FLUID DYNAMICS OF EL NINO VARIABILITY, Henk A. Dijkstra andGerrit Burgers 531

INTERNAL GRAVITY WAVES: FROM INSTABILITIES TO TURBULENCE,C. Staquet and J. Sommeria 559

INDEXESSubject Index 595Cumulative Index of Contributing Authors, Volumes 1–34 627Cumulative Index of Chapter Titles, Volumes 1–34 634

ERRATAAn online log of corrections to the Annual Review of Fluid Mechanics chaptersmay be found at http://fluid.annualreviews.org/errata.shtml

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grupo01-lagrangian investigations of turbulence

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