Analysis of the Lagrangian path structures in fluid turbulenceLipo Wang

Citation: Physics of Fluids (1994-present) 26, 045104 (2014); doi: 10.1063/1.4870702 View online: http://dx.doi.org/10.1063/1.4870702 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/26/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Erratum: “Lagrangian statistics in turbulent channel flow” [Phys. Fluids16, 779 (2004)] Phys. Fluids 19, 059901 (2007); 10.1063/1.2723151 Scalar dispersion by a large-eddy simulation and a Lagrangian stochastic subgrid model Phys. Fluids 18, 095101 (2006); 10.1063/1.2337329 A minimal multiscale Lagrangian map approach to synthesize non-Gaussian turbulent vector fields Phys. Fluids 18, 075104 (2006); 10.1063/1.2227003 Identification and analysis of coherent structures in the near field of a turbulent unconfined annular swirling jetusing large eddy simulation Phys. Fluids 18, 055103 (2006); 10.1063/1.2202648 One- and two-particle Lagrangian acceleration correlations in numerically simulated homogeneous turbulence Phys. Fluids 9, 2981 (1997); 10.1063/1.869409

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PHYSICS OF FLUIDS 26, 045104 (2014)

Analysis of the Lagrangian path structures in fluidturbulence

Lipo Wanga)

UM-SJTU Joint Institute, Shanghai JiaoTong University, Shanghai 200240, China

(Received 4 October 2013; accepted 22 March 2014; published online 15 April 2014)

Because in the Lagrangian frame the time scale separation has a stronger Reynoldsnumber dependence than the length scale case in the Eulerian frame, it is moredifficult to reveal inertial range scaling laws, as predicted from dimensional argu-ments. The present work introduces a newly defined trajectory segment structure totentatively understand Lagrangian statistics. When a fluid particle evolves in space,its Lagrangian trajectory encounters regions of different dynamics, which can becharacterized by the magnitude of material acceleration, i.e., |�a|, in certain timespan. The extrema of |�a| are considered as the representative markers along the La-grangian trajectories. A trajectory segment is defined as the part bounded by twoadjacent extrema of |�a|. The time difference and magnitude of the velocity differenceat the two ends of each segment are chosen as the characteristic parameters. It showsthat such structure reveals interesting turbulence physics, such as the scaling of thestructure function and the quantitative description of the time scale. The correspond-ing explanation and analysis of flow physics are provided as well to improve theunderstanding of some remaining challenging issues. C© 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4870702]

I. INTRODUCTION

Lagrangian statistics is fundamentally important in turbulence research, especially for under-standing the physics of turbulent dispersion and mixing. As a great remaining challenge, availabletheoretical works hereof are very limited. There are few pioneering contributions. Taylor1 studied thesingle particle diffusion at limiting conditions, such as the laminar velocity case at very small timespan and the uncorrelated velocity case at very large time span. In studying the particle pair diffusionproblem, Richardson2 found from existing data that in a very large scale range the diffusivity andthe separation distance roughly satisfy the four-thirds law. After the K41 theory and dimensionalanalysis were developed, these existing results have then been extended or explained from moretheoretical aspect by Obukhov and Landau, including the velocity difference structure function (SF),the pair diffusion coefficient, etc. For more details and literature review, readers are referred to themonograph by Monin and Yaglom.3

From modeling aspect the Lagrangian approach also plays important roles in analyzing tur-bulence related complex systems. For instance for the particle dynamics and passive scalar mix-ing problem, it is possible to have some theoretical solutions by considering the statistics of theLagrangian motion of field variables and the scalar gradient, if the velocity satisfies the so-calledKraichnan model, i.e., Gaussian and white in time.4 The Lagrangian presentation is also widely usedif stochastic models of random forces or other sources can be constructed. Existing results includethe evolution of the passive scalar,5 the velocity gradient tensor,6 and scalar mixing in the even morecomplex turbulent reactive flows.7

Both for the Eulerian and Lagrangian statistics, possible scaling relations are basically derivedbased on the Kolmogorov dimensional analysis. For the Eulerian cases, existing works are relatively

a)Email: Lipo.wang@sjtu.edu.cn. Telephone: +86 21 34206151. Fax: +86 21 34206525.

1070-6631/2014/26(4)/045104/12/$30.00 C©2014 AIP Publishing LLC26, 045104-1

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045104-2 Lipo Wang Phys. Fluids 26, 045104 (2014)

well established, for instance the good agreement of the velocity SF between the prediction andmeasurements.8 With the development of numerical and experimental techniques, many efforts havebeen devoted to study the Lagrangian dynamics.9–11 However, it seems more difficult to claim aconfirmative linkage between the predicative and test results, such as the structure function scalingand the multi-particle correlation. The reasons can be manifold. Assuming the theoretical predictionsare not questionable, it is commonly believed that the achieved Reynolds numbers, both numericallyand experimentally, are still not high enough, which then leads to overwhelming analysis uncertainty.Because the timescale separation in the Lagrangian frame is more Reynolds number dependent thanthe length scale case in the Eulerian frame, the finite Reynolds number influence is stronger forLagrangian statistics.12 Nowadays the available Reλ, the Reynolds number based on the Taylor scaleλ, is already around 1000,9–11 which, however, is still not deemed as “high enough” for Lagrangianstatistics. Under this condition, scaling relations in different regimes may be mixed, making thestatistics very uncertain.

Efforts toward large Reynolds number results are very meaningful and need to be acknowledged,but perhaps not effective. The energy cascade mechanism is inherited and reflected by the turbulencedynamics. Such imprint should be described in some appropriate manner; if not, more likely theproblem may be ascribed to the methodologies adopted for analysis, rather than the flow itself,such as the Reynolds number effect. For instance in defining the conventional SF too strong mixingmay be introduced in averaging the velocity difference, which makes the cascade similarity partlycontaminated.13

An interesting result using empirical mode decomposition by Huang et al.14 shows that theoriginal Lagrangian signal can be decomposed into different intrinsic mode functions. By performingthe Hilbert transform of each, clear inertial range scaling laws can be observed for different orderSFs. Such results cannot be achieved by the standard methods. Such empirical mode decompositioncan generally be applied to other systems as well. For the turbulent scalar case, dissipation elementanalysis13 reveals the enhanced scaling range even at low Reynolds numbers, because by partitioningthe scalar field into the space-filling dissipation elements, regions with different correlation propertiescan effectively be separated for conditional statistics. There are some other interesting methods inthe literature as well, such as exit-time15 and hyper-flatness12 analyses.

The present study focuses on the single fluid particle statistics. Considering a single particlemoving in space, typically its Lagrangian path is unclosed and assumes both chaotic and regularstructures. In the following such a complex system will be analyzed based on a novel approach,i.e., trajectory segment analysis. It shows that quantitative results obtained from this new methodhelp to understand important turbulence physics. Research along this line may also develop relevanttheories.

II. TRAJECTORY SEGMENT STRUCTURE

The dynamical behavior of Lagrangian particles is more dramatically affected by phase spacegeometry (such as the particle-vortex interaction) than by the detailed time history of the velocityfield.16, 17 In this sense, the flow field topology plays especially important roles in studying Lagrangianturbulence.

Both in the Lagrangian and Eulerian frames there are many studies on the structural featuresof the particle path.18–20 Mathematically the curvature and torsion of a Lagrangian path are relatedto the velocity gradient.18 The probability density function (PDF) of curvature shows differentpower laws in the small and large value regimes. In the Eulerian frame the local mean curvatureof streamline-perpendicular surfaces is related to the derivative of the velocity magnitude.19 Onaverage streamlines have unsymmetrical structures, which explains an overall nonzero skewness ofvelocity derivative. Schaefer20 studied the PDF of the streamline curvature and found the same twopower law regimes, as by Braun et al.18 An important progress is that the mean curvature has thesame order of magnitude with the inverse Taylor microscale, which can theoretically be derivedfrom the governing equations,20 by assuming that the velocity magnitude and its derivative areuncorrelated.

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eddy breakup

compression stretching

FIG. 1. Schematic diagram of a Lagrangian trajectory passing regions with different flowing structures.

Considering a Lagrangian fluid particle moving in space, its trajectory encounters differentflow regions, such as the local compression, stretching, and eddy breakup regions, as illustrated inFig. 1. It is an interesting topic to discuss the possibilities of structure characterization.13, 16, 21–24 Inthe Lagrangian frame, there are various ways to describe the structure patterns. In a two-dimensionalspace Haller and Yuan16 put forth a stability criterion of local Lagrangian trajectories by checkingif they will converge or diverge with respect to time. Further the boundaries of different regions aredefined by material lines that are linearly stable or unstable in normal directions for longer times thantheir neighbors. Numerical results show that the boundary topology is closely correlated with thevorticity structure.16 This problem can also be considered in some simplified manner, for instanceto use the velocity gradient, or the strain rate along some specific directions as the characteristicparameters.24 To investigate the more detailed evolution of flow structures, Ghasempour et al.23 triedto follow the predefined structures at a time scale approximately of their lifetimes. This method inprinciple is more capable to characterize the Lagrangian structures; however, due to the analysiscost and some unapplicable difficulties as structure recognition, the results are more illustrative thanquantitative and conclusive.

Physically turbulence dynamics is inherited by the Lagrangian fluid particles. It can be expectedthat the statistical properties of Lagrangian trajectories can be much unlike in regions with differentflow patterns. Numerically, it is very challenging to recognize convincingly different pattern regions.In the present work, to make analysis more applicable, and meaningful as well, we tentatively choose|�a|, the magnitude of the material acceleration �a, to parameterize the turbulence dynamics, andconsider the local extremum points of |�a| as the region separatrices. The similar idea also appears inidentifying the topology of oceanic surface velocity field22 and streamline segment analysis.19

To justify such consideration, Fig. 2 shows the scattered relation between S ≡ | d|�a|dt | and the

second invariant of the velocity gradient Q from randomly chosen fluid particles. Typically Q = 0discriminates the vortex- and stretching-dominated regions. Fig. 2 indicates interestingly that thepoints with | d|�a|

dt | → 0 are inclined to correspond to Q = 0, which suggests that the extrema of |�a|can reasonably function as a simple index of the flowing topology marker.

Once the extrema of |�a| are determined, each Lagrangian path can then be naturally partitionedby these extremal points into segments. We define along a Lagrangian path each part bounded bytwo adjacent extrema of |�a| as a trajectory segment. In principle, the complex Lagrangian paths canbe understood by “integrating” all these segments, which have relatively simple structures.

A. Data analysis

The data set is based on the direct numerical simulation (DNS), implemented for the isotropicturbulence in a 20483 cubic domain. The boundary conditions are periodic along each spa-tial direction and kinetic energy is continuously provided at few lowest wave number compo-nents. A fine resolution of dx ∼ η (the Kolmogorov scale) ensures to resolve the small-scalestructures. The Taylor scale λ based Reynolds number Reλ is about 400. Totally 0.2 × 106

Lagrangian particle samples are collected, each of which has about one integral time life span.

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045104-4 Lipo Wang Phys. Fluids 26, 045104 (2014)

FIG. 2. Scattered relation between Q and S.

During the evolution process, the velocity and velocity derivatives are recorded at each τ η/20, whereτ η is the Kolmogorov time. More numerical details can be found in Ref. 25 and references therein.

As aforementioned, a trajectory segment is the part of a Lagrangian path bounded by twoadjacent extremal points of |�a|. If directly calculated from the velocity difference at two consecutivetime steps, the |�a| profile is shown by the dashed line in Fig. 3(a). Obviously the unphysical noisefrom the raw data is overwhelming, making the further quantitative study very troublesome. Largenoise is from the differentiation operation if inappropriate interpolation was adopted.9 It can beexpected that the noise level from the experimental data will be even more serious. The primitiverequirement for data processing is to extract the skeleton structure from the whole, without muchloss of the physical features. To achieve this the following procedures have been carried out.

1. First smooth |�a| by weighted average of the surrounding points. Here the following seven-pointexpression is adopted: |�a|i = 0.3|�a|i + 0.2(|�a|i+1 + |�a|i−1) + 0.1(|�a|i+2 + |�a|i−2) + 0.05(|�a|i+3 +|�a|i−3), which sweeps all the data points (with different index i). Further replace |�a| by |�a|, themean of |�a| in a finite time span of 0.3τη. Afterwards the |�a| profile is shown as the solid line inFig. 3(a). For comparison the dashed line of the raw data has been artificially shifted up. Suchsmoothing is very effective to depress the noise fluctuation and reduce data sensitivity; at the sametime, fine features can still be preserved.

2. Ideally if |�a|i is a local extremum, then (|�a|i − |�a|i−1)(|�a|i − |�a|i+1) ≥ 0, which, however,cannot work well because of the strong influence from small wrinkles. To improve this, the extremumis searched in a finite span [i − a, i + a] instead of [i − 1, i + 1], where the small integer a ≥ 1.The larger a is, the more stringent the criterion is. For the present data case we set a = 4. Differentvalues of a have been checked, for instance a = 5, 6, 7 and the results are pretty identical. Extremalpoints detected in this way are shown by dots in Fig. 3(a), which suggests that there are still a largenumber of spurious local extrema.

3. Clustering of extrema in Fig. 3(a) is due to the small wrinkles of |�a|. To get rid of thisunphysical feature, if two extrema are very short spanned (�τ < 0.3τη) and the same time thevariation is too small (�|�a| < 0.01A, where A is the overall mean of |�a|), then these extrema pairneed to be removed. The outcome is shown in Fig. 3(b).

4. Maxima and minima must be alternatively located. If several consecutive extrema are ofthe same kind, such as a max(min)imum connected with another max(min)imum, then only themax(min)imal one among the group remains. The obtained result is shown in Fig. 3(c).

5. There may still exist few rare events, for instance a maximum is smaller than its connectedminimum. Under this condition this max-min pair will be removed. The final processed signal isshown in Fig. 3(d).

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045104-5 Lipo Wang Phys. Fluids 26, 045104 (2014)

40 60 800

1

2

3

rawsmoothedextrema

τ/τη40 60 80

0

1

2

3

τ/τη

40 60 800

1

2

3

τ/τη40 60 80

0

1

2

3

τ/τη

(a)

(c) (d)

(b)

FIG. 3. Post-processed results from DNS data: (a) the profiles of |�a| (dashed line) and |�a| (solid line). For comparison, thedashed line has been artificially shifted up. The filled dots are the extremal points detected after step 2; (b) extremal pointsafter step 3; (c) extremal points after step 4; (d) extremal points after step 5.

For each Lagrangian trajectory label all the detected extrema (i = 1, 2,...). We characterizeeach trajectory segment with two parameters: one is the time span between two extrema, i.e., �τ

= τ i − τ i − 1, and the another is the magnitude of the velocity difference at τ i and τ i − 1, i.e.,|��u| ≡ |�uτi − �uτi−1 |. Specifically it needs to mention that |�a| only serves to partition the trajectories,but is not used in further quantitative analysis.

One example of the spatial structure of a Lagrangian trajectory is shown in Fig. 4. Color alongthe path indicates the value of |�a|. Red and blue points are the local maxima and local minima,respectively. The segment length, i.e., the time difference between two subsequent extrema (one redand one blue) varies along the path.

III. RESULTS AND DISCUSSION

The following analysis is a tentative attempt to understand the structural properties of theLagrangian trajectory and trajectory segments. It shows that turbulence physics can be vieweddifferently in such context and the corresponding explanations are also provided.

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045104-6 Lipo Wang Phys. Fluids 26, 045104 (2014)

FIG. 4. Example of the Lagrangian trajectory segment structure. The path color indicates the value of |�a|. Red and bluepoints are the local maxima and local minima, respectively. Each trajectory segment is the part bounded by two adjacentlocal extrema.

A. Joint PDFs of the characteristic parameters

As has been discussed before, �τ and |��u| are defined as the two characteristic parameters todescribe the trajectory segment structure. Fig. 5(a) shows their joint PDF, P(|��u|,�τ ), from all thesegment samples. Here the velocity is normalized by the mean velocity u, which is defined with theturbulent kinetic energy k as u ≡ ( 2k

3 )1/2. In this plot there are two local peaks. The one close to the

(a) (b)

(c) (d)

FIG. 5. (a)–(d) The joint PDF of �τ and |��u| and |�ui|.

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045104-7 Lipo Wang Phys. Fluids 26, 045104 (2014)

origin may be caused unphysically by numerical noise, which can lead to samples with small �τ

and |��u|.In addition from the two ending points of each trajectory segment the difference of the velocity

components �ui (i = 1, 2, 3) can also be calculated. The joint PDFs of �τ and |�ui| are shown inFigs. 5(b)–5(d). Differently from the |��u| case the PDF peaks in Figs. 5(b)–5(d) are located at theorigin.

Physically turbulence dynamics acts on the trajectory structure, which then determines these jointPDFs. In principle, the properties of Lagrangian trajectories can be understood by “integrating” allthe trajectory segments, and thus by these joint PDFs. In this sense the quantitative modeling of thesePDFs and the relations with turbulent dynamics are essentially important. There are several relevantworks. Wang and Peters26 modeled the action of turbulence on the scalar field and formulated thejoint PDF of the feature parameters of the dissipation element structure. Interestingly, it can be arguedthat the inertial scaling of the passive scalar SF is determined by the strength of turbulent randomperturbation on scalar variation. When considering the velocity field, Schaefer et al. constructeda model equation for the joint distribution of the length and velocity difference of streamlinesegments,27 from which the asymmetry of the so-called positive and negative streamline segments,therefore the velocity derivative skewness, can be reproduced.

B. SF and the scaling-moment order relation

The conventional velocity difference SF has been extensively studied. The mth order SF of thevelocity component ui (i = 1, 2, or 3) is defined as

Sm(�t) ≡ 〈[ui (t + �t) − ui (t)]m〉, (1)

where 〈 · 〉 denotes statistical average and �t is an arbitrary time difference. Specifically fromdimensional analysis, the second order SF is supposed to satisfy3

S2(�t) = C0ε�t, (2)

where ε is the rate of energy dissipation per unit mass and C0 is assumed as a universal constant athigh Reynolds numbers.

From the joint PDF P(|��u|,�τ ) the conditional mean of |��u| is defined as

〈|��u||�τ 〉 =∫ ∞

0

P(|��u|,�τ )

P(�τ )d|��u|, (3)

where P(�τ ) is the marginal PDF of �τ , which will be discussed afterwards. The dimensionalargument yields the following relation in the inertial range:

〈|��u||�τ 〉 = Cε1/2(�τ )1/2, (4)

where C is some constant coefficient. 〈|��u||�τ 〉 can be understood as the first order SF. Thedifference of such SF definition from Eq. (1) is that �τ is intrinsically determined by the endingpoints of trajectory segments, so |��u| as well, while �t in Eq. (1) is an independent input. Therefore,the implications of sample averaging based on Eqs. (1) and (4) are different.

Figs. 6(a) and 6(b) demonstrate 〈|��u||�τ 〉 compensated by (�τ )1/2 and the conventional secondorder SF compensated by �t, respectively. It can be seen that the conventional SF in Fig. 6(b) doesnot show any convincing scaling range, which is usually believed as the limited Reynolds numberinfluence. Even at the highest possible Reynolds numbers, both numerically and experimentally,11, 28

there is still large uncertainty in finding such inertial scaling.29 For the high-order moment cases thesituation is worse.30 Verification of Eq. (2) is even believed as impossible in the foreseeable future.28

In comparison a clear inertial scaling range can be observed in Fig. 6(a). The plateau confirms thepredicted 1/2 scaling in Eq. (4) and the precoefficient can be determined as C ∼ 2.18, at the sameorder of magnitude of C0 in Eq. (2).9 Differently from the work by Huang et al.,14 the present scalingrange corresponds to shorter time scales, using the aforementioned definition of �τ .

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045104-8 Lipo Wang Phys. Fluids 26, 045104 (2014)

(a) (b)

FIG. 6. (a) The conditional mean 〈|��u||�τ 〉/u with respect to �τ /τη in the compensated form. The clear plateau indicatesa 1/2 scaling, as from the analytical prediction. (b) The conventional second order SF in the compensated form. No scalingrange can be detected.

In addition from the joint PDF of �τ and |�ui| we can also extract the conditional means ofthe components ui, i.e., 〈|�ui||�τ 〉, which are shown in Fig. 7 in the compensated form. The 1/2scaling can similarly be argued based on Eq. (4).

One possible explanation13 of the difference between Figs. 6(b) and 6(a) and Fig. 7 is that theconventional statistical average used in SF may mix different correlation regions together, for instancethe parts with almost-zero and non-zero time derivatives. Because scalings in different correlationregions behave differently, the overall statistics may then be contaminated by sample average. Aninteresting idea in the literature to attack this problem is decomposing the Lagrangian velocityprofile into the so called intrinsic mode functions,14 which helps to separate the aforementionedmixed turbulence properties to reveal the scaling range.

Moreover, the results in Figs. 6(a) and 7 are surprising if understood from the K41 theory.Because the Cε1/2 term in (4) is not linearly proportional to ε, the intermittency effect will then playa role to alter the scaling, if any, from the dimensional expectation. Even in the Lagrangian frameturbulent fluctuations change more smoothly as compared with the Eulerian case,9 intermittencyis still unavoidable, because it originates from the randomness of the fluid dynamics,31 no matterstrong or weak.

The present outcome can be clarified as follows. In the conventional SF definition (as Eq. (2)),the constancy of C0ε at different time scales arises from the independent input of �t. Differently inEq. (4) the time scale �τ is determined by turbulence dynamics, i.e., the time difference between twoadjacent extrema. In this sense, ε in Eq. (4) has different meanings from the mean energy dissipation

(a) (b) (c)

FIG. 7. (a)–(c) The conditional means of 〈|�ui|〉 (i = 1, 2, 3) with respect to �τ in the compensated form.

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045104-9 Lipo Wang Phys. Fluids 26, 045104 (2014)

FIG. 8. The conditional means of ε and ε1/2, normalized with the overall mean dissipation 〈ε〉, with respect to �τ , normalizedwith τη .

in Eq. (2), neither the same as the so-called refined Kolmogorov quantity.32 It is more reasonable tounderstand Eq. (4) formally as a dimensional balance. From the DNS data, the conditional means ofε and ε1/2 at different �τ are shown in Fig. 8 in the normalized form. It can be seen that both 〈ε|�τ 〉and 〈ε1/2|�τ 〉 change with respect to �τ . The similar behavior has also been observed in dissipationelement analysis of the turbulent scalar flows.33 In the high dissipation regions the strong variationof the flow velocity leads to shorter �τ , while in less dissipative regions the velocity profile becomesmuch smoother, which thus makes �τ larger.

The higher order SFs 〈|��u|n〉 (n > 1) can also be calculated. With respect to the trajectorysegment structure, uncertainty in measuring the scaling values is largely depressed. All the casesshow clearly the enhanced scaling ranges. Fig. 9 plots the variation of the SF scaling vs the momentorder n. As in many other studies, such scaling dependence on n deviates largely from being linear.Because of the different definition of SF and the time scale �τ , the curve in Fig. 9 differs from theexisting results.14

FIG. 9. Scaling of the SF 〈|��u|n〉 with respect to different moment order n, in comparison with the existing results fromHuang et al.14

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045104-10 Lipo Wang Phys. Fluids 26, 045104 (2014)

C. PDF of �τ

Another fundamental problem hereof is the statistic property of �τ . Generically turbulencedynamics can be understood from the interaction of the following mechanisms. On the one hand,random motion of turbulent eddies disturbs the flow field, generating fluctuation at different scales;on the other hand, molecular diffusion smooths the flow field, which then decays the fluctuation.Based on this scenario, the length scale PDF in the turbulent scalar field was first derived in studyingthe dissipation element structure.13 If turbulence is stationary, these two mechanisms finally leadto a dynamical balance. It has been found that such PDF equation is generically appropriate inquantifying different processes, such as the turbulent flame34 and vorticity statistics.35

Physically in partitioning Lagrangian trajectories the time scale �τ shares the similarperturbation-smoothing picture. Therefore following the length scale PDF evolution equation,13

the PDF of �τ is expected to satisfy

∂ P(�τ, t)

∂t+ ∂

∂�τ[v(�τ )P(�τ, t)] = �[

∫ ∞

02P(�τ + z, t)dz︸ ︷︷ ︸

T1

− �τ P(�τ, t)︸ ︷︷ ︸T2

] +

(5)

8∂ P(�τ, t)

∂�τ|�τ=0[

∫ �τ

0

z

�τP(�τ − z, t)P(z, t)dz︸ ︷︷ ︸

T3

− P(�τ, t)︸ ︷︷ ︸T4

].

In Eq. (5), �τ is the time scale normalized with its mean �τm, i.e., �τ = �τ/�τm , and � is aneigenvalue which can be determined from the PDF’s normalization condition, i.e.,

∫ ∞0 P(�τ )d�τ =

1.0. v(�τ ) is the drift velocity describing the motion of extremal points relative to each other. Thedetailed properties of v(�τ ) have been discussed somewhere else.13 The four terms T1–T4 on theright-hand side of Eq. (5) represent generation of �τ from large scale annihilation, removal of �τ

by perturbation, generation of �τ from small scale smoothing, and removal of �τ by reconnection,respectively. Interested readers are referred to Ref. 13 for more technical details.

Fig. 10 shows the comparison of the numerical result of P(�τ ) with the prediction by Eq. (5).In both linear-linear and linear-log coordinate systems, agreements are surprisingly excellent. In thesmall �τ region the PDF depends linearly on τ , which is determined by the drift velocity v(�τ )term, or the molecular diffusion effect; while at large scales the PDF decays exponentially, whichcan be explained from the Poisson process-like random perturbation by turbulent chaotic motions.

FIG. 10. The marginal PDF of �τ .

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045104-11 Lipo Wang Phys. Fluids 26, 045104 (2014)

The order of magnitude of the mean time scale �τm needs to be addressed. Considering the flowstructure in the Eulerian frame, Schaefer20 concluded that on average the streamline curvature scaleswith the inverse of the Taylor microscale. Phenomenologically the streamline curvature, which canbe expressed by the velocity gradient and stain rate,18, 19 will change with different flow patterns, suchas the compressive or extensive strain dominated regions. In this sense the streamline topology andthe material acceleration function in a similar manner. Therefore, it is a reasonable conjecture thatthe length scale of the trajectory segment is the same as the characteristic scale of the streamline, i.e.,the Taylor scale λ. Based on the random sweeping hypothesis, the convective velocity of Lagrangianparticles on average is about u. Therefore, �τm can be estimated as

�τm ∼ λ/u ∼ τη. (6)

Numerical results also suggest such estimation. In addition, the relation between τη and the integraltime τ in, i.e., τη/τin ∼ Re−1

λ , reveals how the trajectory segment scales will depend on the integralflowing quantities.

IV. CONCLUDING REMARKS

In summary, the Lagrangian trajectory segment structure is introduced in analyzing the singlefluid particle properties. Because the scale separation in the Eulerian and Lagrangian frames havedifferent Reynolds number dependence, even at the largest available Reynolds numbers much statis-tical uncertainty still exists in Lagrangian results. To understand better turbulence physics, especiallyscaling invariance and dynamic similarity, new approaches need to be developed.

Phenomenologically, a Lagrangian trajectory encounters flow regions of different properties.An interesting idea proposed in the present work is conditional averaging to prevent mixing ofstatistics in different flow regions. To make the analysis practical, applicable, and meaningful, theextremal points of the magnitude of the material acceleration are treated as the separatrices ofdifferent flow regions. Thus the Lagrangian path can be decomposed into trajectory segments, eachof which is characterized by the time difference and the magnitude of the velocity difference atthe two ending points. The present study indicates that such trajectory segment structure helps tounderstand turbulence physics. Specifically we address the following concluding remarks.

1. In principle, the overall Lagrangian statistics can be reproduced by integrating the trajectorysegment samples. Progress of our knowledge in turbulence needs more quantitative relations otherthan illustrative descriptions. The trajectory segment definition provides a possible way to quantifyLagrangian parameters, for instance the time scale.

2. Physically the time difference between ending points, i.e., �τ , is under the action of bothirregular perturbation from random motions at large scales and molecular smoothing at small scales.Such perturbation-smoothing mechanism is supposed to be generic in describing other processes aswell. The PDF of �τ obeys satisfactorily a theoretical prediction derived based on this scenario.

3. In the conventionally defined Lagrangian SF inertial scalings can hardly be observed, becauseof the regime mixing and scaling contamination. If scaling and similarity are the natural imprintof turbulent inertial motions, such properties should be viewed if they are feasibly presented. Thehigher Reynolds number condition is a very strong requirement and it need to be argued with hugediscretion. In comparison, the trajectory segment approach shows successfully the enhanced inertialscaling range. Especially the first order SF agrees with the theoretically predicted 1/2 scaling. Theseoutcomes may be understood from the separation of conditional statistics in different flow regions,which is related to but not identical with the refined Kolmogorov hypothesis.

ACKNOWLEDGMENTS

The author acknowledges Dr. Y. Huang (Shanghai University, China) for insightful discussionand suggestions. The DNS data and helpful comments from Professor F. Toschi (Eindhoven Univer-sity of Technology, The Netherlands) are greatly appreciated. This research is supported by NationalScience Foundation of China (NSFC) (under Grant No. 11172175).

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045104-12 Lipo Wang Phys. Fluids 26, 045104 (2014)

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