Structures of the vorticity tube segment in turbulenceLipo Wang Citation: Physics of Fluids (1994-present) 24, 045101 (2012); doi: 10.1063/1.3701376 View online: http://dx.doi.org/10.1063/1.3701376 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/24/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Stability of a jet in crossflow Phys. Fluids 23, 091113 (2011); 10.1063/1.3640011 Experimental and numerical investigations of flow structure and momentum transport in a turbulent buoyancy-driven flow inside a tilted tube Phys. Fluids 21, 115102 (2009); 10.1063/1.3259972 Energy dissipation in spiral vortex layers wrapped around a straight vortex tube Phys. Fluids 17, 055111 (2005); 10.1063/1.1897011 Effect of large-scale coherent structures on subgrid-scale stress and strain-rate eigenvector alignments inturbulent shear flow Phys. Fluids 17, 055103 (2005); 10.1063/1.1890425 Large-eddy simulation of low frequency oscillations of the Dean vortices in turbulent pipe bend flows Phys. Fluids 17, 035107 (2005); 10.1063/1.1852573

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PHYSICS OF FLUIDS 24, 045101 (2012)

Structures of the vorticity tube segment in turbulenceLipo Wanga)

University of Michigan-Shanghai JiaoTong University Joint Institute, 200240, Shanghai

(Received 27 September 2011; accepted 1 March 2012; published online 4 April 2012)

To address the geometrical properties of the turbulent velocity vector field, a newconcept named streamtube segment has been developed recently [L. Wang, “On prop-erties of fluid turbulence along streamlines,” J. Fluid Mech. 648, 183–203 (2010)].According to the vectorial topology, the entire velocity field can be partitioned intothe so-called streamtube segments, which are organized in a non-overlapping andspace-filling manner. In principle, properties of turbulent fields can be reproducedfrom those of the decomposed geometrical units with relatively simple structures. Asimilar idea is implemented to study the turbulent vorticity vector field using the vor-ticity tube segment structure. Differently from the conventional vortex tubes, vorticitytube segments are space-filling and can be characterized by non-arbitrary parameters,which enables a more quantitative description rather than just an illustrative expla-nation of turbulence behaviors. From analyzing the direct numerical simulation data,the topological and dynamical properties of vorticity tube segments are explored. Thecharacteristic parameters have strong influence on some conditional statistics, such asthe enstrophy production and the probability density function of vorticity stretching.Consequently the common knowledge in turbulence dynamics that vorticity are morestretched than compressed need to be rectified in the vorticity tube segment context.C© 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3701376]

I. INTRODUCTION

Fluid motions are governed by nonlinear equations. At the turbulent state especially withhigher Reynolds numbers, convective motions, and perturbations from boundary conditions makethe problem strongly nonlinear and nonlocal; together with the complex sources from externalboundary effects, it is prohibitively difficult to obtain self-contained information directly from thegoverning equations. Therefore it is strongly needed to attack turbulence from different viewpointsthan mathematics alone. For instance, the Kolmogorov theory is not based on the Navier-Stokesequations although some connections have been made.1 An important stream of efforts is to focuson the geometrical features of turbulent flows.2

The vorticity vector �ω, defined as the curl of the velocity vector �u, i.e., �ω = ∇ × �u, plays anoutstanding role in turbulence kinematics and dynamics. There is abundant literature in discussingthe mathematical and topological features of �ω.3, 4 Because of the coupling of fluid motions atdifferent scales, large compression regions interact with large stretching regions to make vorticitylocally lumped. High vorticity is closely related to turbulence intermittency.8 Usually it is deemedthat on average vorticity is more likely stretched than compressed.3, 4 Vortex stretching dynamicallycauses velocity fluctuation to spread at different scales.5, 6 This stretching mechanism is believedimportant to understand in 3D turbulence some most essential features.3

Due to stability reasons, regions of significant vorticity are inclined to be organized into tubes,7, 8

named as vortex tubes. Moffatt described vortex tubes as the “sinews” of turbulence.9 Generally thelength and radius of vortices are considered to be of orders of magnitude of the Taylor and dissipativescales, respectively. Topological studies of vortex filaments suggest that the mean curvature radius

a)Electronic mail: Lipo.wang@sjtu.edu.cn.

1070-6631/2012/24(4)/045101/10/$30.00 C©2012 American Institute of Physics24, 045101-1

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045101-2 Lipo Wang Phys. Fluids 24, 045101 (2012)

of the filaments is about the Taylor scale.10 Along the filaments different properties have differentcorrelation lengths. Results from larger direct numerical simulations (DNS) data at higher Reynoldsnumbers indicate that there may still be some uncertainty in determining the scaling of these scales.11

Ruetsch and Maxey12 showed that energy dissipation is correlated with vortex tubes. Regions ofmoderate dissipation tend to surround the tubes, but intensely dissipative regions tend to existbetween two or more neighboring vortex tubes. Usually large dissipation regions do not overlapwith large vorticity regions. Visualization from numerical data shows various evolution modes fromthe interaction of spiral sheets and vortex tubes.13

Although extensively studied, the identification of the vortex structure is still quitecontroversial.14, 15 For instance defining vortex tubes depends on some preset threshold value, whichis in no way objective. Haller suggested14 that the vortex definition should be invariant under ageneral coordinate transform, based on which the new vortex definition is frame independent. Al-though characterized differently, vortex tubes physically address the regions of intense events, whichtake only a very small volume portion (basically less than 10%) of the entire flow field. Thereforethe flow field cannot appropriately be represented by vortex tubes only. In other words, propertiesof vortex tubes may not always be relevant to properties of turbulence. A reasonable remedy asksfor the prerequisites that geometries need to be objectively defined, and cover the entire field, i.e.,space-filling. Which method to use is, however, by no means trivial.

Along this direction, some novel approaches have been developed in recent years to focus onthe topological features of turbulence. For scalar variables for instance, Wang and Peters proposedthe dissipation element structure16 to partition a scalar field into dissipation elements. Each elementconsists of the grid points whose scalar gradient trajectories can share the same pair of local minimumand maximum points of the scalar variable. By spatial partitioning, the original complex system maybetter be understood from these relatively simple decomposed structures. In principle, this methodfunctions for different scalars.17 For the vector variables because of the in nature difference, it isnot straightforward to apply dissipation element analysis in the same vein and there is a need fornew diagnose approaches. An interesting attempt is streamtube segment analysis,18 in which theentire turbulence velocity field can be studied using the so-called steamtube segment structure.Kinematically streamtube segments are classified as either positive or negative. An enlighteningconclusion is that the skewness of velocity derivatives is naturally an evolution consequence becauseof the asymmetry between positive and negative groups. Conditional statistics with respect todifferent segments reveal finer field properties. As dissipation element analysis, in principle thestreamtube segment approach is generally applicable for different vector variables.

Vorticity and velocity vectors are both of essential importance in turbulence kinematics anddynamics. In the present paper, the similar streamtube segment idea is implemented to study theturbulent vorticity field. The non-local properties and conditional statistics are investigated based onthe DNS data.

II. GEOMETRY DEFINITION

As important as the gradient trajectory to scalar variables, the vectorline, which is locallytangent to the vector to be considered, reflects the intrinsic topology of the given vector field. For thevelocity and vorticity cases, vectorline is then named as streamline and vorticityline, respectively.The vorticity equation for the incompressible case is

∂ �ω∂t

+ �u · ∇ �ω = ν∇ · ∇ �ω + �ω · ∇ �u, (1)

where ν is the kinematic viscosity and �u is the velocity vector.Denote the (unit) orientation vector of �ω and �u as t and tu , respectively. Thus the vorticity and

velocity vectors are, respectively, expressed as ωt and utu . Multiplying t on both sides of Eq. (1), ityields

∂ω

∂t+ u

∂ω

∂su= νt · ∇ · ∇ �ω + ω(t · ∂ �u

∂s), (2)

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045101-3 Lipo Wang Phys. Fluids 24, 045101 (2012)

Vorticityline

Extremal surface

Extremal surface(maximal )

(minimal )

ω

ω

Grid point

Vorticitytube segment

FIG. 1. Schematic explanation of (a) vorticityline segments; (b) vorticity tube segments.

where s and su are the curvilinear coordinates along vorticity lines and streamlines, respectively.Multiplying Eq. (2) with ω, we then have the equation of enstrophy ω2/2 as

∂(ω2/2)

∂t+ u

∂(ω2/2)

∂su= νωt · ∇ · ∇ �ω + P, (3)

where

P = ω2t · ∂ �u∂s

(4)

is the enstrophy production, a key parameter in understanding vortex dynamics and it will bediscussed later. While formally different from the Navier-Stokes equation, the voriticity equation is ofcomparable importance in turbulence study. The transformed equations (2) and (3) along vortexlineshave a more clear geometrical meaning.

Differently from the scalar gradient trajectory, the vorticityline or general vectorlines are mostlyunclosed and infinitely long. To well describe the geometry, an idea similar as the streamlinesegment18 is adopted here. As shown in Fig. 1(a), in the vorticity vector field from any spatial pointalong its vorticityline, a local maximum and minimum of the vorticity magnitude ω can be reached.The part of the vorticityline bounded by the two adjacent extremal points (one maximal and oneminimal) is defined as the vorticityline segment with respect to the given point. It may be flexible todivide vorticitylines into segments based on whatever rules. However, the most relevant parameterfor a general vector field is its magnitude and thus it is pertinent to partition segments according tothe extrema of the vector’s magnitude.

The set of extremal points from neighboring vorticitylines constitutes extremal surfaces. Alongthe vorticity vector direction, the vorticity magnitude at the starting point ωs changes monotonouslyto ωe at the ending point. The arclength l between two extremal points and �ω = ωe − ωs areused to parameterize the segments. According to the sign of �ω, segments can be either positive ornegative, if �ω is positive and negative, respectively.

The vorticityline segment structure, although instructive to address the field topology and flowkinematics, is not volumetric and thus irrelevant to mass related dynamics. A remedy hereof is thetube segment concept.18 As shown in Fig. 1(b), a vorticity tube segment is the part of an infinitely thin(but still with nonzero volume) vorticity tube, cross-cut by two adjacent extremal surfaces. All thevorticity tube segments are organized in a non-overlapping and space-filling manner. Although thedemarcation of the tube boundary is indefinite, statistical properties weighted by the tube segmentvolume are independent of the boundary definition.

III. RESULTS AND ANALYSIS

A. Direct numerical simulation

First of all, direct numerical simulations (DNS) for homogeneous shear flow have been per-formed. The flow configuration within a 2π cubic box is illustrated in Fig. 2. A mean velocity

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045101-4 Lipo Wang Phys. Fluids 24, 045101 (2012)

x

x

x

_U=Sx

1

2

2

3

FIG. 2. Flow configuration of the present DNS case in a 2π cubic domain.

gradient S = du1/dx2 is imposed as a free parameter, together with the kinematics viscosity ν, tocontrol the turbulence intensity. In the moving frame attached to the mean flow, governing equationshave the fluctuating parameters as dependent variables. Periodic boundary conditions are adoptedalong three directions xi (i = 1, 2, 3) and the Fourier spectral method is used for spatial deriva-tives. Pressure is solved via a Poisson equation together with the 2/3 dealiasing rule. A third-orderRunge-Kutta algorithm is used for the time advancement and the time step �t is determined by thefollowing Courant condition:

(�t)−1 = 2(|u1|�x1

+ |u2|�x2

+ |u3|�x3

)max . (5)

The continuity equation can be satisfied by the algorithm by Feiereisen et al.19 To ensure theresolution fine enough to extract essential geometrical features, the grid size need to be less than theKolmogorov scale, i.e., �x/η < 1, as that adopted in the existing work.16, 18

A problem with the moving frame is that, with time going on, the calculation domain in physicalspace becomes continuously skewed. Therefore regriding is needed to adjust the deformed frame toexclude over deformation. As depicted in Fig. 3, an original Cartesian mesh (a) deforms more andmore till the status (c). Then regriding shifts it back to (b) and so on.

In principle from an initial condition, the integral length in homogeneous shear turbulence willincrease exponentially with time. In this sense exact homogeneous shear turbulence simulation doesnot exist because of the limited calculation domain. Numerically if the simulation time is longenough largest scales will reach the domain boundaries and turbulent energy will partly releaseto shrink the largest scales, which then increase again and so on. In other words, the eddy sizesfluctuate with time. In data analysis, only the fluctuating variables are taken into consideration.Numerical tests suggest that the boundary interference does not exert sensible effects on statisticalresults. For instance, the parameter Sk/ε, where k and ε are the mean turbulent kinetic energy and

acb

x

x 2

1

FIG. 3. Frame regriding.

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045101-5 Lipo Wang Phys. Fluids 24, 045101 (2012)

TABLE I. Characteristic parameters of DNS.

Grid points du1/dx2 Viscosity k ε λ Reλ

20483 1.5 0.0009 3.5 1.16 0.160 295.0

energy dissipation, respectively, represents the balance between energy production and dissipation.Numerical results show that on average, the parameter Sk/ε is about 4 ∼ 5, which agrees well withthe theoretical prediction. Reλ = √

2kλ/ν, the Reynolds number based on the Taylor scale λ, is 295,together with other characteristic parameters listed in Table I.

B. Topological features

As discussed before, the volumetrical voriticity tube segment structure outperforms the vortic-ityline segment structure in addressing the fluid dynamics. However, numerically it is not straight-forward to identify tubes and their volumes. In theory for a uniformly distributed grid point array,the volume of each tube segment is proportional to the number of grid points it can encompass; thusthe volume weighted statistics is equivalent to that based on vorticityline segments from every gridpoint, by which data analyzing can simply be implemented with respect to vorticityline segments.Results in the following are obtained consistently based on this principle.

The key ingredient to identify vorticityline segments is extrema definition. Similar to the velocityvector case, numerically vorticity extrema are determined by checking the sign of t · ∇ �ω · t,18 whichproves preferred over other criteria.

In fluid dynamics velocity interacts with vorticity. As illustrated in Fig. 4, from any grid point,one can define the corresponding streamline segment and vorticityline segment, whose arclengthsare denoted by lu and l, respectively. The joint probability density function (PDF) of lu and l is shownin Fig. 5 together with 〈lu|l〉 and 〈l|lu〉, the conditional means of lu and l, respectively. Overall thereis no evident correlation between lu and l and the conditional means are approximately constant.

Consider the characteristic parameters, i.e., the arclength l and �ω = ωe − ωs, the differenceof ω between two extrema. Obviously �ω is positive for the positive segments and vice versa.The joint PDF of l and �ω is shown in Fig. 6(a). In terms of the positive and negative �ω part,Fig. 6(a) appears symmetrical. For the streamline segment case with characteristic parameters lu and�u, the counterpart is qualitatively different, as shown in Fig. 6(b). For the positive and negativestreamline segments, the velocity monotonously increases and decreases, respectively, along thevelocity vector direction. Therefore when evolving with time, positive segments with increasingvelocity are inclined to be elongated, while negative segments are likely compressed. This kinematiceffect explains the strong asymmetry of Fig. 6(b) and the negativeness of the velocity derivativeskewness.18 For the vorticity case, this kinematic mechanism is absent and thus the joint PDFs inFig. 6 appear differently.

grid piont

u

umax

min

SS

ω

max

min

ω

VS

FIG. 4. A vortexline segment (VS) connects with a streamline segment (SS) at a joint grid point.

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045101-6 Lipo Wang Phys. Fluids 24, 045101 (2012)

0.0 1.0 2.00.0

1.0

2.0

3.0

4.0

l

l

<l|lu>

<lu|l>

/λu

/λ

FIG. 5. Joint PDF of l and lu, together with the conditional means.

The arclength l is a key parameter in describing geometry and flow physics. Some pioneeringwork has been done in describing the scale of dissipation elements. Wang and Peters16 proposed thefollowing model equation for the PDF of l = l/ lm , l normalized by its mean the lm:

∂ P(l, t)

∂t+ ∂

∂ l(v(l)P(l, t)) = [

∫ ∞

02P(l + z, t)dz − l P(l, t)]

+ 8∂ P(l, t)

∂ l|l=0[

∫ l

0

z

lP(l − z, t)P(z, t)dz − P(l, t)].

(6)

In the above equation, v(l) is the drift velocity describing the motion of extremal points relative toeach other, and is an eigenvalue which can be determined from the PDF’s normalization condition.Physically the evolution of l is a resultant outcome from different effects. First, viscous diffusionmay annihilate the closely located extremal points to smooth the flow profiles, i.e., the smoothingmechanism. Second, turbulent motions will disturb the flow profiles to generate new extremal points,i.e., the perturbation mechanism. Besides these two abrupt effects, at the same time the drift velocityv(l) may slightly drift l in a continuous manner. At the stationary state, the length variation reaches a

0 0.5 1-50

-25

0

25

501.8E+051.7E+051.5E+051.4E+051.3E+051.2E+051.0E+059.1E+047.8E+046.6E+045.4E+044.1E+042.9E+041.6E+044.0E+03

Δω

l/λ0 1 2 3

-3

-2

-1

0

1

2

3

1.5E+061.4E+061.3E+061.2E+061.1E+069.9E+058.8E+057.7E+056.6E+055.5E+054.4E+053.3E+052.2E+051.1E+050.0E+00

Δ u

lu /λ

FIG. 6. Joint PDF of the characteristic parameters for (a) voriticityline segments; (b) streamline segments. The segmentlengths are normalized by the Taylor microscale λ.

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045101-7 Lipo Wang Phys. Fluids 24, 045101 (2012)

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

modelDNS

PD

F

l∼

0 1 2 3 4 510-3

10-2

10-1

100

PD

F

l∼

FIG. 7. PDF of the normalized segment arclength l, compared with the model solution.

dynamic balance under the control of the above factors. In principle, this picture is not only capableto describe the variation of dissipation element lengths, but generally valid for other scale evolutionprocesses as well.

At stationary state, the time derivative term in Eq. (6) vanishes and the corresponding solutionhas been discussed in Ref. 16. Figure 7 shows from DNS data processing the PDF of l, both inthe linear-linear and log-linear coordinates. Compared with the stationary solution of Eq. (6), theagreement is satisfactory.

C. Conditional statistics

By space partition into relatively simple structured units, it is possible to investigate statisticsconditioned on the geometrical features for a deeper understanding of the flow physics.

A well accepted claim is that vorticity are more stretched than compressed.3, 20 For instance inthe spiral vortex model, Lundgren20 considered the strained spiral vortex induced flows and somereasonable features of the energy spectrum can be predicted. The vorticity-strain relation is alsostudied in reactive turbulent flows. It is found in Ref. 21 that the PDF of the orientation angleθ = tan−1 | �ω×S· �ω|

�ω·S· �ω , where S is the strain rate tensor, reaches maximum at cosθ = 1, a consistentoutcome with the vorticity stretching structure.

Consider the velocity projection along vorticityline segments, i.e., ut = �u · t. The difference ofut at two extremal points is �ut = ute − uts and the mean strain along segments is �ut/l. Figure 8(a)shows for the positive vorticityline segments the PDF of �ut, conditioned on groups with different�ω. For negative segments the results are almost identical. Overall it indicates that P(�ut) forsegments with small |�ω| is close to be symmetrical, while more skewed toward the positive sidefor those with larger |�ω|. More relevantly, Fig. 8(b) shows the conditional PDFs of the mean strainrate �ut/l for the same groups.

The mean values of �ut and �ut/l for different groups, i.e., �ut and �ut/ l, are listed in Table II,calculated from Fig. 8. A clear tendency is that both �ut and �ut/ l increase monotonously fromnegative to positive with the increase of the segment parameter |�ω|. Back to the claim that vorticityare more stretched than compressed, the present results manifest that overall vorticityline segments

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045101-8 Lipo Wang Phys. Fluids 24, 045101 (2012)

-6 -4 -2 0 2 4 60

0.2

0.4

0.6 10>30> >1060> >3070> >60

>70PD

F

Δut

ΔωΔωΔωΔω

Δω

-100 0 1000

0.01

0.02

0.03

0.0410>30> >1060> >3070> >60

>70

PD

F

ut /l

ΔωΔω

ΔωΔωΔω

Δ

FIG. 8. PDF of (a) �ut and (b) �ut/l for different categories.

are stretched. However, more precisely segments with small |�ω| are almost unbiased to neitherstretched nor compressed; only those with large |�ω| are inclined to be stretched. The stretchingintensity is strongly dependent on |�ω|, the difference between two vorticity extrema of segments.Physically the mean strain �ut/l is originated from the pressure change. For segments with large|�ω|, if assume a constant background pressure, from the segment starting point to the ending point,centrifugal force will induce a strong pressure difference; thus the mean strain becomes significantunder this condition. Only the value of ω itself cannot lead to the pressure change and the localstrain.

In the enstrophy equation (3) the production term P defined by Eq. (4) is crucial to keep thebalance of total enstrophy. Formally P results from the interaction between vorticity and the localstrain (velocity gradient). Physically P must be positive to balance the enstrophy dissipation, whichis one of the most distinctive features of 3D turbulence from the 2D case. By observing the PDF of theangle between �ω and eigenvetors of the strain rate tensor, Tsinober3 provided a phenomenologicalexplanation of the positiveness of P .

In the vorticity tube segment analysis context, the enstrophy production P as defined in Eq. (4)can be written along vortexlines as

ω2t · ∂ �u∂s

= ω2 ∂(�u · t)∂s

− ω2 �u · ∂t∂s

. (7)

The last term on rhs is a direction fluctuation. Because of randomness of the spatial orientation, onaverage this part is negligibly small. Thus the mean of P can be estimated as

ω2t · ∂ �u∂s

= ω2∂(�u · t)

∂s− ω2 �u · ∂t

∂s∼ ω2

∂(�u · t)∂s

∼ ω2�ut/ l. (8)

Physically because along a vorticityline the ω profile can randomly be cut everywhere and thus it isreasonable to consider ω and �ω as independent, or in other words variables and their gradients arealmost uncorrelated in turbulence. Similar arguments have been verified such as that in fully evolvedisotropic turbulence scalar and scalar dissipation are almost independent,22 especially at higherReynolds numbers. Therefore �ut/l is the dominant factor determining the relative contribution to

TABLE II. Conditional means �ut and �ut / l for different categories.

Parameter 10 > �ω 30 > �ω > 10 60 > �ω > 30 70 > �ω > 60 �ω > 70

�ut −1.381 × 10−2 1.287 × 10−2 6.725 × 10−2 0.115 0.138�ut / l −3.405 × 10−2 −3.611 × 10−3 2.264 × 10−2 3.928 × 10−2 4.670 × 10−2

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045101-9 Lipo Wang Phys. Fluids 24, 045101 (2012)

the sign of P . For segments with small |�ω|, the symmetry of P(�ut), as shown in Fig. 8, leads tosmall �ut/ l and thus small ω2�ut/ l, while the main contribution to the positiveness of P is fromsegments with large |�ω| because the corresponding P(�ut) is largely positively shifted. Overall,the net production must be positive. More generally from the same argument as above the followingrelation can be concluded:

ωn∂(�u · t)

∂s∼ ωn�ut/ l > 0, (9)

where n can take any value of interest. For instance for the n = 1 case, one may check along i = 1,2, 3 directions in the Cartesian frame |ωi|a, where a is the local strain. The calculated results for thepresent DNS case are 4.91, 3.18, and 3.79 along x1, x2, and x3 directions, respectively.

IV. CONCLUSIONS

To step beyond only illustrative explanation, it is necessary to identify geometrical structuresquantitatively in turbulent flows. Furthermore the space-filling condition is an important prerequisiteto make geometry representative of the entire flow field. The streamtube segment structure is such tostudy the turbulent velocity vector filed. The similar idea has been implemented in the present paperto investigate the vorticity vector by introducing the so-called vorticity tube segment structure.

Vorticity tube segments can be parameterized by the segment arclength and �ω, the differenceof two adjacent extremal ω. Based on DNS data analysis, the joint PDF of these two parametersis found to be symmetric, differently from the streamtube segment case. The arclength PDF obeysa general scale evolution equation, which is originally derived in dissipation element analysis.Physically the arclength evolution is under the control of viscous diffusion, perturbation form theconvective motion and the extrema drift process.

Statistics conditioned on the magnitude of �ω have been explored. With respect to the commonclaim that vorticity is more stretched than compressed, present results show that only segments withlarge |�ω| tend to be stretched, while those with small �ω are almost unbiased to neither stretchednor compressed. This property is closely related to the positiveness of the enstrophy production.

ACKNOWLEDGMENTS

The author thanks for the financial support of NSFC (Grant No. 11172175). The help from M.Gauding and J. H. Gobert (RWTH-Aachen) is very appreciated for providing the DNS data.

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