Tetrahedron deformation and alignment of perceived vorticity and strain in a turbulent flow

0.5 MB·English

by  Alain Pumir

Tetrahedron deformation and alignment of perceived vorticity and
strain in a turbulent flow
Alain Pumir,1 Eberhard Bodenschatz,2,3,4 and Haitao Xu2
1)Laboratoire de Physique, Ecole Normale Sup´erieure de Lyon, Universit´e Lyon 1 and CNRS,
F-69007 France
2)Max Planck Institute for Dynamics and Self-Organization (MPIDS), G¨ottingen,
D-37077 Germany
3)Institute for Nonlinear Dynamics, University of G¨ottingen, D-37077 Germany
3 4)Laboratory ofAtomicandSolidState PhysicsandSibleySchoolofMechanicalandAerospace Engineering,
1
Cornell University, Ithaca, NY 14853, USA
0
2 (Dated: 15 January 2013)

n We describe the structure and dynamics of turbulence by the scale-dependent perceived velocity
a gradient tensor as supported by following four tracers, i.e. fluid particles, that initially form a
J
regular tetrahedron. We report results from experiments in a von K´arm´an swirling water flow and

4
fromnumericalsimulations of the incompressible Navier-Stokesequation. We analyze the statistics
1
andthedynamicsoftheperceivedrateofstraintensorandvorticityforinitiallyregulartetrahedron

of size r from the dissipative to the integral scale. Just as for the true velocity gradient, at any
] 0
n instant, the perceived vorticity is also preferentially aligned with the intermediate eigenvector of
y the perceived rate of strain. However, in the perceived rate of strain eigenframe fixed at a given
d
time t=0,the perceivedvorticityevolvesintime such asto alignwith the strongesteigendirection
-
u at t = 0. This also applies to the true velocity gradient. The experimental data at the higher
l Reynolds number suggests the existence of a self-similar regime in the inertial range. In particular,
f
s. thedynamicsofalignmentoftheperceivedvorticityandstraincanberescaledbyt0,theturbulence
c time scale of the flow when the scale r0 is in the inertial range. For smaller Reynolds numbers we
i found the dynamics to be scale dependent.
s
y
h
p
I. INTRODUCTION
[

2 The fundamental question of how small scales are generated in turbulent flows is not fully understood1.
v
From a mathematical point of view, small scales are produced by the temporalamplification of the velocity
7
gradients when following a fluid element2–4. While the velocity gradient tensor is in principle available in
5
8 direct numericalsimulations (DNS), the rare,very intense fluctuations of the gradients make it numerically
5 very challenging to accurately determine their statistical properties5. In laboratory flows, of course the
. physics guarantees that all scales are resolved. However, measurements of the full velocity gradient tensor
4
0 are in practice extremely difficult6–11.
2 Toaddressthequestionofthegenerationofsmallscales,oneneedstoknowtheevolutionoftheflowwhen
1 followingafluidelement. TherecentdevelopmentofLagrangianmeasurementtechnologynowallowsoneto
: experimentallyinvestigatethisissue9. Nevertheless,theprecisemeasurementofthevelocitygradient tensor
v
i athighReynoldsnumberswillfortheforeseeablefuture remainimmenselychallengingdue totemporaland
X
spatial resolution constraints.
r Because of the difficulty in determining reliably the velocity gradient, a fruitful alternative approach
a
consists in investigating a coarse-grainedversion of the velocity tensor over a region of size r . In addition
0
to the possibility of determining the flow evolutionfrom a Lagrangianpoint of view, andthus of addressing
fundamental questions about the generation of small scales, this approach enables the study of the scale
dependence of flow properties, especially in the inertial range of scales where the dynamics is believed to
be determined by inertia only, and dissipation can be considered as negligible. Several approaches can be
used to define a coarse-grainedvelocity gradienttensor. One method consists in measuring the true coarse-
grained tensor by averaging the velocity gradients over a well resolved region7,12. The alternative strategy
followed in this work rests on a reduced description13,14, based on defining a “perceived velocity gradient
tensor” (denoted from now on by M(t) or M (t)) that is supported only by four fluid elements, initially
ij
separated by a distance r from each other, thus forming a regular tetrahedron. The tensor M(t) reduces
0
to the true velocity gradienttensor m(t) when r 0, or more precisely, when the scale r is much smaller
0 0
→
than the Kolmogorov scale, η, the smallest length scale in the flow. One physically important feature of
this description is that it provides insight not only on M, but also on the geometry, i.e., the shape of the
tetrahedron, and on its time evolution. As we see later, the two aspects are strongly coupled13,14. Thus,
2
the approachbased on the perceived velocity gradienttensor and its evolution significantly differs from the
one based on the true coarse-grainedvelocity gradient15. ¿From a theoreticalpoint of view, the description
used here replaces a continuous field by a relatively small number of degrees of freedom. It is one of the
hopes of the present approach that such a conceptual simplification could help in devising simple tractable
models, which would help shed new light on turbulence.
The velocity gradienttensor has been shownto possessthe property that instantaneouslyvorticityaligns
withtheintermediatestretchingdirection(eigenvector)oftherateofstraintensor6,16–18,whoseeigenvalueis
predominantly positive. This was surprisingas vorticity was expected to alignwith the strongest stretching
direction. Already Taylor conjectured1,19 that the turbulent cascade mechanism relies on the amplification
ofvorticesofscaler bystretchingthatsubsequentlyleadstobreakupofthevorticesintosmallerones. Our
0
recentresults14 providenewinsightonthisconjecture. Byanalyzingthetimeevolutionoftetrahedraofsize
r ,we found thatthe perceivedvorticitytends to alignwiththe earlierdirectionofthe strongeststretching,
0
corresponding to the largest eigenvalue of the perceived rate of strain tensor. For r in the inertial range,
0
thisevolutiondependedonlyont/t ,wheret (r2/ε)1/3 isthetimescaleofeddiesofsizer inaturbulent
0 0 ≡ 0 0
flow with energy dissipation rate ε. From here on, we use “vorticity” and ”rate of strain” both for the
perceived as well as real quantities, the difference can be deduced from the scales given.
Here, we analyze the structure and the dynamics of M, and its dependence on scale, thus extending
our previous investigations14. Our experimental results at R =350 demonstrate that the dynamics of the
λ
alignmentofvorticitywiththeeigenvectorsoftherateofstrain,measuredatthetimewherethetetrahedron
is regular,depends onr throughthe rescaledtime t/t . This suggeststhe existence of a self-similarregime
0 0
in the inertial range. At smaller Reynods numbers, we found the dynamics to be scale dependent. The
numerical results show that when r 0 the properties of M extrapolate smoothly to those of the true
0
→
velocity gradienttensor,andthe dynamics of alignmentbetweenvorticityand strainis qualitativelysimilar
with one notable difference: the characteristic time scale for tetrahedra with size r0 . η is of the order of
τ , the Kolmorogovtime scale.
K
We begin, Section II, by describing the constructionof the velocity gradienttensor M. Our experimental
and numerical techniques are explained in Section III. Section IV then presents our results concerning the
evolution of the shape of the tetrahedra. In Section V, we discuss the instantaneous correlations between
strainand vorticity at any instant, and their dependence on the scale r . The dynamics of M, in particular
0
the alignment between vorticity and the initial eigenvectors of strain are presented in Section VI, which
includes a subsection of simplified theoretical analysis of the alignment process at short times. Finally,
Section VII contains our concluding remarks.
II. DEFINITION OF THE PERCEIVED VELOCITY GRADIENT TENSOR M
Givenfluidvelocitiesatfourpointsinaflow,wecomputetheperceivedvelocitygradientMasfollows13,14.
Denoting the positions of the four points as xa(t), and their velocities as ua(t), a= 1,...,4, we express the
positions and velocities with respect to the center of mass as x′a = xa x0, and u′a = ua u0, where
x0 = 1(x1+x2+x3+x4) and u0 = 1(u1+u2+u3+u4). In full genera−lity, M is defined by−minimizing
4 4 ij
the quantity:
4 3 3
2
′a ′a
K = u x M . (1)
Xa=1Xi=1(cid:16) i −Xj=1 j ji(cid:17)
In the equation defining the quantity to be minimized, K, the indices a (=1,...,4) refer to the index of the
point, and i, j (=1,...,3) to the component of the vector. By differentiating with respect to M , one finds
ij
immediately that the condition to minimize K is:
3
g M =W , (2)
ik kj ij
Xk=1
where the matrices g and W are defined by:
4 4
′a ′a ′a ′a
g = x x and W = x u . (3)
ij i j ij i j
Xa=1 Xa=1
3
Under the condition that g is invertible, the matrix M defined by:
M=g−1W, (4)
provides the best fit approximation of M based on four points.
As the flowis incompressible,weimpose theconstraintthattr(M)=0. Thiscanbe doneby determining
M, as explained in the previous paragraph, and subtracting 1tr(M)δ . Alternatively, returning to the
3 ij
minimization condition, Eq. (1), the trace can be imposed by adding a Lagrange multiplier, ξ M to
i ii
Eq. (1). The two definitions, although technically slightly different, lead to equal statistical propePrties, and
to identical physical conclusions.
The construction used here, based on Eq. (4), explicitly requires the matrix g to be invertible. As a
symmetricmatrix, g is diagonalisable. Its eigenvaluesgi arepositive,and√gi characterizethe extentofthe
tetrahedra in the ith eigendirection of g. The matrix g becomes non-invertible when one of its eigenvalue
tendstozero. Thus,theconditionthatgisinvertiblemeansphysicallythatthefourpointsofthetetrahedron
are not coplanar. The technical difficulty associated with the minimization of the quantity K for highly
flattened tetrahedra can be solved using standard Singular Value Decomposition algorithms20.
In this work, we followed tetrahedra that are initially regular (close to regular in experiments), so as to
identify flow properties at a single scale. The alignmentdynamics studied here occurs before the tetrahedra
become strongly deformed21,22 or we did not use the highly deformed tetrahedra in the analysis. As shown
inSectionIVthefractionofhighly-flattenedtetrahedraisnegligibleuptot /4andonlyapproximately10%
0
arehighly-deformedatt /2. As willbe shownin laterchapters,the mostinterestingdynamics occur before
0
t /2. To further reduce the effect of tetrahedron deformation on the accuracy of determination of M, we
0
excluded highly deformed tetrahedrafrom our statistics of M (see details in Section IV). This comes at the
cost that the number of samples in our statistics is slightly reduced.
Similar to the treatment of the true velocity gradient, it is convenient to decompose M as the sum of a
symmetric, strain-like part, S, and an antisymmetric, vorticity-like part, Ω:
1 1
S= (M+MT) ; Ω= (M MT). (5)
2 2 −
The symmetric matrix S describes the local straining motion. It is characterized by three real eigenvalues,
λ , λ and λ , associated with three eigenvectors e . In the following, the three eigenvalues are sorted in
1 2 3 i
decreasing order: λ λ λ . The antisymmetric matrix, Ω, describes the local rotation: Ω = 1ǫ ω ,
1 ≥ 2 ≥ 3 ij 2 ijk k
where ǫ is the completely antisymmetric tensor, so that Ω x= 1ω x. In the following,we characterize
ijk · 2 ×
ω by its norm ω and its direction e , with e = 1. In the limit of very small tetrahedra (r η), the
ω ω 0
| | | | ≪
tensor M reduces to the true velocity gradient tensor m, defined by m ∂ u , and the vector ω defined
ij j i
≡
above reduces to the usual definition of vorticity, ω u.
≡∇×
III. EXPERIMENTS AND NUMERICAL SIMULATIONS
A. Experiments
Usingimage-basedopticalparticletracking,wemeasuredexperimentallythetrajectoriesoftracerparticles
seeded in a swirling water flow between two counter-rotating baffled disks23,24. Measurements were done
in the center of the apparatus, where the residence time of particles is the largest. The maximum Taylor-
microscale Reynolds number is R 103, where R 15u′4/νε with u′ u2 +u2+u2 /3 being the
λ ≈ λ ≡ ≡ h x y zi
fluctuation velocity and ν the kinematic viscosity. Thepsymbol refers to tqhe average of the fluctuating
hi
quantities in the flow. Measurements were carried out by using three high-speed CMOS cameras and high
repetition rate frequency doubled Nd:YAG lasers. Our particle tracking algorithm25 allowed us to follow
simultaneously hundreds of particles in this intense turbulent flow. The measured trajectories, however,
were frequently interrupted due to experimental artifacts such as the fluctuation of the illumination light
intensity, the background and electronic noise. For multi-particle measurements, such as tetrahedra, it is
veryimportantto connectthese interruptedtrajectorysegmentsin orderto obtainbetter statistics at long-
times. We therefore re-connected the trajectories in position-velocity space26, which were then smoothed
and differentiated to obtain instantaneous particle velocities27.
Inthispaper,wefocusonresultsfromameasurementatR =350,atwhichthescaleseparationbetween
λ
the integral scale L = u′3/ε and the dissipative scale η = (ν3/ε)1/4 is L/η 800. As shown previously28,
≈
4
thetemporalandspatialresolutionsaresufficienttoaccuratelymeasurethesecondmomentsofacceleration
statistics.
¿From experimentally measured particle trajectories, we first identified instantaneous configurations of
fourpoints,separatedfromeachotherbyanominaldistancer ,withinatoleranceof 10%. Onceidentified,
0
±
the trajectories of the tetrahedra were followed for as long as all four points remained in the field of view.
Duetothe limitationofparticleseedingdensityinexperiments,thedatareportedaremainlyfortetrahedra
withinitialsizesbetweenL/17 L/5,or50 180η. Inthis rangeofscales,wefollowed107 108 tetrahedra.
− − −
The small tolerance of inter-particle distances at the nominal values results in initially nearly isotropic
tetrahedra21.
The particle trajectoriesused in this workwere measuredin an observationvolume of size approximately
(2.5 cm)3 located at the center of the apparatus, where the flow has been documented to be very close to
isotropic24. Deviationsfromisotropymaybeestimatedfromthewell-knownsecondordervelocitystructure
functions,definedasD (r)= [(u(x+r) u(x)) (r/r)]2 andD (r)= [(u(x+r) u(x))]2 D /2,
LL NN LL
h − · i {h − i− }
wherer = r istheseparationdistance. Forisotropicturbulence, D andD dependonlyonr andthey
LL NN
satisfy the|fo|llowing relations(see e.g.,29): D (r)/D (r)=2 for r η and D (r)/D (r)=4/3. For
NN LL NN LL
≪
the von K´arm´an flow in our experiment, due to the axisymmetric geometry, the statistics could depend on
the angle θ between the axis of rotation and the separationvector r. We found that for 40η r L/3,the
≤ ≤
ratio(3/4)D (r,θ)/D (r,θ)variesasafunctionofθ inthe range0.8 1.3,butaveragesouttobe very
NN LL
−−
close to 1, the isotropic value, when considering all possible orientations of the separation. This provides
evidence both that the dependence on the angle θ is weak, and that isotropy is recovered when averaging
out over all possible directions30,31. Further evidence that the anisotropies in our flow are not a concern
comes from the fact that most experimental statistics derived from tetrahedra are in good agreement with
DNS results using perfectly isotropic tetrahedra, see also14. The small deviation from precise isotropy in
initial shapes, however, did give rise to discrepancies for certain quantities sensitive to initial shape, which
we will discuss in detail later.
B. Direct Numerical Simulations
We used a spectral code to solve the Navier-Stokes equations for the (Eulerian) velocity field u(x,t):
∂ u(x,t)+(u )u(x,t)= p(x,t)+ν 2u(x,t) (6)
t
·∇ −∇ ∇
u(x,t)=0 (7)
∇·
Eqs. (6) and (7) are integrated in a periodic box (size 2π), with up to 3843 modes. Energy is injected into
theflowatlargescalebylettingthelowwavenumbermodes k K (here,K =1.5)evolveaccordingtothe
| |≤
Euler equations truncated to the shell k K. This maintains a constant amount of energy at the largest
scales32. Because the simulated flow is|h|ig≤hly isotropic, we define the turbulence parameters based on the
x component of velocity. The Reynolds number is
′
λ u
g
R , (8)
λ
≡ ν
where the Taylor micro-scale defined as
u′2
λ2 = , (9)
g (∂ u )2
x x
h i
in which u′ u2 1/2. The Kolmogorov length scale is η = (ν3/ε)1/4, where ε = ν ∂ u ∂ u . The
≡ h xi h j i j ii
available resolution has allowed us to simulate reliably turbulent flows up to a Reynolds number R =170.
λ
Quantitatively,the largestwavenumberfaithfully simulated, k , is such that the productk η 1.4.
max max
Defining the integrallengthscaleby L=u′3/ε,the ratioL/η is approximately300atR =170×. We≈note
λ
that L defined here is roughly twice as large as the correlation length scale of the Eulerian velocity field:
∞
π
L k−1E(k)dk L/2, (10)
corr ≡ 2u′2 Z ≈
0
where E(k) is the energy spectrum.
5
With the Eulerianvelocity field determined by solving numerically Eqs. (6) and (7), we also followedthe
motion of tetrahedra, whose vertices are tracer fluid particles, evolving according to:
dx(t)
=u(x(t),t). (11)
dt
Tothisend,thevaluesofu,knownaftereachtimesteponaregulargridofcollocationpoints,areinterpolated
at the location x(t) of the Lagrangiantracers, using accurate third order schemes33.
To study the dynamics using tetrahedra, we initialized tetrahedra of particles mutually separated by a
distance r0, which we varied overthe full range of availablescales, η .r0 .L. The statistical properties at
a given scale r have been determined by following at the minimum 125,000 tetrahedra.
0
IV. DEFORMATION OF THE TETRAHEDRA
Weconsiderherethedeformationoccuringoverashorttimescale,whiletheprocessofalignmentbetween
theperceivedvorticityandstrainistakingplace. Toquantifythedeformationoftetrahedra,itisconvenient
to introduce the tensor g, defined in Eq. (3), and its eigenvalues14,21,22,34,35. The trace of g, which is the
square of the radius of gyration of the four points, quantifies the overall size of the object. The tensor
I g/tr(g), whose trace is 1, characterizes the shape of the object. Specifically, because I is a symmetric
≡
matrix, it can be diagonalized in an orthogonal basis formed by its eigenvectors e . In the following, we
Ii
arrangethe eigenvalues of I in decreasing order: I I I . Since I +I +I =1, a regular tetrahedron
1 2 3 1 2 3
≥ ≥
corresponds to I =I =I =1/3, while a nearly co-planar configuration gives I 0.
1 2 3 3
≈
The matrix g is related to the moment of inertia tensor, J, familiar in the context of mechanics of solid
bodies, by14:
J tr(g)δ g . (12)
ij ij ij
≡ −
A. Alignment of tetrahedron geometry with the eigenvectors of the rate of strain
The dynamics of tetrahedra is strongly influenced by the local rate of strain. In particular, it has been
demonstrated14 that the stretching in the direction of the largest eigenvalue of the strain, e , leads to an
1
elongationofthe tetrahedroninthis direction,and,byconservationofangularmomentum, toanalignment
of the vorticity ω(t) in the direction e (0). The results presented here generalize this observation: at short
1
times, the deformation of the tetrahedra is very well aligned with the eigendirection of the eigenvectors of
the rate of strain tensor.
Fig. 1 shows how the principal axes of initially regular tetrahedra, e (t), align with the eigenvectors of
Ii
the rate of strain, e (0). For initially isotropic tetrahedra, as in DNS, the matrix I is diagonal at t = 0. A
i
very small deformation of the tetrahedra due to the strain breaks the isotropy, and leads to the immediate
alignment of the principal axes of the tetrahedra with the strain eigenvectors. This is reflected in Fig. 1
by the perfect alignment between e (0) and e (t) at very small times t (t = 0+). By t = 0+, we refer
k Ik
to a strictly positive, but very small value of time. The tetrahedra studied experimentally are not exactly
isotropic initially. Instead, the eigenvalues of I are slightly different from each other: I = 1 +δI , where
i 3 i
δI 1. At t=0, the corresponding three eigenvectors of I are uncorrelatedwith the strain eigenvectors,
i
i|.e.,|≪[e (0) e (0)]2 1/3. This explains the difference at short times seen in Fig. 1 between the curves
Ik j
h · i ≈
obtainedfromDNSandfromexperiment. Onthe otherhand,under the actionofthe strain,the tetrahedra
deform quickly and their principal axes align with the eigenvectorsof the strain in a time of approximately
t /10, after which the tetrahedron evolution is independent of initial shape. We note that over the range
0
of scales shown here, L/16 r L/2, the dynamics of alignment is essentially self-similar, once expressed
0
≤ ≤
in units of t . In a time of order t , the axes e (t) lose any memory of the initial alignment with e (0);
0 0 Ik j
all values of [e (t) e (0)]2 relax towards 1/3. This is consistent with the fact that t , which is the time
Ik j 0
h · i
introducedinKolmogorovtheory,isthepropercorrelationtimesofanobjectinitiallywithascaler . Small
0
deviations are visible for DNS data of r =L/16 20η, which is knownto be the lower limit of the inertial
0
≈
range.
6
1 1
DNS, r=L/16 DNS, r=L/16
0 0
0.9 EXP, r≈L/9 0.9 EXP, r≈L/9
0 0
0.8 e1 DEXNPS,, rr0≈=LL//58 0.8 e DEXNPS,, rr0≈=LL//58
0.7 0 0.7 2 0
DNS, r=L/4 DNS, r=L/4
2〉0)] 0.6 DNS, r00=L/2 2〉0)] 0.6 DNS, r00=L/2
〈[e(t).e(I1i00..45 〈[e(t).e(I2i00..45
0.3 e2 0.3 e1
0.2 0.2
e
3
0.1 0.1
e
3
0 0
0 0.5 1 1.5 0 0.5 1 1.5
t/t t/t
0 0
(a) (b)
1
DNS, r=L/16
0
0.9 EXP, r≈L/9
0
0.8 e DNS, r0=L/8
3 EXP, r≈ L/5
0.7 0
DNS, r=L/4
2〉0)] 0.6 DNS, r00=L/2
〈[e(t).e(I3i00..45
0.3 e
2
0.2
0.1 e1
0
0 0.5 1 1.5
t/t
0
(c)
FIG. 1. (Color online) Deformation of tetrahedra: alignment of the principal axes of the tetrahedra with the
eigenvectors of the strain. The ensemble averages h[e (t)·e (0)]2i as a function of time for (a) i=1(longest axis),
Ii j
(b) i = 2 (intermediate axis) and (c) i = 3 (shortest axis of the tetrahedron). As expected, the ith axis of the
tetrahedron, e (t), aligns perfectly at very short times with theith eigenvector of the strain, e (0). After a time of
Ii i
order∼t ,thealignment relaxes and theaxesof thetetrahedron donot show any particular alignment with any of
0
theeigenvectors of thestrain. TheDNS data correspond to R =170, whereas theexperiments to R =350.
λ λ
B. Flattening of the tetrahedra
Along with the stretching of the tetrahedra in the direction e (0), documented in particular in14, one
1
expects a strong compression in the direction e (0), corresponding to the most negative eigenvalue of the
3
perceivedstrain. Thishasbeenshowntoleadtoverysignificantlyflattenedtetrads21,22,35. Inparticular,the
probabilitydistributionfunction(PDF)ofI3issharplypeakedaround0atlatetimes,t/t0 &1. Asexplained
in Section II, this can be the cause of a serious limitation in our construction of the perceived velocity
gradient, as implied by Eq. (4). A good understanding of the dynamics leading to flattened tetrahedra is
thusofinterestnotonlyforitsownsake,butalsoinrelationtothedynamicsofalignmentbetweenperceived
vorticityandstrain. This subsectionthereforediscusses the flattening oftetrahedra,anddemonstratesthat
the formation of very flat configurations is not faster than the alignment process between vorticity and
strain.
To this end, Fig.2 showsthe evolutionofthe PDFofI (a), andthe evolutionofthe probabilitythatthe
3
value of I is less than a very small number, I 10−3 (b). Initially, the probability of I is peaked near
3 3 3
≤
I = 1/3. In DNS, which starts with precisely isotropic tetrahedra, the PDF is originally a δ-function at
3
I = 1/3. In experiments, where we select particles that form nearly isotropic tetrahedra, the initial PDF
3
7
15 DNS, r = L/16 EXP, r≈L/17
0 0
EXP, r ≈ L/9 t/t=0.07 DNS, r=L/16
DNS, r0 = L/2 0 EXP, r00≈L/12
0 EXP, r≈L/9
0
10 0.1 DNS, r=L/8
Pdf(I)3 t/t0=0.21 t/t0=0.13 P( I < 0.001 )3 DDNNSS,, rr000==LL//42
5 0.05
0 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5
I t/t
3 0
(a) (b)
FIG. 2. (Color online) Flattening of tetrahedra. (a) PDF of I at several times, t/t = 0.07, 0.13 and 0.21, for
3 0
different initial tetrahedron sizes (r from L/16 to L/2) in both DNS (R =170) and experiment (R =350). The
0 λ λ
observedpeak of thePDFdecreases when timeincreases. Att=0.21t , anon-zeroPDFat I ≈0isobserved. The
0 3
probabilitythatI ≤10−3 isshownin(b)forseveralvaluesofr ,asafunctionofthenormalizedtimet/t . Asteep
3 0 0
growthoftheprobabilityP(I3 ≤10−3)isseenfort&t0/4. Thegrowthoftheprobabilitydoesnotdependmuchon
thescaler overtheinertialrange: fromr =L/16≈20η inDNSandr =L/17≈50η inexperimentstor =L/2.
0 0 0 0
is not a δ-function, but is still narrowly distributed around I 0.3, as shown before21. As time increases,
3
≈
the peak of the PDF shifts towards smaller values of I (Fig. 2a), reflecting a tendency of the tetrahedra
3
to flatten. The evolution of the PDF does not depend very much on the initial size of the tetrahedron, as
seen when comparing DNS and experimental data ranging from r = L/16 20η in DNS to r = L/2. A
0 0
≈
noticeable fraction of the tetrahedra becomes flattened after t & t0/4 (see Fig. 2(b)), consistent with the
sharply peaked distribution of I observed at later times21,22,35. ¿From a flat configuration, with all points
3
in the same plane, it is not possible to extract any information on the variation of the velocity in the plane
transverse to the plane. As explained in Section II however, this does not prevent us from computing M
using Singular Value Decomposition. However, it points to a limitation on the accuracy of the procedure.
Therefore,in the results reportedhere, we excluded tetrahedrawith I 10−3 fromthe statistics ofM. As
3
≤
seenfromFig.2(b), this leadstoatmaximuma10%reductioninthe numberofsamplesattime t/t =0.5,
0
while most of the interesting dynamics occurs before this time.
The results of this section thus show a very strong alignment between the axes of the tetrahedra and
those of strain, see Fig. 1, and that the dynamics of flattening of the tetrahedra happens over a time of the
order of t /4, see Fig. 2. In addition, we find that the dynamics is essentially self-similar, insofar as the
0
time-dependent correlations presented here are independent of scale, once time has been expressed in units
of t . This is consistent with earlier results21 demonstrating that the evolution of the shape factors, I (t),
0 i
is also self-similar. The ensuing picture is thus that in a statistical sense, initially regular tetrahedra of size
r evolves with a time scale (r2/ε)1/3, when r in the inertial range.
0 ∼ 0 0
V. PROPERTIES OF M: SCALE DEPENDENCE (INSTANTANEOUS STATISTICS)
Previousinvestigations6,10,16,17,36 ofthestructureofthetruevelocitygradienttensor,m,revealtwomain
properties of the instantaneous statistics of m:
(i) The direction of the vorticity e is preferentially aligned with e , the direction of the intermediate
ω 2
eigenvalue of the rate of strain, s = 1(m+mT), but does not show any particular alignment with e , the
2 1
strongest eigen-direction of the rate of strain.
(ii)Theintermediateeigenvalueoftherateofstrain,λ ,ismostlypositive,whichimpliesthattheproduct
2
of the eigenvalues of s is negative and thus ensures vortex stretching ω s ω 0, an important property
of turbulent flows37. h · · i≥
It is natural to ask what are the instantaneous properties of the perceived velocity gradient M.
8
On general grounds, one expects that the properties of M will depend on the size r0. For r0 . η, M
coincides with the true velocity gradienttensor. As the separationr betweenthe points of the tetrahedron
0
increase,the velocities atthe four points become less and less correlated. For r L, M is not expected to
0
≫
reflect any particular property of the turbulent dynamics. On the other hand, one may expect that in the
inertial range, some properties become scale-independent, consistent with the notion that the flow shows
some degree of self-similarity.
A. Alignment between vorticity and strain
The PDFs of the cosines of the instantaneous angles between e and e , are shown in Fig. 3. The PDF
ω i
for i = 1 (Fig. 3(a) and (b)) hardly varies through scale, and remains nearly uniform: P(e e ) 1,
1 ω
consistent with previous observations with the true velocity gradient tensor6,17. In Fig. 3(c|) an·d (|e),≈the
alignment corresponding to the true velocity gradient tensor m (i.e., r = 0) are the ones with the largest
0
variations. As scale increases, the alignment between e and e becomes less pronounced, as shown by the
2 ω
observed decrease of the PDF at e e = 1. Similarly, the tendency of e to be perpendicular to e
2 ω ω 3
| · |
diminishes when scale increases. As the scale r approaches L, the PDF becomes flat. This tendency can
0
be clearlyseeninFig.4,whichshowsthe averagedvalues (e e )2 . Whenr increasesfromdissipativeto
i ω 0
integral scales, (e e )2 continuously decreases from thhe la·rger miagnitute for the true velocity gradient
ω 2
h · i
tensor (shown by the horizontal dashed lines in Fig. 4) towards 1/3, which corresponds to two randomly
selected vectors. In the same spirit, (e e )2 continuously increases from the lower magnitude of the
ω 3
h · i
velocity gradient tensor to 1/3 when r increases. In Fig. 4 are shown both the numerical data (R =170,
0 λ
open symbols) and the experimental data (R = 350, filled symbols). Remarkably, in both Figs. 3 and
λ
4, the lower Reynolds number data show a continuous variation of the statistics of the cosines e e and
2 ω
·
e e with scale. The variation is particular strong when r decreases towards the dissipative range, i.e.,
3 ω 0
·
for r0 . L/16 20η, In comparison, the higher Reynolds number data suggest that over the accessible
≈
rangeofscales,the propertiesoftheflowareessentiallyscaleindependent. Thissuggeststhatatsufficiently
large Reynolds numbers, the dynamics of M may show some self-similarity, as far as the relation between
vorticityandstrainis concerned. This remainsto be studiedfurther withdata atlargerReynolds numbers.
The results summarized in Figs. 3 and 4 thus show that the alignment properties of e and e , or the
ω 2
orthogonalitybetweene ande ,areobservablethroughouttheentireinertialrange. Thisistobecontrasted
ω 3
with the lack of alignment between e and e , which is observed over all the scales that we have studied.
ω 1
B. The intermediate eigenvalues of the rate of strain
We now turn to the statistics of λ , the eigenvalues of the rate of strain. Whereas it is clear, from the
i
relations λ λ λ and λ +λ +λ = 0 that λ 0 and λ 0, the sign of λ can be either positive
1 2 3 1 2 3 1 3 2
≥ ≥ ≥ ≤
or negative. To quantify the relative value of λ compared to λ and λ , it is convenient to introduce the
2 1 3
dimensionless variable β, defined by17:
√6 λ
2
β . (13)
≡ λ2+λ2+λ2
1 2 3
p
This definition ensures that 1 β 1.
− ≤ ≤
The PDF of β is shown in Fig. 5a for several values of r . At small Reynolds number, the PDFs show
0
a clear evolution as a function of the scale r . In the dissipative range (r = 0, i.e., for the true velocity
0 0
gradient tensor), a strong bias towards high values of β is observed. As r increases, the PDFs show a
0
reduced bias towards high values of β. For the highest Reynolds number experimental flow, the PDF of β
changes little throughout the limited range of measured scales. As shown in Fig. 5b, the mean β remains
h i
positive throughoutthe entire inertialrange,droppingfrom β 0.285forr =0,to β 0.07for r L
0 0
h i≈ h i≈ ≈
forthelowReynoldsnumberdata. Onthecontrary,thevalueforthehigherReynoldsnumberexperimental
flow, in the range 1/17 r /L 1/5 is close to 0.13. Figure 5 suggests that the dependence of β onscale
0
≤ ≤ h i
isrelativelyweakintheinertialrangeL/16.r0 .L/2,inparticularatthehighestReynoldsnumber. This
is consistentwith the trendobservedinsubsectionVA,whichpoints to a scale-independentstructure ofM
in the inertial range at sufficiently high Reynolds numbers.
The results of this sectionsuggesta smooth change of the structure of M throughoutthe rangeof scales.
Whereas no particular alignment between e and e is observed,irrespective of the size of the tetrahedron,
ω 1
9
4 DNS, r0=0 DNS 4 EXP, r0≈L/17 EXP
3.5 DNS, r0=L/64 3.5 EXP, r0≈L/9
DNS, r0= L/32 EXP, r0≈L/5
3 DNS, r0=L/8 3 EXP, r0≈L/3
DNS, r=L/2
2.5 0 2.5
|)ω |)ω
P(|e.e1 2 P(|e.e1 2
1.5 1.5
1 1
0.5 0.5
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
|e.e | |e.e |
1 ω 1 ω
(a) (b)
4 DNS, r0=0 DNS 4 EXP, r0≈L/17 EXP
3.5 DNS, r0=L/64 3.5 EXP, r0≈L/9
DNS, r0=L/32 EXP, r0≈L/5
3 DNS, r0=L/8 3 EXP, r0≈L/3
DNS, r=L/2
2.5 0 2.5
|)ω |)ω
P(|e.e2 2 P(|e.e2 2
1.5 1.5
1 1
0.5 0.5
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
|e.e | |e.e |
2 ω 2 ω
(c) (d)
4 DNS DNS, r0=0 4 EXP EXP, r0≈L/17
3.5 DNS, r0=L/64 3.5 EXP, r0≈L/9
DNS, r0=L/32 EXP, r0≈L/5
3 DNS, r0=L/8 3 EXP, r0≈L/3
DNS, r=L/2
2.5 0 2.5
|)ω |)ω
P(|e.e3 2 P(|e.e3 2
1.5 1.5
1 1
0.5 0.5
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
|e.e | |e.e |
3 ω 3 ω
(e) (f)
FIG. 3. (Color online) Scale dependence of the alignment of vorticity and strain (instantaneous statistics): The
PDFs of |e ·e | from both DNS (panels a,c,e) and experiments (panels b,d,f) for tetrahedra with different sizes r
i ω 0
from dissipative (r = 0 in DNS) to integral (L/2) scales. Panels (a) and (b) are for i = 1; (c) and (d) for i = 2;
0
and (e) and (f) for i = 3. The lack of alignment of vorticity with e is nearly independent of scale. The PDF of
1
|e ·e |sharplypeaksat1atverysmallscales,inparticularforthetruevelocitygradienttensor(r =0). Theeffect
ω 2 0
weakens as the scale increases. Similarly, the maximum at 0 of the PDF of |e ·e | at very small scales becomes
ω 3
milder as scale increases. The DNS data correspond to R =170, whereas theexperiments toR =350.
λ λ
10
0.6
e
1
e
2
0.5
e
3
2〉
0)) 0.4
(ω
e
0).
e(i0.3
〈(
0.2
0.1
2−7 2−6 2−5 2−4 2−3 2−2 2−1 20
r /L
0
FIG.4. (Coloronline)Themeanvaluesh(e (0)·e (0))2i. TheopensymbolsareDNSdata(R =170)andthefilled
ω i λ
symbols are experimental results (R =350).
λ
the tendency ofe to alignwith e (respectively to be perpendicular to e ) is the strongestfor r 0, i.e.,
ω 2 3 0
→
for the true velocity gradienttensor m, and decreases for M with increasing r . Similarly, the intermediate
0
eigenvalue of the rate of strain is comparatively more positive for the true velocity gradient tensor m than
for M at values of r in the inertial range. As the scale r increases towardsthe inertial length L, the PDF
0 0
of β becomes more symmetric about β =0, as expected from a random matrix.
Thus, the perceived velocity gradient tensor M in the inertial range shares many essential properties
with the true velocity gradient tensor m. Our observations thus suggest a continuity in the behavior from
the dissipative to the inertial range. Remarkably, the data obtained with the higher Reynolds number
experimental flow suggest that the structural properties of M become scale independent in the inertial
scale. This should be confirmed by studying flows at even higher Reynolds numbers.
VI. PROPERTIESOF M: DYNAMICS (TEMPORAL CORRELATIONS)
The main observation in our recent study14 was the dynamical alignment of vorticity with the leading
eigenvector of the rate of strain, which we were able to study conveniently by investigating the correlation
between e (0) and e (t), i.e., with a time delay. Hence the lack of alignment between e (t) and e (t)
1 ω ω 1
(Figs.3(a)and(b))isaconsequenceofthede-correlationofthevectore itselfinatimet,whichcounteracts
1
the alignment of e (t) with e (0).
ω 1
Aconsequenceoftheprevioussectionisthatitisnota priori clearthatthealignmentofe (t)withe (0)
ω i
(i = 2 or i = 3) is going to be self-similar, i.e., that the angle between e (t) and e (0) (i = 2 or i = 3) is
ω i
going to depend only on t/t , as it had been found for the angle between e (t) and e (0)14. This is due
0 ω 1
to the fact that the PDFs of e (0) e (0), for i = 2 and i = 3 change with the scale r for small scale
ω i 0
| · |
separation L/η.
A. Averaged alignment of the direction of vorticity with the eigenvectors of the rate of strain
Fig. 6(a) shows that, as time increases, the vector e (t) tends to become better aligned with e (0). As
ω 1
reportedpreviously14,thestatisticscharacterizingthealignmentofthetwovectorsisessentiallyindependent
of the scale r , once time t is rescaled by t . The characteristic time scale for this process, Fig. 6(a) is of
0 0
order t /5. For the DNS at R =170, the alignment of e (t) with e (0), and to a lesser extend with e (0),
0 λ ω 2 3
showsasignificantdependence onscale,see Fig. 6(b) and(c). Note thatfort=0,the difference wasshown
as a function of r /L in Fig. 4. In comparison, the experimental data obtained at R =350 shows a much
0 λ
reduced variation as a function of scale (see also Fig. 4). We also notice that [e (0) e (t)]2 is almost
2 ω
h · i