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Viscous tilting and production of vorticity in homogeneous turbulence
M. Holzner1, M. Guala2, B. Lu¨thi1, A. Liberzon3, N. Nikitin4, W. Kinzelbach1 and A. Tsinober3
1 Institute of Environmental Engineering, ETH Zurich, CH 8093 Zurich, Switzerland
2 Galcit, California Institute of Technology, Pasadena CA 91125, USA
3 School of Mechanical Engineering, Tel Aviv University, Ramat Aviv 69978, Israel
4 Institute of Mechanics, Moscow State University, 119899 Moscow, Russia
(International Collaboration for Turbulence Research)
(Dated: February 8, 2010)
Viscous depletion of vorticity is an essential and well known property of turbulent flows, balancing,
in the mean, the net vorticity production associated with the vortex stretching mechanism. In this
letter we however demonstrate that viscous effects are not restricted to a mere destruction process,
but play a more complex role in vorticity dynamics that is as important as vortex stretching. Based
on results from particle tracking experiments (3D-PTV) and direct numerical simulation (DNS) of
homogeneous and quasi isotropic turbulence, we show that the viscous term in the vorticity equation
can also locally induce production of vorticity and changes of its orientation (viscous tilting).

2In turbulent flows, the energy is injected at large scales by some forcing mechanism and dissipated into heat through
the effect of viscosity at the smallest scales of motion, e.g. Ref. [1]. The main physical mechanisms that control fluid
turbulence at the smallest scales are commonly described in terms of strain and vorticity, quantities that represent
the tendency of fluid parcels to deform and rotate, respectively.
One of the most prominent processes occurring at small scales is the so-called ‘vortex stretching’: following a
common argument,1 if a vortical fluid element is stretched by the surrounding flow, the rotation rate should increase
to conserve angular momentum. However, Lu¨thi et al.2 showed that this does not hold true point-wise and the
dynamics are significantly influenced by a viscous contribution. The enstrophy balance equation,
D
Dt
ω2
2 = ωiωjsij + νωi∇
2ωi, (1)
where the squared vorticity magnitude ω2 denotes the enstrophy, sij the rate of strain tensor and ν the kinematic
viscosity of the fluid, contains a production term ωiωjsij and a viscous term νωi∇2ωi. The two terms in the mean
(hereinafter mean values < · > are obtained by spatial and temporal averaging) approximately balance each other,
i.e., 〈ωiωjsij〉 ' −〈νωi∇2ωi〉, see Ref. 1. The presence of a viscous contribution in Eq. (1) shows that the effect of
molecular viscosity is not limited to energy dissipation through deformation work, expressed as ε = 2νsijsij , but,
among other things, it controls also vorticity growth. The effects of vortex stretching and viscous destruction are
usually captured in the well-known picture that in turbulence at small scales the nonlinearities increase gradients,
whereas the viscosity depletes them, e.g. Refs. 1,3 and references therein. However, as noted already by, e.g. Tennekes
and Lumley,1 viscous effects are not restricted to vorticity destruction only. For example, viscosity may tilt vorticity,
see, e.g. Refs. 1,3,4,5,6 and is believed to be responsible for vortex reconnection, e.g. Ref.s 3,4 and 5. It is reminded
that this ‘classical’ reconnection mechanism (due to viscosity) is fundamentally different from reconnection events in
quantum fluids, which take place due to a quantum stress acting at the scale of the vortex core without changes of
total energy.7,8 However, direct experimental evidence for the occurrence of tilting and production of vorticity due to
viscosity is still missing in the literature, also because up to now it was difficult to measure the associated small scale
quantities experimentally. Derivatives of the velocity became accessible through particle tracking experiments since
the developments in, e.g. Ref.s 2,9,10. Holzner et al.10 recently measured viscous production of vorticity in proximity
of turbulent/nonturbulent interfaces, which raised the question about the role of positive νωi∇2ωi in fully developed
and homogeneous turbulence.
In this letter we present the first measurements of tilting, depletion and considerable production of vorticity through
viscosity in a turbulent flow through particle tracking velocimetry (Ref.s 2,9,10). The main goal is to unfold viscous
effects on vorticity dynamics at the small scales of turbulence, with an emphasis on genuine (i.e. intrinsic to Navier
Stokes turbulence as opposed to kinematic) effects. The results discussed hereafter are based on higher order deriva-
tives and are challenging to obtain, both experimentally and numerically, which is why we compare the experimental
results with those obtained through direct numerical simulation.
We measured the flow velocities and its gradients in a laboratory experiment of homogeneous, quasi isotropic and
statistically stationary turbulence by using particle tracking velocimetry, see Ref.s 2 and 11 for details. Particle
tracking velocimetry is based on high speed imaging of the motion of small buoyant tracer particles seeded into the

3−20 −10 0 10 20
10−3
10−2
10−1
100
ωiωjsij/〈ωiωjsij〉, νωi∇2ωi/〈ωiωjsij〉
P
D
F
−5 0 5
100
FIG. 1. PDFs of ωiωjsij (—, •) and νωi∇2ωi (− −, +) normalized with 〈ωiωjsij〉. Symbols are from PTV, lines from DNS.
The inset shows the analogous results from a random Gaussian velocity field, ωiωjsij (—), ωi∇2ωi (− −), the vertical reflection
of the PDF corresponding to negative events, ωiωjsij< 0 (− · −), demonstrates the symmetry.
flow. The experiment was carried out in a glass tank filled with water and the flow was forced mechanically from two
sides by two sets of rotating disks as in Ref. 11. The observation volume of approximately 15 x 15 x 20 mm3 was
centered with respect to the forced flow domain, mid-way between the disks. The turbulent flow is characterized by
an r.m.s velocity of about 10 mm/s, a Taylor-based Reynolds number of Reλ = 50 and the Kolmogorov length and
time scales are estimated at η =0.5 mm and τη =0.25 s, respectively. The Laplacian of vorticity, ∇2ω, is obtained
indirectly from the local balance equation of vorticity in the form ∇ × a = ν∇2ω by evaluating the term ∇ × a
from the Lagrangian tracking data. Through this indirect method only one derivative in space is needed instead of
three, but particle positions have to be differentiated twice in time in order to get Lagrangian acceleration. For the
numerical simulation we used an open source turbulence database12 that was developed at Johns Hopkins University,
see Ref.s 13,14 for details. The data are from a direct numerical simulation of forced isotropic turbulence on a 10243
periodic grid, using a pseudo-spectral parallel code. The Taylor Reynolds number is Reλ = 434. After the simulation
had reached a statistically stationary state, 1024 frames of data, which includes the 3 components of the velocity
vector and pressure, were generated and stored into the database. The time interval covered by the numerical data
set is thus only one large-eddy turnover time, whereas it is O(10) turnover times for the experiment. For comparison
to a random velocity field, divergence-free Gaussian white noise was generated as in Ref. 15.
First, we statistically analyze effects of viscosity on the vorticity magnitude. One of the most basic phenomena of
three dimensional turbulence is the predominant vortex stretching, which is manifested in a positive net enstrophy
production, 〈ωiωjsij〉 >0, e.g., Refs. 1,3 and references therein. A strong positive skewness of the Probability Density
Function (PDF ) of the term ωiωjsij is indeed visible in Fig. 1, in agreement with earlier results, e.g. Ref. 3. For
statistically stationary turbulence the growth of enstrophy is balanced by viscous effects, i.e., the two terms on the
RHS of Eq. (1) balance in the mean. Consistently, the term νωi∇2ωi shows an opposite distribution, being strongly
negatively skewed (Fig. 1). Although viscosity mostly depletes enstrophy, we note that also events where νωi∇2ωi> 0

4−1 0 1
10−0.5
10−0.1
100.3
cos(ω,W )
P
D
F


−1 0 110
−4
10−2
100
102
cos(ω,∇2ω)
(a) (b)
FIG. 2. PDFs of the cosine between vorticity and the vortex stretching vector (a) and between vorticity and its Laplacian (b),
as obtained from DNS (—), PTV (− −) and random Gaussian field (− · −).
are statistically significant. In fact, about one third of all events represent viscous production of enstrophy. The
experimental curves qualitatively agree with the numerical ones, the PDFs obtained from DNS are slightly more
skewed. It is important to note that, while the reasons for the positiveness of the mean enstrophy production
term are dynamical and due to interaction between vorticity and strain, the destructive nature of the viscous term
〈νωi∇2ωi〉 < 0 arises also for kinematical reasons: one can decompose the viscous term as, e.g.
ωi∇2ωi = −∇ · (ω × (∇× ω))− (∇× ω)2, (2)
where the first term on the RHS is a divergence of a vector and vanishes in the mean for homogeneity, whereas the
second is a (always negative) dissipation term16. Indeed, while for a Gaussian random field 〈ωiωjsij〉=0 and the PDF
of ωiωjsij becomes symmetric, the PDF of the viscous term is strongly negatively skewed, see the inset in Fig. 1.
This means that the destructive nature of the viscous term is also recovered in a random field and does not represent
a genuine property of turbulent flow fields. However, from the same inset, we estimate that for a random gaussian
field, the events with ωi∇2ωi > 0 are statistically far less significant (about 2% of all events) compared to the same
events in a Navier Stokes field (about 30%). We therefore conclude that considerable viscous production of vorticity
is a genuine characteristic of Navier Stokes turbulence.
The positiveness of the mean enstrophy production is associated with the predominant alignment between vorticity
and the vortex stretching vector. The enstrophy production can be expressed as the scalar product of vorticity and
the vortex stretching vector, ωiωjsij=ω·W, where Wi = ωjsij . In real turbulent flows, the two vectors are strongly
aligned. Thus, the PDF of the cosine between ω and W is asymmetric (Fig.2a), in conformity with the prevalence
of vortex stretching over vortex compression, whereas it is symmetric for a random Gaussian field (Fig. 2a), see also
Ref. 3 and references therein. Analogously, we show the alignment between ω and ∇2ω in Fig. 2b. The figure shows
high probabilities (much higher for the random field) of pronounced anti-alignment between ω and ∇2ω, consistent
with the negative skewness of the PDF of νωi∇2ωi, but we also note that with some smaller probability the two
vectors can attain any orientation and, in particular, they can also be strongly aligned. This reminds of the results
in Ref. 10, who measured cos(ω,∇2ω) ' 1 in the proximity of the interface between turbulent and irrotational flow
regions. The fact that the two vectors are not always strictly anti-aligned implies that the term ν∇2ω does not

5−20 −10 0 1010
−3
10−2
10−1
100
νωi∇2ωi/〈ωiωjsij〉
P
D
F

(a)
−1 −0.5 0 0.5 10
0.5
1
1.5
2
2.5
cos(ω, ν∇2ω)
P
D
F

(b)
FIG. 3. PDFs of ωiωjsij and νωi∇2ωi (a) and of the cosine between vorticity and its Laplacian (b) for different (ω-λi)
alignments from DNS (lines) and PTV (symbols), ω aligned with λ1 (—, •), λ2 (− − , +) and λ3 (− · −, ∗).
act exclusively in the direction of the vorticity vector (mostly dampening and sometimes increasing the vorticity
magnitude), but also normally to it, thus contributing to altering the orientation of vorticity. Since the negative
skewness of the PDF is much stronger for the random velocity field than for the turbulent one, we may infer that
viscous tilting is characteristic of fluid turbulence. The observation that the viscous term can effectively influence
the orientation of vorticity is important, also because this will affect the relative orientation between ω and λi and
therefore indirectly influence the vortex stretching (compression) mechanism.
The inviscid tilting of vorticity was measured by Guala et al.6 and found to be sensitive to the alignments between
vorticity and the strain eigenvectors. With the present data it is possible to estimate for the first time both the inviscid
and the viscous contribution to the tilting of vorticity and to quantify the influence of the relative (ω-λi) alignments.
We adopt the approach of Ref. 6 and condition the data on situations of different alignment of vorticity with the
principal axis of the strain eigenframe. Note that in a Gaussian field no differences are observed when conditioning
on such alignments and therefore the expected effects in turbulent flow are explicitly dynamical.
Fig. 3a depicts the PDFs of the two terms divided into the three subsets depending on the local alignment between
ω and λi. The subsets are divided according to the condition cos2(ω,λi) ≤ 0.7, corresponding to a cone of roughly
33◦, as in Ref. 6. It is visible that, while for the case of alignment with the intermediate eigenvector, λ2, the PDF
becomes more skewed, i.e. νωi∇2ωi contributes more to the reduction of ω2, whereas in the case of alignment with
λ1, the skewness decreases and even more so when vorticity is aligned with λ3. Again, the main qualitative trends
are the same both for the numerical and experimental results, with the curves obtained from DNS showing a stronger
skewness.
In Fig. 3b we analyze how this qualitatively different behavior of the term νωi∇2ωi is reflected in the alignment
between ω and ∇2ω. The PDF of the cosine between the two vectors is strongly negatively skewed for the cases when
ω is aligned with λ1 and λ2. In the case of ω aligned with λ3 the distribution changes dramatically becoming very
flat in conformity with the reduced skewness of the PDF of νωi∇2ωi. Therefore, in this case viscosity contributes
less to the destruction of enstrophy, but still plays a role, e.g. for the tilting of the vorticity vector.

6100
10−4
10−2
100
η2inv/〈Ω2〉, η2vis/〈Ω2〉, Ω2/〈Ω2〉
P
D
F
100
10−2
100
η2inv/〈Ω2〉, η2vis/〈Ω2〉, Ω2/〈Ω2〉
(a) (b)
FIG. 4. PDFs of inviscid (—), viscous (− −) and total (− · −) tilting from DNS (left) and PTV (right).
The inviscid and the viscous contribution to the total tilting Ω of vorticity can be written as follows,
Ωk =
Dω̂k
Dt = η
i
ωk + η
v
ωk , (3)
where ηiωk =
ωjskj
ω −
ωlωjslj
ω3 ωk and ηvωk =
ν∇2ωk
ω −
νωj∇2ωj
ω3 ωk represent the inviscid and viscous tilting respectively.
Fig. 4 shows PDF s of the squared magnitudes of total, inviscid and viscous tilting and it appears that viscous
tilting is typically smaller than the inviscid one, but at large magnitudes both contributions to the total tilting are
comparably significant. The PDFs of viscous and total tilting obtained from PTV appear to be somewhat higher
at the tails compared to the numerical result, but the experimental scatter is considerable at high magnitudes. In
order to appreciate the dependence of the tilting magnitudes on geometrical properties introduced before, it is useful
to write the following equations,
(ηiω)2 =
√
W2/ω2sin2(ω,W) (4)
= Λ2kcos2(ω,λk)− (Λkcos2(ω,λk))2 (5)
and
(ηvω)2 =
√
(ν∇2ω)2/ω2sin2(ω,∇2ω). (6)
From Eq. (5) one can see that the inviscid tilting vanishes identically, when ω is strictly aligned with λi. This
alignment can then only be changed in two ways: through viscous tilting and/or through a change of the orientation
of the strain eigenframe.
In summary, in this letter we have shown that viscosity in two thirds of all events depletes enstrophy and that there
is an essential contribution of kinematic nature to this effect. Viscous tilting and production of vorticity, which occur
in one third of all events, are instead characteristic features of turbulent flows. Our results demonstrate that viscosity
influences enstrophy production by changing vorticity in magnitude and direction. The observed effects are sensitive
to the (ω-λi) alignments and thus to the local vortex stretching (compression) regime. When ω is aligned with λ3
the purely destructive contribution of νωi∇2ωi is strongly suppressed. From the technical point we note that the
experimental and numerical results agree well with each other on the qualitative level. Some quantitative discrepancies

7might be attributed to the fact that experimental measurements are affected by limited spatial resolution, noise and
to the difference in Reynolds numbers. Finally, we propose a plausible postulate regarding the role of viscosity for
the predominant ω-λ2 alignment so typical for turbulent flows, e.g. Refs. 17 and 18. In these situations the vectors
ω and ∇2ω are predominantly anti-aligned. Strong ω-λ2 alignment stalls inviscid tilting, while the anti-alignment of
ω and ∇2ω points to reduced viscous tilting, i.e., both mechanisms could work towards maintaining the alignment.
In future work we hope to pursue these questions that are intimately related to moderation of enstrophy growth
and to prevention of finite time singularities.19 This will also require to address the tilting mechanisms of the strain
eigenframe.
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16 It is noteworthy that the above decomposition of νωi∇2ωi - though useful - has a limitation since it is not unique and there is an
infinite number of possibilities to represent it as a sum of a dissipation and a flux term (i.e. as a divergence of some vector). There is no
way to define dissipation (i.e. to choose one among many purely negative expressions) of enstrophy as it is not an inviscidly conserved
quantity, unlike the kinetic energy.3 .
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