12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

STUDY OF FINE-SCALE VORTICAL STRUCTURES IN A VON KÁRMÁN
MIXING FLOW

Farid AligolzadehMarkus Holzner
Department of Energy and Process Engineering
Swiss Federal Institute of Forest,
Norwegian University of Science and Technology Snow and Landscape Research WSL
NO-7491, Trondheim, Norway
8903 Birmensdorf,Switzerland
farid.aligolzadeh@ntnu.no
Swiss Federal Institute of Aquatic Scienceand Technology Eawag
8600 Dübendorf,Switzerlandholzner@ifu.baug.ethz.ch

James R. Dawson
Department of Energy and Process Engineering
Norwegian University of Science and Technology
NO-7491, Trondheim, Norway
james.r.dawson@ntnu.no

ABSTRACTIntroduction/Motivation

Intense small-scale vortical structures also known as ‘filaments´ or ‘worms’ have been studied in a wide range of turbulent flows, mostly using DNS. In the present study, we investigate vorticity dynamics of vortex filaments at the dissipation scale in a fully resolved three-dimensional experimental
data set of a turbulent mixing flow measured at the center ofa large von Kármán mixing tank at a Reλ = 179. To avoid
arbitrariness inherent to threshold-dependent detection criteria and dependence of the results on the observer, an objective
vortex detection method proposed by Haller et al. (2016) is
implemented. One thousand instantaneous 3D velocity fields
are studied. These fields were measured at random times witha spatial resolution of 1η, where η is the Kolmogorov lengthscale. About 12500 structures were detected having an average radius of 5.1η, which is similar to previous findings on
vortex filaments in HIT and turbulent jets and channels. Local features related to the structures and global features of theflow were investigated and compared. Structures are characterized by high vorticity and low strain and the vorticity vector is
predominantly aligned with the intermediate strain eigenvector
that has a positive eigenvalue on average. The vorticity vector
is predominantly oriented normally to the compressive and extensional strain eigenvectors suggesting that the structures are
quasi one-dimensional and shows that enstrophy production
inside the structures results from vortex stretching. We further
investigate the mechanisms that sustain the vortical structuresby treating them as turbulent structures embedded in a less turbulent ambient flow, analogous to a turbulent flow separated
by a turbulent/nonturbulent interface from its quiescent surroundings, and we analyze the entrainment/detrainment acrosstheir boundaries. This analysis shows that the structures are
entraining ambient fluid on average in radial direction and that
this entrainment is a result of the competing effects of nonviscous and viscous phenomena consistent with Burgers’ vortex model.

In turbulent flows, vortical structures are defined as regions of concentrated enstrophy with a life time larger thanthe characteristic time scale of the flow (Dubief & Delcayre,
2000). Intense vortical structures (IVSs) at the dissipation
scale, often called worms or filaments, have shown universal features among a variety of different turbulent flows suchas homogeneous isotropic turbulence (Jiménez et al., 1993;
Jiménez & Wray, 1998), jets (Ganapathisubramani et al.,2008; da Silva et al., 2011), channel flows (Kang et al., 2009),
stratified flows (Neamtu-Halic et al., 2021). These studies
have shown that worms or vortex filaments have a radius of3
about 5η, where η = ( νε )1/4 is the Kolmogorov length scale.
In the DNS study of Jiménez et al. (1993) it was shown thatstretching, α = ωi ω j si j /ω 2 , is relatively low in small-scalevortices which indicates lack of self-amplification. The lackof self-amplification means that the vortices are passive anddecoupled from the straining field of the flow. The worms are
sustained by the strain field of the flow but the strain field is
not affected by the presence of the worms. So, the worms are
the consequence of turbulence dynamics and not important inevolution of the dynamics. For larger size vortices the strain
field and the vortices show a two-way interaction with eachother. The strain field is modified by the presence of the large
scale vortices and the modified strain field modifies back thevortices. So on the contrary to small scale vortices, the large
scale vortices are dynamically important in evolution of turbulence (Tsinober, 2009). Despite the many efforts to understandthe importance and interactions of vortical structures with thesurrounding flow, our understanding is still incomplete. In particular, the way vortical structures exchange mass, momentum
and vorticity with the surrounding fluid has remained obscure.
We show that fine-scale vortical structures are by no means
passive in the sense of exchange, i.e. the interact with the background flow by entraining (detraining) mass radially (axially)
and are thus not frozen to the flow or passively advected.

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
1.6m

In addition to the inherent difficulty in obtaining fully resolved
volumetric time-resolved measurements, there is the additional
challenge of robust identification of high enstrophy structures.
The intense vortical structures are usually detected by usinga threshold on the norm of the vorticity field or the intensityof vorticity relative to the strain field, which are not objective
detection methods. That is, besides the dependence on a userdependent threshold, the results also depend on the observer,
i.e. a change of reference system will yield different structures.
In this study, we use a recent method proposed by Haller et al.
(2016) based on the vorticity deviation to objectively detect thestructures in our experimental data set. The method isolates acoherent fluid volume, in which fluid elements complete equalbulk material rotation relative to the mean rotation. The resulting coherent structures do not change with reference system
and are truly objective. Recently, the method has been successfully applied to 3D turbulence (Neamtu-Halic et al., 2019).
However, it has not yet been used in a more fundamental HIT
set up with a moderately large Reynolds number forcing exhibiting a broad range of vortex sizes.
Our aim here is to study the interaction of the worms with thesurrounding flow based on an objective detection method thatprovides the boundaries of the structures and allows quantifying the exchange of mass, momentum and vorticity across
them.

2m

1.25m

2m

Figure 1.
cility.

Schematic of the big von Kármán mixing flow fa-

implemented. The definition of OECS is based on Instantaneous Vorticity Deviation (IVD) scalar field.

Description of the experimental data set
IV D(x,t) = |ω(x,t) − ω(t)|The experimental data set that is analyzed in this study
is from the scanning particle image velocimetry (PIV) measurement of homogeneous turbulent flow between a pair ofcounter-rotating impellers in a large von Kármán mixing tankfacility (Lawson & Dawson, 2014, 2015). A two-dimensionalsketch of the facility with its dimensions and the theoreticalflow pattern inside the tank is shown in figure 1. The facility
is a dodecagonal tank made of Perspex with 2m height and 2m
width. The diameter of the impellers is 1.6m and the verticaldistance between the two impellers is about 1.25m. The axisymmetric shear generated by the revolution of the impellers
induces a secondary flow pattern because of the centrifugal
pumping effect. The superposition of the primary flow pattern(axisymmetric shear) and the secondary flow pattern (centrifugal pumping) makes homogeneous turbulence at the center of
the tank with almost zero mean velocity and high level of turbulent fluctuations. The Reynolds number of the flow basedon the impeller radius ReRI = ΩI R2I /ν is 23, 000 where ΩI is
the rotational speed of the impellers, RI is the radius of theimpellers, and ν is the kinematic viscosity of the fluid. The
Reynolds number based on Taylor micro-scale of the flow is
Rλ = 179 and the Kolmogorov length scale is η = 0.926mm.
The spatial resolution of the data set is about 1η. The nondimensional volume of the flow is Lx /η × Ly /η × Lz /η =135 × 134 × 25.4. In total, the data set consists of 1003 statistically independent volumes with long and random separation
times between consecutive acquisitions. The detailed description of the volumetric scanning-PIV measurement techniquethat is used to achieve the velocity fields can be found in Lawson & Dawson (2014).

(1)

where ω = ∇ × u is the vorticity vector and ω is the spatially averaged vorticity vector at each time step. OECS is defined as a nested family of level sufaces of IVD. The value of
IVD does not increase when marching outwards from the center. The center has the maximum value of IVD and the boundary is defined as the outermost convex level surface. This definition of vortical structure is observer-independent and ensuresinstantaneous coherence of the rate of material bulk rotation
relative to the mean background rotation (Haller et al., 2016).
An algorithm based on the presented definition has been developed and implemented to detect the three dimensional vorticalstructures. The algorithm is explained in details in NeamtuHalic et al. (2019).

Burgers’ vortex model
Since some features of the small-scale vortical structures
are compared to the Burgers’ vortex model, a short descriptionof it is provided here. The Burgers’ vortex model is an exact solution of incompressible Navier-Stokes equation by assuming that the vorticity field is unidirectional. It means thatthe vorticity field is either one dimensional or both the vorticity and strain fields are axisymmetric (Saffman, 1995). Burgers’ vortices are stable as their radii do not change. Vorticity
is produced by the inviscid phenomenon of vortex stretching(ω · ∇u) inside the vortex and is diffused outward by the viscous phenomenon of vorticity diffusion (ν∇2 ω). The competing effects of these phenomena lead to entrainment of fluid inradial direction into the vortex and detrainment of fluid along
the axial direction to conserve rotational energy and mass (figure 2). Since an exact solution exists for the flow of Burgers’
vortex, all the parameters related to the velocity field and velocity gradient tensor can be calculated analytically.

Objective Eulerian Coherent Structure definition and the detection method
To detect the fine-scale vortical structures objectively inthe flow field the definition of Objective Eulerian Coherent
Structure (OECS) proposed by Haller et al. (2016) has been2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Vortex stretching

Vorticity diffusion(outward)

ur

uθ

RB

that the vorticity vector and the intermediate eigenvector are
well-aligned with each other. The alignment between thevorticity vector and the compressive eigenvector was foundand shows that these two vectors are mostly perpendicular toeach other. However, the PDF of the alignment between thevorticity vector and the extensional eigenvector showed no
preferential alignment. The preferential alignment between
the vorticity and the intermediate strain eigenvector seems to
be a universal feature of turbulent flows (Elsinga & Marusic,
2010). On the other hand, for the case of inside the structures
it can be seen that the vorticity vector is also aligned with theintermediate eigenvector but it is normal to both extensionaland compressive eigenvectors. So, one can interpret thesestructures as quasi 1-D with weak curvature because thevorticity vectors inside them have a strong preference to bealigned with the intermediate eigenvector and to be normal
to the rest of the eigenvectors (Tsinober, 2009). As it was
discussed earlier, the intermediate eigenvalue σ2 can take either positive or negative values. For the case of the structures,
since the vorticity vector has a strong preference to be onlyaligned with the intermediate eigenvalue the sign that σ2 takes
determines if the dominant topological phenomenon is vortex
stretching (ωi ω j si j > 0) or vortex compression (ωi ω j si j < 0).
Figure 8 shows PDFs of the three eigenvalues of the rate ofstrain tensor for both the cases of whole volume and inside
the structures. The PDFs of the eigenvalues for the case of
inside the structures have wider tails compared to the wholevolume of flow. This means that these structures are intense
realization of vortex stretching/compression in the flow field.
As it is expected the value of σ1 is only positive and the valueof σ3 is only negative. For σ2 it can be seen that both negativeand positive values are probable but on average it is positive
for the both cases of whole volume and inside the structures(⟨ ⟨ωσ2 2⟩0.5 ⟩ > 0). It can be concluded that these structures are
stretched on average by the strain field of the flow.

Vorticity diffusion(outward)uθurVorticity convection(inward)

Vorticity convection(inward)Vortex stretchingzθr
Figure 2.

Burgers’ vortex model.

RESULTSIn figure 3 the intense vortical structures in a snapshotof the experimental data set are shown. The black curvedlines show the center of the structures and the pink surfaces
show the boundary of each structure. In total, about 12500
structures have been detected over the 1003 snapshots of thedata set. The average radius of the structures is 5.1η (figure 4)
and were found to occupy 1.4% of the measurement volume
which is in agreement with the DNS study of Jiménez et al.
(1993).

Since these intense vortex filaments tend to be embedded in a more quiescent (i.e. mostly characterized by rather
weak enstrophy) flow, we can treat the boundary of the vortex
filaments as an internal interface (Eisma et al., 2015; Ishihara
et al., 2013), analogous to the turbulent/non-turbulent interfaceof free shear flows and evaluate whether they entrain and/or
detrain flow. To investigate this, a formula for calculating theentrainment velocity, vn , derived by Holzner & Lüthi (2011) isused (equation 2):

The J-PDFs of enstrophy (ω 2 = ω.ω where ω isthe vorticity vector) and dissipation (ε = 2νsi j si j where∂usi j = 21 ( ∂∂ xui + ∂ x j ) is the rate of strain tensor) are calcuj
ilated for the whole volume of flow (figure 5) and within
the structures (figure 6), separately. By comparing these
J-PDFs one can see a shift towards the high-vorticity andlow-strain quadrant for the points within the structures. Also,a noticeable preference can be seen for the J-PDF within
the structures to get aligned with the diagonal line of the
enstrophy-dissipation figure. This preference shows that theextreme events of enstrophy and dissipation scale together
inside the structures.

2

2

∂ ω
2ν ∂∂ ωx i ∂∂ ωx i
2ωi ω j si j ν ∂ x j ∂ x jjj−+
vn = −|∇ω 2 ||∇ω 2 ||∇ω 2 |

Further insight can be gained by considering thealignment between the vorticity vector ω = (ω1 , ω2 , ω3 )and the rate-of-strain eigenvectors e = (e1 , e2 , e3 ), whereσ1 ≥ σ2 ≥ σ3 are the corresponding eigenvalues, as it affects
the vortex stretching phenomenon (production/destructionof enstrophy, ωi ω j si j ) (Tsinober, 2009). For incompressible
flows, from the continuity equation ( ∂∂ uxi = 0) it can beiconcluded that σ1 + σ2 + σ3 = 0. This means than σ1 > 0
and σ3 < 0 everywhere in a flow field and that the valueof σ2 can be either positive or negative depending on thevalue that the sum of σ1 and σ3 has (σ2 = −(σ1 + σ3 )). Theω, are calculated for the wholealignments, cos(θi ) = ei · |ω|volume of the flow and within the structures and shown infigure 7. For the case of the whole volume it can be seen

(2)

Here the entrainment velocity, vn , is defined as
vn n = us − u where us is the velocity of an isosurface element,∇ω 2
u is the flow velocity at that isosurface, and n = |∇ω
2 | is thenormal vector to the isosurface. According to this definition,
when vn ≤ 0 fluid elements at the isosurface are entrained into
the structure and when vn > 0 fluid elements are detrainedout of the structure (Mistry et al., 2019). Figure 9 showsthe entrainment velocity and its components in equation 2
evaluated on the boundary of the structures. The entrainment
velocity is negative on average which means that the structures
are entraining ambient fluid on average. To conserve mass thesame amount of fluid needs to be detrained axially in orderto preserve the total volume of the vortices. Furthermore,by comparing the different terms in the entrainment velocity

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 3.

Example of OECSs in a snapshot of the experimental data set. Dimensions are normalized by the Kolmogorov length scale.

0.25

106

10-1

104

0.2

10-2

0.15
!2
<! 2 >P DF (R$ )

102

100

10-3

0.1
10!2
10-4

0.05

0

10!4

10!6

0

5

Figure 4.

R$ = R2

10

10!5

15

PDF of normalized radius of the structures.

100

0
<0>

105

Figure 6. J-PDF of normalized enstrophy and dissipation forthe structures.

106
100

104

greater than the local vortex stretching. This behaviour isconsistent with a stable Burgers’ vortex model where theambient fluid is entrained to the vortex from the boundary inradial direction, ur = − α2 r < 0, and that the vortex remains
stable because of the balance between inviscid and viscousphenomena acting on it.

2

10

!2
<! 2 >

10-1

100
10-2

10!2

To further investigate the importance of different physicalphenomena in the vortical structures and to compare them with
the Burgers’ vortex model, all the terms in enstrophy transport
equation (equation 3) are calculated along the radial lines ofthe structures. This equation implies that the rate of change
2
∂ ω2
∂ ω2
of enstrophy (the material derivative, Dω
Dt = ∂t + u j ∂ x j )desscribes a competition between inviscid vortex stretching
2 2(2ωi ω j si j ) , the viscous diffusion of enstrophy (ν ∂∂x ω∂ x ), and

10!4
10-3

10!6

10!5

100 0

105

<0>

Figure 5. J-PDF of normalized enstrophy and dissipation for
the whole volume.

jj

enstrophy dissipation (−2ν ∂∂ ωx i ∂∂ ωx i ). Figure 10 plots the avj
jeraged values of these terms conditioned on radial directionsof the structures. The terms are normalized by tη3 where

equation we can see that the average entrainment results froma competition between inviscid and viscous effects. Figure 9shows that local entrainment ( uvηn ≤ 0) happens where local
vortex stretching is greater than vorticity diffusion. On theother hand, detrainment ( uvηn > 0) occurs when the summationof local effects of vorticity diffusion and dissipation are

3tη = ( νε )1/2 is the Kolmogorov time scale. The horizontal
axis of the plot represents the normalized radius of the structures. For each radial line, the radial distance from the center(r) is normalized by the local radius of the structure (R).

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
4.5

P DF (cos(3i ))3.5
3P DF (vn and its components)4

101

cos(31 ) inside structurescos(32 ) inside structurescos(33 ) inside structurescos(31 ) whole volumecos(32 ) whole volumecos(33 ) whole volume

2.5
2
1.5
1

entrainment velocity
vortex stretching
enstrophy dif f usion
enstrophy dissipation

100

10!1

10!2

10!3

0.5
0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2
!
cos(3i ) = ei : j!j

0.4

0.6

0.8

10!4

1

0

vn
u2

2

4

6

<3 inside structures<2 inside structures<1 inside structures<3 whole volume<2 whole volume<1 whole volume

V ortex stretchingDif f usion
Dissipation

2

D! 2
Dt

1

2.5

#t32

P DF ( <!<2 >i 0:5 )

-2

3

4

3

-4

Figure 9. PDFs of entrainment velocity, vn , at the boundaryof the structures and its components.

Figure 7. Alignment between the vorticity vector and theeigenvectors of the rate of strain tensor.

3.5

-6

0

2
1.5

-1

1
-2

0.5
0
-3

-3

-2

-1

0

1
<i
<! 2 >0:5

2

3

0.2

0.4

0.6

0.8

1

r 1.2
R

1.4

1.6

1.8

2

Figure 10. Averaged enstrophy transport equation terms conditioned on radial lines inside the structures.

Figure 8. PDFs of eigenvalues of the rate of strain tensor for
the whole volume of the flow and inside the vortical structures.

Dω 2∂ 2ω 2∂ ωi ∂ ωi= 2ωi ω j si j + ν− 2ν
Dt∂ x j∂ x j∂xj ∂xj

0

observed although with some small changes in their magnitude which shows that, on average the Burgers’ vortex model
reproduces the dominant physics of the small-scale structuresquite well.

(3)

Figure 10 shows that near the center of the structures( Rr = 0), enstrophy diffusion is more dominant than both vortex stretching and enstrophy dissipation, the latter of which has
2relatively small values resulting in Dω
Dt < 0. By marching towards the boundary from the center both vortex stretching andenstrophy diffusion become weaker as dissipation gradually
increases. The rate of change of diffusion is faster than vortex
2rstretching which results in Dω
Dt > 0 after R ≈ 0.2. Dissipation reaches a maximum near the boundary of the structures.
As Rr > 1, all the terms become constant (flat). To compare
the behaviour with Burgers’ vortex model, equivalent Burgers’ vortices are considered and the same terms of the equation are calculated and plotted in figure 11. The equivalent
Burgers’ vortices are achieved by calculating stretching (α0 )and enstrophy (ω02 ) at the center of the structures. By knowing α0 and ω02 and using the analytical relations available for
Burgers’ vortex model all the desired terms can be calculated.
By comparing figures 10 and 11 a similar competition between
the different terms in the enstrophy transport equation can be

SUMMARY & CONCLUSIONSIn this study, features of small-scale vortical structures(’worms’) in a fully resolved 3D-3C experimental data setof homogeneous axisymmetric turbulence with an Reλ = 179
measured at at the center of a large-scale von Kármán mixing
tank are investigated. To avoid the arbitrariness in the definition of vortices (thresholding), an objective definition basedon Objective Eulerian Coherent Structure (OECS) proposed
by Haller et al. (2016) is implemented to detect the vortices
in the volumetric velocity fields. In total, 12500 vortices weredetected in 1003 volumes of the turbulent flow. The averageradius of the structures is ⟨R⟩ = 5.1η that is in agreement with⟨R⟩ ≈ 5η widely reported in the literature for DNS studies.
Small-scale vortex structures were found to occupy 1.4% of
the volume of the flow in agreement with ∼ 1% reported in
the literature from DNS. Joint PDFs of enstrophy and dissipation within the volume and conditioned inside the structures
were showed that the values of enstrophy and dissipation areconcentrated in the small-scale vortices compared to the whole5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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rit E. 2015 Interfaces and internal layers in a turbulent
boundary layer. Physics of Fluids 27 (5).
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Mechanics 598, 141–175.
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Defining coherent vortices objectively from the vorticity.
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Thin shear layers in high reynolds number turbulence—dns
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V ortex stretchingDif f usion
Dissipation

2

D! 2
Dt

#t32

1

0

-1

-2

-3

0

0.2

0.4

0.6

0.8

1

r
R

1.2

1.4

1.6

1.8

2

Figure 11. Averaged enstrophy transport equation terms of
the equivalent Burgers’ vortex model conditioned on radiallines inside the structures.
flow field and that they scale inside the structures. The alignment of the vorticity vector and eigenvectors of the rate ofstrain tensor showed the vorticity vector has a strong tendency
to be aligned with the intermediate eigenvector and be normal
to the compressive and extensional eigenvectors as found inprevious turbulent flows. Since the average value of the intermediate eigenvalue is positive (⟨σ2 ⟩ > 0) vortex stretching isthe dominant topological phenomenon.
To investigate the exchange of mass by the small-scalevortices, the entrainment velocity equation was solved on theboundary of the structures showing that they, on average, entrain relatively ambient flow from the surroundings. The total
volume of the structures does not change in homogeneous stationary turbulence, the mass entrainment is compensated by
the mass outflow along the axial direction. It is also shownthat the local entrainment/detrainment at the boundary is theresult of a competition between inviscid and viscous phenomena. This exchange of mass is accompanied with exchangeof momentum, energy, and enstrophy with the ambient fluid.
Therefore the structures are not passive in this sense. Finally,
the enstrophy transport equation conditioned on radial lines ofthe vortices was studied and compared to those of equivalent
Burgers’ vortices. The results indicate that Burgers’ vortex
model on average reasonably captures the flow physics of thevortex structures.

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identification in turbulence. Journal of Turbulence 1.
Eisma, Jerke, Westerweel, Jerry, Ooms, Gijs & Elsinga, Ger-

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