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

TURBULENCE WITH SPATIALLY FIXED ANISOTROPIC VISCOSITY

Hanna Berning, Daniel Stalder, Tommaso Ciuti and Thomas Rösgen
Institute of Fluid Dynamics
ETH Zurich
Sonneggstr. 3, 8092 Zurich, Switzerland
berningh@ethz.ch, dstalder@student.ethz.ch, tommaso.ciuti@hotmail.it, roesgen@ethz.ch

Davide Gatti
Institute of Fluid Mechanics
Karlsruhe Institute of Technology
Kaiserstraße 10, 76131 Karlsruhe, Germany
davide.gatti@kit.edu

ABSTRACTnExperimentally and numerically, we investigate the impact of a spatially homogeneous, direction-dependent tensorial
viscosity on a turbulent flow. The anisotropy is induced by themagnetic alignment of prolate tracer particles in an external
magnetic field. In the experiment, a spatial fixation of global
flow modes in a von Kármán flow is observed. In the direct
numerical simulation, the anisotropy dominates at small scales
and causes field-parallel alignment of vortices and a decreaseof the energy spectra in the same direction.

ν⊥
H

νk

Figure 1: Schematic of anisotropic viscosity induced by thealignment of elongated particles with an external magnetic
field H. The average alignment direction can be described witha director n.

EXPERIMENT IN A VON KÁRMÁN FLOW
Setup

INTRODUCTION

As an experimental configuration, we choose a vonKármán flow due to its strong turbulence in a small flow volume (Fig. 2).
The anisotropic viscosity is introduced by a magnetic induction field B of up to 80 mT which acts on magnetic micro
particles dispersed in water. These consist of a magnetite core
and a silica coating to prevent agglomeration and chain formation (Fig. 3). With an average length of 1.9 µm = 0.03 η andan aspect ratio of 5.7, the particles have a Stokes number ofaround 1 × 10−5 and act like passive tracers. The Mason number Mn ≈ 3.5 × 10−3 indicates that the magnetic moment dominates over the hydrodynamic influence. A dilute suspension
approaching the semi-dilute state (volume fraction Φ ≈ 0.02
with dimensionless number density nL3 ≈ 0.78) allows to observe maximal anisotropic effects while still limiting the interaction between the particles.
The turbulence properties are investigated using ultrasonic techniques. An ultrasonic profiler (UDV, DOP3010 from
Signal Processing SA) as an Eulerian measurement captures
global effects throughout the cell (Fig. 2a). It records the velocity component in the emission direction along transverselines through the cell centre. In addition, a custom-built ultrasound particle tracking system, Fig. 2b, allows to spatially andtemporally resolve Lagrangian trajectories in the centre of theWell established relations such as Kolmogorov’s first similarity hypothesis are based on a scalar viscosity ν and the assumption of small-scale isotropy, so that turbulence statistics at
small scales are uniquely determined by this viscosity and thedissipation rate ε. Following the work by Grünberg (2016), we
ask how turbulent statistics are modified if the viscosity is oftensorial nature. In particular, we pose the question of whether
and how small-scale anisotropy affects the larger-scale features of the flow. As opposed to frequently studied flows withvarious anisotropic particles (Butler & Snook, 2018; Winkler
et al., 2004; Voth & Soldati, 2017), geometrical constraints(Antonia et al., 1994) or anisotropic forcing (Julien et al.,2018) where anisotropy at the small scales is induced by theflow itself, here the viscous anisotropy is imposed a priori witha fixed orientation in space. This can be achieved with magnetic, anisotropic particles much smaller than the Kolmogorov
length η which are aligned with their long axis in the directionof a magnetic field. The resulting viscosity will be lower inthe direction parallel than in the direction perpendicular to theparticle long axis (Fig. 1, ν⊥ > νk ).
We tackle the problem both numerically and experimentally.

1

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

H

uk

u⊥

(a) Eulerian ultrasound velocity measurement along a transverse line.

Figure 3: TEM images of the nanorods including an element
analysis showing silica (blue) and iron (red) content.

H

u

w
v

This finding is supported by quasi-simultaneous recordings of the perpendicular transducers. Fig. 5 shows the joint
probability of the projection of the instantaneous velocity profiles onto the inflow profile. A positive correlation coefficient
P indicates an inflow state while a negative coefficient represents an outflow observation. While in the field-free casethe narrow negative correlation proves near-equal probabilityof the modes for both transducers, with the alignment of themagnetic particles the distribution becomes asymmetric andinflow prevails in the alignment direction. We hypothesize thatthe flow structure, originally rotating in azimuthal direction,orients as in the sketch in Fig. 6 in the transverse meridional
plane.
This large-scale anisotropy leaves a footprint on smallscale quantities in the centre of the cell, such as the Lagrangian Taylor microscale, which is depicted Fig. 7. The
small-scale correlations are larger in the field-parallel direction
compared to the remaining directions. For the randomly forced
flow, however, no significant small-scale anisotropy is detected
within the measurement uncertainty. The slight trend observedin the Taylor microscales may as well be a large-scale imprint due to the remaining 10% outflow probability. In agreement with the moment profiles, the ratio of the Lagrangian
second-order structure functions, Fig. 8, reveals a small initial
anisotropy already without the application of a magnetic field.
The ratio increases towards large lags, confirming the scaledependence of the anisotropy, which is primarily emerging atlarge scales.

(b) Lagrangian ultrasound particle tracking in centre of cell.
Figure 2: Measurement techniques in a von Kármán flow withsuperimposed magnetic field H.

cell (Grünberg, 2016). We compare the velocity statistics inthe direction of the magnetic field and perpendicular to it.
We apply both a constant forcing (C), where the discs arecontinuously counter-rotating, and a pseudo-random forcing(PRBS), where the sense of disc rotation is reversed irregularly. In the former case, a large scale structure emerges, which
is characterized by an in- and outflow in the transverse plane(compare gray curves in Fig. 4 and left sketch in Fig. 6). The
random forcing suppresses this mean-flow instability and ensures isotropy in the transverse plane (Berning et al., 2017).

Results
In Fig. 4, the central moments of the transverse velocitiesof the continuously forced flow show that the roughly isotropic
moment profiles diverge for the two directions under the influence of a magnetic field. A pinning of the global modes
occurs, where the inflow orients in the field-parallel direction.
The limiting case of pure in- or outflow mode, indicated withgray lines, is approached for all moments.

DIRECT NUMERICAL SIMULATION
Setup
Complementary to the experiment, the phenomenon is investigated numerically. The anisotropy can be modeled using
Leslie and Ericksen’s director theory for liquid crystals which212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

C

0.0

−0.1
−0.2

1.0
0.5

⟨𝑢 4 ⟩/⟨𝑢 2 ⟩2 − 3⟨𝑢 3 ⟩/⟨𝑢 2 ⟩3/2

⟨𝑢 2 ⟩1/2 [m/s]

0.1

0.0

1.5
1.0
0.5
0.0

−0.5

0.0
x/R

C

|𝐁| [mT]

C

0.5

0

−0.5 −0.5−1.0 −1.0−0.5 −0.50.0 0.0 0.5
x/R x/R

PRBS

0.125
0.100
0.075

1.0

PRBS

0.0 −0.5 0.5 0.0
x/R
x/R

PRBS

C

|𝐁| [mT]

800.5

40

C

0.150

−0.5 1.5⟨𝑢 4 ⟩/⟨𝑢 2 ⟩2 − 3

𝑢 [m/s]

0.2

0.175

PRBS

0.0

0.5

0

40

PRBS

0.0
x/R

0.5

0.0
x/R

0.5

80

−0.5

0.5

−1.0
−0.5 −0.5
0.0 −0.50.0
0.5 0.0 0.5 0.5x/Rx/R x/R

0 40

4080

|𝐁| [mT]|𝐁| [mT]
0

−0.5

80

|𝐁| [mT]

0

−0.5

40

80

Figure 4: Profiles of velocity moments of transverse velocity components for different magnetic induction field strengths |B|. Coloured
solid line: direction parallel to magnetic field, dotted line: direction perpendicular to magnetic field. Gray lines: inflow (dashed) andoutflow (dash-dotted) profiles without magnetic field.

Figure 5: Joint probability density function of instantaneous projections of the transverse velocity components parallel and perpendicular to the magnetic field onto the inflow profile. Results are presented without and with influence of the magnetic field for constantforcing.

reduces the unknown viscosity tensor of 4th order to only six
Leslie coefficients αi (Leslie, 1966). The governing equations
for an incompressible fluid are:

1∇u + ∇uT2
N = ṅ − ω × nD=∂u+ ρ(u · ∇)u = −∇p + ∇ · τ∂tτ =α1 (nn : D)nn + α2 nN + α3 Nn + α4 D+ρ

α5 nn · D + α6 D · nn

(1)

(3)
(4)

Ilg & Kröger (2002) derived an expression for the Leslie
coefficients accounting for the effect of non-interacting rigid
magnetic ellipsoids that are aligned in a magnetic field (Fig.
1). With a strong magnetic field H the parameters approach the

(2)

with

3

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

|B| = 0 mT

Statistical steadiness is imposed via a linear low-wavenumber
forcing (Lamorgese et al., 2005).
The stress tensor is implemented in the velocity-vorticity
formulation of ISODNS and validated at Reλ ,eff = 35 against
simulations with the spectral element solver Nek5000, where
the analytically simpler formulation in natural variables anda random low-wavenumber forcing (Eswaran & Pope, 1988;
Weiss et al., 2019) are employed.
Next to the experimental volume fraction Φ = 0.02, alarge volume fraction, Φ = 0.2 is studied. In a real experiment,
the latter would violate the assumption of non-interacting particles, but numerically it allows to study enhanced effects. For
comparison with the experiment, the particle aspect ratio ischosen to be 5.7 and 6.5, respectively. The ratios between thelargest and the smallest Miesowicz-viscosities, resulting fromparticle-alignment in streamwise and gradient direction of a
two-dimensional flow, are 1.6 and 6.5. The field and director
are pointing in the second direction (y-axis).

|B| = 80 mT

Figure 6: Sketch of the mode pinning under alignment of
anisotropic magnetic particles in the transverse plane of a continuously forced von Kármán flow.

𝑡𝜆 [ms]22
20

Results

18
16

0
C, ∥
C, ⟂

40
|𝐁| [mT]The results presented here are based on studies at low
to moderate effective Reynolds numbers, Reλ ,eff = 35 − 92,
where an effective Reynolds number can be defined using themean transverse Taylor microscale (λT,eff ) and an effective viscosity νeff :80

Reλ ,eff =PRBS, ∥
PRBS, ⟂

νeff =

𝐷2,∥ /𝐷2,⟂

1.0|𝐁| [mT]

1
0

𝜏 /𝑇𝐿
40

2

PF
2hsi j si j i(6)with PF as the constant energy injection rate of the forcing
and si j the rate-of-strain tensor. We focus on the results at
Φ = 0.2. The lower particle volume fraction exhibits the same
trends, albeit with such small magnitude that effects might bedifficult to resolve experimentally. The Reynolds number does
not affect the trends in the investigated range.
In Fig. 9, we present the spatial microscales λ ii, j , extracted from the autocorrelation function of the i-th velocity
component in the j-th direction, normalised by the respective mean in transverse and longitudinal direction. The spatial
Taylor microscales are much larger in the field-parallel direction. Exemplary, the results from both solvers are shown. The
isotropy in the field-free case is better achieved with Nek5000
using the random forcing. The anisotropic simulations with
ISODNS, however, yield qualitatively and quantitatively comparable results. The good agreement of both codes – with improved isotropy resulting from the random forcing – is also
found in the remaining analyses.
In addition to the autocorrelation functions, also the related one-dimensional energy spectra contain direction- andscale dependent information. They are computed for the i-th

1.2

0

(5)

The effective viscosity is defined as

Figure 7: Temporal taylor microscales τλ of transverse velocity components aligned in field-parallel or perpendicular direction using constant (C) and random (PRBS) forcing.

0.8

urms λT,eff
νeff

3
80

Figure 8: Effect of magnetic field strength on the ratio of the
Lagrangian second-order structure functions in transverse directions using constant forcing. T L denotes the Lagrangian
integral time averaged over the transverse directions.

limiting case of a fixed director n, corresponding to a spatiallyhomogeneous and constant anisotropic viscosity. The Lesliecoefficients can then be computed a priori1 .
A direct numerical simulation (DNS) allows to fully resolve all scales of the problem and to characterize the effect of
the anisotropy on the various scales of the flow. We performa DNS with the spectral solver ISODNS (Carini & Quadrio,2010) to study a periodic box of homogeneous turbulence.

1 In some of the equations of Ilg & Kröger (2002) typographicalerrors were found. In agreement with the authors, the corrected equations are:21(28) α4 − 2η0,r = −4η0Φ ( Q32 S2 + Q23 S4 )7358(29) α5 + α6 = η0Φ (3Q32 S2 + Q23 S4 )7

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
1.2

1.2

transverse

1.1
𝜆𝑖𝑖,𝑖 /𝜆L,eff

𝜆𝑖𝑖,𝑗 /𝜆T,eff

1.1
1.0
0.9

longitudinal

𝜆

𝜆

11,2

𝜆

11,3

isotropic

22,1

anisotropic, 𝜆⟂,∥

𝜆

𝜆

22,3

33,1

𝜆

1.0
0.9

33,2

anisotropic, 𝜆⟂,⟂

𝜆11,1

𝜆22,2

isotropic
anisotropic, 𝜆⟂,⟂anisotropic, 𝜆∥,⟂𝜆33,3anisotropic, 𝜆∥,∥Figure 9: Spatial transverse and longitudinal Taylor microscales λ ii, j at Reλ ,eff = 35 − 40 for directions parallel and perpendicular to
the magnetic field. Circles: ISODNS, crosses: Nek5000. The values are normalized by their respective means.

5 1/4
𝐸𝑖𝑖 (𝜅𝑗 )/(𝜖𝜈eff)

103

isotropic

anisotropic

101
10−1
10−3
10−1

100

𝜅𝑖 𝜂eff

𝐸11 (𝜅1 )
𝐸22 (𝜅2 )

𝐸22 (𝜅1 )
𝐸11 (𝜅2 )10−1
𝜅𝑖 𝜂eff

100

𝐸11 (𝜅3 )𝐸(𝜅)Figure 10: Selected one-dimensional energy spectra from ISODNS with Φ = 0.2 at Reλ ,eff = 72 − 96. Yellow: longitudinal spectra,
green and blue: transverse spectra. Solid lines: in second wavenumber direction. Left: isotropic, right: anisotropic simulation. The
gray line indicates the boundary of the forcing shell. Additionally, E11 (κ1 ) from grid turbulence measurements at Reλ = 72 is shown(CBC 1971 after Pope (2000)).

velocity component in the j-th wavenumber direction from thevelocity spectrum tensor Φi j , which is the Fourier transformof the two-point correlation (Pope, 2000).

pronounced for the transverse quantities compared to the longitudinal ones.
Finally, the structure of the flow is of interest. The eigenvalues of the velocity gradient tensor are used to compute thevortex orientation vr and the swirling strength λci employing
an enhanced criterion (Zhou et al., 1999; Chakraborty et al.,2005). The probability density functions of the vortex orientation show a strong dependence on the swirling strength (Fig.
11). At higher swirling strengths, which correspond to smaller
eddies, the vortices are increasingly aligned in the direction of
the director. Interestingly, an opposite effect appears for low
swirling strengths.

ZZ ∞Eii (κ j ) = 20κ )dκm dκnΦ(i)(i) (κ(7)

Here, j, m, n are non-equal directional indices and no summation over indices (i)(i) is employed. The energy spectra exhibit
a strong wavenumber-dependent anisotropy (Fig. 10). Regardless of the velocity component, the kinetic energy decreases inthe direction of the director and the longitudinal and transverse
component approach each other in this direction.
For both energy spectra and Taylor microscales, the influence of the direction exceeds the impact of the velocity component. Variations in the field-parallel direction are much larger
than in the perpendicular directions, irrespective of the velocity component analysed. Furthermore, the effects are more

CONCLUSIONSThe effects of a spatially fixed anisotropic viscosity on
turbulence properties are investigated in fully resolved numerical simulations and experiments. The anisotropy is induced

5

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

𝜆𝑐𝑖 /𝜆𝑚𝑎𝑥

4

1.0

𝑝𝑑𝑓

3

0.6

2

0.3

1
0

0.15
−1

0
𝑣𝑟,𝑥 /|𝐯𝑟 |10

𝑣𝑟,𝑦 /|𝐯𝑟 |

1

−1

0
𝑣𝑟,𝑧 /|𝐯𝑟 |

1

0.01

Figure 11: PDFs of vortex orientation for different swirling strength ranges computed with the enhanced criterion for Φ = 0.2 andISODNS at Reλ ,eff = 72 − 96.

via the alignment of prolate magnetic particles in a magnetic
field. In the experiment, we observe a directional fixation of a
global flow structure. This mode pinning leaves an imprint on
all scales. For a randomly forced flow without dominant flow
structure, no small-scale anisotropy can be detected within themeasurement uncertainty. On the other hand, the simulations
exhibit a strong anisotropy emerging directly from the highest wavenumbers, propagating and decreasing towards lower
wavenumbers. In contrast to the commonly implied energy
cascade where isotropy is regained at small scales even if theenergy input is anisotropic, here, the opposite effect is therefore present. In the direction of the smaller viscosity, energy
decreases and vortices align at small scales. This is reflectedin an increase of the Taylor microscales in this direction.

Chakraborty, P., Balachandar, S. & Adrian, R. J. 2005 On therelationships between local vortex identification schemes.
Journal of Fluid Mechanics 535, 189–214.
Eswaran, V. & Pope, S.B. 1988 An examination of forcing in
direct numerical simulations of turbulence. Computers &Fluids 16 (3), 257–278.
Grünberg, Tim 2016 Response of a turbulent flow toanisotropy at the molecular scale. PhD thesis, ETH Zurich.
Ilg, P. & Kröger, M. 2002 Magnetization dynamics, rheology,
and an effective description of ferromagnetic units in dilute
suspension. Physical Review E 66 (2).
Julien, K., Knobloch, E. & Plumley, M. 2018 Impact of domain anisotropy on the inverse cascade in geostrophic turbulent convection. Journal of Fluid Mechanics 837.
Lamorgese, A. G., Caughey, D. A. & Pope, S. B. 2005 Direct numerical simulation of homogeneous turbulence withhyperviscosity. Physics of Fluids 17 (1), 015106.
Leslie, F. M. 1966 Some constitutive equations for anisotropic
fluids. The Quarterly Journal of Mechanics and Applied
Mathematics 19 (3), 357–370.
Pope, S.B. 2000 Turbulent Flows. Cambridge University
Press.
Voth, G. A. & Soldati, A. 2017 Anisotropic Particles in Turbulence. Annual Review of Fluid Mechanics 49 (1), 249–276.
Weiss, P., Oberle, D., Meyer, D. W. & Jenny, P. 2019 Impact of
turbulence forcing schemes on particle clustering. Physics
of Fluids 31 (6), 061703.
Winkler, R. G., Mussawisade, K., Ripoll, M. & Gompper, G.
2004 Rod-like colloids and polymers in shear flow: a multiparticle-collision dynamics study. Journal of Physics: Condensed Matter 16 (38), S3941–S3954.
Zhou, J., Adrian, R. J., Balachandar, S. & Kendall, T. M. 1999
Mechanisms for generating coherent packets of hairpin vortices in channel flow. Journal of Fluid Mechanics 387, 353–
396.

Acknowledgments
The authors gratefully acknowledge Dr. Robin Schäublinof ScopeM for support and assistance with the Transmission
Electron Microscopy (TEM) work.

References
Antonia, R. A., Djenidi, L. & Spalart, P. R. 1994 Anisotropy ofthe dissipation tensor in a turbulent boundary layer. Physics
of Fluids 6 (7), 2475–2479.
Berning, H., Grünberg, T. & Rösgen, T. 2017 Suppression of
Large Scale Structures in a Turbulent von Kármán Flow using Random Forcing. 10th International Symposium on Turbulence and Shear Flow Phenomena .
Butler, J. E. & Snook, B. 2018 Microstructural Dynamics andRheology of Suspensions of Rigid Fibers. Annual Review of
Fluid Mechanics 50 (1), 299–318.
Carini, M. & Quadrio, M. 2010 Direct-numerical-simulationbased measurement of the mean impulse response of homogeneous isotropic turbulence. Physical Review E 82 (6),066301.

6