Statistical model for the orientation of non-spherical particles settling in turbulence
K. Gustavsson1 , J. Jucha2 , A. Naso3 , E. Lévêque3 , A. Pumir4 and B. Mehlig1
1
Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden
Projektträger Jülich, Forschungszentrum Jülich GmbH, D-52425 Germany
3
LMFA, Ecole Centrale de Lyon and CNRS, F-69134 Ecully, France
Laboratoire de Physique, Ecole Normale Supérieure de Lyon and CNRS, F-69007 Lyon, France
2
The orientation of small anisotropic particles settling in a turbulent fluid determines some essential
properties of the suspension. We show that the orientation distribution of small heavy spheroids
settling through turbulence can be accurately predicted by a statistical model, thus providing a
quantitative understanding of the orientation distribution on the problem parameters. This opens
the way to a parameterisation of the distribution of ice-crystals in clouds, and potentially leads
to an improved understanding of radiation reflection, or particle aggregation through collisions in
clouds.
The settling of particles in turbulence is important in
a wide range of scientific problems. In clouds, the settling of water droplets or ice crystals plays an essential
role for precipitation [1–4]. A second, quite different example is the settling of plankton in the turbulent ocean
[5–7]. Settling can induce patchiness in a population of
plankton, affecting biological processes such as mating,
feeding, and predation [8].
The settling of a single spherical particle in turbulence
has been intensively studied. Maxey [9] found that turbulence increases the settling speed of a small spherical
particle. This work has led to a large number of experimental and numerical studies, summarised in Ref. [10]. A
related question, where substantial progress was recently
achieved, is how two particles settling together in turbulence move relative to each other and collide [11–15].
In the absence of settling, the orientation of nonspherical particles in a turbulent fluid is determined by
fluid vorticity and strain acting on the particle [7, 16–24].
Much less is known about the settling of non-spherical
particles. In still air, the orientation of a slowly settling needle is determined by fluid-inertial torques [25–
28]. Siewert et al. [29] analysed the orientation of
small spheroids settling in turbulence by direct numerical
simulations (DNS) of homogeneous isotropic turbulence.
They found that settling induces a bias in the orientation distribution of the particles. This bias affects light
reflection from the cloud, and the rate at which settling
crystals may grow by sweeping up supersaturation [30].
The mechanisms that cause the orientation bias are not
known, and it is not understood how the effect depends
on the dimensionless parameters of the problem, the turbulent Reynolds number Reλ , the Stokes number (a dimensionless measure of particle inertia), the Froude number (a dimensionless measure of settling), and the particle shape. Also, how significant are non-Gaussian, intermittent, small-scale features of the turbulent flow [31],
such as intense vortex tubes [2], in aligning the particles?
To answer these questions we have analysed a statistical
4
P (ng )
PACS numbers: 05.40.-a,47.55.Kf,47.27.eb
P (ng )
arXiv:1702.07516v1 [physics.flu-dyn] 24 Feb 2017
4
a
3
4
2
2
1
1
0
0
0.2
0.4
0.6
ng
0.8
1
b
3
0
0
0.2
0.4
0.6
ng
0.8
1
FIG. 1.
Orientational bias of thin oblate spheroids settling in turbulence. Distribution P (ng ) of ng ≡ |n · ĝ|; particle symmetry-vector n and direction ĝ of gravity. a DNS
results for P (ng ) for aspect ratio λ = 1/50, and different
dissipation rates ε = 1 cm2 s−3 (H), ε = 16 cm2 s−3 (•), and
ε = 256 cm2 s−3 (J). Statistical-model simulations, see text,
open symbols. b Same, but for different aspect ratios at
ε = 16 cm3 s−2 : λ = 1/100 (), 1/50 (•), and 1/20 (). The
isotropic distribution P (ng ) = 1 is shown as a dashed line.
model for the orientation of small heavy spheroids settling in homogeneous isotropic turbulence. Fig. 1 shows
the predicted bias in the distribution of the vector n that
points along the symmetry axis of the particle (disk-like
spheroids settle edge first, consistent with Ref. [29]). The
predictions of the statistical model (open symbols) agree
very well with the DNS results (full symbols). This allows
us to conclude that only the small-scale properties of the
turbulence matter for the bias, and that non-Gaussian
fluctuations due to intense vortex tubes are not important. Second, the statistical model explains the sensitive
parameter dependence of the DNS results. This is important because it allows us to parametrise the bias, and
to understand quantitatively the radiative cloud properties, or the aggregation process leading to precipitation.
Third, we analyse the statistical model by an expansion
in the ‘Kubo number’ Ku, a dimensionless measure of
the persistence of the flow [32]. Padé-Borel resummation
yields excellent agreement with numerical simulations at
Ku = 0.1, and qualitative agreement with DNS of turbulence. At larger Ku the theory fails to converge, but it
still explains qualitatively the underlying mechanisms.
2
Formulation of the problem. The equation of motion
for translation and rotation of a particle reads
ṅ = ω ∧ n ,
mẍ = f + mg ,
d
dt
J(n)ω = T .
(1)
Here g is the gravitational acceleration, x is the particle
position, n is the particle symmetry vector, m is its mass,
ω is its angular velocity, and J(n) its inertia tensor in the
laboratory frame. In the point-particle approximation
the force f and torque T on a spheroid with translational
velocity v and angular velocity ω are [20, 33, 34]:


(t)
u−v
f
M
0
0
Ω − ω  .
=γ
T
0 M(r1) M(r2)
S
(2)
In Eq. (2), u(x, t) is the turbulent velocity field, Ω ≡
1
2 ∇ ∧ u is half the turbulent vorticity, S is the strainrate matrix, the symmetric part of the matrix A of fluidvelocity gradients, and M are translational and rota(t)
(t)
(t)
tional resistance tensors: M(t) ≡ C⊥ I+(Ck −C⊥ )nnT ,
(r)
(t)
(t)
M(r1) ≡ C⊥ I+(Ck −C⊥ )nnT , and M(r2) is a third-rank
tensor. For a fore-aft symmetric particle, the equations
of motion (1,2) are invariant under n → −n, so that
only the magnitude ng ≡ |n · ĝ| can play a role in the
dynamics. The form of M(r2) and of the C-coefficents are
known for a spheroidal particle (Supplemental Material
[35]). For crystals with discrete rotational symmetry the
tensors are of the same form [36], but the numerical values of the C-coefficients are not known in general. The
parameter γ ≡ 9νρf /(2abρp ) is Stokes constant, ν is the
kinematic viscosity of the fluid, ρf and ρp are fluid and
particle mass densities, a is the major semiaxis length of
the spheroid, and b is the length of its minor semiaxis.
Our DNS of turbulence use the code described in [37].
Details are given in the Supplemental Material [35]. The
Kolmogorov scales uK , ηK , and τK are determined by
the dissipation rate ε ≡ νhTr AAT i (the average is along
steady-state Lagrangian trajectories), and by ν. The particle aspect ratio is λ ≡ a/b for prolate particles, and
λ ≡ b/a for oblate particles. The simulations were done
for oblate spheroids with a = 150µm, much smaller than
ηK for values of ε pertaining to mixed-phase clouds (DNS:
ε ≈ 1, 16, and 256 cm2 s−3 ). Particle inertia is measured
by the Stokes number StK ≡ (γτK )−1 . The mass-density
ratio is R ≡ ρp /ρf ≈ 1000 (ice crystals in air), and
FK ≡ gτK /uK measures the importance of settling.
Statistical model. The model is appropriate for particles smaller than ηK . We approximate the universal
[31] dissipative-range turbulent fluctuations by an incompressible, homogeneous, isotropic Gaussian random
velocity field u(x, t) with zero mean, correlation length
`, correlation time τ , and typical speed u0 [32]. In the
persistent limit [32], for Ku ≡ u0 τ /` > 1, the statisticalmodel parameters
St ≡ (γτ )−1 and F ≡ gτ /u0 map to
√
StK = 5 Ku St and FK = [F/(5 Ku)]`/ηK . Here `/ηK
is the ratio between the size of the dissipation range and
the Kolmogorov length. In turbulence this ratio depends
1/2
weakly on the Reynolds number Reλ [38], ` = cηK Reλ .
For the data shown in Fig. 1 we have Reλ = 151 (J),
Reλ = 95 (•), Reλ = 56 (H). We find good agreement
between the statistical-model results at large Ku and the
DNS for c ≈ 1.3, Fig. 1 (open symbols). But in this
limit [32] the statistical-model results depend on two parameter combinations only, Ku St and F/ Ku. Expressed
in terms of the DNS parameters this means that the orientation alignment depends only on the two parameter
−1/2
combinations StK and FK Reλ .
Perturbation theory. Eqs. (1,2) can be solved iteratively, by refining approximations for the particle path
through the turbulence [32, 39]. This yields expansions
of steady-state averages in powers of Ku. We outline
the essential steps below, details are given in the Supplemental Material [35]. We use dimensionless variables:
t0 ≡ t/τ, r 0 ≡ r/`, u0 ≡ u/u0 , and drop the primes.
To calculate the steady-state distribution of ng ≡ |n·ĝ|
we must evaluate the fluctuations of the fluid-velocity
gradients along particle paths. In the statistical model
(d)
this can be achieved by an expansion in δxt ≡ xt − xt
(d)
around the deterministic solution xt of the equations of
motion (1,2) for u = 0. As shown in Ref. [32] this gives
expansions in powers of Ku. For nt we find to order Ku2 :
Z t
(r)
2
nt = n0 + Ku dt1 [1 − e(t1 −t)C⊥ / St ] (δxt1 · ∇) O(x, t1 )n0 + λλ2 −1
(3)
+1 S(x, t1 )n0 − n0 · S(x, t1 )n0 n0 x=x(d)
0
t1
Z t
(r)
2
T
+ Ku
dt1 (1 − eC⊥ (t1 −t)/ St ) O(t1 )n0 + λλ2 −1
+1 S(t1 )n0 − (n0 S(t1 )n0 n0 )
0
Z t
Z t1
(OO)
2
+ Ku
dt1
dt2 cijkl (n0 ; t, t1 , t2 )Oij (t1 )Okl (t2 ) + similar terms with OS and SS .
0
0
Here O is the antisymmetric part of A. All O and S are
(d)
evaluated along deterministic paths xt = x0 + v s (n0 ) t
(Fig. 2a) with settling velocity (denoting g ≡ F ĝ):
v s (n0 ) = F St
h I
+
(t)
C⊥
n0 nT
0
(t)
Ck
−
i
n0 nT
0
(t)
C⊥
· ĝ .
(4)
3
1
0.6
v s (n0 )
0.6
b
hn2p
g i1
The paths depend on the initial orientation n0 . Eq. (4)
is the lowest-order solution of Eqs. (1,2).
The terms in Eq. (3) that do not involve δxt depend
only on the history of the fluid-velocity gradients along
these paths (‘history contribution’). The c-coefficients
contain at most five powers of n0 , and one must sum
over all tensor products allowed by symmetry (Einstein
convention). See Supplemental Material [35].
The first integral shown in Eq. (3), by contrast, depends on δxt . It is therefore sensitive to how turbulence
modifies the settling paths (‘preferential sampling’ [32]).
We determine the steady-state moments h(nt · ĝ)p i∞
by first calculating the moments conditional on the initial
orientation n0 , using Eq. (3) and the relation
n0 a
0.4
c
0.4
PSfrag replacements
0.2
0.2
PSfrag replacements
0
10!1
u(x, t)
100
101
F
0
10!1
100
101
F
FIG. 2. a Deterministic settling path along v s (n0 ). The
path bears no relation to the instantaneous fluid-velocity field
u(x, t). b, c Moments of ng = |n · ĝ|, for p = 1 (red,◦), p = 2
(green,), p = 3 (blue,♦), and p = 4 (magenta,M). Dashed
lines: moments of isotropic orientation distribution. Thin
solid lines: Eq. (6) to O(G2 ). Thick solid lines: order 8-by-8
Padé-Borel resummation of Eq. (6) to order G32 . Parameters:
Ku = 0.1, St = 10, λ2 = 0.1 (b) and λ2 = 10 (c).
(1)
h(nt · ĝ)p in0= (n0 · ĝ)p +p Ku(n0 · ĝ)p−1 hnt · ĝin0 (5)
(2)
(1)
+p2 Ku2 (n0 · ĝ)p−2 2(n0 · ĝ)(nt · ĝ)+(p−1)(nt · ĝ)2 n
Padé-Borel resummation. Now consider higher orders
in the G-expansion. The series (6) is asymptotically divergent
and must be resummed.
Fig. 2 demonstrates
that
(i)
1.08
1
1.08
where nt is the coefficient of Kui in Eq. (3). Eq. (5)
Padé-Borel
resummation
[32,
40]
of
the
series
yields
excel1.06
0.8
1.06
is valid to order Ku2 . We average over the fluid-velocity
1.04
0.6
lent1.04results. Shown are results
from a resummation
of (6)
fluctuations as described in Ref. [32]. Since the flow is
replacements 1.02
0.4 lines). PSfrag
1.02
to order
G34PSfrag
(thick
solid
These
results
agree
very
replacements
homogeneous, the conditional moments do not depend
1
0.2
1
well
with
numerical
simulations
of
the
statistical
model
PSfrag replacements
0.98
0
0.98
on the initial position x0 . We expect that the effect of
10
10
10
10
10
10
10
10
for Ku
= 0.1
and St
=10 10 (symbols).
The10 resummed
the initial velocity v 0 and the initial angular velocity ω 0
theory works up to G = 10, and in this range the bias
decay exponentially, so that neither v 0 nor ω 0 affect the
increases with increasing G. The resummed theory also
steady state. We therefore set both to zero. Only the n0 predicts that the moments increase as St increases, for
dependence matters. In this way we obtain expressions
fixed G. A more detailed analysis of the recursion leadfor h(nt ·ĝ)p in0 , which involve secular terms that increase
ing to Eq. (6) reveals, however, that the limit G → ∞ is
linearly with time as t → ∞. But these terms must
delicate. Perfect alignment requires λ = ∞ [35].
vanish since nt is a unit vector. This condition yields
In summary, perturbation theory in Ku shows that
a recursion relation for the steady-state averages h(n ·
turbulence
gives rise to an orientation bias (Fig. 2), in
p
ĝ) i∞ , independent of n0 . This recursion is valid for
excellent agreement with statistical-model simulations at
(t)
0
arbitrary values of G ≡ Ku F St /C⊥ , and to order Ku .
Ku = 0.1 and in qualitative agreement with DNS (Fig. 1).
Note that G can be large for small values of Ku. We solve
The calculations leading to Eq. (6) reveal that the mothe recursion by a perturbation expansion in small G:
ment h(nt · ĝ)2p i∞ is a sum of two contributions, for
Pi
(2i)
all values of p, that stem from the ‘preferential sam∞
X
G2i j=1 pj Aj (St, λ)
1
2p
pling’ and ‘history’ terms in Eq. (3), respectively. For
h(n · ĝ) i∞ =
+
. (6)
Qi+1
2p+1 i=1
small Ku the history effect is the dominant mechanism,
k=1 (2p + 2k − 1)
the orientation bias is entirely determined by the his(2i)
tory of fluid-velocity gradients along straight determinThe coefficients Aj (St, λ) depend on the shape and inistic paths, Fig. 2a. Decomposing the leading-order
ertia of the particle, but not on G or p. From Eq. (6) we
(2)
(2)
(2)
obtain the Fourier transform of the probability distribucontribution as A1 = A1,pref. + A1,hist. we find that
(2)
(2)
tion of ng = |n · ĝ|. Inverse Fourier transformation yields
A1,pref. A1,hist. (Fig. S1b in the Supplemental Mathe distribution. To order G4 we find:
terial [35]). Fig. 3a leads to the same conclusion. It
0
!1
1
(2)
(4)
1
2(1−n2g )(5n2g −1)A1
P (ng ) = 1+ (3n2g −1)A1 G2 + 32
4
(4) + (1 − 18n2g + 25n4g )A2 G4 + . . . .
(7)
The lowest-order term corresponds to a uniform distribution of nt . Let us examine the G2 -term. It turns out
(2)
that A1 is negative for disks and positive for rods (see
Fig. S1 in the Supplemental Material [35]). This explains
that the orientation of settling disks is biased: disks tend
to fall edge on (as in Fig. 1), and rods settle tip first.
0
1
!1
0
1
2
!1
0
1
shows the distribution P (ng ) for Ku = 0.1. Also shown
is P (ng ) computed for particles falling with the constant
velocity (4), averaged over the steady-state distribution
of initial orientations. Here the average hn0 nT
0 i∞ is evaluated using the small-Ku theory. This corresponds to
keeping just the history contribution to hn2p
g i∞ . We observe excellent agreement with the full statistical-model
simulations. This shows that it is the history effect that
causes the orientation bias at small values of Ku.
Large Kubo numbers. At large Ku we use numeri-
102
1 4
4
0.4
b
Ku = 10
hn2p
g i1
3
a
Ku = 0.1
P (ng )
P (ng )
6
2
p=1
Ku = 10
c
p=2
0.2
2
0
1
0
0.2
0.4
0.6
ng
0.8
1
0
p=3
0
0.2
0.4
ng
0.6
0.8
1
0
0.01
St 0.1
1
FIG. 3. History effect causes orientation bias. a Distribution of ng based on full statistical-model
simulations (symbols),
√
and based on straight deterministic paths (shown in Fig. 2a), solid lines. Parameters: λ = 1/ 10, Ku = 0.1, St = 10, F = 1
(◦), F = 10 (). b Same for Ku = 10. Parameters correspond to red and green data in Fig. 1b, λ = 0.02 (◦), λ = 0.05 ().
Preferential-sampling contribution is hatched. c Moments hn2p
g i∞ from statistical-model simulations (p = 1, ◦; p = 2, ; p = 3,
√
♦). Parameters λ = 1/ 10, Ku = 10, F = 10. Also shown are simulations based on straight deterministic paths (solid lines).
cal simulations to analyse the orientation bias in the
same way as for small Ku. The result is shown in
Fig. 3b (parameters correspond to red and green curves
in Fig. 1b). We plot the full statistical-model distribution for Ku = 10. Also shown are the distributions obtained from straight deterministic paths (4), neglecting
preferential sampling. For the data in Fig. 3b, the average hn0 nT
0 i∞ is computed using statistical-model simulations. We see that the history effect makes a substantial
contribution to the P (ng ). But since the distributions
do not match, we infer that preferential sampling also
contributes. This contribution is hatched in Fig. 3b.
Limit of large settling speeds. Fig. 3c shows the moments hn2p
g i∞ for p = 1, 2, 3 and Ku = 10 as a function
of the Stokes number (other parameters correspond to
the blue curve in Fig. 1b). Open symbols denote full
statistical-model simulations, solid lines correspond to
simulations based on straight deterministic paths. At
intermediate Stokes numbers we see a clear difference
between the two simulations, preferential sampling is important in this region. As the Stokes number grows, however, the Figure demonstrates that preferential sampling
ceases to play a role. In this limit the orientation bias is
entirely caused by the history effect. At very small St,
by contrast, preferential sampling dominates. But the
bias is very small in this case. The bias shown in Fig. 3c
increases as St increases. But as the perturbation theory
indicates, the limit of large G is quite subtle. Statisticalmodel simulations for Ku = 1 show that the degree of
alignment starts to decrease as G becomes very large.
Conclusions. We analysed a statistical model for the
orientational dynamics of small heavy spheroids settling
in turbulence. The predictions of the model are in good
agreement with our own numerical results based on the
DNS of homogeneous isotropic turbulence (Fig. 1). Our
statistical-model analysis shows that there are two distinct competing mechanisms causing the orientation bias:
preferential sampling and the history effect. The latter
dominates for large settling speeds, but it makes substantial contributions also for other parameter regimes.
Preferential sampling dominates only when the bias is
negligibly small. We find that the history effect explains
at least about 50% of the orientation bias observed in
Fig. 1, when the bias is significant.
We have shown that the orientation alignment depends on combinations of dimensionless numbers: StK
−1/2
and FK Reλ . Our analysis shows that it is the smallscale properties of the flow that determine the orientation alignment. The Reλ -dependence arises only because
it determines the ratio between the smooth scale ` to ηK .
−1/2
We note that FK Reλ
equals the ratio of the settling
velocity and the rms turbulent velocity fluctuations.
Our results pertain to small ice crystals settling in turbulent clouds, and allow us to model the sensitive dependence of the effect upon particle shape, size, and the turbulence intensity. This is important since turbulent dissipation rates vary widely in clouds. Our results predict
strongly varying degrees of alignment. That the statistical model is in excellent agreement with the DNS opens
a way to parametrise the orientation distribution of icecrystals in clouds. This potentially leads to an improved
understanding of the radiative properties of clouds, and
of particle aggregation through collisions in clouds.
The present work is based on the point-particle approximation which neglects the effect of fluid inertia. This requires the particle Reynolds number Rep ≡ avc /ν to be
small. Estimating the slip velocity vc by the Stokes settling speed, we find that Rep is of order unity for the data
shown in Fig. 1, so the condition is marginally satisfied.
Our model should work better for smaller particles (the
orientation bias can still be substantial if the turbulent
dissipation rate is small). The shear Reynolds number
Res ≡ a2 s/ν must also be small, where s is an estimate of
the turbulent strain rate. Res is of order (a/ηK )2 in turbulence [41], so that this condition is satisfied for small
particles. Simulations fully resolving particle and fluid
motion [42, 43] and experiments [44–46] for micron-sized
particles make it possible to test our predictions, and
to determine the orientational dynamics of much larger
particles where fluid inertia must matter [47].
5
Acknowledgments. This work was supported by Vetenskapsrådet [grant number 2013-3992], Formas [grant
number 2014-585], and by the grant ‘Bottlenecks for particle growth in turbulent aerosols’ from the Knut and
Alice Wallenberg Foundation, Dnr. KAW 2014.0048.
The numerical computations used resources provided by
C3SE and SNIC.
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6
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E28.00002, 69th Annual Meeting of the APS Division of
Fluid Dynamics, Portland, Oregon (2016).
Supplemental Material: Angular dynamics of non-spherical particles settling in
turbulence
K. Gustavsson,1 J. Jucha,2, 3 A. Naso,4 E. Lévêque,4 A. Pumir,2 and B. Mehlig1
1
2
Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden
Laboratoire de Physique, Ecole Normale Supérieure de Lyon and CNRS, F-69007, Lyon, France
3
Projektträger Jülich, Forschungszentrum Jülich GmbH, D-52425, Germany
4
LMFA, Ecole Centrale de Lyon and CNRS, F-69134, Ecully, France
arXiv:1702.07516v1 [physics.flu-dyn] 24 Feb 2017
PACS numbers: 05.40.-a,47.55.Kf,47.27.eb
I.
DETAILS ON EQUATIONS OF MOTION
We begin by recalling the form of the resistance tensors, defined by Eq. (2) in the main text. The tensor M(t)
relates the force on the particle to the slip velocity, the difference between the fluid and the particle velocities. The
superscript refers to the translational (t) degrees of freedom. The tensor is expressed as:
(t)
(t)
(t)
(t)
Mij ≡ C⊥ δij + (Ck − C⊥ )ni nj .
(S1)
For the rotational degrees of freedom, superscript (r), the resistance tensor is decomposed into two terms, M(r1) and
M(r2) . The first term relates the torque to the angular slip velocity, the difference between the rotation rate of the
fluid and that of the particle. The second term relates the torque to the local strain in the fluid. The expression for
M(r1) is similar to Eq. (S1):
(r1)
Mij
(r1)
(r1)
≡ C⊥ δij + (Ck
(r1)
− C⊥ )ni nj .
(S2)
The expression for M(r2) is of the form:
(r2)
Mijk = C (r2) ijl nk nl .
(S3)
Here ijl is the completely anti-symmetric Levi-Civita tensor, and the summation convention is used: repeated indices
are summed from 1 to 3. For spheroidal particles, the coefficients in Eqs. (S1), (S2), and (S3) are known [1]:
(t)
8(λ2 − 1)
4(λ2 − 1)
(t)
, Ck =
,
2
3λ((2λ − 3)β + 1)
3λ((2λ2 − 1)β − 1)
40(λ2 − 1)
20(λ2 − 1)
(r)
=
, Ck =
,
2
9λ((2λ − 1)β − 1)
9λ(1 − β)
√
2
ln[λ + λ2 − 1]
(r2)
r1 λ − 1
√
Cijk = −C⊥ 2
, β=
.
λ +1
λ λ2 − 1
C⊥ =
(S4a)
(r)
(S4b)
C⊥
(S4c)
Here λ is the aspect ratio of the spheroid. Prolate
√ spheroids have λ > 1. Oblate spheroids correspond to λ < 1. In
this case, Eq. (S4c) reduces to β = arccos(λ)/[λ 1 − λ2 ]. The motion of the particle is fully characterised by the
knowledge of v, the velocity of the particle, and the angular velocity of the particle in the laboratory frame, ω. The
direction of the particle symmetry vector, n, is related to the angular velocity ω by the kinematic equation:
dn
= ω ∧ n.
dt
(S5)
The equation for the angular velocity reads in the laboratory frame:
d
J(n)ω = T .
dt
(S6)
Here T is the torque on the particle, and J(n) is the inertia tensor of the particle. For a spheroid its elements are
given in Ref. [1]. A difficulty is that the inertia tensor depends on the instantaneous orientation n(t). In the DNS we
therefore express Eq. (S6) in the particle frame, at the cost of an extra term in the equations of motion, due to the
rotation of the axes [2].
2
II.
DETAILS ON DIRECT NUMERICAL SIMULATIONS OF TURBULENCE
The Navier-Stokes equations read:
ρf [∂t u + (u · ∇)u] = −∇p + νρf ∇2 u + Φ
(S7)
where u is the velocity field of the flow, which is assumed to be incompressible:
∇ · u = 0,
(S8)
p is the pressure field, Φ is a forcing term, ρf is the density of the fluid, and ν is the kinematic viscosity. The equations
are solved in a triply periodic domain of size L3 , using a pseudo-spectral method. The code has been described
elsewhere [3]. Briefly, the fluid is forced in the band of lowest wave-numbers by imposing an energy injection of ε.
With the viscosity chosen here, ν = 0.113cm2 /s, ε is chosen so that he Kolmogorov length, ηK = (ν 3 /ε)1/4 is of the
order of the grid spacing. The code is fully dealiased, using the 2/3-rule. Specifically, if N is the number of grid
points in each direction, the nonlinear term is computed by using only Fourier modes 0 ≤ n ≤ N/3. The number
of points taken here was N = 384 (with an energy injection rate of ε ≈ 1 cm2 /s3 ), N = 784 (ε ≈ 16 cm2 /s3 ), and
N = 1568 (ε ≈ 256 cm2 /s3 ). We chose the units so the size L of the box is equal to 8π cm. The equation of motion of
the particles is implemented by interpolating the velocity and velocity gradients at the location of the particle, using
tri-cubic schemes. The method has been carefully checked, by systematically comparing the results in the absence of
motion u = 0, with the code, and with an elementary solver (Mathematica) of the equations of motion of the particle.
III.
DETAILS ON THE PERTURBATION THEORY FOR SMALL VALUES OF Ku
A.
Implicit solution of equations of motion
We follow the method outlined in Ref. [4] and expand the solution of the equations of motion for the settling
spheroid, Eqs. (1) and (2) in the main article, in powers of Ku:
v(t) =
∞
X
n
n
(−1) Ku
Z
t
dt1 e
(t)
M0 (t1 −t)
0
n=0
(t)
∆M (t1 ) · · ·
Z
tn−1
dtn e
(t)
M0 (tn −tn−1 )
(t)
∆M (tn )
0
Z
tn
n=0
×
n(t) = n0 + Ku
Z
0
t
1
2M
tn
(r)
dtn+1 eM0
(tn+1 −tn )
0
0
(r)
(tn+1 −tn )
0
× Fĝ + M(t) (tn+1 )u(x(tn+1 ), tn+1 )
Z
Z tn−1
Z t
∞
X
(r)
(r)
dtn eM0 (tn −tn−1 ) ∆M(r) (tn )
ω(t) =
(−1)n Kun
dt1 eM0 (t1 −t) ∆M(r) (t1 ) · · ·
0
(t)
dtn+1 eM0
(S9)
(r)
(tn+1 )Ω(x(tn+1 ), tn+1 ) − ΛC⊥ (S(x(tn+1 ), tn+1 )n(tn+1 )) ∧ n(tn+1 )
+ Ku Λ(ω(tn+1 ) ∧ n(tn+1 ))(n(tn+1 ) · ω(tn+1 ))
dt1 ω(t1 ) ∧ n(t1 ) .
Here Λ ≡ (λ2 − 1)/(λ2 + 1), and we have decomposed the translational resistance tensor into a part that does not
(t)
depend on the Kubo number, and a Ku-dependent part, M(t) = M0 + Ku ∆M(t) , with
(t)
(t)
(t)
(t)
M0 = IC⊥ + n0 nT
0 (Ck − C⊥ )
and
(t)
(t)
T
T
∆M(t) = [∆n nT
0 + n0 ∆n + Ku ∆n ∆n ](Ck − C⊥ ) .
(S10)
Here ∆n is a rescaled orientation displacement vector
∆n(t) ≡
n(t) − n0
=
Ku
Z
0
t
dt1 ω(t1 ) ∧ n(t1 ) .
(S11)
The rotational resistance tensors are decomposed in a similar way. Eq. (S9) is an exact, but implicit, solution to
Eqs. (1) and (2) in the main article, provided that the initial conditions of the particle velocity and angular velocity
are set to zero. The solution (S9) depends implicitly on orientation n, the angular velocity ω, and on the centre-ofmass position x of the particle. If the Kubo number is small we can terminate the solution (S9) at some order in Ku
and solve the resulting set of equations iteratively. Details are given in Ref. [4].
3
B.
Explicit solution of equations of motion for small values of Ku
To eliminate the implicit dependence on x through the flow velocity u(x(t), t) and through the flow velocity
gradients A(x(t), t), we consider a decomposition of particle trajectories x(t) into two parts:
x(t) ≡ x(d) (t) + δx(t) .
(S12)
The first part, x(d) (t), is a deterministic part that is obtained by setting the turbulent velocity u to zero in the particle
equation of motion. The second part is the remainder, the flow-dependent fluctuating part, δx(t) ≡ x(t) − x(d) (t).
The deterministic part of the trajectory is obtained by integrating the solution for the particle velocity in Eq. (S9)
with u = 0 and A = 0. For large values of t, the corresponding particle velocity approaches the settling velocity
v s (n0 ) in a quiescent fluid
v s (n0 ) = F St [M(t) (n0 )]−1 ĝ = F St
h I
(t)
C⊥
+
n0 nT
0
(t)
Ck
−
i
n0 nT
0
(t)
C⊥
ĝ .
(S13)
The corresponding deterministic trajectory approaches x(d) (t) = x0 + v s (n0 )t. Here n0 is the constant orientation
at which the particle settles. In the steady state, the orientation n0 must be chosen from a stationary distribution of
particle orientations that must be determined self consistently.
For small values of Ku, the flow gives rise to small deviations δx from the deterministic settling trajectories x(d) .
Following the procedure outlined in Ref. [4], we attempt a series expansion of the fluid velocity experienced by particles
in terms of small δx
u(x(t), t) = u(x(d) (t), t) + δxT (t)∇u(x(d) (t), t) + . . . ,
(S14)
and similarly for the fluid-velocity gradients. We insert these expansions into the implicit solutions (S9), and iterate
the solution to find an expression for the orientation n valid for small values of Ku. To second order in Ku we find
Eq. (3) in the main article. Writing out all terms explicitly this equation reads in vector notation:
Z t
(r)
2
n(t) = n0 + Ku ds [1 − e(s−t)C⊥ / St ] (δxs · ∇) O(x, s)n0 + λλ2 −1
+1 S(x, s)n0 − n0 · S(x, s)n0 n0 x=x(d)
s
0
Z t
(r)
+ Ku
ds(1 − eC⊥ (s−t)/ St )[Os n0 + Λ(Ss n0 − (nT
0 Ss n0 )n0 )]
0
Z t
Z t1
h
T
T
+ Ku2
dt1
dt2 d1 nT
0 Ot1 Ot2 + d2 (n0 Ot1 Ot2 n0 )n0 + d3 (n0 Ot1 St2 n0 )n0 + d4 (n0 St1 n0 )(n0 St2 n0 )n0
0
0
T
+ d5 O(t2 )S(t1 )n0 + d6 (nT
0 St1 Ot2 n0 )n0 + d7 (n0 St1 St2 n0 )n0 + d8 (n0 St2 n0 )Ot1 n0 + d9 (n0 St1 n0 )Ot2 n0
+ d10 Ot1 Ot2 n0 + d11 Ot1 St2 n0 + d12 (n0 St2 n0 )St1 n0 + d13 (n0 St1 n0 )St2 n0 + d14 St1 Ot2 n0 + d15 St1 St2 n0
i
(S15)
4
(d)
(d)
with Ot ≡ O(xt , t), St ≡ S(xt , t). The coefficents are:
(r)
C⊥ (Λ − 1)e
d1 = −
d2 = (1 − Λ)e
+
Ck (Λ − 1)e
(r)
(r)
C⊥
(r)
+
(r)
(r)
Ck
C⊥ (Λ − 1)Λe
(r)
(r)
(r)
C⊥
(r)
(r)
(r)
− (Λ + 1)ΛeC⊥
+ Λ2 e
+
(r)
(r)
(t1 −t)/ St
(r)
d8 =
d9 =
C⊥ (Λ − 1)Λe
−
d10 =
(t1 −t)/ St
d11 = −
(r)
− ΛeC⊥
(r)
(r)
− Λ2 e
(−2t+t1 +t2 )/ St
2
d12 = Λ e
(r)
(r)
d14 =
d15
(r)
+ Λ2 eC⊥
+ Λ2 e
(r)
C⊥ (Λ − 1)Λe
(r)
(r)
(t2 −t)/ St
− ΛeC⊥
(r)
(r)
− 2Λ2 eC⊥
(t2 −t1 )/ St
+ 3Λ2
(r)
(r)
(r)
C⊥ (t1 −t)/ St +Ck (t2 −t1 )/ St
−
(r)
+ ΛeC⊥
(r)
(r)
(r)
+ Λ 2 eC ⊥
2
(t2 −t)/ St
(r)
− Λ2 eC⊥
(r)
(t2 −t)/ St
(r)
(r)
(r)
(r)
(r)
(t2 −t1 )/ St
(r)
C⊥ + Ck
(r)
(t2 −t1 )/ St
(r)
(C⊥
−
−
(r)
(r)
C⊥ (t1 −t)/ St +Ck (t2 −t1 )/ St
−Λ
(r)
(r)
+ Ck Λ)eC⊥
(r)
(r)
C⊥
+
(r)
(t2 −t1 )/ St
(r)
Ck
(r)
Λ(C⊥ + Ck Λ)eC⊥
(r)
(t2 −t1 )/ St
(r)
(t2 −t1 )/ St
+ Λ2 eC⊥
+ Λ2 eC⊥
C⊥ (t1 −t)/ St +Ck (t2 −t)/ St
(r)
(r)
C⊥ + Ck
(r)
+
(r)
− eC⊥
Ck (Λ − 1)Λe
(t1 −t)/ St
(r)
Ck (t2 −t1 )/ St
(r)
(r)
C⊥ + Ck
(r)
+ ΛeC⊥
(t2 −t)/ St
(t2 −t1 )/ St
(r)
C⊥ + Ck
+Λ
(r)
C⊥ (t2 −t)/ St
−Λ e
(r)
Ck (t2 −t1 )/ St
− 2Λ2
Λ(C⊥ + Ck Λ)eC⊥
(r)
(r)
Ck (Λ − 1)Λe
− 2Λ
(r)
C⊥ (t2 −t1 )/ St
− (Λ − 1)Λe
C⊥ (t2 −t)/ St +Ck (t1 −t)/ St
(r)
(t2 −t1 )/ St
(r)
(r)
(t2 −t)/ St
Ck (t2 −t1 )/ St
(t2 −t1 )/ St
(r)
(r)
C⊥ (t1 −t)/ St +Ck (t2 −t)/ St
(t1 −t)/ St
(r)
(t2 −t)/ St
+ (Λ − 1)Λe
(r)
C⊥ + Ck
d13 = Λ2 eC⊥
(r)
(r)
+ (−Λ − 1)eC⊥
C⊥ + Ck
+ ΛeC⊥
(r)
C⊥ (−2t+t1 +t2 )/ St
(r)
(r)
C⊥ (t1 −t)/ St
(r)
(−2t+t1 +t2 )/ St
(r)
(r)
(r)
(r)
(r)
C (t −t)/ St +Ck (t1 −t)/ St
C⊥ Λ)e ⊥ 2
(r)
(r)
C⊥ + Ck
(t1 −t)/ St
(r)
Λ(C⊥ + Ck Λ)eC⊥
(r)
C⊥ (t2 −t)/ St +Ck (t1 −t)/ St
(r)
(r)
+ eC ⊥
(r)
− ΛeC⊥
C⊥ (Λ − 1)Λe
(r)
C⊥ + Ck
C⊥ (t1 −t)/ St +Ck (t2 −t)/ St
C⊥ + Ck
−
(r)
Ck
Ck (Λ − 1)Λe
+
(r)
(r)
C⊥ + Ck
(r)
(r)
+ ΛeC⊥ (t1 −t)/ St + Λ2 −eC⊥ (t2 −t)/ St − Λ
(r)
(r)
C⊥ + Ck
(t2 −t)/ St
C⊥ + Ck
C⊥ (Λ − 1)Λe
(r)
(C⊥
(r)
(r)
(r)
(t2 −t1 )/ St
(r)
(r)
(r)
C⊥ (t1 −t)/ St
(r)
(r)
C⊥ + Ck
C⊥ (t1 −t)/ St +Ck (t2 −t)/ St
(r)
(r)
C⊥ (−2t+t1 +t2 )/ St
C⊥ (t1 −t)/ St +Ck (t2 −t1 )/ St
(r)
+ ΛeC⊥
+
(r)
(r)
(r)
C⊥ (t1 −t)/ St
C⊥ (Λ − 1)Λe
(r)
+
+
C⊥ (t2 −t)/ St +Ck (t1 −t)/ St
(−2t+t1 +t2 )/ St
d5 = (Λ − 1)Λe
+ 2ΛeC⊥
(r)
(r)
C⊥ (Λ − 1)e
(C⊥ + Ck Λ)eC⊥
C⊥ + Ck
d4 = −3Λ2 e
d7 = 2Λ2 e
(r)
C⊥ + Ck
Ck (t2 −t1 )/ St
(r)
d6 = −
(r)
(r)
C⊥ (t2 −t)/ St +Ck (t1 −t)/ St
+
(r)
−
(r)
(r)
(r)
+ ΛeC⊥
+ (Λ − 1)e
(r)
Ck (Λ − 1)e
C⊥ (t1 −t)/ St +Ck (t2 −t1 )/ St
C⊥ (Λ − 1)e
d3 =
(r)
C⊥ + Ck
(r)
(r)
C⊥ (t1 −t)/ St +Ck (t2 −t1 )/ St
(r)
Ck (t2 −t1 )/ St
(r)
(r)
(r)
C⊥ (t1 −t)/ St +Ck (t2 −t)/ St
− Λ2
− Λ2
(r)
(r)
(r)
+ Λ −eC⊥ (t1 −t)/ St + ΛeC⊥ (t2 −t)/ St − ΛeC⊥ (t2 −t1 )/ St + Λ
C⊥ + Ck
(r)
(r)
(r)
= Λ2 −eC⊥ (t1 −t)/ St + Λ2 eC⊥ (t2 −t)/ St − Λ2 eC⊥ (t2 −t1 )/ St + Λ2
+1
5
where Λ = (λ2 − 1)/(λ2 + 1). To obtain the first term in the last row of Eq. (3) in the main text, for example, we
must add the terms involving d1 , d2 , and d10 .
C.
Moments of alignment
Eq. (S15) relates the orientation dynamics to the fluid velocity and its gradients. This expression makes it possible
to calculate the moments h(n · ĝ)2p in0 conditional on n0 (odd-order moments vanish because spheroids are fore-aft
symmetric). By raising Eq. (S15) to the power 2p, terminating the resulting expressions at order Ku2 , and taking the
average using the known stationary statistics of the fluid velocity and its gradients, we obtain the moments of n · ĝ
conditional on initial alignment n0 · ĝ.
However, the resulting expressions contain secular terms. These terms are proportional to Ku2 and grow linearly
with time. We know that such secular terms must vanish, because the vector n must remain normalised to unity. Its
components cannot continue to grow. We therefore demand that the initial alignment n0 · ĝ is distributed according
to a stationary distribution of alignments that causes all secular terms to vanish. This procedure gives rise to one
condition per value of p, and this is sufficient to determine the desired stationary distribution of alignments.
The dependence of the secular terms upon (n0 · ĝ)2p is not only through algebraic powers, but also functional
through exponentials and error functions. As a consequence, the conditions that make sure that the secular terms
vanish are hard to solve in general. We therefore seek a perturbative solution and expand the secular conditions and
(t)
the moments mp ≡ h(n0 · ĝ)2p i in terms of the parameter G ≡ Ku F/C⊥ , for instance:
mp =
∞
X
i
m(i)
p G .
(S16)
i=0
To lowest order in G we find the recursion
(0)
(1 + 2p)m(0)
p + (1 − 2p)mp−1 = 0 .
(S17)
(0)
Using the boundary condition m0 = 1 we find
m(0)
p =
1
.
2p + 1
(S18)
These moments correspond to a uniform distribution of n0 . Higher-order contributions in G to the moments can
(i)
be recursively determined by series expansion of the secular terms. We find that moments mp with odd values of i
(i)
vanish, and that moments mp with even values of i are on the form given by Eq. (6) in the main article:
h(n · ĝ)2p i∞ =
(2i)
The coefficients Aj
order G2 we have
Pi
(2i)
∞
X
G2i j=1 pj Aj (St, λ)
1
.
+
Qi+1
2p+1 i=1
k=1 (2p + 2k − 1)
(S19)
(St, λ) are quite lengthy in general. We therefore only give the lowest-order expression here. To
(2)
h(n · ĝ)2p i∞ =
G2 pA1 (St, λ)
1
+
.
2p+1 (2p + 1)(2p + 3)
(S20)
(2)
Decomposing A1 (St, λ) in one contribution due to preferential sampling and one due to the history effect, we have
(2)
(2)
(2)
A1 (St, λ) = A1,pref. (St, λ) + A1,hist. (St, λ) .
(S21)
The preferential-sampling contribution reads:
h
i
(t)
(t)
(t)
(t) 2
(t) 2
(t) (t)
(t)
(t)
3
+
C
+
C
C
+
3
St
C
+
C
+
3
St
4C
St
(C
−
C
)
C
⊥
⊥
⊥
⊥
⊥
k
k
k
k
7−Λ
(2)
A1,pref. (St, λ) =
.
(t)
(t)
(t)
5 + 3Λ2
3(C⊥ + St)3 Ck (Ck + St)3
(S22)
(2)
(2)
|A
|A1,preferential
1,pref. /A1/A|1 |
6
(2)
0.02
(2)
A1
(2)
10
0.01
0
−10
100
10−2
0
102
100
10−2
λ
102
λ
(2)
FIG. S1: Left: Coefficient A1 (St, λ) [Eqs. (S21)–(S23)] plotted against λ for St = 0.1 (red), 1 (green), 10 (blue), 100 (magenta).
(2)
(2)
Right: Corresponding plot for the relative contribution due to preferential sampling |A1,pref. (St, λ)/A1 (St, λ)|.
The contribution due to the history effect is
1
(2)
A1,history (St, λ) =
(t) 2
(t)
(t)
(t)
(t)
(t)
h
3Ck (3Λ2 + 5) (10Ck + 3St)3 (10C⊥ Ck − 3C⊥ St + 6Ck St)3
(t) 2
(t) 3 (t) 6
(t) 2
(t) (t)
19Λ2 − 42Λ + 35
− 2000000 C⊥ Ck
−4C⊥ 4Λ2 + 7Λ + 7 + C⊥ Ck 3Λ2 + 28Λ − 7 + Ck
(t) 2 (t) 6
(t) 2
(t) (t)
(t) 2
− 3600000 StC⊥ Ck
−4C⊥ 4Λ2 + 7Λ + 7 + C⊥ Ck 3Λ2 + 28Λ − 7 + Ck
19Λ2 − 42Λ + 35
(t) (t) 4
(t) 2
(t) (t)
(t) 2
(t) 2
(t) (t)
− 540000 St2 C⊥ Ck
C⊥ − 2C⊥ Ck − 4Ck
4C⊥ 4Λ2 + 7Λ + 7 + C⊥ Ck −3Λ2 − 28Λ + 7
(t) 2
+ Ck
−19Λ2 + 42Λ − 35
(t) 2 (t) 2
(t) 3 (t)
32Λ2 − 119Λ + 105
15Λ2 + 22Λ + 23 − 21C⊥ Ck 11Λ2 + 24Λ + 13 − 4C⊥ Ck
(t) 4
(t) (t) 3
30Λ2 − 49Λ + 49 + 4Ck
19Λ2 − 42Λ + 35
+ 8C⊥ Ck
(t) 3 (t) 2
(t) 2
(t) 5
(t) 4 (t)
−41Λ2 − 210Λ + 119
− 48600 St4 Ck
C⊥ − 23Λ2 + 28Λ + 21 + 14C⊥ Ck 7Λ2 + 12Λ + 5 + C⊥ Ck
(t) 5
(t) (t) 4
(t) 2 (t) 3
19Λ2 − 42Λ + 35
3Λ2 + 28Λ − 7 + 8Ck
22Λ2 + 7Λ + 49 + 8C⊥ Ck
− 8C⊥ Ck
(t) 2
(t)
(t)
(t) 2
(t) (t)
(t) 2
+ 29160 St5 Ck (C⊥ − 2Ck )2 14C⊥ Λ2 + 3Λ + 2 + C⊥ Ck −3Λ2 − 28Λ + 7 + Ck
−19Λ2 + 42Λ − 35
i
(t)
(t)
(t) 2
(t) 2
− 1458 St6 (C⊥ − 2Ck )3 C⊥ 9Λ2 + 28Λ + 35 + Ck
−19Λ2 + 42Λ − 35
.
(S23)
(t) 4
− 108000 St3 Ck
(t) 4
7C⊥
Here we have used the identities
(r)
C⊥ =
(t)
(t)
to simplify. Eq. (S4) relates C⊥ and Ck
10 (t)
C
3 k
(t)
(r)
and Ck
(2)
to λ, so that A1
=
(t)
10 C⊥ Ck
3 2C (t) − C (t)
k
(S24)
⊥
is a function of St and λ only. Fig. S1 shows how the
(2)
A1
(2)
(2)
expressions for
and for the relative preferential-sampling contribution |A1,pref. /A1 | depend on the aspect ratio
λ, for different values of St. We see that the preferential contribution is small compared to the contribution due to
the history effect for all parameter values.
D.
Large-G limit
In the previous Section we summarised how the moments of the orientation distribution can be calculated by
using perturbation expansions in the parameter G. Now consider the opposite limit of large values of G. In this
7
limit we obtain a recursion equation for the moments h(n · ĝ)2p i that is independent of preferential effects. This is
expected because the particles fall rapidly through the turbulence in this limit, experiencing the turbulent gradients
approximately as a white-noise signal. The history contribution is obtaind from a fourth-order recursion that is hard
to solve in general.
One might expect that the orientation of rod-like particles approaches perfect alignment with gravity, that is
h(n · ĝ)2p i = 1 for all values of p as G → ∞. However, when we insert this limiting expression into the fourth-order
recursion equation, we obtain a condition that is only satisfied when λ = ∞. This shows that perfect alignment is
only possible when λ = ∞. For other values of λ the moments must approach limiting values as G → ∞. We have
not yet managed to calculate these limits from our statistical-model theory.
[1]
[2]
[3]
[4]
S. Kim and S. J. Karrila, Microhydrodynamics: principles and selected applications (Butterworth-Heinemann, Boston, 1991).
L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon Press Ltd., Oxford, 1969).
M. Voßkuhle, A. Pumir, E. Lévêque, and M. Wilkinson, J. Fluid Mech. 749, 841 (2014).
K. Gustavsson and B. Mehlig, Adv. Phys. 65, 1 (2016).