PHYSICS OF FLUIDS 21, 113301 共2009兲
A unified sweep-stick mechanism to explain particle clustering
in two- and three-dimensional homogeneous, isotropic turbulence
S. W. Colemana兲 and J. C. Vassilicos
Department of Aeronautics and Institute for Mathematical Sciences, Imperial College,
London SW7 2BY, United Kingdom
共Received 24 February 2009; accepted 23 September 2009; published online 3 November 2009兲
Our work focuses on the sweep-stick mechanism of particle clustering in turbulent flows introduced
by Chen et al. 关L. Chen, S. Goto, and J. C. Vassilicos, “Turbulent clustering of stagnation points and
inertial particles,” J. Fluid Mech. 553, 143 共2006兲兴 for two-dimensional 共2D兲 inverse cascading
homogeneous, isotropic turbulence 共HIT兲, whereby heavy particles cluster in a way that mimics the
clustering of zero-acceleration points. We extend this phenomenology to three-dimensional 共3D兲
HIT, where it was previously reported that zero-acceleration points were extremely rare. Having
obtained a unified mechanism we quantify the Stokes number dependency of the probability of the
heavy particles to be at zero-acceleration points and show that in the inertial range of Stokes
numbers, the sweep-stick mechanism is dominant over the conventionally proposed mechanism of
heavy particles being centrifuged from high vorticity regions to high strain regions. Finally, having
a clustering coincidence between particles and zero-acceleration points, both in 2D and 3D HIT,
motivates us to demonstrate the sweep and stick parts of the mechanism in both dimensions. The
sweeping of regions of low acceleration regions by the local fluid velocity in both flows is
demonstrated by introducing a velocity of the acceleration field. Finally, the stick part is
demonstrated by showing that heavy particles statistically move with the same velocity as
zero-acceleration points, while moving away from any nonzero-acceleration region, irrespective of
their Stokes number. These results explain the clustering of inertial particles given the clustering of
zero-acceleration points. © 2009 American Institute of Physics. 关doi:10.1063/1.3257638兴
I. INTRODUCTION
Turbulence, as is well known, is a very efficient mixer of
suspended substances. This mixing is one of the most important properties of turbulence as it is of great importance in
geophysical, astrophysical, and industrial processes. Turbulence, however, can also cause coagulation when the suspended impurity has significant inertia. This phenomenon
has long been well known in the case of particles heavier
than the surrounding fluid and is often called preferential
concentration.1,2 This phenomenon continues to attract widespread attention from many communities because it is a crucial factor in such diverse fields as aerosol physics 共crucial in
climate models兲, planet formation in the early solar system,
and understanding the concentration of chemical or biological species in turbulent flows.
There have been several mechanisms proposed to predict
where heavy particles go in turbulent flows, depending on
the scales of interest. For particle relaxation times larger than
the integral time of the turbulence, it has been shown that
caustics can cause an unbounded increase in the concentration of particles in finite time.3,4 Meanwhile, for particle relaxation times and distances smaller than the respective
Kolmogorov time and length scales, it has been proposed
that the motion of inertial particles is governed by fluid
strain, namely, heavy particles are centrifuged out of eddies
and accumulate in low vorticity and high strain rate
a兲
Electronic mail: [email protected].
1070-6631/2009/21共11兲/113301/10/$25.00
regions.5,6 This centrifuging phenomenology has also been
implicitly extended to particle relaxation times within the
inertial range of the turbulence and is widely believed to
account for clustering in this range.1,7,8 While it seems that
these centrifuging effects are indeed predominant at subdissipative scales or in cases where the energy spectrum is
mostly concentrated around a single length scale, such as in
low Reynolds number turbulence, at higher Reynolds numbers the clustering is not a single scale phenomemon but
rather has a multiscale nature.9 At higher Reynolds numbers
not only do the smallest eddies play a role in segregating the
heavy particles but there are also larger coherent eddies
which effect the process.
One mechanism proposed to explain this multiscale clustering structure is the so-called sweep-stick mechanism,10,11
whereby in two-dimensional 共2D兲 inverse energy-cascading
homogeneous, isotropic turbulence 共HIT兲, the spatial distribution of heavy particles mimics that of the clustering of
points where the acceleration of the fluid is zero, over the
entire range of length scales where the energy spectrum has
the Kolmogorov ⫺5/3 power-law shape. The sweep-stick
mechanism explains this coincidence in terms of the sweeping of regions of low acceleration by the local fluid velocity
and the fact that particles on zero-acceleration points move
together with these points 共stick兲, whereas they move away
from nonzero-acceleration points. The sweep-stick mechanism is different from centrifugal effects which are indeed
predominant in flows where the energy spectrum is mostly
concentrated around a single length scale such as in low
21, 113301-1
© 2009 American Institute of Physics
113301-2
Phys. Fluids 21, 113301 共2009兲
S. W. Coleman and J. C. Vassilicos
TABLE I. Statistics for the two 2D turbulent velocity fields considered. N is
the number of grid points, L, the integral scale, ␩, the forcing scale, u⬘, rms
velocity, ␶␩, the Kolmogorov time scale determined via the rms vorticity, ␶␩,
and T ⬅ L / u⬘ is the eddy turnover time.
TABLE II. Statistics for the two 3D turbulent velocity fields considered.
Notation is as in Table I and Re␭ is the Reynolds number based on the
Taylor scale. kmax = 共冑2 / 3兲N is the largest wavenumber of the DNS dealiased
by the phase-shift method.
L
L/␩
u⬘
␶␩
T / ␶␩
N
Re␭
L
␩
T
␶␩
kmax␩
20482
0.24
26.0
1.12
0.0057
37.6
3843
2
0.23
30.0
1.30
0.0068
25.0
5123
139
187
1.009
1.87
0.0109
5.55⫻ 10−3
0.615
2.38
0.0398
4.89⫻ 10−2
1.97
1.34
N
4096
Reynolds number turbulence. The difference is twofold: The
sweep-stick mechanism takes into account the sweeping of
small eddies by larger ones and also the fact that eddies of
very different length scales coexist in a single flow, thus
causing the particle clustering to have a multiscale spatial
distribution.
There are several outstanding questions to be resolved in
order to establish the sweep-stick mechanism on a solid footing. Much of the work on this mechanism has been qualitative and it is therefore now desirable to introduce quantitative measures of the coincidence of the two clusters
共particles and zero-acceleration points兲. Furthermore, a
modified sweep-stick mechanism has been proposed for
three-dimensional 共3D兲 HIT 共Ref. 12兲 which, by taking account of the compressibility of the particle velocity field,
proposes that heavy particle clusters mimic those of a suitably chosen component of the acceleration. Thus, there currently exist two different sweep-stick mechanisms, one for
2D inverse cascading HIT and one for 3D HIT. It is important to determine if either of these mechanisms can be applied to both cases, so as to have general applicability of the
mechanism. Finally, it is crucial to the mechanism that low
acceleration regions are swept by the local fluid velocity,
something which has been firmly established in 2D inverse
energy-cascading HIT 共Ref. 13兲 but has not been studied in
3D HIT. This paper aims to answer these questions. After
introducing numerical techniques in Sec. II, Sec. III shows
that inertial particle clusters mimic those of zero-acceleration
clusters in both 2D and 3D and there is no need for a modified sweep-stick mechanism in 3D. Section IV quantifies the
magnitude of the preferential concentration of heavy particles at zero-acceleration points via use of statistics sampled
at inertial particle positions and correlation functions. Similar quantification is performed on statistics relating to the
centrifuging phenomenology and it is shown that this mechanism cannot account for clustering at higher particle inertias.
Finally, in Sec. VI we validate both the sweep and stick
components of our phenomenology.
II. NUMERICAL SIMULATIONS
A. Equations of motion
The particles considered are passive, small, heavy, and
rigid spheres all of the same size. Their mass density ␳ p is
much greater than the fluid density ␳ f , i.e., ␳ p Ⰷ ␳ f . The radius of the particles a is sufficiently small for the Reynolds
number based on the particle radius to be much less than
unity, allowing the use of the Stokes flow approximation for
the local flow around the particles. The particle radius is also
assumed to be smaller than the smallest length scale of the
turbulence. Under these assumptions, the equation of motion
for a particle, neglecting gravity, is14
1
d
v p共t兲 = 兵u关x p共t兲,t兴 − v p共t兲其,
dt
␶p
共1兲
where v p共t兲 is the particle velocity at time t and u共x , t兲 is the
fluid velocity at position x at time t. The equation implies
that the particle velocity relaxes to that of the fluid within a
time scale ␶ p, where ␶ p = 2␳ pa2 / 共9␮兲, with ␮ being the fluid
viscosity. A nondimensional time scale can be obtained by
normalizing with a time scale of the turbulence, which is
chosen to be the Kolmogorov time ␶␩. This normalized time
is called the Stokes number and is defined by S␩ ⬅ ␶ p / ␶␩.
Although the system is much simplified, in many cases particles are so massive that the assumptions made here are
justified. Equation 共1兲 is integrated together with the Navier–
Stokes equations for an incompressible fluid using a fourth
order Runge–Kutta scheme. The fluid velocity at the particle
position x p共t兲 is obtained using a sixth order Lagrange
interpolation.
The fluid velocity field is obtained using a pseudospectral direct numerical simulation 共DNS兲 of the Navier–Stokes
equations. In the 2D case, the Navier–Stokes equations are
integrated with an external small scale energy source and a
large scale energy sink. Periodic boundary conditions are applied in two orthogonal directions. Full details of the simulation can be found in Ref. 15. There are two different cases,
the parameters of which are summarized in Table I. By using
20482 and 40962 grid points, there is a region between the
forcing scale ␩ and the integral length scale L, spanning
approximately two decades of wavenumber space where the
energy spectrum E共k兲 has the form E共k兲 ⬃ ⑀2/3k−5/3.
In the 3D case a Fourier spectral method is used with a
fourth order Runge–Kutta–Gill time scheme. Dealiasing is
achieved using the phase-shift method. In the high resolution
5123 simulation the amplitudes of Fourier components of
velocity in a low-wavenumber range are kept constant in
time to realize a statistically stationary state.16 In the lower
resolution case, a constant low-wavenumber forcing is applied. In both cases, periodic boundary conditions are applied in all three orthogonal directions. The parameters of the
3D simulations are also given in Table II.
113301-3
Phys. Fluids 21, 113301 共2009兲
A unified sweep-stick mechanism
B. Finding zero-acceleration points
Crucial to the assessment of the range of validity of the
sweep-stick mechanism is a robust, accurate, and fast determination of the positions of zero-acceleration points in the
acceleration field. Previous works in Refs. 10 and 11 have
used a modified Newton–Raphson method to locate the
points. While this is an all purpose method, it can be expensive in more than two dimensions as it requires a good initial
guess for starting points due to the fact that the method has
unpredictable global convergence properties. Here, an algorithm proposed in Ref. 17 for the detection of unstable periodic orbits is used. This method is based on a universal set of
linear transformations that transform unstable fixed points to
stable ones while maintaining their positions. For the sake of
completeness, a summary of the method is given here.
First, the problem of finding the roots of the
m-dimensional acceleration field a共x兲 is changed into that of
finding the fixed points of the m-dimensional dynamical system A : xn+1 = a共xn兲 + xn. Clearly the roots of a共x兲 and the
fixed points of this dynamical system coincide. Now, any
initial point iterated under this dynamical system will be attracted to stable fixed points of the dynamical system. The
key to the method is to use a set of transformations such that
all fixed points are stabilized by one of these transformations. This allows the detection of all fixed points, both stable
and unstable, simply by finding all of the stable fixed points
of each of the transformed systems in turn by a simple fixed
point iteration. To achieve this a series of different dynamical
systems Sk is then constructed which possesses the same
number of fixed points in the same positions as the original
dynamical system. This requirement imposes that any transformation of the original system must be linear. Therefore Sk
must have the form17
xn+1 = xn + ␭Ck关xn+1 − xn兴,
共2兲
xn+1 = xn + ␭Ck关a共xn兲兴,
共3兲
where Ck is an invertible m ⫻ m matrix with Ckij = O共1兲 and ␭
is a constant such that 1 Ⰷ ␭ ⬎ 0. The condition on ␭ is necessary as this allows a very restrictive set of matrices Ck to
be considered, of which there exists at least one matrix
which transforms a given unstable fixed point of the original
dynamical system to a stable fixed point in the transformed
system. In fact, it is possible to show that all fixed points can
be stabilized, i.e., their stability matrices have eigenvalues
with an absolute value less than 1, via the symmetry transformations of the m-dimensional regular polygon.17 The
number of such matrices is m ! 2m and consists of matrices Ck
whose elements are Ckij 苸 兵0 , ⫾ 1其, with each row and column
containing only one element which is different from zero.
The algorithm has several advantages over Newton–
Raphson techniques. Most importantly, the stabilization
method is of global character. Even initial guesses lying far
from the linear neighborhood of the stabilized fixed point are
attracted to it after a few iterations. This allows for a cruder
set of initial guesses to be used to find the zeros of the acceleration points. This reduction in initial guesses is enhanced by the fact that the same initial guess can converge to
(a)
(b)
(c)
(d)
FIG. 1. 共a兲, 共b兲, and 共c兲 show spatial distribution of heavy particles for
S␩ = 0.1, S␩ = 1.6, and S␩ = 51.2, respectively, at t ⬇ 4T in 2D HIT. 共d兲 shows
the position of zero-acceleration points at the same time. The side length of
the plots is approximately 800␩.
several different fixed points when iterated with different dynamical systems Sk. The global convergence properties also
allows efficiencies in the implementation of the algorithm.
Iterating an initial guess x0, two possible behaviors are observed. Either the guess iterates to a stabilized fixed point
with continuously decreasing steps or the trajectory evolves
chaotically. Therefore a simple check on the magnitude of
the step size allows termination of initial guesses which are
not evolving to fixed points.
III. CLUSTERING OF HEAVY PARTICLES
AND ZERO-ACCELERATION POINTS
It was pointed out by several authors that the clustering
of heavy particles strongly resembles that of the acceleration
field, although mostly visual comparisons were made.5,10,11
We begin by showing in Fig. 1 snapshots of heavy particle
positions with three different Stokes numbers at t ⬇ 4T in 2D
turbulence at a resolution of 40962 and comparing this to the
position of zero-acceleration points in the fluid, of which we
find approximately 106.
As noticed in Ref. 10 initially uniformly distributed inertial particles with different values of S␩ develop holes in
the same regions, and these holes become bigger with increasing S␩. Comparison of the inertial particle distributions
with the distribution of zero-acceleration points of the fluid
shows a clear visual correlation between the clustering of
particles and that of zero-acceleration points, although the
sharpness of this similarity clearly depends on the Stokes
number.
113301-4
Phys. Fluids 21, 113301 共2009兲
S. W. Coleman and J. C. Vassilicos
(a)
(b)
FIG. 3. 共a兲 shows the positions of particles with S␩ = 2 in a thin layer of
dimensions 500␩ ⫻ 500␩ ⫻ 5␩ in 3D HIT. 共b兲 shows the set of a = 0 points in
the same layer.
FIG. 2. Spatial distribution of e1 · a = 0 points at t ⬇ 4T in 2D HIT. The side
length of the plots is 1/4 of that in Fig. 1, i.e., 200␩.
Similar clustering of heavy particles can be observed in
3D HIT. Reference 12 attempted to explore the similarity
between heavy particle clusters and zero-acceleration points
in this flow. In fact, using a Newton–Raphson technique to
locate the acceleration field zeros, it was found that these
points were very rare, which was justified by considering the
tubular nature of coherent eddies. Instead, a modified mechanism that took account of the particle compressibility was
proposed in Ref. 12. Many authors, beginning with Ref. 1,
explained particle clustering by appealing to the leadingorder asymptotic expansion of Eq. 共1兲,
v p共t兲 ⬇ u关x p共t兲,t兴 − ␶ pa关x p共t兲,t兴,
共4兲
with ␶ p Ⰶ 1. Therefore, ⵜ · v p ⬇ −␶ p ⵜ · a, which implies that
particles converge in regions where ⵜ · a ⬎ 0. This convergence will happen predominantly along the eigenvector e1
associated with the largest positive eigenvalue ␭1 of the symmetric part of the acceleration gradient tensor ⵜa. This led to
the proposal that instead of heavy particles clustering at
a = 0 points, they cluster at points where
e1 · a = 0
and
␭1 ⬎ 0.
共5兲
Of course regions of space where Eq. 共5兲 is satisfied are
lines in 2D and surfaces in 3D and are both far more numerous and easier to locate than a = 0 points. The spatial distribution of e1 · a = 0, which will be labeled A, in a box a quarter of the side length of Fig. 1 in 2D HIT is shown in Fig. 2.
Unsurprisingly, set A is far denser than the set of a = 0
points.
We now turn our attention to finding zero-acceleration
points in 3D HIT using the zero finding algorithm described
above. Figure 3共a兲 shows the spatial distribution of particles
with S␩ = 2 in a thin layer of thickness 5␩ and Fig. 3共b兲
shows the distribution of zero-acceleration points in the same
layer. Not only do we find substantially more a = 0 points
than found in Ref. 12 which reported finding less than 100
points in a slice of the same dimensions 共we find approximately 4 000 000 in the entire box, we also find that there is
a strong visual correlation between the void areas of the particle and a = 0 distribution, especially at larger length scales.
This result lends considerable support to the original sweepstick mechanism proposed in Refs. 10 and 11 for 2D HIT,
whereby heavy particle clustering mimics that of zeroacceleration points.
IV. QUANTIFYING THE VALIDITY
OF THE SWEEP-STICK MECHANISM
Clearly, only limited conclusions can be drawn from visual inspection of the clustering of heavy particles and sets
of zero points related to fluid quantities. There is a clear need
to develop a systematic tool, which quantifies the extent to
which two distributions are similar. There have been many
previous attempts across a wide range of disciplines to quantify the distance between two distributions.18–20 It is important to realize that when finite sets of data drawn from a
given distribution are compared, even sets from the same
distribution will display differences. Therefore, any comparison of the distributions of finite sets of data will depend on
the scale r on which the data are observed. In order to examine the k-dimensional distributions at different observation scales r, the whole “volume” of the domain is split into
N共r兲 equisized boxes. In this work, we choose to use the
correlation function Cab共r兲 to measure the correlation between
two sets of data a and b,
Cab共r兲
=
N共r兲
关共nai − 具na典兲共nbi − 具nb典兲兴
兺i=1
N共r兲
关共nai − 具na典兲2共nbi − 具nb典兲2兴其1/2
兵兺i=1
,
共6兲
where ni␣ is the number of particles of type ␣ in box i and
具n␣典 is the mean number of particles of type ␣ in a box of
size r. Cab共r兲 is bounded between ⫺1 and 1 with Cab共r兲 = −1
corresponding to anticorrelation, Cab共r兲 = 1 to positive correlation and Cab共r兲 = 0 to uncorrelated data sets a and b. The use
of a segregation length scale based on a Kolmogorov-type
distance is possible,21 although this provides no advantage
over Cab共r兲, as both require a coarse graining scale to be
introduced.
113301-5
Phys. Fluids 21, 113301 共2009兲
A unified sweep-stick mechanism
0.7
0.7
a = 0 - Sη = 0.8
e1.a = 0 - Sη = 0.8
uniform - Sη = 0.8
0.6
a = 0 - Sη = 3.2
e1.a = 0 - Sη = 3.2
a = 0 - Sη = 0.8
e1.a = 0 - Sη = 0.8
0.6
0.5
0.5
0.4
C(r)
C(r)
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
-0.1
0
1
10
1
10
r/η
r/η
FIG. 4. Cab共r兲 as a function of r in 2D for three different cases of the set a
as indicated by the legend 共a = 0 points, the set A, and a homogeneous
distribution of the same number of points as there are a = 0 points兲. All three
cases are compared to the set b as the positions of heavy particles with
S␩ = 0.8.
Initially, let us discuss the correlation functions in 2D.
The first point of note, as shown in Fig. 4, is that both
a = 0 points and set A show positive correlation to the distribution of heavy particles at all observation scales r. Meanwhile, the same number of points distributed homogeneously
at random show approximately zero correlation with the positions of heavy particles at all r up to r ⬇ L. This result
clearly establishes that heavy particles cluster in a similar
way to zero-acceleration points when compared to a uniform
distribution, which could physically represent passive fluid
tracers. It is important to stress that this positive correlation
would not be obtained for any clustered set of data as Cab共r兲
is sensitive to the location of the clusters and can detect
anticorrelated clusters.
Also interesting is that set A shows positive correlation
with the positions of heavy particles, although this is not
surprising as trivially a = 0 points are a subset of A. The
correlation function also acts as a way of measuring the
Stokes dependency of the similarity between the zeroacceleration clustering and that of the heavy particles. Visu-
FIG. 6. Cab共r兲 as a function of r in 2D for two different Stokes numbers
compared to both a = 0 points and A. The correlation between heavy particles and a = 0 points is stronger at length scales larger than a crossover
length scale which is a decreasing function of S␩ and is O共␩兲.
ally, the similarity between the heavy particle clusters and
the zero-acceleration points seems strongest at S␩ = O共1兲; at
smaller Stokes numbers the particles are spread more uniformly and at higher Stokes numbers the clusters of heavy
particles become “fuzzy.”11 In Fig. 5共a兲 the variation of Cab共r兲
with r is shown for the distributions of heavy particles at
S␩ = 0.1, 0.8, 3.2, and 51.2 compared to a = 0 points, while
Fig. 5共b兲 gives the variation when the particle distributions
are compared to set A. The values of the correlation functions at all scales agree with the visual comparison in that
when S␩ is of O共1兲, the correlation function is significantly
higher than when S␩ Ⰷ 1 or S␩ Ⰶ 1.
The correlation function also allows a direct comparison
of whether the particle clustering more closely mimics that
of a = 0 points or set A at a given resolution r. Surprisingly,
Fig. 6 shows that there is a crossover length scale below
which the correlation between particles and A is stronger
than that between the particles and zero-acceleration points.
This crossover length scale is a decreasing function of S␩,
implying that as the particles become more clustered, they
mimic the a = 0 over an ever wider range of length scales.
0.7
0.5
Sη = 0.1
Sη = 0.8
Sη = 3.2
0.45
0.6
0.4
0.5
C(r)
Cba(r)
0.35
0.4
0.3
0.3
0.25
0.2
0.2
Sη = 0.1
Sη = 0.8
Sη = 3.2
Sη = 51.2
0.1
0
1
10
0.15
0.1
0.05
1
10
r/η
(a)
r/η
(b)
FIG. 5. Cab共r兲 as a function of r in 2D for S␩ = 0.1, 0.8, 3.2, and 51.2 compared to 共a兲 zero-acceleration points and 共b兲 the set A.
113301-6
Phys. Fluids 21, 113301 共2009兲
S. W. Coleman and J. C. Vassilicos
0.18
uniform - Sη = 2
a = 0 - Sη = 2
e1 . a - Sη = 2
0.16
0.14
0.12
Cab (r)
0.1
0.08
0.06
0.04
0.02
0
-0.02
1
10
r/η
FIG. 7. Cab共r兲 as a function of r in 3D for heavy particles with S␩ = 2 compared to both a = 0 points, A, and a homogeneous distribution of the same
number of points as there are a = 0 points. The homogeneous, random, uniform distribution is plotted for comparison.
The crossover can be explained by noting that at S␩ ⬃ O共1兲,
the crossover length scale is O共␩兲. The fact that the particles
are better correlated with A below this length scale is to be
expected as the correspondence to the zero-acceleration clustering is only expected to occur at length scales that fall in
the inertial range ␩ Ⰶ r Ⰶ L. The higher correlation between
A and particles at smaller length scales is likely an artifact of
the fact that points A are more numerous than a = 0 points.
A similar analysis can also be performed in 3D and the
difference in correlation between the heavy particle positions
and the distribution of a = 0 points and A is much clearer, as
shown in Fig. 7. Over all length scales the correlation is
stronger between heavy particles and the set of a = 0 points
then it is between heavy particles and A. As in the 2D case
the correlation between particle positions and a = 0 is weaker
at smaller length scales. More noteworthy is that the correlation is lower than in the 2D case. The exact reasons for this
depletion in the correlation have not been quantified, but are
likely a combination of increased dimensionality and decreased Reynolds number.
All of our results lead us to conclude that the sweepstick mechanism proposed in Ref. 10, whereby heavy particles cluster in a way which mimics that of the clusters of
zero-acceleration points of the underlying fluid over the
range of length scales constituting the inertial range, is sufficient to explain particle clustering in both 2D and 3D HIT.
There is no need for the modified sweep-stick mechanism of
Ref. 12 which takes account of the compressibility of the
particle velocity field. With this unified sweep-stick mechanism in mind we investigate statistics of fluid quantities
sampled at inertial particle positions. This will enable us to
quantify the tendency of heavy particles to cluster at zeroacceleration points as a function of their Stokes number. The
most natural quantity to investigate is the acceleration itself.
Figure 8共a兲 shows the relative probability distribution function 共pdf兲 of a single component of the acceleration in 2D,
conditionally sampled at heavy particle positions and also at
positions homogeneously and randomly distributed throughout the fluid.
The pdfs for inertial particles are all clearly peaked
around zero, with the highest peak occurring for S␩ = 0.8.
Even at S␩ = 0.1, there is still a clear peak around ax = 0 when
compared to that of a homogeneous, random distribution.
The pdf becomes more peaked as S␩ approaches unity and
then broadens again at higher Stokes number. These results
are consistent with the results for the correlation function
displayed in Fig. 5, which showed that the maximum correlation between zero-acceleration points and heavy particles
occurred at S␩ = 0.8. This phenomenon can also be observed
in 3D HIT, as shown in Fig. 8共b兲, which shows the pdf of a
single component of the acceleration clearly peaked when
sampled at the positions of heavy particles. Again, the pdf
becomes more peaked as S␩ approaches unity, but still remains significantly more peaked than for fluid elements even
at S␩ = 5. The effect is not as strong in the 3D case as in 2D.
This can be attributed to the smaller inertial range in the 3D
simulation due to the presence of a large intermediate dissipation region.
It is also possible to confirm our assertion that there is no
need for a modified sweep-stick mechanism in 3D HIT by
examining the pdfs of ei · a, where we remind the reader that
ei is the eigenvector associated with the ith eigenvalue ␭i of
the symmetric part of the acceleration gradient tensor ⵜa,
such that ␭1 ⬎ ␭2 ⬎ ␭3. The modified sweep-stick mechanism
relies on the fact that particles will converge primarily along
the direction e1, resulting in the heavy particle clustering
being the same as that of set A. This, in turn, implies that the
particles are not as compressed along the directions e2 and
e3, and in fact may even statistically diverge along these
directions. Therefore one would expect the pdf of ei · a
sampled at heavy particle positions to be peaked around zero
for i = 1, but for the pdf to broaden for i = 2 and i = 3. Figure
8共c兲 shows that this is not the case, rather the pdf of ei · a
sampled at heavy particle positions is peaked relative to the
pdf at fluid positions for all i. Due to the orthogonality of the
eigenvectors ei, this result implies that heavy particles reside
preferentially at zero-acceleration points.
V. CENTRIFUGING
It is interesting to compare the proposed sweep-stick
mechanism with the well-known phenomenology of the centrifuging of heavy particles from vortical regions. From taking the divergence of Eq. 共4兲, particles are expected to cluster
in regions where strain overwhelms vorticity.1 This can be
quantified by examining the Okubo–Weiss parameter22
Q ⬅ 兩S共x,t兲兩2 − 21 兩␻共x,t兲兩2 ,
␻ ⬅ 21 关ⵜu − 共ⵜu兲T兴
T
共7兲
is the vorticity tensor and
where
1
S ⬅ 2 关ⵜu + 共ⵜu兲 兴 is the strain rate tensor. It would be expected that heavy particles reside preferentially in regions
where Q Ⰷ 0. Figure 8共d兲 shows the pdf of Q for two different Stokes numbers and the fluid in 2D HIT. For S␩ ⬍ 1, the
pdf of the Okubo–Weiss parameter is significantly positively
skewed, in agreement with the centrifuging phenomenology.
At higher Stokes numbers though, such as the S␩ = 3.2 shown
113301-7
Phys. Fluids 21, 113301 共2009兲
A unified sweep-stick mechanism
1
0.9
0.8
0.8
0.7
0.6
0.5
0.4
0.3
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
-1.5
fluid
Sη = 0.1
Sη = 0.5
Sη = 2
Sη = 5.0
0.9
Relative frequency
Relative frequency
1
fluid
S η= 0.1
S η= 0.8
S η= 3.2
0
-1
-0.5
0
ax/arms
0.5
1
1.5
-3
-2
-1
(a)
1
0.7
0.8
0.5
0.4
0.3
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
-3
-2
-1
0
ai/arms
3
1
2
fluid
S η= 0.8
S η= 3.2
0.9
0.6
-4
2
(b)
Relative frequency
Relative frequency
0.8
1
1
e1 . a - fluid
e1 . a - Sη = 2.0
e2 . a - fluid
e2 . a - Sη = 2.0
e3 . a - fluid
e3 . a - Sη = 2.0
0.9
0
ai/arms
3
4
(c)
0
-1.5
-1
-0.5
0
Q/Qrms
0.5
1
1.5
(d)
FIG. 8. pdf of a single component of acceleration for 共a兲 2D HIT and 共b兲 3D HIT, conditionally sampled at heavy particle positions and for comparison at
positions homogeneously and randomly distributed throughout the fluid 共labeled fluid兲. The pdfs for inertial particles are all clearly peaked around zero. In 2D
HIT the highest peak occurs for S␩ = 0.8, before dropping again for higher Stokes numbers. 共c兲 shows the pdf of ei · a共i = 1 , 2 , 3兲, conditionally sampled at the
positions of heavy particles with S␩ = 2 in 3D HIT, compared to homogeneous sampling in the fluid. The pdf at particle positions is peaked for all i implying
that particles reside at zero-acceleration points. 共d兲 shows the pdf of the Okubo–Weiss parameter Q. Although this is clearly positively skewed for S␩ ⬍ 1, for
S␩ = 3.2 it has exactly the same form as at uniformly random fluid positions, despite the particles still exhibiting strong clustering.
in the figure, the pdf of Q is indistinguishable from that of
the fluid, despite the fact that the heavy particles still show
significant clustering. This should be compared to Fig. 8共a兲,
which shows that the pdf of a component of acceleration is
only slightly peaked around zero for all S␩ Ⰶ 1, while showing much stronger peaks when S␩ is closer to inertial scales.
This weaker peak can be accounted for by considering
the stickiness of a zero-acceleration point as a function of S␩.
The relative velocity between a heavy particle with S␩ Ⰶ 1
and a zero-acceleration point being swept with the local fluid
velocity is simply −␶ pa from Eq. 共4兲. It can be expected that
the heavy particle and zero-acceleration point will approximately move together 共stick兲 as long as v p − u共x p , t兲 is less
than some characteristic velocity of the turbulent fluid.
Clearly, at very small Stokes numbers the particle can stick
to low intensity, yet nonzero-acceleration points, without
violating this condition. This makes large regions of the turbulent fluid sticky at small Stokes numbers, resulting in the
broadened pdf in the acceleration sampled at heavy particle
positions. Given that a large amount of the fluid is sticky for
S␩ Ⰶ 1, the classical phenomenology of accumulation in high
strain and low vorticity regions is the dominant mechanism
at these small Stokes numbers, while the sweep-stick mechanism is dominant when the stickiness of the zeroacceleration point is sufficiently focused at 1 ⬍ S␩ ⬍ T / ␶␩.
It is possible to imagine that the coincidence between the
pdfs of Q for fluid elements and particles with S␩ = 3.2 is due
to the contamination of the pdfs by small scale activity
which is not relevant due to the larger time scale of the
particle. To see the impact of small scales, we have filtered
the velocity field using a sharp low-pass filter whereby the
amplitudes of all Fourier modes with a wavenumber
兩k兩 ⬎ kc are set to zero. These pdfs are shown in Fig. 9 for
S␩ = 0.1 and 3.2.
For S␩ = 0.1, for all values of kc the pdf of Q is positively
skewed when compared to that for fluid elements. On the
other hand, for S␩ = 3.2, there is some, but little difference
between the pdf of Q for inertial particles or for fluid elements at any cutoff wavenumber. The results in 3D are even
less clear due to the limited extent of the inertial range 共due
to both lower numerical resolution and the use of regular as
opposed to hyperviscosity, which leads to a large intermediate dissipation region兲. This makes it very difficult to select
cutoff wavenumbers which clearly lie in the inertial range.
Phys. Fluids 21, 113301 共2009兲
S. W. Coleman and J. C. Vassilicos
0.018
0.014
Relative frequency
0.016
kc = 400
kc = 400
kc = 1200
kc = 1200
No cutoff
No cutoff
0.016
kc = 400
kc = 400
kc = 1200
kc = 1200
No cutoff
No cutoff
0.014
0.012
Relative frequency
113301-8
0.012
0.01
0.008
0.006
0.01
0.008
0.006
0.004
0.004
0.002
0.002
0
0
-1
-0.5
0
Q/Qrms
0.5
1
-1
-0.5
(a)
0
Q/Qrms
0.5
1
(b)
FIG. 9. The pdf of the Okubo–Weiss parameter Q at two different values of kc for 共a兲 S␩ = 0.1 and 共b兲 S␩ = 3.2, compared to fluid elements. Heavy particles
are denoted by points and fluid elements by lines.
Nevertheless, a clear advantage of the sweep-stick mechanism is that there is no need for any filtering to describe and
explain particle clustering at any value of S␩ in the inertial
range.
To conclude this section we present results examining
the relationship between the Q parameter and regions of the
fluid with zero acceleration. While there is evidence that acceleration correlation statistics are dominated by pressure
gradient correlation statistics,23 the connection between the
acceleration and strain/vorticity fields in a turbulent flow is
less clear. Figure 10 shows the pdf of Q sampled at zeroacceleration points in both 2D and 3D. Although there is a
clear difference between the two pdfs, the figure clearly
shows that there is no trivial relationship between the values
of Q and a 共of course ⵜ · a = Q兲, as all values of Q can be
found at zero-acceleration points.
An implicit assumption in the centrifuging phenomenology is that eddies are persistent enough so that the particles
remain in these eddies for a sufficient time for the centrifuging to cause inhomogeneity in the particle concentration. In
fact, it has already been shown in Ref. 10 that this temporal
correlation is crucial to the nature of heavy particle clustering. By constructing a synthetic velocity field with the same
0.016
⫺5/3 energy spectrum as a DNS velocity field but with different temporal correlations 共no sweeping兲, it was observed
that the nature of particle clustering was fundamentally
changed. Properly taking account of temporal correlations is
a necessity to describe clustering of inertial particles in any
velocity field. The sweep-stick mechanism does exactly this
for the specific case of high Reynolds flows described by the
Navier–Stokes equations, where sweeping determines the
frequency spectrum of the velocity field in the inertial
range.24
VI. THE SWEEP AND STICK COMPONENTS
OF THE MECHANISM
Having a clustering coincidence between particles and
zero-acceleration points both in 2D and 3D HIT motivates us
to demonstrate both the sweeping of zero-acceleration points
by the local fluid velocity in both flows and the stick component of the mechanism, namely, that particles move together with a = 0 points and away from any region with nonzero acceleration. Both parts of the mechanism have been
partially demonstrated before; it was demonstrated in Refs.
10 and 13 that regions of low acceleration statistically move
0.16
fluid
a=0
0.014
0.12
Relative frequency
0.012
Relative frequency
a=0
fluid
0.14
0.01
0.008
0.006
0.1
0.08
0.06
0.004
0.04
0.002
0.02
0
0
-2
-1.5
-1
-0.5
0
Q/Qrms
(a)
0.5
1
1.5
2
-2
-1.5
-1
-0.5
0
Q/Qrms
0.5
1
1.5
2
(b)
FIG. 10. The pdf of the Okubo–Weiss parameter Q sampled at zero-acceleration points compared to that sampled randomly in the fluid for 共a兲 2D and 共b兲 3D.
Phys. Fluids 21, 113301 共2009兲
A unified sweep-stick mechanism
−3
−3
x 10
2.5
2
1
2
1
0
1.5
1.5
1
−1
1
−2
0.5
−2
0.5
−3
0
−3
−3
−1
0
1
2
3
0.1
2
0
−1
−2
Sη = 0.1
Sη = 0.4
Sη = 1.6
-0.1 τη ai
-0.4 τη ai
2.5
2
−3
0.15
x 10
3
ξ/u’
ξ/u’
3
−2
−1
0
1
2
3
u/u’
V /u’
0.05
<vpi-ui|ai>
113301-9
0
a
(a)
-0.05
(b)
−3
x 10
3
2
u/u’
1
-0.15
1.5
0
-3
-2
1
−1
-1
0
ai/a
1
2
3
0.5
−2
−3
-0.1
2.5
2
−3
−2
−1
0
1
2
FIG. 12. The average difference in particle velocity and local fluid velocity
conditioned on the value of acceleration. The solid lines show −␶ pa as
predicted by Eq. 共4兲.
3
V /u’
a
(c)
FIG. 11. 共Color online兲 Joint pdfs of 共a兲 Va and ␰, 共b兲 u and ␰, and 共c兲 Va
and u sampled at zero-acceleration points.
with the local fluid velocity in 2D inverse cascading HIT and
Ref. 12 demonstrated that heavy particles statistically move
with the local fluid velocity at zero-acceleration points and
with a velocity different to that of the fluid at all other points
in 3D HIT.
Concentrating first on sweeping, we introduce the acceleration velocity Va = ds共t兲 / dt, where s共t兲 is the position vector of a point which possesses a constant acceleration with
time, i.e., da关s共t兲 , t兴 / dt = 0.10 It was noted in Ref. 10 that
applying the Kolmogorov scaling to ␰ = 共u − Va兲 gives
具共u − Va兲2兩共a = 0兲典 ⬃ u⬘2共L/␩兲−2/3 ,
共8兲
where u⬘ is the rms of the fluid turbulence. This result implies an asymptotic statistical correspondence between u and
Va in the limit of infinite Reynolds number. In order to
strengthen this result, Ref. 13 examined the joint pdf of u
and Va and showed them to be statistically dependent at
zero-acceleration points; moreover that the likeliest value of
Va at zero-acceleration points is equal to u. The joint pdfs of
both Va and ␰ and u and ␰ were also plotted and showed an
approximate statistical independence. We adopt the same approach here to demonstrate the sweeping of zero-acceleration
points in 3D turbulence by calculating the quantities Va, ␰,
and u at zero-acceleration points. The joint pdfs of these
quantities are shown in Fig. 11 and can be seen to very
similar for the results found for 2D HIT in Ref. 13.
The most statistically dependent of the velocities are
clearly u and Va. This result confirms statistically that zeroacceleration points are swept by the local fluid velocity in 3D
HIT, as required by the sweep-stick mechanism.
Finally, we turn our attention to the stick part of the
mechanism. In order for the sweep-stick phenomenology to
hold, it is required that heavy particles move with a velocity
equal to that of the local fluid velocity when they reside at
zero-acceleration points in the flow and to move with a velocity different to that of the local fluid at all other points.
Thus, a heavy particle which encounters a zero-acceleration
point travels with it for a long time with a velocity equal to
the local fluid velocity, while a heavy particle at a point
where the fluid has any nonzero acceleration moves away
from this point. To verify this, we plot the average value of
the ith component of v p共t兲 = u关x p共t兲 , t兴 conditioned on the
value of ai at x p for S␩ = 0.1, 0.4, and 1.6 in 2D HIT in
Fig. 12.
Despite the fact that the asymptotic approximation of
Eq. 共4兲 only holds at very small Stokes number, even at
higher Stokes numbers the average difference between the
particle velocity and the local fluid velocity is equal to zero
when the local fluid acceleration is also zero. Hence an inertial particle at a zero-acceleration point moves with velocity
u关xp共t兲 , t兴, which as shown above, is statistically the velocity
with which the zero-acceleration point moves. A heavy particle at any point with finite acceleration moves away from
this point until it encounters a zero-acceleration point. These
observations are sufficient to establish the sweep-stick
mechanism of particle clustering given the clustering of the
zeros of the underlying fluid acceleration.
VII. CONCLUSIONS
The main purpose of this paper has been to investigate
the validity of the sweep-stick mechanism in 2D and 3D
HIT. Although this mechanism has been proposed previously, much of the work has been of a qualitative nature. We
have firmly established that heavy particles with a Stokes
number lying in the inertial range of time scales cluster preferentially at zero-acceleration points in both 2D and 3D HIT,
and have introduced two ways of quantifying this tendency
共correlation functions and pdfs sampled at heavy particle positions兲. The key to reaching this conclusion was the discovery that zero-acceleration points are numerous in 3D HIT,
whereas previously they were thought to be very rare due to
the tubular nature of eddies. The modified sweep-stick
mechanism of Ref. 12 is no longer necessary now these zeroacceleration points are known to exist in 3D HIT. We have
also demonstrated that the classical picture of heavy particles
being centrifuged from vortical regions is only valid for
113301-10
S␩ ⬍ 1, i.e., at dissipative scales, by examining the pdf of the
Okubo–Weiss parameter at heavy particle positions and observing that it is indistinguishable from the pdf for the same
quantity for massless tracers for S␩ in the inertial range. Finally, we have extended previous results showing the sweeping of zero-acceleration points in 2D HIT to 3D HIT and also
verified the stick part of the mechanism, namely, that heavy
particles at zero-acceleration points move with the local fluid
velocity, in 2D HIT.
The advantage of the sweep-stick mechanism relative to
other mechanisms is twofold: 共i兲 it gives a clear prediction of
where heavy particles will cluster without the need for a
Lagrangian integration and 共ii兲 it seems readily extendable to
inhomogeneous flows. This final point is of great interest for
future work and is currently under investigation.
ACKNOWLEDGMENTS
The authors particularly acknowledge S. Goto for supplying the data for the analysis contained in this paper. We
would also like to acknowledge T. Faber and M. Priego
Wood for additional 2D and 3D HIT data, respectively.
The authors thank the EPSRC for funding under Grant No.
EP/E029973/1.
1
Phys. Fluids 21, 113301 共2009兲
S. W. Coleman and J. C. Vassilicos
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