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Flatness of tracer density profile produced by a point source in turbulence

01 Oct 2003-Physics of Fluids (American Institute of Physics)-Vol. 15, Iss: 11, pp 3546-3557

Abstract: The average concentration of tracers advected from a point source by a multivariate normal velocity field is shown to deviate from a Gaussian profile. The flatness (kurtosis) is calculated using an asymptotic series expansion valid for velocity fields with short correlation times or weak space dependence. An explicit formula for the excess flatness at first order demonstrates maximum deviation from a Gaussian profile at time t of the order of five times the velocity correlation time, with a t–1 decay to the Gaussian value at large times. Monotonically decaying forms of the velocity time correlation function are shown to yield negative values for the first order excess flatness, but positive values can result when the correlation function has an oscillatory tail.

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Flatness of tracer density profile produced by a point source in turbulence
James P. Gleeson
a)
Applied Mathematics, University College Cork, Cork, Ireland
D. I. Pullin
Graduate Aeronautical Laboratories, Caltech, Pasadena, California 91125
Received 25 November 2002; accepted 18 August 2003; published 1 October 2003
The average concentration of tracers advected from a point source by a multivariate normal velocity
field is shown to deviate from a Gaussian profile. The flatness kurtosis is calculated using an
asymptotic series expansion valid for velocity fields with short correlation times or weak space
dependence. An explicit formula for the excess flatness at first order demonstrates maximum
deviation from a Gaussian profile at time t of the order of five times the velocity correlation time,
with a t
1
decay to the Gaussian value at large times. Monotonically decaying forms of the velocity
time correlation function are shown to yield negative values for the first order excess flatness, but
positive values can result when the correlation function has an oscillatory tail. © 2003 American
Institute of Physics. DOI: 10.1063/1.1616558
I. INTRODUCTION
The pioneering work of Taylor
1,2
on dispersion problems
in turbulent flows has led to the widespread use of Gaussian-
plume models for the prediction of mean concentration of
passive tracers or pollutants. In isotropic turbulence, for ex-
ample, the mean tracer concentration may be defined as the
probability distribution function PDF of particles released
from the same point in space, with the statistical ensemble
consisting of either independent experiments, or of indepen-
dent particles released from the source at widely spaced time
intervals. This PDF is usually assumed to have a Gaussian
form, with variance determined from Taylor’s formula.
3
Tay-
lor also argued that the Gaussian form is asymptotically cor-
rect for large times, as the particles have effectively executed
a random walk through uncorrelated eddies. Moreover, if the
turbulent velocity is modeled by a Gaussian i.e., multivari-
ate normal velocity field, it immediately follows that the
concentration is Gaussian at small times also. Thus it is only
at intermediate times that any deviation from a Gaussian
distribution might be observed, but few attempts have been
made to examine this case.
Kraichnan
4
investigated single-particle diffusion in
Gaussian velocity fields using kinematic simulations and the
direct-interaction approximation DIA. His Fig. 9 shows de-
viations from the Gaussian distribution in numerical experi-
ments, quantified by the flatness factor or kurtosis, which
dips below its Gaussian value at intermediate times. Direct-
interaction approximations of the flatness were not attempted
in Ref. 4, but Koch and Shaqfeh
5
report that DIA calcula-
tions lead to an incorrect small-time limit for the flatness in a
Gaussian velocity field.
Sawford and Borgas
6
investigated a variety of stochastic
models for the Lagrangian velocity in turbulent flow, and
showed that a multifractal model
7
and a Markovian jump
model with discontinuous velocities both predict leptokur-
tic density functions, i.e., with flatness factors larger than the
Gaussian value, although they note the magnitude of the de-
viation from the Gaussian form depends on the model cho-
sen. Data from wind tunnel experiments is better fitted by a
Gaussian distribution than by a leptokurtic distribution, al-
though the difference is not large.
In this paper we utilize an asymptotic series expansion
of the mean concentration to derive quadrature formulas for
the flatness, and show that simple forms of the velocity time
correlation predict a platykurtic sub-Gaussian flatness dis-
tribution, in agreement with Kraichnan’s numerical simula-
tions, but contrary to the models discussed by Sawford and
Borgas. The small parameter of our asymptotic series is
u
k
0
, 1
where u is the root mean square velocity,
is the velocity
correlation time, and k
0
is a characteristic wavenumber of
the energy spectrum see Eqs. 13 and 14 for full defini-
tions. We first demonstrate that the concentration is exactly
Gaussian in the limit of vanishing
: This limit corresponds
to either a white-noise in time velocity field
0,
3
or to a
space-independent velocity (k
0
0). We then calculate the
flatness using the first few terms in an asymptotic series for
small
Sec. III, and examine some simple examples in
Sec. IV. Simplified formulas for terms in the asymptotic se-
ries are listed in Appendix A, and in Appendix B Pade
´
ap-
proximants are used to show that our results are not restricted
to infinitesimally small values of
.
II. EXACT RESULTS
The advection of a tracer from a point source at the
origin by a random velocity field is described by the solution
of the advection equation
t
u
0,
x,0
x
. 2
a
Telephone: 353 21 490 3410; fax: 353 21 427 0813. Electronic mail:
j.gleeson@ucc.ie
PHYSICS OF FLUIDS VOLUME 15, NUMBER 11 NOVEMBER 2003
35461070-6631/2003/15(11)/3546/12/$20.00 © 2003 American Institute of Physics
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Here
dx is the probability, for one realization of the random
velocity u that a marked particle which was at the origin at
time t 0 will be in the volume element dx at time t. Taking
the average over the velocity statistics yields the mean prob-
ability density function or ‘concentration’
x,t
. 3
In isotropic turbulence the PDF is a function only of time
and the distance r from the source, and so (r,t)dr is the
probability of finding a tracer which was released at the ori-
gin at time t 0 in the spherical (d 3) or circular (d 2)
shell with radius between r and r dr.
In the following, the velocity will be assumed to be iso-
tropic, with Gaussian multivariate normal statistics and
mean zero. The effects of molecular diffusion are ignored for
clarity, so tracer particles follow the fluid exactly. It is well
known that the concentration profile spreads in an approxi-
mate Gaussian shape—in particular the width of the cloud is
often measured by the dispersion in this paper we use the
isotropic dispersion as defined in 4, which is three times the
one-dimensional dispersion used in Ref. 8
D
t
r
2
x
x
x,t
dx, 4
with the integral being over all of space, and repeated indices
summed from 1 to d, the number of space dimensions (d
2or3. Taking r as the distance from the origin, we have
r
2
x
x
,sor
2
x
2
y
2
in two dimensions, and r
2
x
2
y
2
z
2
in for d 3. In a previous paper
8
we addressed the
calculation of the dispersion by means of an asymptotic se-
ries for small velocity correlation times and confirmed the
theoretical results by calculating
r
2
in numerical simula-
tions. It has been noted, however, that the average concen-
tration does not have an exact Gaussian shape for all times.
1,5
The deviation from a Gaussian shape may be measured by
the flatness or kurtosis, defined as
f
t
r
4
r
2
2
,
with
r
4
defined similarly to 4
r
4
x
x
x
x
x,t
dx. 5
The flatness of the distribution (x,t) is defined as
f
t
r
4
r
2
2
x
x
x
x
x,t
dx
x
x
x,t
dx
2
. 6
The flatness of a d-space-dimensional Gaussian distribution
is (2 d)/d for all times, as may be confirmed by calculating
the integrals in 6 for the general isotropic distribution with
zero mean and variance
2
(t):
x,t
1
2
␲␴
2
t
兲兲
d/2
exp
x
x
2
2
t
, 7
to obtain
r
2
d
2
and
r
4
d(2 d)
4
. Note that for a
one-dimensional Gaussian distribution the corresponding
flatness
y
4
/
y
2
2
equals 3: It can readily be demonstrated
for an isotropic distribution in d space dimensions that
r
2
d
y
2
and
r
4
d
y
4
d(d 1)
y
2
2
. Thus the isotropic
flatness and the one-dimensional flatness are related by the
equation
r
4
r
2
2
1
d
y
4
y
2
2
d 1
d
. 8
All our results are expressed in terms of the isotropic flat-
ness.
As an example of an exactly Gaussian concentration pro-
file, consider Eq. 2 when the velocity field is independent
of space. The velocity statistics are then fully specified by
the covariance
u
t
u
t
1
d
u
2
␣␤
R
t t
.
We call R(t) the time correlation function of the velocity;
note it is symmetric about t 0, with R(0) 1. In this rather
unusual example R must be independent of spatial arguments
since the velocity depends only on time; note that in general
R is defined through Eq. 12 below. The time correlation is
usually assumed to decay to zero as t increases, with a char-
acteristic decay time
called the correlation time. The solu-
tion for the average concentration can then be shown to be
precisely 7 with variance given by
2
t
2u
2
0
t
0
t
1
R
t
1
t
2
dt
1
dt
2
. 9
As 7 is an exact Gaussian form, its flatness is (2 d)/d for
all time. In the following we examine how weak space de-
pendence in the velocity field results in a concentration dis-
tribution with flatness less than (2 d)/d, with maximum
deviation near t 5
. This non-Gaussian flatness is not
present in the limit of vanishing correlation time
0, and
so is not seen in models with white-noise in time velocity
fields.
3
III. ASYMPTOTIC SERIES EXPANSION
A. Series expansion
We begin by Fourier-transforming all space-dependent
variables such as
k,t
x,t
e
ik"x
dx. 10
Henceforth only such Fourier-transformed variables are em-
ployed, so the same symbol is used as in physical space. For
an isotropic, stationary, and incompressible velocity field in d
space dimensions, the covariance is given by
u
k,t
u
p,t
k p
Q
␣␤
k,t t
, 11
with
Q
␣␤
k,t
E
k
R
t,k
2
d 1
k
d 1
␣␤
k
k
k
2
. 12
Here E(k) is the usual energy spectrum and R(t,k) is the
time correlation function of the velocity. The velocity corre-
lation time
may be defined by
3547Phys. Fluids, Vol. 15, No. 11, November 2003 Flatness of tracer clouds
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0
R
t,k
dt, 13
and k
0
is chosen to be the wavenumber where the spectrum
E(k) has its peak. The r.m.s. velocity u is defined by
u
2
d 1
d
0
E
k
dk. 14
We seek to solve Eq. 2 when the parameter
defined in
terms of
, k
0
and u by Eq. 1兲兴 is significantly smaller than
unity.
Equation 2 is transformed to
t
k,t
i
dpk"u
p,t
k p,t
0,
15
k,0
1,
which may be recast as an integral equation
k,t
1 i
0
t
dt
1
dpk"u
p,t
1
k p,t
1
. 16
We seek a formal solution of 16 by iteration
0
1,
1
1 i
0
t
dt
1
dpk"u
p,t
1
0
k p,t
1
1 i
0
t
dt
1
dpk"u
p,t
1
,
17
2
1 i
0
t
dt
1
dpk"u
p,t
1
1
k p,t
1
1 i
0
t
dt
1
dpk"u
p,t
1
0
t
dt
1
0
t
1
dt
2
dp
dqk"u
p,t
1
k p
u
q,t
2
.
]
We thus formally construct an infinite series solution to Eq.
15, involving multiple integrals over wavevectors and time.
The usefulness of this approach lies in the fact that each term
in the infinite series is stochastic only through the appearance
of multiple velocity terms, and so the series may be averaged
term-by-term to yield a series expansion for
of the
form
k,t
q
0
q
1
2
q
2
3
q
3
¯, 18
where
is a bookkeeping parameter whose power equals the
number of velocity terms in the corresponding integral, q
n
represents the multiple wavevector and time integrals whose
integrands depend on the velocity field, and q
0
1. For a
Gaussian velocity field, all even moments may be expressed
in terms of the covariance 11, and all odd moments are
zero:
q
1
q
3
q
5
¯ 0.
Thus q
2
, for instance, is given by
q
2
⫽⫺
0
t
dt
1
0
t
1
dt
2
dp
dq
k"u
p,t
1
k p
u
q,t
2
⫽⫺
0
t
dt
1
0
t
1
dt
2
dpk"Q
p,t
1
t
2
k p
⫽⫺
0
t
dt
1
0
t
1
dt
2
dpk"Q
p,t
1
t
2
k, 19
where we have used 11 and the incompressibility of the
velocity field. Contributions to q
4
come from the average of
four velocity terms, which factors to yield
q
4
0
t
dt
1
0
t
1
dt
2
0
t
2
dt
3
0
t
3
dt
4
dp
dq
k"Q
p,t
1
t
2
k
兴关
k"Q
q,t
3
t
4
k
0
t
dt
1
0
t
1
dt
2
0
t
2
dt
3
0
t
3
dt
4
dp
dq
k"Q
p,t
1
t
3
k q
兴关共
k p
Q
q,t
2
t
4
k
0
t
dt
1
0
t
1
dt
2
0
t
2
dt
3
0
t
3
dt
4
dp
dq
k"Q
p,t
1
t
4
k
兴关共
k p
Q
q,t
2
t
3
k p
. 20
3548 Phys. Fluids, Vol. 15, No. 11, November 2003 J. P. Gleeson and D. I. Pullin
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The factorization of averages of the Gaussian field into prod-
ucts of velocity covariances leads to a sum of (2n)!/2
n
n!
terms contributing to q
2n
.
We remark here that the various terms may be accounted
for using a diagram expansion method, such as is commonly
employed in perturbation expansions over Gaussian fields.
9
First, define the diagrams of order n to be 2n-polygons with
dotted lines joining pairs of vertices. For example, the dia-
gram of order 1 representing Eq. 19 is shown in Fig. 1,
with the three diagrams of order 2 representing q
4
) in Fig.
2. The expressions for the q
2n
may be recovered from the
diagrams of order n by applying the following diagram rules.
Consider the middle diagram of Fig. 2, which represents the
second term on the right hand side of 20:
0
t
dt
1
0
t
1
dt
2
0
t
2
dt
3
0
t
3
dt
4
dp
dq
关共
k p
Q
p,t
1
t
3
k q
兴关共
k p q
Q
q,t
2
t
4
k
. 21
Observe that 21 may be deduced from the diagram by ap-
plying the following rules:
1 Vertex labels are the time integration variables.
2 The wavevector integration variables are the
wavevectors labeling the internal dotted lines; these integrals
are over all wavevector space.
3 The vector sum of wavevectors at each vertex is
zero, except for the first vertex labeled t
1
) which has sum
k, and the final vertex which has sum k.
To compose the integrand, we multiply the factors result-
ing from each of the following rules:
4 For each internal dotted line, consider the start and
end vertices. In the diagram example above, for the internal
dotted line labeled p, the start vertex is labeled t
1
and the end
vertex is labeled t
3
. Both the start and the end vertex have
solid lines emanating from them; suppose the wavevector
labels on these lines are a and b, respectively. Then the fac-
tor we seek is a"Q(p,t
s
t
e
)b where p is the dotted line
label and t
s
and t
e
are the start and end vertex labels. If the
end vertex is the last vertex, then let bk. In the example,
akp and bkq, so that the factor is (k p)
Q(p,t
1
t
3
)(kq). By applying this rule again to the sec-
ond dotted line, we find another factor of (k p q)
Q(q,t
2
t
4
)k. Further simplification may be possible due
to incompressibility.
These rules form an algorithm for finding the q
2n
terms
in the iteration expansion of and so may be implemented
using a symbolic manipulation program like
MATHEMATICA.
B. Renormalization
The behavior of each diagram as time increases deter-
mines the quality of the approximation to resulting from
truncating the infinite series. If the time correlation function
R(t,k) decays sufficiently quickly e.g., exponentially to
zero as t , it can be shown that connected diagrams, i.e.,
those which cannot be split into two separate parts by cutting
one solid line, grow linearly in time and so their contribution
to
/
t remains bounded as t . On the other hand, un-
connected diagrams such as the first term on the right hand
side of 20 grow faster than linearly, and their contribution
to
/
t is unbounded. With a view to renormalizing the
infinite series 18 to eliminate these secular effects, we con-
sider simple equivalent equations describing the evolution of
the PDF . For example, the functional-derivative closure
FDC method advanced in Ref. 8 leads to an integrodiffer-
ential equation for
t
0
t
K
1
k,s
k,s
ds, 22
and an asymptotic expansion is derived for the kernel K
1
.
However, following the method of cumulant expansion,
9,10
we suggest that a simpler ansatz
t
K
k,t
k,t
, 23
is equally as effective a renormalization, and indeed gener-
ates the same results as the FDC method with a significant
reduction in the complexity of algebraic manipulations. Not-
ing that the solution to 23 satisfying the initial condition is
k,t
exp
0
t
K
k,T
dT
, 24
it remains only to find an expression for K as a cumulant
expansion.
We seek an expansion for K in even powers of
K K
0
2
K
2
4
K
4
¯, 25
and utilize this and 18 into 23 to match coefficients of
powers of
term-by-term
2
q
2
t
4
q
4
t
¯
K
0
2
K
2
4
K
4
¯
q
0
2
q
2
4
q
4
¯
,
26
FIG. 1. Diagram of order 1.
FIG. 2. The three diagrams of order 2.
3549Phys. Fluids, Vol. 15, No. 11, November 2003 Flatness of tracer clouds
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yielding the K
n
in terms of the known q
n
K
0
0, K
2
1
q
0
q
2
t
,
27
K
4
1
q
0
q
4
t
q
2
q
0
q
2
t
.
]
Thus, for example, K
2
is found from 19 to be
K
2
⫽⫺
0
t
dt
2
dpk"Q
p,t t
2
k, 28
and again, each K
n
may be calculated using symbolic ma-
nipulation computer packages. Moreover, it is found that the
undesirable growth of terms as t noted above is not
present in the expansion for K, thus allowing us to use 24
as an approximation to over all time.
C. Flatness of the PDF
Having found an expansion for the probability density
function of tracers 24, it remains only to use this to calcu-
late the flatness of the distribution, according to Eq. 6.In
terms of the Fourier transformed variables, the moments
such as 5 may be written as
r
4
4
k
k
k
k
k,t
k 0
, 29
and for an isotropic distribution i.e., depending only on
magnitude k of k, independent of orientation in d dimen-
sions this reduces to
r
4
d
2 d
3
4
k
4
k,t
k 0
. 30
Similarly, the isotropic second moment is
r
2
⫽⫺d
2
k
2
k,t
k 0
,
and so the flatness 6 is
f
t
2 d
3d
4
k
4
k 0
2
k
2
k 0
2
, 31
which is written in terms of K using 24
f
t
2 d
3d
3
4
k
4
0
t
K
k,T
dT
k 0
2
k
2
0
t
K
k,T
dT
k 0
2
. 32
Using the expansion 25 of K derived above, it is straight-
forward to calculate the derivatives in 32 term-by-term; for
convenience we introduce the notation
D
n
⫽⫺
2
k
2
0
t
K
n
k,T
dT
k 0
,
33
F
n
4
k
4
0
t
K
n
k,T
dT
k 0
,
and so 32 becomes
f
t
2 d
3d
3
n 1
2n
F
2n
t
n 1
2n
D
2n
t
兲兲
2
. 34
Noting that F
2
F
4
0, we list in Appendix A the formulas
for F
6
and D
2
to D
6
, having performed all angular integrals,
and so reducing the expressions to multiple integrals over
time and wavenumbers.
IV. TIME CORRELATION FUNCTIONS
The calculation of the tracer flatness for a given pertur-
bation parameter
has been reduced to the evaluation of the
quantities D
n
and F
n
as in Eq. 34. In this section some
simple time correlation functions are chosen to demonstrate
the lowest-order perturbation results. Pade
´
approximants are
employed in Appendix B to support the claim that the results
presented here are qualitatively correct for noninfinitesimal
.
A. Exponential time correlation
To simplify the analysis we take the time correlation
function to have the following form:
R
t,k
e
t
, 35
where
1
is the inverse of the velocity correlation time.
This is a rather unrealistic approximation to the time corre-
lation of turbulent velocity fields, chiefly because it is not
differentiable at t 0, and also due to its lack of dependence
on the wavenumber k see Refs. 8 and 11. However, it re-
sults in a number of simplifications of our analysis which
enable the structure of the expansion to be clearly shown; we
note further that numerical computation by quadrature is al-
ways possible in the general case. Such a quadrature compu-
tation is performed for a correlation function which is
smooth at t0 in the next section, and the flatness behaves
similarly to the analytical results derived here.
With R independent of wavenumber, the integrals over p,
q, and r reduce to moments of the energy spectrum, for
which we introduce the notation
m
i
d 1
d
0
k
i
E
k
dk. 36
The factor of (d 1)/d ensures the simple identification m
0
u
2
. Note that if R is wavenumber-dependent, then the full
quadrature expressions given in Appendix A must be evalu-
ated, whereas assuming R to depend only on the time differ-
ence allows us to evaluate all wavenumber integrals in terms
of the moments 36 of the energy spectrum.
3550 Phys. Fluids, Vol. 15, No. 11, November 2003 J. P. Gleeson and D. I. Pullin
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Citations
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Journal ArticleDOI
TL;DR: Analytical predictions for the dependence of the angle between the direction of motion and the external force on a number of model parameters for periodic as well as random surfaces are provided.
Abstract: There has been a recent revolution in the ability to manipulate micrometer-sized objects on surfaces patterned by traps or obstacles of controllable configurations and shapes. One application of this technology is to separate particles driven across such a surface by an external force according to some particle characteristic such as size or index of refraction. The surface features cause the trajectories of particles driven across the surface to deviate from the direction of the force by an amount that depends on the particular characteristic, thus leading to sorting. While models of this behavior have provided a good understanding of these observations, the solutions have so far been primarily numerical. In this paper we provide analytic predictions for the dependence of the angle between the direction of motion and the external force on a number of model parameters for periodic as well as random surfaces. We test these predictions against exact numerical simulations.

23 citations


References
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01 Jan 1981
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7,842 citations


01 Jan 1992
Abstract: Preface to the first edition. Preface to the second edition. Abbreviated references. I. Stochastic variables. II. Random events. III. Stochastic processes. IV. Markov processes. V. The master equation. VI. One-step processes. VII. Chemical reactions. VIII. The Fokker-Planck equation. IX. The Langevin approach. X. The expansion of the master equation. XI. The diffusion type. XII. First-passage problems. XIII. Unstable systems. XIV. Fluctuations in continuous systems. XV. The statistics of jump events. XVI. Stochastic differential equations. XVII. Stochastic behavior of quantum systems.

6,887 citations


Book
01 Jan 1973
TL;DR: This website becomes a very available place to look for countless perturbation methods sources and sources about the books from countries in the world are provided.
Abstract: Following your need to always fulfil the inspiration to obtain everybody is now simple. Connecting to the internet is one of the short cuts to do. There are so many sources that offer and connect us to other world condition. As one of the products to see in internet, this website becomes a very available place to look for countless perturbation methods sources. Yeah, sources about the books from countries in the world are provided.

5,321 citations



Journal ArticleDOI

2,537 citations


"Flatness of tracer density profile ..." refers background in this paper

  • ...concentration does not have an exact Gaussian shape for all times [ 1 , 5].The deviation from a Gaussian...

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  • ...The pioneering work of Taylor [ 1 ], [2] on dispersion problems in turbulent flows has led to the widespread...

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  • ..., F [1, 1 ](t )= F 6 + α2 F...

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  • ...� D[0,1] � 2 , f [1,1] = F [ 1 ,1] � D[1,1] � 2 ,...

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  • ...� D[0,0] � 2 , f [0,1] = F [0, 1 ] � D[0,1] � 2 ,...

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Metrics
No. of citations received by the Paper in previous years
YearCitations
20061