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

共Received 25 November 2002; accepted 18 August 2003; published 1 October 2003兲

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 共kurtosis兲 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. 关DOI: 10.1063/1.1616558兴

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 共PDF兲 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 共i.e., multivari-

ate normal兲 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 共DIA兲. 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 共with discontinuous velocities兲 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 共sub-Gaussian ﬂatness兲 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

, 共1兲

where u is the root mean square velocity,

is the velocity

correlation time, and k

0

is a characteristic wavenumber of

the energy spectrum 关see Eqs. 共13兲 and 共14兲 for full deﬁni-

tions兴. 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 共

→0兲,

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

␣

共Sec. III兲, 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

⫹ ⵜ•

共

u

兲

⫽ 0,

共

x,0

兲

⫽

␦

共

x

兲

. 共2兲

a兲

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

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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 共or ‘‘concentration’’兲

⌰⫽

具

共

x,t

兲

典

. 共3兲

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 共multivariate normal兲 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 关in this paper we use the

isotropic dispersion as deﬁned in 共4兲, which is three times the

one-dimensional dispersion used in Ref. 8兴

D

共

t

兲

⫽

具

r

2

典

⫽

冕

x

␣

x

␣

⌰

共

x,t

兲

dx, 共4兲

with the integral being over all of space, and repeated indices

summed from 1 to d, the number of space dimensions (d

⫽ 2or3兲. 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

具

r

2

典

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

共

t

兲

⫽

具

r

4

典

具

r

2

典

2

,

with

具

r

4

典

deﬁned similarly to 共4兲

具

r

4

典

⫽

冕

x

␣

x

␣

x

x

⌰

共

x,t

兲

dx. 共5兲

The ﬂatness of the distribution ⌰(x,t) is deﬁned as

f

共

t

兲

⫽

具

r

4

典

具

r

2

典

2

⫽

兰

x

␣

x

␣

x

x

⌰

共

x,t

兲

dx

共

兰

x

␣

x

␣

⌰

共

x,t

兲

dx

兲

2

. 共6兲

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 共6兲 for the general isotropic distribution with

zero mean and variance

2

(t):

⌰

共

x,t

兲

⫽

1

共

2

2

共

t

兲兲

d/2

exp

冉

⫺

x

␣

x

␣

2

2

共

t

兲

冊

, 共7兲

to obtain

具

r

2

典

⫽ d

2

and

具

r

4

典

⫽ d(2⫹ d)

4

. Note that for a

one-dimensional Gaussian distribution the corresponding

ﬂatness

具

y

4

典

/

具

y

2

典

2

equals 3: It can readily be demonstrated

for an isotropic distribution in d space dimensions that

具

r

2

典

⫽ d

具

y

2

典

and

具

r

4

典

⫽ d

具

y

4

典

⫹ d(d⫺ 1)

具

y

2

典

2

. Thus the isotropic

ﬂatness and the one-dimensional ﬂatness are related by the

equation

具

r

4

典

具

r

2

典

2

⫽

1

d

具

y

4

典

具

y

2

典

2

⫹

d⫺ 1

d

. 共8兲

All our results are expressed in terms of the isotropic ﬂat-

ness.

As an example of an exactly Gaussian concentration pro-

ﬁle, consider Eq. 共2兲 when the velocity ﬁeld is independent

of space. The velocity statistics are then fully speciﬁed by

the covariance

具

u

␣

共

t

兲

u

共

t

⬘

兲

典

⫽

1

d

u

2

␦

␣

R

共

t⫺ t

⬘

兲

.

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. 共12兲 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 共7兲 with variance given by

2

共

t

兲

⫽ 2u

2

冕

0

t

冕

0

t

1

R

共

t

1

⫺ t

2

兲

dt

1

dt

2

. 共9兲

As 共7兲 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

共

k,t

兲

⫽

冕

共

x,t

兲

e

⫺ ik"x

dx. 共10兲

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

具

u

␣

共

k,t

兲

u

共

p,t

⬘

兲

典

⫽

␦

共

k⫹ p

兲

Q

␣

共

k,t⫺ t

⬘

兲

, 共11兲

with

Q

␣

共

k,t

兲

⫽

E

共

k

兲

R

共

t,k

兲

2

共

d⫺ 1

兲

k

d⫺ 1

冉

␦

␣

⫺

k

␣

k

k

2

冊

. 共12兲

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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⫽

冕

0

⬁

R

共

t,k

兲

dt, 共13兲

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

冕

0

⬁

E

共

k

兲

dk. 共14兲

We seek to solve Eq. 共2兲 when the parameter

␣

关deﬁned in

terms of

, k

0

and u by Eq. 共1兲兴 is signiﬁcantly smaller than

unity.

Equation 共2兲 is transformed to

t

共

k,t

兲

⫹ i

冕

dpk"u

共

p,t

兲

共

k⫺ p,t

兲

⫽ 0,

共15兲

共

k,0

兲

⫽ 1,

which may be recast as an integral equation

共

k,t

兲

⫽ 1⫺ i

冕

0

t

dt

1

冕

dpk"u

共

p,t

1

兲

共

k⫺ p,t

1

兲

. 共16兲

We seek a formal solution of 共16兲 by iteration

共

0

兲

⫽ 1,

共

1

兲

⫽ 1⫺ i

冕

0

t

dt

1

冕

dpk"u

共

p,t

1

兲

共

0

兲

共

k⫺ p,t

1

兲

⫽ 1⫺ i

冕

0

t

dt

1

冕

dpk"u

共

p,t

1

兲

,

共17兲

共

2

兲

⫽ 1⫺ i

冕

0

t

dt

1

冕

dpk"u

共

p,t

1

兲

共

1

兲

共

k⫺ p,t

1

兲

⫽ 1⫺ i

冕

0

t

dt

1

冕

dpk"u

共

p,t

1

兲

⫺

冕

0

t

dt

1

冕

0

t

1

dt

2

⫻

冕

dp

冕

dqk"u

共

p,t

1

兲

共

k⫺ p

兲

•u

共

q,t

2

兲

.

]

We thus formally construct an inﬁnite series solution to Eq.

共15兲, 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 ⌰⫽具

典 of the

form

⌰

共

k,t

兲

⫽ q

0

⫹

␣

q

1

⫹

␣

2

q

2

⫹

␣

3

q

3

⫹ ¯, 共18兲

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 共11兲, and all odd moments are

zero:

q

1

⫽ q

3

⫽ q

5

⫽ ¯⫽ 0.

Thus q

2

, for instance, is given by

q

2

⫽⫺

冕

0

t

dt

1

冕

0

t

1

dt

2

冕

dp

冕

dq

具

k"u

共

p,t

1

兲

共

k⫺ p

兲

•u

共

q,t

2

兲

典

⫽⫺

冕

0

t

dt

1

冕

0

t

1

dt

2

冕

dpk"Q

共

p,t

1

⫺ t

2

兲

•

共

k⫺ p

兲

⫽⫺

冕

0

t

dt

1

冕

0

t

1

dt

2

冕

dpk"Q

共

p,t

1

⫺ t

2

兲

•k, 共19兲

where we have used 共11兲 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

⫽

冕

0

t

dt

1

冕

0

t

1

dt

2

冕

0

t

2

dt

3

冕

0

t

3

dt

4

冕

dp

冕

dq

关

k"Q

共

p,t

1

⫺ t

2

兲

•k

兴关

k"Q

共

q,t

3

⫺ t

4

兲

•k

兴

⫹

冕

0

t

dt

1

冕

0

t

1

dt

2

冕

0

t

2

dt

3

冕

0

t

3

dt

4

冕

dp

冕

dq

关

k"Q

共

p,t

1

⫺ t

3

兲

•

共

k⫺ q

兲

兴关共

k⫺ p

兲

•Q

共

q,t

2

⫺ t

4

兲

•k

兴

⫹

冕

0

t

dt

1

冕

0

t

1

dt

2

冕

0

t

2

dt

3

冕

0

t

3

dt

4

冕

dp

冕

dq

关

k"Q

共

p,t

1

⫺ t

4

兲

•k

兴关共

k⫺ p

兲

•Q

共

q,t

2

⫺ t

3

兲

•

共

k⫺ p

兲

兴

. 共20兲

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. 共19兲 is shown in Fig. 1,

with the three diagrams of order 2 共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 共20兲:

冕

0

t

dt

1

冕

0

t

1

dt

2

冕

0

t

2

dt

3

冕

0

t

3

dt

4

冕

dp

冕

dq

关共

k⫺ p

兲

•Q

共

p,t

1

⫺ t

3

兲

•

共

k⫺ q

兲

兴关共

k⫺ p⫺ q

兲

•Q

共

q,t

2

⫺ t

4

兲

•k

兴

. 共21兲

Observe that 共21兲 may be deduced from the diagram by ap-

plying the following rules:

共1兲 Vertex labels are the time integration variables.

共2兲 The wavevector integration variables are the

wavevectors labeling the internal dotted lines; these integrals

are over all wavevector space.

共3兲 The vector sum of wavevectors at each vertex is

zero, except for the ﬁrst vertex 共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:

共4兲 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. 共If the

end vertex is the last vertex, then let b⫽k.兲 In the example,

a⫽k⫺p and b⫽k⫺q, so that the factor is ⫺ (k⫺ p)

•Q(p,t

1

⫺ t

3

)•(k⫺q). 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 共e.g., exponentially兲 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 共20兲 grow faster than linearly, and their contribution

to

⌰/

t is unbounded. With a view to renormalizing the

inﬁnite series 共18兲 to eliminate these secular effects, we con-

sider simple equivalent equations describing the evolution of

the PDF ⌰. For example, the functional-derivative closure

共FDC兲 method advanced in Ref. 8 leads to an integrodiffer-

ential equation for ⌰

⌰

t

⫽

冕

0

t

K

1

共

k,s

兲

⌰

共

k,s

兲

ds, 共22兲

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

共

k,t

兲

⌰

共

k,t

兲

, 共23兲

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 共23兲 satisfying the initial condition is

⌰

共

k,t

兲

⫽ exp

冋

冕

0

t

K

共

k,T

兲

dT

册

, 共24兲

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

⫹ ¯, 共25兲

and utilize this and 共18兲 into 共23兲 to match coefﬁcients of

powers of

␣

term-by-term

␣

2

q

2

t

⫹

␣

4

q

4

t

⫹ ¯⫽

共

K

0

⫹

␣

2

K

2

⫹

␣

4

K

4

⫹ ¯

兲

⫻

共

q

0

⫹

␣

2

q

2

⫹

␣

4

q

4

⫹ ¯

兲

,

共26兲

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

yielding the K

n

in terms of the known q

n

K

0

⫽ 0, K

2

⫽

1

q

0

q

2

t

,

共27兲

K

4

⫽

1

q

0

冋

q

4

t

⫺

q

2

q

0

q

2

t

册

.

]

Thus, for example, K

2

is found from 共19兲 to be

K

2

⫽⫺

冕

0

t

dt

2

冕

dpk"Q

共

p,t⫺ t

2

兲

•k, 共28兲

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 共24兲

as an approximation to ⌰ over all time.

C. Flatness of the ⌰ PDF

Having found an expansion for the probability density

function of tracers 共24兲, it remains only to use this to calcu-

late the ﬂatness of the distribution, according to Eq. 共6兲.In

terms of the Fourier transformed variables, the moments

such as 共5兲 may be written as

具

r

4

典

⫽

4

k

␣

k

␣

k

k

⌰

共

k,t

兲

兩

k⫽ 0

, 共29兲

and for an isotropic distribution 共i.e., ⌰ depending only on

magnitude k of k, independent of orientation兲 in d dimen-

sions this reduces to

具

r

4

典

⫽

d

共

2⫹ d

兲

3

4

k

4

⌰

共

k,t

兲

兩

k⫽ 0

. 共30兲

Similarly, the isotropic second moment is

具

r

2

典

⫽⫺d

2

k

2

⌰

共

k,t

兲

兩

k⫽ 0

,

and so the ﬂatness 共6兲 is

f

共

t

兲

⫽

2⫹ d

3d

4

⌰

k

4

冏

k⫽ 0

冉

⫺

2

⌰

k

2

冏

k⫽ 0

冊

2

, 共31兲

which is written in terms of K using 共24兲

f

共

t

兲

⫽

2⫹ d

3d

冋

3⫹

4

k

4

兰

0

t

K

共

k,T

兲

dT

兩

k⫽ 0

冉

⫺

2

k

2

兰

0

t

K

共

k,T

兲

dT

冏

k⫽ 0

冊

2

册

. 共32兲

Using the expansion 共25兲 of K derived above, it is straight-

forward to calculate the derivatives in 共32兲 term-by-term; for

convenience we introduce the notation

D

n

⫽⫺

2

k

2

冕

0

t

K

n

共

k,T

兲

dT

兩

k⫽ 0

,

共33兲

F

n

⫽

4

k

4

冕

0

t

K

n

共

k,T

兲

dT

兩

k⫽ 0

,

and so 共32兲 becomes

f

共

t

兲

⫽

2⫹ d

3d

冋

3⫹

兺

n⫽ 1

⬁

␣

2n

F

2n

共

t

兲

共

兺

n⫽ 1

⬁

␣

2n

D

2n

共

t

兲兲

2

册

. 共34兲

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. 共34兲. 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

共

t,k

兲

⫽ e

⫺

兩

t

兩

, 共35兲

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 共see Refs. 8 and 11兲. 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 t⫽0 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

冕

0

⬁

k

i

E

共

k

兲

dk. 共36兲

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 共36兲 of the energy spectrum.

3550 Phys. Fluids, Vol. 15, No. 11, November 2003 J. P. Gleeson and D. I. Pullin