Rev Chem Eng 26 (2010): 13–27
2010 by Walter de Gruyter • Berlin • New York. DOI 10.1515/REVCE.2010.004
2010/004
Article in press - uncorrected proof
Review
Diffusion of scalar concentration from localized sources in
turbulent flows
Roi Gurka1,*, Alex Liberzon2, Partha Sarathi3
and Paul J. Sullivan4
1 Department of Chemical Engineering, Ben-Gurion
University, Beer-Sheva, Israel, e-mail: gurka@bgu.ac.il
2 School of Mechanical Engineering, Tel-Aviv University,
Ramat-Aviv, Israel
3 Department of Civil Engineering, University of Western
Ontario, London, Canada
4 Department of Applied Mathematics, University of
Western Ontario, London, Canada
*Corresponding author
Abstract
This review paper discusses the dispersion of scalar released
from a localized source in turbulent flows. In particular, the
review emphasizes the distribution of scalar concentration in
conjunction to environmental and industrial applications.
Here, both general models of scalar concentration distribu-
tions, which appear in the literature and studies on specific
aspects of the problem, are presented. Diverse subjects relat-
ed to statistical descriptions of the scalar diffusion such as:
location, size and distribution of the released scalar are dis-
cussed. These are characterized through models of probabil-
ity density function of the scalar concentration and expected
mass fraction. Experimental methods such as PIV and PLIF
for measuring velocity and scalar concentration in turbulent
flows as well as examples of applications are summarized.
Keywords: diffusion; dispersion; modeling; PLIF; scalar
transport; turbulence.
Introduction
Mixing and transport of passive scalar in turbulent flows are
important processes occurring in many natural and engi-
neered environments. The study of dispersing contaminants
in turbulent flows is important for industry (e.g., chemical
mixing in reactors) and for the assessment of the hazards
that result from pollutants released into atmosphere or the
ocean (e.g., Figure 1).
Ruptured gas pipes, toxic spills, stack gas emissions and
sewage discharges are some common examples of practical
releases of contaminants that occur in many domestic and
industrial processes. In the case of a ruptured gas pipe one
is interested in evaluating the risk of exceeding a dangerous
concentration of toxic gas level. In the planning of industrial
plants, attention is on the distribution of airborne contami-
nants near a heavily populated area and in the study of sew-
age discharge in rivers, one is interested in minimizing the
high levels of effluents downstream. The assessment of the
potential hazards from an accidental spill of toxic contami-
nant is seldom simple because of the complex nature of tur-
bulent flows involved in the turbulent diffusion process. The
transport of scalar is crucial for animals too, such as bees,
moths, and crabs, among others, that use olfactory sense to
navigate towards food sources or finding mates.
Most of the environmental and industrial flows are in a
turbulent state of motion, and vary with time and space.
These features of turbulent flows play a dominant role during
the spread of contaminants (Shraiman and Siggia 2000,
Warhaft 2000).
In many practical problems, such as hazard assessment
and responding to accidents, one is concerned with contam-
inant at particular concentration levels. For example, the con-
taminant within the flammable limits is of interest when
dealing with combustion. When dealing with odor, one is
concerned with contaminant concentrations that are above a
noxious level, as it is in the example shown in Figure 2.
Instantaneous concentration values can be 10–1000 times
higher than the average. Therefore, it is often asked about
the risk involved in exceeding a dangerous level of contam-
inants. In this case, the use of a probability density function
for the contaminant concentration is applicable. Since the
concentrations of a contaminant at some position and time
in the turbulent flow are in general random variables, one
will be able to make probabilistic predictions. The probabil-
ity density function of concentration is very difficult to meas-
ure in turbulent flows due to the large number of realizations
required for estimating ensemble averages and is difficult to
predict due to the complexity of the equations that govern
its evolution. Therefore, there is a need to provide sufficient
knowledge to develop simplified models for predicting the
contaminant concentration levels due to a release of contam-
inant (see Figure 1), which would give the information need-
ed in hazard assessment, industrial planning and other
relevant problems. The use of probability density function
(PDF) for describing and predicting phenomena associated
with scalar transport in turbulent flows (such as combustion)
have been addressed by Pope (1979). Furthermore, PDFs are
essential to quantify hazards, such as toxicity, flammability
and malodor, associated with gases dispersing in the atmos-
phere or in a more recent case of the BP oil spill in The Gulf
of Mexico. However, theoretical analysis is limited by the

14 Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows
Article in press - uncorrected proof
Figure 1 Smoke released to the atmospheric boundary layer from
in an industrial zone. Reproduced from Library of Congress, Prints
and Photographs Division, Washington, D.C. 20540 USA, hdl.loc.
gov/loc.pnp/pp.print.
Figure 3 Conceptual framework for contaminant cloud dispersion.
Figure 2 (Left) Instantaneous concentration distribution in a plane of the centerline of a plume and (right) time-averaged concentration
distribution, reproduced from Roberts and Webster (2002, Figures 14 and 15).
ubiquitous closure problem and while this may be circum-
vented (Smyth 1999) through accurate numerical simulations
this is not feasible for most realistic flows and geometries as
yet. Therefore, there is considerable scope for simple mathe-
matical models based on the physical insight and available
experimental results.
It is noteworthy that this review is limited to applications
where the flow scales are weakly influenced by stratification.
For the cases containing larger scales where stratifications
play a key role, such as atmospheric stably stratified, con-
vective boundary layers, etc, the reader is referred to the
relevant works in the field, for example, Elperin et al. (1996),
Sofiev et al. (2009), among others.
Diffusion of scalar in turbulent flow
Here the problem of turbulent diffusion is discussed, with
particular emphasis on diffusion of scalar fields released
from a concentrated source. A schematic view of the problem
is outlined in Figure 3. A distinction will be drawn between
the turbulent motions that contribute to the location wabsolute
dispersion, denoted as x(t)x, size of the cloud wrelative dis-
persion, denoted as s(t)x and state (concentration field dis-
tribution) of concentration fields (shown as color levels in
the inset of Figure 3).
In addition, some discussion will be devoted to the high-
concentration tails of the probability density function. This
range of (high) concentration values, with possibly low prob-
ability of occurrence, can be of inordinately high conse-
quence, as discussed in the Introduction. Specifically, the
modeling approaches reviewed here, use lower-order
moments of the concentration distributions to approximate
the PDF and expected mass fraction (EMF) functions. Fur-
thermore, several closure schemes aiming at relatively gen-
eral solutions for the equation governing the evolution of
moments of the concentration PDF, and which has received
some limited but encouraging qualitative comparison with
data, are reviewed. The spectral approach of passive scalar
(see e.g., Celani et al. 2005, Danaila and Antonia 2009) in
a given turbulent flow based on the Kolmogorov theory is
not in the scope of this review.
There is an inevitable problem of adequate spatial and
temporal resolution in measuring concentration values in a
turbulent flow. Advanced experimental all-field optical imag-
ing methods of particle image velocimetry (PIV) and laser-
induced fluorescence (LIF) are reviewed with a focus on the
measurements of concentration fields in turbulent flows.

Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows 15
Article in press - uncorrected proof
Diffusion of scalar in shear flows
Common examples of confined shear flows are found in pipe
and channel flows. Other types are free shear flow. Both
exhibit relatively strong velocity gradients in the direction
normal to the streamwise direction. For confined shear flow,
the boundaries usually will be solid while for free shear flow,
such as air-sea interaction, mixing layers, etc. the boundaries
will not be solid. One can also consider the important case
of the natural atmospheric boundary layer, the log layer of
mean velocity (typically 10–100 m above the ground in the
atmosphere). Here everything is determined by the friction
velocity and time t (Batchelor 1964). Batchelor extended this
for the case of release from an elevated source. The exten-
sion was: ‘‘the statistical properties of the velocity of a
marked particle at time t after release at height a are the
same as those of a particle released at the ground at the
instant – t1, provided , where t1 is expected to be of thet4t1
order of magnitude of the time scale of the turbulence at
height ‘a’, that is, of order of a/u.’’ (Batchelor 1964). Thus,
the mean concentration of a small cloud released in the con-
stant stress layer is found from the superposition of statistics
of fluid elements released at appropriate earlier times and
following becomes self-similar (Labropulu and Sulli-t4t1
van 1995). The spread of a cloud of contaminant in atmos-
pheric conditions cannot be accurately described by a
constant diffusivity, and therefore, use of a variable diffusi-
vity is necessary (Saffman 1962). Elperin et al. (2000) gave
a more generalized description, using mean field equations.
There are other factors that are responsible for making the
spread of contaminant complex in an environmental flow,
like the time-dependent properties of the flow-field and also
the presence of a solid boundary leading to inhomogeneities
in the flow-field. These are evident from changing wind
speeds during different times of the day and changing gra-
dient of velocity that occur in the boundary layer. The
unsteady feature, which is likely to be the most dominant
feature, can complicate the interpretation of the experimental
data when the sample time is larger than the time scale of a
typical eddy. The turbulent length scales appear to have a
functional dependence on the vertical direction. In the region
away from the solid boundary, the cloud of contaminant
grows in size because of the large scales of motion, which
occur much faster than would be the case without a solid
boundary and continuously affect it. Near the solid boundary,
the length scales of the turbulent motion are very small rel-
ative to the length scales in the region away from the bound-
ary. In a flow over a smooth wall, the small turbulent length
scales adjacent to the wall cause a buildup of contaminants
near the solid boundary (Csanady 1973). For real fluids, the
no-slip boundary condition at the wall is likely to compound
this buildup. Townsend (1976) showed that the flow over a
rough wall has a structure quite similar to the flow pattern
over a smooth wall at heights greater than the characteristic
scale of the rough elements, i.e. roughness height. Hence, it
is possible to use the properties in smooth flows to describe
the spread of contaminants in rough wall flows. However,
the thin layer adjacent to the wall is a critical region for
evolution of a cloud of contaminant, which requires a solu-
tion scheme that is capable of incorporating semi-empirical
information, since good theoretical models do not exit for
this complex flow region. For a flow over a rough wall there
will be a circulating region in which the fluid particles are
trapped, and for a smooth wall the viscous dominated region
adjacent to the wall delays the asymptotic stage of dispersion
(Taylor 1954). The presence of an absorbent boundary can
also have a significant effect on the longitudinal dispersion
coefficient (Smith 1983). Similar behavior will occur for oth-
er types of flows where shear is dominant.
Statistical description of scalar diffusion
Statistical description of scalar concentration field is based
on the concentration G(x, t), in units of mass per unit vol-
ume, of a miscible contaminant fluid at a position located
by vector x at time t, within the turbulent host fluid. The
concentration of dispersing scalar obeys the advection-dif-
fusion equation:
≠G 2qU=Gsk= G (1)
≠t
where k is the molecular diffusivity.
Concentration G(x, t), is a random variable in turbulent
flows such that the only reproducible quantity is an ensemble
average. Usually, for steady flows, ensemble averages are
approximated experimentally by taking time averages. The
important case of the sudden release of contaminant resulting
in a contaminant cloud requires a number of repeated releas-
es to approximate the ensemble average.
In assessing the hazards due to an accidental release, the
greatest danger from the exposure of hazardous gases dis-
persing in the atmosphere often comes from the highest con-
centrations within the cloud of the contaminant gas (Davies
1989, Griffiths 1991). In order to evaluate, the risk involved
in exceeding a dangerous level of contaminants, it is natural
to investigate the probability density function, p(u;x,t), for
the contaminant concentration. The probability density func-
tion is defined as
µ ∂p u;x,t dusprob uFG x,t -uqdu . (2)Ž . Ž .
The equation for the evolution of the PDF can be derived
from the convective diffusion equation (Chatwin 1990),
2≠p ≠ 22Z Zq=ØŽpNu GsuM sk= p-k ŽpN =G GsuM (3). .Ž .2≠t ≠u
where denotes the conditional expected value. The sec-NØZ ØM
ond term on the right hand side of eq. (3) is the so-called
‘‘small scale mixing term’’ (Pope 1994) which depends on
the joint PDF of G and =G. This term represents the effect
of molecular diffusion in reducing the maximum concentra-
tion from the concentration at the source. This sometimes,
plays a role in determining the PDF shape (Munro et al.
2001).
In general, knowledge of all of the moments mn defined
as

16 Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows
Article in press - uncorrected proof
`
n nm x,t sNG x,t Ms u p u;x,t du (4)Ž . Ž . Ž .n |
o
is equivalent to the PDF. It is also expected to obtain a rea-
sonable approximation to the probability density function
from the inversion of joint set of low order moments (e.g.
Derksen and Sullivan 1990). The equation for the moments
is
≠mnq1 2nq1 n-1 2q=ØNuG Ms-n nq1 kNG =G Mqk= m . (5)Ž . Ž . nq1≠t
The diffusive term, k=2mnq1, is generally assumed neglec-
ted in respect to the convective term . It wouldnq1=ØNu9G M
appear that the equations for the evolution of the moments
are less complex than those for the PDF; however, it still
remains intractable due to the lack of closure of the two
terms.
The only agency to reduce concentration is the molecular
diffusivity k. Without molecular diffusivity the only outcome
at any position x and time t is either 0 or, for a uniform
source, the initial uniform concentration uo. In this case, the
PDF is
p u;x,t s 1-p x,t d u qp x,t d u-u (6)Ž . Ž Ž .. Ž . Ž . Ž .o
where p(x,t) is the probability of being in marked fluid. That
is C(x,t)sp(x,t)uo, where we denote the mean concentration,
m1(x,t), as C(x,t) hereafter as a matter of convention and
convenience, and all the moments are
nm x,t su C x,t . (7)Ž . Ž .nq1 o
The total moments, when ks0 are conserved
n nm x,t dxsu C x,t dxsu Q (8)Ž . Ž .nq1 o o| |
while the volume integral is taken over all space.
Qs C x,t dxŽ .|
a.s.
is the total release mass in the cloud.
When a ‘‘blob’’ of contaminant is released in a turbulent
flow the turbulent convective motions stretch the contami-
nant into ever thinning sheets and strands until the thinning
(due to stretching) is balanced by thickening due to molec-
ular diffusivity. This occurs at the conduction cut-off length
where n is the kinematic viscosity and ´ is1y42h s nk /´Ž .B
the rate of turbulent energy dissipation per unit mass. hB is
of the order 10-3-10-5 m in most flows. Experimental evi-
dence of both Dahm et al. (1991) and Corriveau and Baines
(1993) show virtually all of the contaminant to be confined
within such sheets and strands. This fine scale texture is
manifest in fixed-point measurements where the ‘spiky’ con-
centration record is typically observed (see e.g., Mylne and
Mason 1991). These high concentration spikes, frequently
many standard deviations above the mean, lead to significant
measurement inaccuracy.
The mean concentration is relatively insensitive to either
molecular diffusivity or experimental resolution. In the equa-
tion for the mean concentration wns0 in (2.5)x, the last term
on the right-hand side, k=2mnq1, is small with respect to the
other terms. Therefore the mean concentration, although the
easiest to predict and to measure, does not reflect appropri-
ately the concentration reduction. Higher order moments are
extremely sensitive to both k and experimental resolution.
The instantaneous gradients that appear in the right-hand side
of the moment eq. (5) are large over the thin sheets and
strands. Coarser spatial and temporal resolutions limit the
extreme values of concentration near 0 and the maximum
concentration and the moments are increasingly reduced with
the moment order. For example, in a laboratory experiment
on the centerline of a contaminant jet, Sakai et al. (2008)
showed that the mean square value of concentration, when
the spatial resolution is improved over the normally used
values, to be increased by a factor of 2.
Given the complexities of contaminant diffusion in envi-
ronmental flows, which are to some extent intrinsically
unsteady and inhomogeneous, making approximations for
ensemble averages is problematic, and particularly so, in the
case of a contaminant cloud. It is worth noting that in the
well-controlled laboratory experiments of Hall et al. (1991),
where a tent full of contaminant was released in a boundary
layer only the mean concentration could be measured con-
fidently with 100 repeat releases.
In order to obtain a simple measure of the contaminant
concentration reduction, it is proposed to model or measure
the three basic parameters: location of the center-of-mass of
the concentrated field; the size of the field and the concen-
tration distribution within the field. Each of the three major
parameters depends on different (and sometimes independ-
ent) ranges of scales of turbulent motion and, hence, presents
different problems in determining approximations for ensem-
ble averages.
The location of the released concentrated field
The location of the concentration field at some time t fol-
lowing release at ts0, is defined by the location of the cen-
ter-of-mass . This is the so-called ‘one fluid particle’N Mx(t)
analysis (Batchelor 1949) that assumes that the initial
released field is made up of elemental fluid particles, which
do not interfere with each other or change shape during their
travel. Superposition of trajectories from all of the fluid par-
ticles from the initial concentration field are then used to
compile the probable arrival of a single particle at the loca-
tion (within an infinitesimally small volume element cen-
tered on) x at time t. It is noteworthy that the differential
equation governing the mean concentration weq. (5) with
ns0x is linear and as such admits a superposition of solu-
tions. A weighting is used to reflect the initial cloud con-
centration when it is non-uniform and the number of particles
located at a particular position, normalized by the conserved

Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows 17
Article in press - uncorrected proof
release mass, provides the mean concentration C(x,t) from
which the center-of-mass is derived asN Mx(t)
-1
Nx t MsQ xC x,t dv. (9)Ž . Ž .|
a.s.
Often symmetry in the turbulent flow structure will obvi-
ate the location of . For example, in homogeneous tur-N Mx(t)
bulence, statistics are determined from the repeated release
of one typical fluid particle irrespective of release position.
In another practically interesting application of contaminant
dispersion in atmospheric boundary layers the statistical
properties of the particles depend only on the friction veloc-
ity u* and time t by the Lagrangian similarity analysis of
Batchelor (1964), in the initial release is at the ground level
in the constant stress region (typically 10–100 m above the
ground), i.e., the three components of the center-of-mass are
given by
Ny t Ms0Ž .
dNzM
sbuUdt
B EdNx t M c zŽ . Ž .uU
C Fs ln
D Gdt k zo
(10)n
where x is the streamwise y is the lateral and z is the vertical
component of the displacement vector; b and c are universal
constants, which Batchelor estimated to be about 0.1–0.2; k
is von Karman constant (ks0.4), u* is the friction velocity;
and zo is the roughness height. Another special case is the
flow within a lengthwise uniform conduit such as a pipe or
channel (Lavertu and Mydlarski 2005) used ubiquitously in
industry. A fluid particle will sample all of the velocity var-
iation over the flow cross-section during its migration due to
the cross-stream components of turbulent motion (as shown
by Bakosi et al. 2007 among others). After a sufficient period
of time the particle will ‘forget’ its release position on the
flow cross-section and move downstream with the flow dis-
charge velocity whereN Mx(t) sUt
-1UsA u y,z dA (11)Ž .|
A
where A denotes area of the conduit cross-section.
The size of the concentrated field
The size of the concentration field is quantified by means of
its spatial variance. The classical approach of Taylor (1921)
in isotropic turbulence and the further extensions of Batche-
lor (1952) provide the right framework for this analysis. The
analysis is readily generalized to shear flows, even with the
inclusion of molecular diffusivity.
A purely statistical treatment of resolving diffusion from
a continuous source was first introduced by Taylor (1921).
His discussion includes a demonstration that the usual laws
of differentiation may be applied to the mean values of fluc-
tuating variables and their products. If X(t) and u9(t) are the
Lagrangian displacement and velocity respectively of a typ-
ical particle after time t , the ensemble average of the2N MX
mean square values of X(t), is found from
t
2dNX M dX
s2NX Ms2NXu9Ms2 Nu9 t u9 tqt Mdt. (12)Ž . Ž .|dt dt
0
For homogeneous and stationary turbulence, the average
properties are uniform in space and steady in time. Hence,
the velocity product may be replaced by , where2N Mu9 R(t)
Nu9 t u9 tqt MŽ . Ž .
R t s (13)Ž . 2
Nu9 M
is the Lagrangian autocorrelation coefficient, which results
in
t
2dNX M 2s2Nu9 M R t dt (14)Ž .|dt
0
and
T t
2 2
NX Ms2Nu9 M R t dtdt. (15)Ž .||
0 0
Here would represent the standard deviation of spa-1/22N MX
tial displacement of the particles at time T. Hence, the mean
square of the deviations of the particle is finally expressed
in terms of the mean square velocity of the particle and the
Lagrangian correlation coefficient between the velocity of
the particle at time t and that at time tqt. The correlation
should be unity when ts0, and is effectively 0 for large t.
Hence,
2 2 2
NX MsNu9 MT for small T (16)
T
B E
2 2
NX Ms2 R t dt Nu9 MT for large T, (17)Ž .
C F|
D G0
the integral in the brackets is also known as Lagrangian time
scale tL.
Batchelor (1949) showed that for the diffusion in homog-
enous turbulence at short time, when the correlation coeffi-
cient becomes unity, particle velocity fluctuations of all
frequencies contribute to the dispersion exactly as they do to
the turbulent energy. For longer times, the slower fluctuations
(larger scales) progressively dominate the dispersion; in
effect the high frequency components merely oscillate the

18 Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows
Article in press - uncorrected proof
Figure 4 Apparent eddy diffusivity as per equation (18). wAdapted
from Csanady (1973).x
position of the particle, whereas the low frequency compo-
nents tend to displace it in a more sustained way. One can
define an apparent diffusivity (K) at large time as one half
the spatial variance growth rate (Pasquill 1974),
t
21 dNX M 2K t s sNu9 M R t dt. (18)Ž . Ž .x |2 dt
0
The value of K, initially 0, increases with time, at first
approximately linearly and then more slowly, finally tending
to the constant value of eq. (18). This is shown in Figure 4.
The observed spread of smoke plumes in the atmosphere
over short distances show an increase in K with distance of
travel. Sullivan and Yip (1985) showed that for a continuous
source of contaminant, the time dependence was an impor-
tant feature in describing dispersion in the natural atmos-
pheric boundary layer where the use of even a spatially
dependent ‘eddy diffusivity’ was inadequate.
The main feature to be noted is that the largest scales of
turbulent motion make the dominant contribution to the var-
iance growth rate at all times. This is particularly relevant,
for example, in finding average values in field experiments
where the relevant length and time scales for averaging pur-
poses are many times those scales in the flow. It is also worth
noting that the variance growth rate only becomes a constant
(as in molecular diffusivity, where the constant is k when
the integral over the Lagrangian autocorrelation function
becomes constant).
The formulation can be readily generalized to shear flows
such as the flow in a channel or pipe. Here, though, the
Lagrangian particle displacement and velocities depend on
release position. The motion is treated relative to an axis
moving with the discharge velocity, i.e., the streamwise com-
ponent x9(t)sx(t)-Ut and u9(t)su(t)-U. The direct contribu-
tion in the lateral and longitudinal direction due to
fluctuating turbulent motions is comparable to u*d, where d
is the length scale of the cross-section, for example, the
diameter of a pipe. By far the largest contribution to variance
growth rate is in the streamwise direction and comes from
the interaction between the cross-stream mixing and the
mean velocity gradient. That is, the longitudinal growth rate
is reduced by cross-stream mixing from what would be the
case if released particles continue to travel with the release
mean velocity without lateral movement. Thus, neglecting
the direct contribution from longitudinal fluctuating motions,
for a particular release position on the cross-section, we have
t2dNx9 t MŽ .1
s Nu9 t u9 tqt Mdt. (19)Ž . Ž .|2 dt
0
After normalizing with the relevant mean velocity scale,
U, and the turbulent cross-stream mixing time scale, d/u*,
t2 2dNx9 t M Nu9 t u9 tqt M
B EŽ . Ž . Ž .1 U d tuU
C Fs d . (20)| 2
D G2 dt u U dU
0
Following a long period of time from release, when the
location of release is ‘forgotten’ and the particle has sampled
the variation over the cross-section many times the integral
converges to
22 2 UdŽ .1 dNx9 M U d
; s . (21)
2 dt u u dU U
That is, the variance growth rate is given by the ratio of
two diffusivities, the square of a diffusivity based on the
mean velocity to the cross-stream diffusivity due to the fluc-
tuating turbulent motion. An argument can be made based
on an extension of the central limit theorem that displace-
ments will have a Gaussian distribution about an axis moving
with the discharge velocity. That is, for the concentration,
integrated over the cross section, is
˜C x,t s C x,t dA (22)Ž . Ž .|
A
21 x-Ut
-Q ž /2 s
˜C x,t s e (23)Ž . ys 2p
2 21 ds U d
sT (24)L2 dt uU
where TL is the Lagrangian integral constant. One notes that
when the typical marked fluid particles sample the viscous
dominated region adjacent to the walls (or equivalent recir-
culation region around roughness elements applies for non-
smooth walls), the motion is reduced and leading to a long
upstream tail in , and also delays the approach to sym-˜C x,tŽ .
metrical Gaussian form for a long time (Dewey and Sullivan
1977).

Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows 19
Article in press - uncorrected proof
In the special case of the atmospheric boundary layer treat-
ed by Batchelor (1964), the Lagrangian similarity theory
gives the components of the variance as
2 2 2
N x t -Nx t M MAu tŽ Ž . Ž . . U
2 2 2
Ny t MAu tŽ . U
2 2 2
N z t -Nz t M MAu tŽ Ž . Ž . . U
(25)n
We now consider the case where scales of turbulent
motion are large enough to transport the cloud in its entirety.
This is graphically apparent with a case of a meandering
plume from a continuous release. In this case, the main con-
cern is the probable location of the center-of-mass of
individual concentration fields and the distribution of con-
centration in the center-of-mass reference frames. This is
referred to as ‘relative diffusion’ in contrast to ‘absolute dif-
fusion’ when displacements are referred to a fixed point or
inertial reference frame system. The variance growth rate
here is determined from a two fluid particle analysis in which
the fluid particles comprising the initial cloud are taken two
at a time over all possible pairs (Batchelor 1952). The sep-
aration between a typical pair of particles y(t) contributes to
the variance as
t2dNy t MŽ .1
s Nv t v tqt Mdt (26)Ž . Ž .|2 dt
0
where v(t) is the relative velocity of the two particles.
A very practical advantage in center-of-mass coordinates
is that at any time t it is the length scales of turbulence that
are of the size of particle separation that are important. Scales
that are much larger just move both particles together and
do not contribute to their separation. Smaller scales just jig-
gle the particles about and are less effective in separating
them than scales of the separation size. In absolute diffusion
the largest scales are always the most important, whereas in
relative diffusion and at small separations only small scales
matter and taking averages are much less problematic.
Another big advantage in relative diffusion approach is in
the treatment of the turbulent flows for extremely high Reyn-
olds numbers, typical for engineering and environmental pro-
blems. The energy transfer process in turbulence suggests
that large scale eddies that contain non-homogeneities such
as information on container size etc. interact with smaller
scales and so on to the smallest scales of turbulence where
energy is taken out of the system by viscosity. That is, the
inertial terms pump energy down the system without redis-
tribution, while the pressure term acts to make the turbulence
structure isotropic. For a sufficiently high Reynolds number,
there could appear a range of scales between the large energy
containing scales and the small dissipation scales, the so-
called ‘inertial range’, that can be treated as dependent on
the energy transfer rate (the rate of energy dissipation per
unit mass) only. On simple dimensional grounds Batchelor
(1952) showed that if the contaminant field size is in this
range, then
3
B E22
C FNy Ms at (27)
D G3
Richardson (1926) pointed out that relative dispersion is
an accelerating process in which an initial marked volume
of fluid is spread at a rate dependent upon its size and arrived
at the ‘4/3 power law’ for the relative diffusion as
21 dNy M 4/3say (28)
2 dt
where asc´1/3; c is an universal constant of order 1, and ´
is the rate of energy dissipation. This was observed with
floating particles over a range of 2 m to 2 km in the ocean
(Richardson and Stommel 1948, Ozmidov 1957, 1960). It
should be noted that generally, to get a substantial inertial
sub-range (usually in the ocean and atmosphere); a very high
Reynolds number is needed.
The relationship between the mean concentration distri-
butions in absolute diffusion is derived from that in relative
diffusion with a convolution integral. If we let r(x9,t) be the
probable location of the center-of-mass x9 and C9(y,t) the rel-
ative mean concentration, the mean concentration in absolute
diffusion is (Munro et al. 2003)
`
C x,t s r x9,t C9 x-x9,t dx9 (29)Ž . Ž . Ž .|
-`
One can note that the probability of the center-of-mass
location can be found from C and C9 by using a Fourier
transform, for example. It may be equally compelling to
combine the probable center-of-mass location and cloud size
from a relative framework and consider the location and size
in an inertial framework.
Distribution of the concentration field
The probability density function p(u;x,t) of a potentially haz-
ardous contaminant dispersed in turbulent environment was
shown to be well-represented by various models. The main
idea is to use a few lower order moments to represent the
PDF or one of its models. This is because moments are more
easily measured and the equations governing the moment
evolution are simpler. The modeling procedure typically pro-
vides a reasonable approximation for the bulk of the PDF
but does not necessarily perform well in the higher concen-
tration tails.
For example, a four-parameter PDF model by Lewis and
Chatwin (1995) is also known as exponential and generalized
Pareto distribution (EGPD) by Mole and Clarke (1995) is
given by:
-u/l (1/z )-1
B Ee ´ zu
C Fp(u;x,t)dus(1-´) q 1- (30)
-n/lz
D Gw xl 1-H(z )e n n

20 Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows
Article in press - uncorrected proof
where l(x,t), v(x,t) are concentration scales, ´(x,t), z(x,t) are
shape factors and H is the Heaviside step function (e.g. Lew-
is and Chatwin 1995). Applications of the different PDF
models are very similar. Here we present briefly an example
of application, based on another model, using the so-called
expected mass fraction (EMF) function. The EMF is defined
as:
`
-1q u;t sQ u p u;x,t dx, q u;t dus1 (31)Ž . Ž . Ž .| |
0
where the integral in eq. (31) is calculated over all space.
This has the simple and straightforward interpretation that
the expected fraction of the release mass Q i.e., between ua
and ub is given by
ub
q u;t du. (32)Ž .|
ua
The effect of molecular diffusivity k to reduce concentra-
tion values is observed as the area under q(u,t) shifts to lower
values of u as time increases. The EMF approach was pro-
posed with the expectation to obtain a more simple form than
would be observed of the PDFs. The moments of q(u,t) sim-
ply relate to the moments of the PDF:
umax
n -1M t s u q u;t duQ s m x,t dx. (33)Ž . Ž . Ž .n nq1| |
0 a.s.
There is a maximum value of concentration umax which is
the largest initial value of concentration that can never be
exceeded. The evolution of the moments of the EMF is given
by the following differential equation (Sullivan and Ye
1997):
≠ 2
-1 n-1M s-n nq1 kQ NG x,t =G x,t Mdv. (34)Ž . Ž .Ž Ž ..n |≠t
a.s.
In a similar manner the higher order moments are derived
and applied to represent the EMF (or PDF) of the scalar
concentration field. Special care is devoted to the high con-
centration tails, which are important in some applications as
mentioned in the Introduction.
Moment closure approximations
In order to solve the moment equations shown above, there
is a need for closure approximations for the (i) convective
terms and (ii) dissipative terms. For example, a convective
closure approximation of Sullivan (2004) is proposed to be
≠Cnq1 n=ØNu9G Ms-u (35)o ≠t
disregarding the molecular diffusion. Later an improved ver-
sion was proposed taking into account the effect of molecular
diffusion. For instance, for an axisymmetric plume and cylin-
drical coordinates the expression for the point source is
w z≠C 1 ≠ 1 ≠nq1 nq1x |- A Nu9G M q yNv9G M (36)Ž . Ž .nU≠x U u ≠x y ≠yy ~Ž .o
and for the line source (2D approximation):
w z≠C 1 ≠ ≠nq1 nq1x |- A Nu9G M q Nv9G M . (37)Ž . Ž .nU≠x U u ≠x ≠yy ~Ž .o
The second, dissipative closure was proposed by Moseley
(1991) and partially confirmed by Mole (1995):
2B G-GŽ .t2=G s (38)Ž . 2hB
where hB is the conduction cut-off length, B is a proportion-
ality constant and Gt is a threshold concentration (which is
set to be 0). The solution of the dissipative approximation
for plumes behind both the line source and the point source
can be written as
2 2
B E B E≠G ≠Gn-1 n-1
C F C FNG MqNG M
D G D G≠x ≠y
2
B E≠G Bn-1 nq1
C FqNG Ms NG M (39)2
D G≠z hB
The moment equation that uses the two closures is then
becomes,
≠m ≠Cnq1 nqn nq1 m su (40)Ž . nq1 o≠t ≠t
where . The solution for a line source, where hB2tskBtyhB
is assumed to be constant throughout the flow, is:
n n -n(nq1)tm x,t su C x,t -n nq1 u eŽ . Ž . Ž .nq1 o o
t
n(nq1)z= C x,z e dz. (41)Ž .|
o
Sullivan (2004) has shown that these closures lead to a
good qualitative comparison with the measurements in the
plume from a line source in grid turbulence for the distrib-
uted four lowest order central moments.
Geometry of the scalar field interfaces
Turbulent mixing and transport of passive scalars can be
described as a combination of folding and wrinkling pro-
cesses (Catrakis 2000). The effect of these processes can be
studied by quantifying the geometric scale distributions of

Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows 21
Article in press - uncorrected proof
Figure 5 Schematic of the scalar interface when the Batchelor
scale is resolved (from Sreenivasan 1991).
the convoluted structure of passive scalar interfaces within
the fluid (e.g., Gonzalez 2009).
In turbulence research, an emphasis is given to the small-
scale structure and the fractal geometry of turbulent passive
scalar fields that is built on several amendments to Kolmo-
gorov hypothesis. The phenomenological model of the small-
scale passive scalar, introduced by Obukhov (1949) and
Corrsin (1951), was based on similarity arguments. Accord-
ing to them, there exists an inertial range where the turbulent
structure of the passive scalar field is independent of the
large scales. This length scale is know as Obukhov-Corrsin
length scale, hocs(k3/´)1/4. Later, Batchelor (1959) showed
that the Obukhov-Corrsin cut-off length scale was appropri-
ate only for low Schmidt number passive scalars (Scs
n/k--1) and that the strain rate of the fluctuating velocity
determines the cut-off length scale for scalars with higher
Schmidt numbers (Sc))1). This is called the Batchelor
scale (hBs(nk2/´)1/4).
Passive scalars in turbulent flows are rapidly stretched into
thin sheet-like structures by the turbulent convective motion.
The geometry of the interfacial surface has significant prac-
tical importance in a number of applications. In non-reactive
mixing processes, the molecular diffusive flux occurs across
concentration gradients at interfaces (Schumacher and Sree-
nivasan 2005). Estimating the resulting mixing is necessary
for predicting the dilution of the pollutants into the atmos-
phere or the discharge of wastewater into a stream. Molec-
ular diffusion happens within the smallest scales (hB), where
the concentration gradients are the highest. Sreenivasan
(1991) argued that at a high Schmidt number, the convolu-
tions of the scalar interface are space filling on scale between
the Kolmogorov scale and the Batchelor scale and that the
surface of the scalar, where diffusion happens, features frac-
tal scaling.
Sreenivasan et al. (1989) suggested that the large scale
structure of the turbulence determines the precise amount of
mixing; however, it is eventually the diffusive action at the
molecular level that performs the actually mixing. It was
argued that the gradients across the scalar interface are of
order DG/hB for a unit Schmidt number, where DG is the
concentration difference at the interface. Later, Sreenivasan
(1991) developed a relation between the surface area and the
resolution of measurement with high Schmidt number sca-
lars. The physical picture that was depicted in this study is
reproduced here as Figure 5.
Experimental techniques
Simultaneous measurements of PLIF and PIV provide a
comprehensive mapping of the scalar fields along with the
velocity fields in turbulent flows. The mapping of the flow
field will provide turbulent quantities associated with scalar
transport ( , ). These represent unresolved terms in theu9c9 y9c9
Reynolds averaged scalar transport equation. Batchelor
(1949) proposed a generalization to the gradient transport
hypothesis that defines turbulent diffusivity tensor (Gij), such
as:
—–
-u9c9
G s (42)12 ≠Cy≠y
—–
-v9c9
G s (43)22 ≠Cy≠y
where G22 is called the turbulent diffusivity GT (Tavoularis
and Corrsin 1981).
Particle image velocimetry
The use of particle image velocimetry (PIV) technique in the
context of scalar measurements is required according to the
diffusion equation where scalar concentration and velocity
are coupled. The description of the PIV technique and its
utilization is beyond the scope of this review, however, a
short description of the method principle and its extensions
is provided.
PIV is a non-intrusive technique for measuring complex
flow fields in a two-dimensional plane providing two com-
ponents of the velocity vector with high spatial resolution.
In PIV measurements, two consecutive images are obtained
to capture the light scattered by tracer particles from the laser
pulses whose timing is precisely controlled. Two-dimension-
al cross-correlations of a small interrogation area of the
corresponding two images are obtained to deduce the dis-
placement of particles in the interrogation areas. Raffel et al.
(2007) describes the technique in detail.
PIV is a two-dimensional technique. Since most of the
turbulent flows are three-dimensional, several modifications
have been proposed and utilized to extend the technique
towards volumetric measurements.
Amongst these are the stereoscopic viewing using two
cameras located at given angles in respect to the light sheet,
providing two-dimensional velocity field with three velocity
components (Soloff et al. 1997). Defocusing principle (Perei-
ra et al. 2000) is used in order to obtain a few planes while
using one camera. Multiple-plane stereoscopic PIV is sug-
gested to integrate stereoscopic viewing with defocusing

22 Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows
Article in press - uncorrected proof
Figure 6 Simplification of Jablonski diagram with focus on flu-
orescence energy levels (Walker 1987).
principles and provided three-dimensional velocity fields
with limited spatial resolution (Liberzon et al. 2004). Anoth-
er technique utilizes tomography to extract the volumetric
information while using volumetric illumination and analyz-
ing the images based on tomography principles. A detailed
review of the 3D extensions can be found in Arroyo and
Hinsch (2008).
Planar laser induced fluorescence
The use of planar laser induced fluorescence (PLIF) has
become widespread in fluid mechanics experiments (Dimo-
takis et al. 1983, Catrakis and Dimotakis 1996, Houcine et
al. 1996, Distelhoff et al. 1997). Planar laser induced fluo-
rescence (PLIF) is a non-intrusive technique for measuring
the spatial scalar concentration field in continuous flow sys-
tems. The underlying physical principle of this technique is
based on the absorption and subsequent re-emission of pho-
tons by fluorescent dye tracers. This allows, detecting small
changes in the concentration very accurately. The rapidity of
the fluorescent emission compared to velocities in the flow
allows the possibility of following rapid concentration fluc-
tuations. In the case of the passive (non-reacting) tracer, the
effect on the existing flow condition such as pollutant dis-
solution and dispersion can be studied.
The physical principle underlining the planar laser induced
fluorescence (PLIF) technique in fluids is based on the prin-
ciples of fluorescence spectroscopy. Fluorescence is induced
when a molecule releases light due to electron excitation by
an incident light beam. In the first step, the molecule is excit-
ed due to the absorption of photon from the incident light
beam. It reaches a higher singlet (electronic) state, and than
it has three pathways, based on Jablonski diagram, as shown
schematically in Figure 6.
According to Figure 6, the electron has three possible
pathways to return to the ground state; radiationless collision,
crossing through the triplet state, or the desired fluorescence
path. Energy is lost through non-radiative processes associ-
ated with the vibration levels present within each of the
excited and ground states, therefore a shift (Stokes’ shift) to
longer, lower energy wavelengths occurs in the fluorescent
emission. Fluorescent material can have broad absorption/
emission spectra and usually it is more desirable to use a
tracer dye with emission in the visible region.
To be able to utilize the properties of a fluorescent dye in
the flow field, information relating intensity to a parameter
about the flow is desired. The Beer-Lambert law is used to
relate the absorption to the concentration of the medium:
dF
s-´CF
dx
l
lnF-lnF s-´ C(x,t)dx (44)0 |
0
where F is the photon flux intensity (W/m2), ´ the extinction
coefficient (m2/mol), and C the concentration (mol/m3). With
the average concentration over the length
l
C(x,t)dx|
– 0Cs . (45)
l
Equation (45) can be used in eq. (44) giving
–
-´ClFsF e . (46)o
Similarly, analyzing emitted light, the fluorescent emission
intensity Ie (W/m3)
dF
I s-V (47)e dx
with V representing the quantum yield; the fraction of light
emitted from that which is absorbed. Considering the meas-
uring volume, the measured intensity l(W), can then be
determined by
lsxI V (48)e m
with x representing the optical efficiency of the system tak-
ing into account both filtering and solid angle. Filtering
wavelengths in the range of the excitation beam is necessary
as to limit scattering in the measurement. Equations (44),
(47) and (48) can be combined to produce
lsVxV ´CF (49)m
and with varying concentration over space x and time t
giving
–
-(´ Iq´ I )Ci i e elsVxV ´CF e (50)m o
with (´iIiq´eIe) taking into account the attenuation of the
incident beam to the measurement point, and also the atten-
uation of the emitted beam both with respect to distance and

Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows 23
Article in press - uncorrected proof
Table 1 Properties of common passive tracers for PLIF imaging.
Fluorescent tracer Quantum la,max lf,max ´ (cm-1M-1)
yield V (nm) (nm)
Disodium fluorescein 0.92 491 515 88,000
Rhodamine B 0.60 555 574 115,000
Rhodamine 6G 0.78 538 564 116,000
Figure 7 Acetone absorption spectrum (Lozano et al. 1992).
time scales. Therefore, through eq. (50), it is seen that inten-
sity and concentration are proportional with the remaining
parameters characterizing the optical system, experimental
setup, and average concentration (Gaskey et al. 1990). This
now allows us to predict concentration variation from inten-
sity variation within the measuring volume.
Some popular tracers in water are fluorescein and diso-
dium salt, Rhodamine B and Rhodamine 6G. The properties
of the fluorescence are listed in Table 1. Tracers with high
absorption coefficient and high quantum efficiency are gen-
erally favorable.
One-dimensional LIF Walker (1987) and Gaskey et al.
(1990) were among the first to use the fluorescence spec-
troscopy to study the local concentration in flowing liquids.
In general, the 1D LIF system contains laser source, optical
components, receiving optics, photomultiplier, A/D converter
and signal processing unit. It is noteworthy to comment that
in this early stage of LIF development, a camera was not
required.
The LIF source was a laser beam emitted from continuous
Argon-Ion laser (ls488 nm). The power of the lasers used
to create incident beam varied from 25 to 400 mW. The
incident beam was split before entering the measurement sec-
tion in order to monitor the stability of the laser power. It
was found that 1% drift in the laser power could cause a
significant change in the fluorescence signal. The beam is
then introduced into the fluid. The measurement point is
determined by the intersection of the non-focused incident
beam with the image of a pinhole that is focused into the
liquid. The receiving optics and the photomultiplier are locat-
ed perpendicular to the direction of the incident beam. The
system is robust but it suffers from two disadvantages. First,
the fluorescent signal is collected from a large measurement
volume (100 mm), which is comparable to the size of the
turbulent eddies. Second, the laser flux that passes through
the measurement volume is low. To overcome these problems
the incident beam was focused with a lens within the fluid.
This reduces the measurement volume further and increases
significantly the laser flux. Unfortunately, the increase of the
laser flux resulted in undesirable photo bleaching reaction
between the fluorescent dye and the dissolved oxygen. The
photo bleaching can be avoided by selecting different dye
tracer. One common choice is the use of Rhodamine 6G dye
(see Table 1).
Another problem is associated with the alignment of the
laser focal point with the image of the pinhole. Because of
the small size of the measuring volume the focusing of the
photomultiplier becomes extremely difficult. Small shifts in
the equipment as well as thermal expansion can cause further
misalignment between the measuring volume and photomul-
tiplier. However, in spite of these experimental problems, the
focused system offers much greater resolution in frequency,
with a smaller measurement volume.
Gaskey et al. (1990) employed the 1D LIF to study the
concentration fluctuations in a continuous stirred tank reac-
tor. They measured the concentration spectra up to the begin-
ning of the viscous-diffusive subzone, which is the region
where concentration fluctuations rapidly decay.
Two-dimensional PLIF The evolution of the LIF tech-
nique is directly connected to the development of the digital
solid-state sensor cameras, especially the charged coupled
device (CCD) cameras. These devices replaced photomulti-
plier and receiving optics making possible to measure simul-
taneously concentration not only in a single point but also
along the line and later along the 2D plane. Both lens and
the CCD chip are protected by filters, which block ultraviolet
and infrared light. An additional optical filter is required with
a cut-off wavelength similar to the wavelength of the inci-
dent light beam. This combination of filters passed only the
fluoresced light and hence reduced the noise to signal ratio
of the image recording.
The laser used in planar laser induced fluorescence must
have part of its power band within the absorption band of
the dye being used (Crimaldi 2008). This makes laser selec-
tion important, as it is desired to utilize the peak absorption
spectrum of the fluorescent dye. For example, acetone has
an absorption band that lies between 225 and 320 nm. As
seen in Figure 7, peak absorption for acetone occurs within
the flat region of 270–280 nm, making it suitable for exci-
tation by a XeCl excimer (ls308 nm), KrF (ls248 nm),

24 Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows
Article in press - uncorrected proof
Figure 9 Three-dimensional PLIF experimental setup (Van-Vliet et al. 2004).
Figure 8 Simultaneous measurement of velocity and scalar using
PLIF and PIV.
or a frequency quadruple Nd:YAG (ls266 nm) laser (Loza-
no et al. 1992).
The laser beam emitted from a laser source is passed
trough a cylindrical lens to form a thin laser sheet which will
illuminate the investigated flow field. Because the required
thickness of the laser sheet is smaller than the laser beam
diameter, it is necessary to focus the laser beam to the size
desired. Placing a spherical lens in front of the cylindrical
lens does this. Generally, a long focusing length, combined
with a large beam diameter, which also makes focusing eas-
ier, enables a small focal point diameter and a minimally
varying beam diameter. The focal point should be located in
the center of the image plane.
The light sheet created with a cylindrical lens has a Gaus-
sian intensity distribution and therefore corresponding gray
value corrections are necessary. The discussion of advantages
and disadvantages of the PLIF with a cylindrical lens can be
found in van Cruyningen et al. (1991).
Figure 8 demonstrates the utilization of 2D-PLIF in
applied research. It shows a contaminant plume captured
with Planar Laser Induced Fluorescence in a water tunnel.
Particle image velocimetry yields simultaneous velocities
(arrows), revealing the interaction between the velocity field
and the concentration of contaminants (Sarathi et al. 2010).
Three-dimensional PLIF There are various methods for
performing three-dimensional analysis to resolve the scalar
field. One method involves the simultaneous imaging of the
two-dimensional scalar field using multiple close space laser
sheets. This method has no temporal delay between imaging
of the planes, which yields the true value of the three-dimen-
sional scalar field. This information can then be manipulated
in a single plane in the flow to obtain the three-dimensional
scalar gradient field and its associated scalar energy dissi-
pation rate. Disadvantages of this method are that multiple
cameras must be used which translates to multiple scattering
methods for interrogation between planes. Also the three-
dimensional scalar gradient from the two-dimensional plane
cannot be resolved to a three-dimensional volume (Van-Vliet
et al. 2004).
Another method uses a single swept laser sheet in the
normal direction to obtain consecutively spaced, two-
dimensional images from which the three-dimensional con-
centration field is reconstructed (Figure 9).
This method has the benefit of not having restrictions to
the quantity of images taken in the normal direction; there-
fore visualization of the structure in all three-dimensions is
constructible. One drawback, however, is that unlike the first
method, the images from each plane are separated temporar-
ily which places restrictions on the time spacing between
each captured plane. These restrictions will be based on the
smallest needed resolvable turbulent time scales; the Kol-
mogorov time and advection time scale.
PLIF image corrections and calibration procedure The
PLIF technique relies on the relationship between the fluo-
resced light intensity, the concentration of tracer dye and
the intensity of the laser. At low concentrations levels
(-120 mg/l) there is a linear relationship between the dye
concentration and emitted light intensity.
Calibration for basic concentration measurements is car-
ried out using a similar method. A solution is prepared at a

Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows 25
Article in press - uncorrected proof
known concentration or temperature. The solution is isotro-
pic such that it is calibrated for only each individual value
of the scalar field. The fluorescent intensity of the solution
is then measured and calibrated to the known value of the
system. Since the image of the control volume is divided
into pixels, each pixel must be calibrated based on the gray
scale value of the intensity related to the scalar measurement,
where local intensity of the laser is disregarded.
The background emission could be found by averaging the
intensity level over number of images. Each of the calibra-
tion images should to be corrected for laser attenuation due
to the presence of the dye. The corrected intensity can be
calculated using Beer-Lambert law (Ferrier et al. 1993) as:
µ ∂IsI exp aC x-x (51)Ž .o o
where I is the intensity (gray scale) at location x, Io is the
intensity at location xo, C is the dye concentration (mg/l) and
a is the attenuation coefficient. The attenuation coefficient
for clear water is a constant and equal to 0.00023 cm-1
(mg/l)-1 (Daviero et al. 2001). The calibration should to be
performed for all the pixels in the CCD camera. The least-
square procedure can be used to obtain the slope and the
intercept of the calibration curve. The linear fit is forced to
pass through zero dye concentration in order to avoid erro-
neous negative concentrations in the instantaneous images.
After obtaining the slope and the intercept for each pixel the
raw images are converted to concentration field data.
Although PLIF has been extensively used in non-stratified
environments, its use in stratified environments has been lim-
ited by refractive index fluctuations caused by density dif-
ferences. This is because the distance that the laser travels
through the fluid combined with internal waves can cause
significant laser intensity variations (Daviero et al. 2001).
Variations in refractive index must therefore be very small
if quantitative information is to be obtained. These difficul-
ties can be overcome to some extent by the use of refractive
index matching in which liquids of different density but
equal refractive index are used.
Summary
This review presented and analyzed the topic of turbulent
diffusion of scalar concentration field released from a point
source. The main physical mechanism of turbulent and
molecular diffusion, absolute and relative dispersion and
models of the probability density functions of the scalar
fields have been summarized in the paper.
The literature survey covers models such as expected mass
fraction, where their goal is to predict adequately the distri-
bution of scalar in turbulent flow. In particular, it can assess
the high concentration events that are associated with the
tails of the probability density function. It is emphasized that
the extreme concentration is of special importance for pro-
blems of hazardous materials released in to the environment
and the health risk assessment of chemical processes.
Valuable results have been obtained experimentally using
the expected mass fraction model for which the point-wise
correlation of turbulent velocity and concentration were
measured. This data became available due to technological
progress in non-intrusive imaging methods such as particle
image velocimetry (PIV) that measures spatially resolved
two-dimensional velocity fields and planar laser induced flu-
orescence that quantifies scalar concentration field while
using the same imaging principles for both techniques. This
experimental data enables to calculate directly the coupling
between velocity and concentration at instantaneous events
in a plane.
The quality of data reported in literature were sufficient
to estimate the moments of the probability density function
up to 5th order and establish a set of parameters for the
expected mass fraction model.
One of the challenges in predicting the scalar transport in
a given turbulent flow is our ability to measure or calculate
the diffusion coefficients. In addition, the coefficients vary
for different type of flow conditions, which makes it hard to
form a generalized receipt of calculations. Therefore, it will
be necessary to perform a field experiment and develop
means to measure it directly.
In order to control the transport of scalars, one needs to
fully characterize the small-scale physics of turbulence. One
of the arguments suggests that the high concentration tail of
the PDF ought to have a universal character which will be
a function of the flow and the scalar; would this be valid
and if so, can we scale it appropriately? In conjunction to
the former question, can we deduce similarity laws for the
scalar transport phenomena? Answering these questions will
provide better tools for modeling these flows and shed light
on the physical mechanisms associated with turbulence.
References
Arroyo, M.P. and K.D. Hinsch. 2008. Recent developments of PIV
towards 3D measurements. Top. Appl. Phys. 112: 127–154.
Bakosi, J., P. Franzese and Z. Boybeyi. 2007. Probability density
function modeling of scalar mixing from concentrated sources
in turbulent channel flow. Phys. Fluids 19: 115106.
Batchelor, G.K. 1949. Diffusion in a field of homogeneous turbu-
lence – Eulerian analysis. Aust. J. Sci. Res. 2: 437.
Batchelor, G.K. 1952. Diffusion in a field of homogeneous turbu-
lence – II. The relative motion of particle. Proc. Camb. Phil.
Soc. 48: 345–362.
Batchelor, G.K. 1959. Small scale variation of convected quantities
like temperature in turbulent fluid. Part 1. General discussion
and the case of small conductivity. J. Fluid Mech. 5: 113–133.
Batchelor, G.K. 1964. Diffusion from sources in a turbulent bound-
ary layer. Arch. Mech. Sws. 3, 16: 661–670.
Catrakis, H.J. 2000. Distribution of scales in turbulence. Phys. Rev.
E 62: 564–578.
Catrakis, H.J. and P.E. Dimotakis. 1996. Mixing in turbulent jets:
scalar measures and isosurface geometry. J. Fluid Mech. 317:
369–406.
Celani, A., M. Cencini, E. Vergassola and D. Vincenzi. 2005. Shear
effects on passive scalar spectra. J. Fluid Mech. 523: 99–108.
Chatwin, P.C. 1990. Hazards due to dispersing gases. Environme-
trics 1: 143–162.

26 Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows
Article in press - uncorrected proof
Corriveau, A.F. and W.D. Baines. 1993. Diffusive mixing in tur-
bulent jets as revealed by pH indicator. Exp. Fluids 16: 129–136.
Corrsin, S. 1951. On the spectrum of isotropic temperature fluctu-
ations in an isotropic turbulence. J. Appl. Phys. 22: 469–473.
Crimaldi, J.P. 2008. Planar laser induced fluorescence in aqueous
flows. Exp. Fluids 444: 851–863.
Csanady, G.T. 1973. Turbulent diffusion in the environment. Reidel
Publishing Company, Dordrecht, Holland.
Dahm, W.J., K.B. Southland and K.A. Buch. 1991. Direct, high
resolution, four-dimensional measurements of the fine scale
structure of Sc))1 molecular mixing in turbulent flows. Phys.
Fluids A3, 5: 1115–1127.
Danaila, L. and R.A. Antonia. 2009. Spectrum of a passive scalar
in moderate Reynolds number homogeneous isotropic turbu-
lence. Phys. Fluids 21: 111702.
Daviero, G.J., P.J.W. Roberts and K. Maile. 2001. Refractive index
matching in large-scale stratified experiments. Exp. Fluids 31:
119–126.
Davies, J.K.W. 1989. The application of box models in the analysis
of toxic hazards by using the probit dose-response relationship.
J. Hazard. Mater. 22: 19–329.
Derksen, R.W. and P.J. Sullivan. 1990. Moment approximation for
probability density functions. Combust. Flame 81: 378–391.
Dewey, R. and P.J. Sullivan. 1977. The asymptotic stage of longi-
tudinal turbulent dispersion within a tube. J. Fluid Mech. 80:
293–303.
Dimotakis, P.E., R.C. Miake-Lye and D.A. Papantoniou. 1983.
Structure and dynamics of round turbulent jets. Phys. Fluids 26:
3185–3192.
Distelhoff, M.F.W., A.J. Marquis, J.M. Nouri and J.H. Whitelaw.
1997. Scalar mixing measurements in batch operated stirred
tanks. J. Chem. Eng. 75: 641–652.
Elperin, T., N. Kleeorin and I. Rogachevskii. 1996. Turbulent ther-
mal diffusion of small inertial particles. Phys. Rev. Lett. 76:
224–228.
Elperin, T., N. Kleeorin, I. Rogachevskii and D. Sokoloff. 2000.
Passive scalar transport in a random flow with a finite renewal
time: mean-field equations. Phys. Rev. E 61: 2617–2625.
Ferrier, A.J., D.R. Funk and P.J.W. Roberts. 1993. Application of
optical techniques to the study of plumes in stratified fluids.
Dynam. Atmos. Oceans 20: 155–183.
Gaskey, S., P. Vacus, R. David and J. Villermaux. 1990. A method
for the study of turbulent mixing using fluorescence spectros-
copy. Exp. Fluids 9: 137–147.
Gonzalez, M. 2009. Kinematic properties of passive scalar gradient
predicted by a stochastic Lagrangian model. Phys. Fluids 21:
055104.
Griffiths, R.F. 1991. The use of probit expressions in the assessment
of acute population impact of toxic releases. J. Loss Prevent.
Proc. 4: 49–57.
Hall, D.J., R.A. Waters, G.W. Marsland, S.L. Upton and M.A.
Emmott. 1991. Repeat variability in instantaneously released
heavy gas clouds – some wind tunnel model experiments.
Report LR 804, Warren Spring Laboratory, UK.
Houcine, I., H. Vivier, E. Plasari, R. David and J. Villermaux. 1996.
Planar laser induced fluorescence technique for measurements
of concentration fields in continuous stirred tank reactors. Exp.
Fluids 22: 95–102.
Labropulu, F. and P.J. Sullivan. 1995. Mean-square values of con-
taminant cloud dilution. Environmetrics 6: 619–625.
Lavertu, R.A. and L. Mydlarski. 2005. Scalar mixing from a con-
centrated source in turbulent channel flow. J. Fluid Mech. 528:
135–172.
Lewis, D.M. and P.C. Chatwin. 1995. A new model PDF for con-
taminant dispersing in the atmosphere. Environmetrics 6:
583–593.
Liberzon, A., R. Gurka and G. Hetsroni. 2004. XPIV-multi-plane
stereoscopic Particle Image Velocimetry. Exp. Fluids 36:
355–362.
Lozano, A., B. Yip and R.K. Hanson. 1992. Acetone: a tracer for
concentration measurements in gaseous flows by planar laser-
induced fluorescence. Exp. Fluids 13: 369–376.
Mole, N. 1995. The a-b model for concentration moments in tur-
bulent flows. Environmetrics 6: 559–569.
Mole, N. and E.D. Clarke. 1995. Relationships between higher
moments of concentration and of dose in turbulent dispersion.
Bound-Lay. Meteorol. 73: 35–53.
Moseley, D.J. 1991. A closure hypothesis for contaminant fluctua-
tion in turbulent flow. M.Sc. Thesis. The University of Western
Ontario.
Munro, R.J., P.C. Chatwin and N. Mole. 2001. High concentration
tails of the probability density function of a dispersing scalar in
the atmosphere. Bound-Lay. Meteorol. 98: 315–339.
Munro, R.J., P.C. Chatwin and N. Mole. 2003. Some simple statis-
tical models for relative and absolute dispersion. Bound-Lay.
Meteorol. 107: 253–271.
Mylne, K.R. and P.J. Mason. 1991. Concentration fluctuation meas-
urements in a dispersing plume at a range of up to 1000 m.
Q.J.R. Meteorol. Soc. 117: 253–271.
Obukhov, A.M. 1949. Structure of the temperature field in a tur-
bulent flow. Izv. Akad. Nauk SSSR, Ser. Fiz. 6: 59–63.
Ozmidov, R.V. 1957. Experimental investigation of horizontal tur-
bulent diffusion in the sea and in a shallow artificial reservoir.
Izv. Akad. Nauk SSSR, ser. geofiz. 6: 756–764.
Ozmidov, R.V. 1960. On the rate of dissipation of turbulent energy
in sea currents and on the dimensionless universal constant in
the ‘‘4/3-power law’’ (in Russian). Izv. Akad. Nauk SSSR, ser.
geofiz. 8: 1234–1237. American Geophysical Union Translation,
English edition, Bulletin of the Academy of Sciences of the
USSR. Geophysics Series 8: 821–823.
Pasquill, F. 1974. Atmospheric diffusion. Ellis Horwood Ltd.,
Chichester, UK.
Pereira, F., M. Gharib, D. Dabiri and D. Modarress. 2000. Defo-
cusing digital particle image velocimetry: a 3-component
3-dimensional DPIV measurement technique. Application
to bubbly flows. Exp. Fluids 29: 78–89.
Pope, S.B. 1979. The statistical theory of turbulent flames. Philos.
Tr. R. Soc. A. 219: 529–568.
Pope, S.B. 1994. Lagrangian PDF methods for turbulent flows.
Annu. Rev. Fluid Mech. 26: 23–63.
Raffel, M., C.E. Willert, S.T. Wereley and J. Kompenhans. 2007.
Particle Image Velocimetry: A Practical Guide, Springer.
Richardson, L.F. 1926. Atmospheric diffusion shown on a distance-
neighbour graph. Proc. Roy. Soc. A. 110: 709–737.
Richardson, L.F. and H. Stommel. 1948. Note on eddy diffusion in
the sea. J. Meteorol. 5: 238–240.
Roberts, P.J.W. and D.R. Webster. 2002. ‘‘Turbulent Diffusion’’. In:
(H. Shen, A. Cheng, K.-H. Wang, M.H. Teng and C. Liu, eds.)
Environmental Fluid Mechanics-Theories and Application.
ASCE Press, Reston, Virginia.
Saffman, P.G. 1962. The effect of wind shear on horizontal spread
from an instantaneous ground source. Q.J.R. Meteorol. Soc. 88:
382–393.
Sakai, Y., K. Uchida, T. Kubo and K. Nagata. 2008. Statistical fea-
tures of scalar flux in a High-Schmidt-number turbulent jet.
IUTAM Symposium on Computational Physics and New Per-
spectives in Turbulence 4: 209–214.
Sarathi, P., R. Gurka, P.J. Sullivan and G.A. Kopp. 2010. Experi-

Gurka et al.: Diffusion of scalar concentration from localized sources in turbulent flows 27
Article in press - uncorrected proof
mental measurements of expected mass fraction in a contaminant
plume, submitted to Bound-Lay. Meteorol.
Schumacher, J. and K.R. Sreenivasan. 2005. Statistics and geometry
of passive scalars in turbulence. Phys. Fluids 17: 125107.
Shraiman, B.I. and E.D. Siggia. 2000. Scalar turbulence. Nature
405: 639–646.
Smith, R. 1983. Effect of boundary absorption upon longitudinal
dispersion in shear flows. J. Fluid Mech. 134: 161–177.
Smyth, W.D. 1999. Dissipation-range geometry and scalar spectra
in sheared stratified turbulence. J. Fluid Mech. 401: 209.
Sofiev, M., V.F. Sofieva, T. Elperin, N. Kleeorin, I. Rogachevskii
and S. Zilitinkevich. 2009. Turbulent diffusion and turbulent
thermal diffusion of aerosols in stratified atmospheric flows. J.
Geophys. Res. 114: 1–19.
Soloff, S.M., R.J. Adrian and Z.C. Liu. 1997. Distortion compen-
sation for generalized stereoscopic particle image velocimetry.
Meas. Sci. Tech. 8: 1455–1464.
Sreenivasan, K.R. 1991. Fractals and multifractals in fluid turbu-
lence. Annu. Rev. Fluid Mech. 23: 539–600.
Sreenivasan, K.R., R. Ramsankar and C. Meneveau. 1989. Mixing,
entrainment and fractal dimension of surfaces in turbulent flow.
Proc. Roy. Soc. A. 421: 79–108.
Sullivan, P.J. 2004. The influence of molecular diffusion on the distri-
bution moments of a scalar PDF. Environmetrics 115: 173–191.
Sullivan, P.J. and H. Ye. 1997. The need for a new measure of
contaminant cloud concentration reduction. Il Nuovo. Cimento.
20: 413–423.
Sullivan, P.J. and H. Yip. 1985. A solution-scheme for the convec-
tive diffusion equation. J. Appl. Math. Phys. 36: 596–608.
Tavoularis, S. and S. Corrsin. 1981. Experiments in nearly homo-
geneous turbulent shear flow with a uniform mean temperature
gradient. Part 1. J. Fluid Mech. 104: 311–347.
Taylor, G.I. 1921. Diffusion by continuous movements. Proc. Lond.
Math. Soc. 2: 196–212.
Taylor, G.I. 1954. The dispersion of matter in turbulent flow though
pipe. Proc. R. Soc. Ser. A 223: 446–468.
Townsend, A. 1976. The structure of turbulent shear flow, 2nd edi-
tion. Cambridge University Press.
Van-Cruyningen, P., A. Lozano and R.K. Hanson. 1991. Quantita-
tive imaging of concentration by planar laser-induced fluores-
cence. Exp. Fluids 10: 41–49.
Van-Vliet, E., M. Van Bergen, J.J. Derksen, L.M. Portela and A.
Vander Akker. 2004. Time resolved, 3D laser induced fluores-
cence measurements of fine structure passive scalar mixing in a
tubular reactor. Exp. Fluids 37: 1–21.
Warhaft, Z. 2000. Passive scalar in turbulent flows. Annu. Rev. Flu-
id Mech. 32: 203–240.
Walker, D.A. 1987. A fluorescence technique for measurement of
concentration in mixing liquids. J. Phys. E Sci. Instrum. 20:
217–224.