Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 9.7.2 - Multi-Scale Analysis of a 2-D Liquid Atomization Mechanism
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 Material Information
Title: 9.7.2 - Multi-Scale Analysis of a 2-D Liquid Atomization Mechanism Collision, Agglomeration and Breakup
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Dumouchel, C.
Grout, S.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: liquid spray
atomization process
multi-scale analysis
scale entropy diffusion model
 Notes
Abstract: This paper presents a new lead in analyzing liquid atomization process by investigating the temporal evolution of the liquid system shape from the nozzle down to the spray. The atomization of the liquid sheet produced by a single orifice triple-disk nozzle is investigated as a function of the liquid physical properties. This liquid atomization process is essentially 2-D and is apprehended by image analysis. The description of the liquid system shape makes use of the cumulative surface-based scale distribution. It is found that this distribution continuously evolves during the atomization process suggesting modeling the whole process by the description of this distribution. This modelization is achieved by applying the scale entropy diffusion model initially developed to investigate turbulent interfaces. In turbulence, as well as in atomization, the diffusion of the scale entropy (quantity related to the cumulative surface-based scale distribution) is associated to a scale- and time-dependent fractal dimension. The scale entropy diffusion model introduces the scale diffusivity and the scale entropy flux sink. The scale diffusivity is measured and reports a clear organization with the injection pressure as well as with the liquid physical properties. Furthermore, it is demonstrated that the scale entropy flux sink at initial time is dependent on the issuing liquid flow characteristics that impose the initial perturbations on the liquid sheet. Therefore, this work demonstrates that the application of the scale entropy diffusion model is physically representative to describe liquid atomization processes and spray formation. It offers a new lead in apprehending and investigating this complex phenomenon that has been often disregarded because of a lack of relevant approaches.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Multi-scale Analysis of a 2-D Liquid Atomization Mechanism


Christophe Dumouchel and Sebastien Grout

CNRS UMR 6614 CORIA, Universite et INSA de Rouen
Avenue de l'Universite BP. 12, 76801 Saint Etienne du Rouvray, France
Christophe.Dumouchel coria.fr


Keywords: Liquid spray, Atomization process, Multi-scale analysis, Scale entropy diffusion model



Abstract

This paper presents a new lead in analyzing liquid atomization process by investigating the temporal evolution of the liquid
system shape from the nozzle down to the spray. The atomization of the liquid sheet produced by a single orifice triple-disk
nozzle is investigated as a function of the liquid physical properties. This liquid atomization process is essentially 2-D and is
apprehended by image analysis. The description of the liquid system shape makes use of the cumulative surface-based scale
distribution. It is found that this distribution continuously evolves during the atomization process suggesting modeling the whole
process by the description of this distribution. This modelization is achieved by applying the scale entropy diffusion model
initially developed to investigate turbulent interfaces. In turbulence, as well as in atomization, the diffusion of the scale entropy
(quantity related to the cumulative surface-based scale distribution) is associated to a scale- and time-dependent fractal
dimension. The scale entropy diffusion model introduces the scale diffusivity and the scale entropy flux sink. The scale
diffusivity is measured and reports a clear organization with the injection pressure as well as with the liquid physical properties.
Furthermore, it is demonstrated that the scale entropy flux sink at initial time is dependent on the issuing liquid flow
characteristics that impose the initial perturbations on the liquid sheet. Therefore, this work demonstrates that the application of
the scale entropy diffusion model is physically representative to describe liquid atomization processes and spray formation. It
offers a new lead in apprehending and investigating this complex phenomenon that has been often disregarded because of a lack
of relevant approaches.


Introduction

The most commonly encountered process to produce a liquid
spray consists in ejecting a liquid flow in a gaseous
environment. Free of any parietal constraints, the liquid
system deforms thanks to the growth of initial perturbations
that can be initiated by the liquid flow itself or result from the
air liquid interaction. The perturbation growth stretches the
liquid until fragments detach from the bulk flow to form a
spray. During the whole mechanism, the ratio of surface to
mass in the liquid is increased (Mansour and Chigier, 1991).
The atomization process designates the stage from the
atomizer exit section down to the spray region. This stage is
of paramount importance since it is the vital link between the
liquid emerging from the nozzle and the fully developed
spray (Chigier, 2005). However, in many investigations
dedicated to the spray formation, this important stage is often
excluded mainly because of a lack of appropriate approaches
to apprehend this complex phenomenon. The work presented
in this paper intends participating to fill this gap.
Atomization mechanism visualizations abound in the
literature (see Dumouchel, 2008, for instance) and show that
the shape of the liquid system continuously evolves during
the process. This suggests that an atomization process could
be described from the evolution of the liquid system shape.
Note that from a physical point of view this idea is attractive
since such an approach should report a quantitative
information on the liquid-gas interface generation during the


process. The difficulty lies in the fact that atomizing liquid
system are objects with complex shapes and boundaries and
require sophisticated tool to be described such as the concept
of fractal dimension introduced by Kolmogorov (Hunt and
Vassilicos, 1991) generalized by Mandelbrot (1982) and
extensively used to investigate fluid turbulence (Sreenivasan
and Meneveau, 1986). The fractal dimension is an extension
of the Euclidean dimension and allows describing complex
boundaries. It is a measure of the tortuosity or roughness of a
surface or a line that is self-similar over a scale range, i.e.,
that presents similar structure when observed at different
magnifications. The concept of fractal has received little
attention so far to investigate atomizing liquid flow.
We can quote the first attempt due to Shavit and Chigier
(1995) that treated the case of an air assisted liquid jet.
Second and third attempts concerned the atomization of the
liquid flow produced by a simplified low-injection pressure
compound injector (Dumouchel et al. 2005b, Grout et al.
2007). Despite the difference of the investigated cases, these
studies clearly evidence that an atomizing liquid flow has
fractal properties. In Shavit and Chigier's investigation the
fractal nature of the air assisted liquid jet interface is a
consequence of the interaction of the air turbulence and its
eddy structure with the liquid, whereas in the two other
investigations (Dumouchel et al. 2005b, Grout et al. 2007)
the fractal characteristic of the interface is related to the
liquid flow turbulence. However, Grout et al. (2'" I) pointed
out that the fractal characteristic provided an incomplete





7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


liquid system description since the scale range representative
of the whole liquid system shape was larger than the one
where self-similarity was found. Such a result suggested
generalizing the traditional fractal analysis by considering
scale- and time-dependent fractal dimension (Queiros-Conde,
2003). This approach is called a multi-scale analysis.
Multi-scale analysis was performed on liquid sprays and
introduced the surface-based scale distribution (Dumouchel
et al. 2008). This distribution is an alternative to the diameter
distribution traditionally used to characterize spray drop size.
The advantage of the scale distribution is that it is explicitly
dependent on the shape of the droplets. The other advantage
of this distribution is that it can be applied to characterize
objects of any shape such as, for instance, a liquid flow
experiencing an atomization mechanism.
The objectives of the present work are to describe a liquid
atomization process by measuring the temporal evolution of
the surface-based scale distribution of the liquid system and
to examine the possibility of modeling this evolution by the
scale entropy diffusion model developed by Queiros-Conde
(2' 11) to describe turbulent interface. This model is
appropriate to systems that show a scale- and time-dependent
fractal dimension such as turbulent interfaces or atomizing
liquid flows.
The article is organized as follows. Section 2 introduces the
analysis considerations, namely, the surface-based scale
distribution and the scale entropy diffusion model. Section 3
is dedicated to the presentation of the experimental set-up,
diagnostic and working conditions. Then, the results of the
analysis are presented and discussed in the last section.

Nomenclature

d diameter (vtm)
dAw Analyzing Window position (m)
D observation scale (= 2r) (vtm)
Do reference scale (vtm)
LT total interface length (ktm)
N total number of droplets (-)
r observation scale (= D/2) (vtm)
S(r) delimited surface at scale r (ulm-i
ST total surface area (u 1I
S cumulative surface-based scale distribution (-)
s surface-based scale distribution (=dS(D)/dD)
(pm-1)
t time (s)
V, liquid flow mean velocity (m/s)
WeL liquid Weber number (-)
Greek letters
8 fractal dimension (-)
AP, injection pressure (MPa)
( scale entropy flux (-)
Y scale entropy (-)
oo local scale entropy flux sink (-)
X scale diffusivity (s-1)
PL liquid density (kg/m3)
a surface tension coefficient (N/m)
r7o initial amplitude (vtm)
IUL liquid dynamic viscosity (kg/ms)
A0 characteristic length scale (m)
Subsripts
BU breakup


G
Max
or
Spray


droplet
maximum
orifice
related to the spray


Analysis considerations

The surface-based scale distribution

A detailed definition of the surface-based scale distribution is
available in Dumouchel et al. (2" I') and is only summarized
in this section. The surface-based scale distribution has been
initially developed to characterize liquid spray droplets. This
distribution is an alternative to the traditional drop-diameter
distribution. Instead of attributing a single characteristic
length to each droplet, the scale distribution provides a
multi-scale description of each element as follows. Let us
consider a 2D image containing Nobjects of any shape as the
one illustrated in Fig. 1. Each object on the image is
described as follows. We consider the line defined by the
inner point located at a given distance r from the boundary of
the object (dash line in Fig. 1). For each distance r, called the
observation scale, the delimited surface S(r) (gray surface in
Fig. 1) is calculated. When the observation scale covers the
whole object, the delimited surface S(r) is equal to the object
total surface area Sr and the delimited surface S(r) is kept
equal to ST for any greater observation scale. For the set of N
objects, the cumulative surface-based scale distribution S(r)
is defined by:


N



ST,
s()


Figure 1: Description of an object of any shape at scale r

The first derivative of the cumulative surface-based scale
distribution is called the surface-based scale distribution and
is noted s(r) in the following. As for the traditional drop-size
distribution, the dimension of the function s(r) is equal to the
inverse of a length. In the following the observation scale r is
replaced by the parameter D = 2r. (Thus, the observation
scale that allows a circular object to be fully covered is equal
to its diameter.)
The definition of the cumulative surface-based scale
distribution is based on the application of the Euclidean
Distance Mapping (EDM) method which is a sausage
technique to determine the fractal dimension of a contour
(Grout et al. 2007). Thus, the representation of S(D) in a
logxlog coordinate system is identical to a


Paper No





7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Richardson-Mandelbrot plot from which the system fractal
characteristic can be obtained. According to EDM, the local
fractal dimension (D) in the scale space is given by:

d(ln(S(D)))
d(ln(D))

If 8(D) is a constant over a scale range, the contour is a fractal
in this scale range and 8(D) is its fractal dimension.
Contrary to the traditional drop-diameter distribution, the
surface-based scale distribution s(D) has not a bell shape. It is
a continuously decreasing function showing a maximum at
D = 0. This maximum is equal to half of the liquid-gas
interface length per unit liquid surface area, i.e.:


s(0)= (3)
2s,

The advantage of the scale distribution compared to the
traditional drop-diameter distribution is its dependence with
the droplet shape. For non-spherical droplets, several
equivalent diameters can be defined among which the
projected-area diameter, i.e., the diameter of the circle that
has the same projected area as the one of the droplet, is
commonly used. It was demonstrated that sets of droplets
having the same projected-area diameter distribution report
different scale distribution if the droplet shape is different
from one set to another (Dumouchel et al. 2008). In particular,
more and more deformed droplets are characterized by a
decreasing reference scale Do and an increasing amount of
interface, i.e., an increasing s(0).
Another advantage of the scale distribution is that its
application is not restricted to sets of dispersed objects such
as liquid spray droplets. This distribution can be used to
describe any system whose shape is so complex that its
complete description is impossible with a single
characteristic length. In particular, it can be applied to
provide a multi-scale description of an atomizing liquid flow
which is the objective of the present work. The scale
distribution of the liquid flow is locally measured on
visualizations from the nozzle exit down to the spray region.
This measurement provides the temporal evolution of the
scale distribution during the atomisation process that we
intend modeling using the scale entropy diffusion model
developed to investigate the evolution of turbulent interface
(Queiros-Conde 2003).

The scale entropy diffusion model

The scale entropy diffusion has been developed to model
temporal shape evolution of complex systems. First, the
system morphology is described thanks to the scale entropy 2,
which is a function of the scale D. The scale entropy is a
global quantity that decreases when the observation scale
increases towards the system reference scale Do, smallest
scale for which the scale entropy is equal to zero, i.e.,
2(Do) = 0. When the embedding dimension is 2 (as it is the
case in the present work), the scale entropy is related to the
cumulative scale distribution:


(D)= In S(D)


According to Eq. (2), the evolution of the scale entropy in the
logarithm scale space evolution that provides a multi-scale
description of the system is proportional to the local (in the
scale space) fractal dimension. In order to quantify how scale
entropy cascades through scale space, one introduces the
scale entropy flux (. Using the scale logarithm as a new
variable x (x = ln(D/Do)) as well as Eq. (2), the scale entropy
flux is given by:


dx
O(x)= = ( (x)- 2
dr


The flux variation throughout the scale space and expressed
by Eq. (5) results from a sink of scale entropy flux which is
quantified by introducing a scale entropy flux sink cx)
defined by unit of scale logarithm (it is in fact a scale entropy
flux density). Throughout the scale space, the entropy flux
continuity between x and x + dx is expressed by:


(x + dx)- (x)- w(x)dx= 0


And the combination of Eqs.(5) and (6) leads to the following
diffusion equation for the scale entropy:


d2(x)
S-cox)= 0
dx


A purely fractal system is characterized by a constant scale
entropy flux (see Eq. (5)) and therefore corresponds to a scale
entropy flux sink Ox) = 0 (all over the scale range for which
the system is fractal). In other words, systems with a
scale-dependent fractal dimension are characterized by the
existence of a function gx) whose form is not imposed and
can remain general. Queiros-Conde (2 1"') emphasizes the
analogy between Eq. (7) and the one-dimensional conduction
equation: the scale entropy 2(x) would correspond to the
temperature, the scale entropy flux (x) to a quantity
proportional to a heat flux and Ox) to a quantity proportional
to a volumetric heat sink, which would be space dependent.
He then considered the more general situation where the
scale entropy 2(x) of the system is also time dependent and
becomes 2(x,t). This case occurs when the local fractal
dimension is time dependent ((x,t)) due to non-stationary
fluctuations or for some experiments where multi-scale
construction is observed in time. In this case, the scale
entropy flux sink becomes also time dependent ((ox,t)) and
carrying the analogy with heat equation, Queiros-Conde
(2 1111) suggests describing time-dependent scale entropy
with the following diffusion equation:


a2(x,t) xt 1 I2 (x,t)
ax 2 at


where the scale diffusivity X characterizes diffusion of scale
entropy through scale space. This parameter is related to a
scale range quantified by a value of Ixl and a diffusion time


Paper No





Paper No


Z*, i.e., =|xl' /* z* can be interpreted as the
characteristic time needed by a perturbation to propagate
over the scale range quantified by x. Queiros-Conde (2i 1 1)
examined the simplified case where the term 21(x,ft)/x2 is
small compared to the two other terms for a given scale range
characterized by x. In this scale range, the fractal dimension
becomes mainly time dependent: it indicates a local fractal
feature of the object (8(x,t) = 8t)) and the term 2lx,t)/,x2
can be omitted in Eq. (8). This reduced diffusion equation
indicates that the temporal decrease of the scale entropy flux,
which is conditioned by the scale diffusivity X, is balanced by
the scale entropy flux sink. (In a heat diffusion problem, this
corresponds to a small Biot number.) By modeling the scale
entropy flux sink as c(x,t)= (8- (t))/|x| where 8 is the
fractal dimension imposed at scale Do by an external
mechanism, and by using Eqs. (5) and (8), Queiros-Conde
(2 11".) obtained the following fractal dimension temporal
evolution in this simplified case:


where the characteristic time z* = I 12/X. Queiros-Conde
(2 1"'.) found this simplification appropriate to characterize
scalar passive turbulent interface.

Description of the Rayleigh instability with the scale
entropy

As an illustration, the initial and final scale entropy
corresponding to the 2-D analysis of the Rayleigh instability
is presented in this section. The Rayleigh instability occurs
on low velocity cylindrical liquid jet. It is characterized by
the growth of a axisymmetric sinusoidal wave whose wave
length is equal to 4.51d where d is the jet diameter. When the
amplitude of the wave is equal to the radius of the jet, a drop
is formed. One of the characteristics of the Rayleigh
instability is that all produced drops have the same diameter.
By applying the mass conservation principal from the initial
to the final stage, it can be demonstrated that the diameter of
the drop is d = 1.89d.
Considering a volume of the jet that produces a single drop,
the projection of its initial state gives a rectangular that is d in
width and 4.51 d in height and the projection of its final state
gives a circle with a diameter equal to 1.89d. In both
situations, the cumulative scale distribution is easy to
calculate and the scale entropy can be deduced from Eq. (4).
Using the variable x = ln(D/d) where d is actually the system
initial reference scale, it can be shown that:


A(x)= -

I(x)=- x-ln(1.89)+ln 2- .


att=0

att=tf


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

scale space. This diffusion lasts until a stable state is reached.
This state is associated to the stationary diffusion equation,
i.e. Eq. (6). By integrating this equation over the whole scale
range, namely for x ranging from -oo to +c0, we note that the
final stable state is characterized by:


Sco(xx 1=


Scale entropy 2x) (-)


-4 -3 -2 -1 0 1


Scale x = In (D/d)
Figure 2: Initial (t = 0) and final (t = tf) scale entropy of the
2-D analysis of the Rayleigh instability (Eq. (10))

In the small scale range, Fig. 2 shows that the scale entropy
diffusion model beneficiates of an entropy flux condition at
any time, i.e., (x) = -1 when x -* -o. This condition comes
from the fact that whatever the shape of the system, there is
always a small scale under which the system boundary is
seen as a straight line, i.e., no perturbation is visible at this
scale. For the scale range for which the scale entropy flux
condition applies, the exchange term a2'/,x2 in Eq. (8) is
equal to zero. Furthermore, since the scale entropy flux is
independent of the scale and of the time, the scale entropy
temporal derivative is independent of the scale and so is the
scale entropy flux sink. Thus, for the small scale range in
which the scale entropy flux boundary applies, the diffusion
equation reduces to:


1 Ia(- o,t)
Z at


Equation (12) shows that the slight reduction of the scale
entropy in the small scale range during the whole process is
controlled by the scale entropy flux sink, which is a function
of time only, and by the scale diffusivity. Using Eq. (4) it can
be demonstrated that:


aS(- ,ot)
at


Figure 2 shows the scale entropy at initial and final time
given by Eq. (10). At t = 0, we note a scale entropy flux
discontinuity at x = 0: for x < 0, (x) = -1 and for x > 0,
(x) = 0. According to Eq. (6), this flux discontinuity
imposes an infinite value of a0). These conditions are
representative of the instable characteristic feature of the
initial system and generate scale entropy diffusion in the


s(0,t)
s(0,t)


where the dot indicates a temporal derivative. Considering
the analytical case examined in this section, Eq. (13) can be
checked by using Eq. (3) and Eq. (10). This demonstration is
not provided here.
The scale diffusivity X for the Rayleigh instability can be
estimated by considering the time required by the system


8(t) = So (S, (t = 0))e -l






Paper No


reference scale to increase from d to 1.89d. This time is the
breakup time. According to Rayleigh (1878), it can be
calculated by using the temporal growth rate of the most
unstable wave, i.e.:


In d
t n23 (14)
Bu 0.97 V


where qo is the initial amplitude of the perturbation. Thus, the
Rayleigh instability scale diffusivity can be estimated by:


0.971n2 (1.89)

in d p) Ld3
270 _


We see that the scale diffusivity is dependent on the jet initial
shape (through the parameters d and qo) and on the liquid
physical properties and that it is independent of the jet
velocity. This makes sense since the Rayleigh mechanism is a
capillary instability: the initial perturbation as well as the
physical mechanism that controls its growth are both
independent of the velocity. In a more general situation, we
expect the scale diffusivity to be dependent of the liquid
velocity. This is illustrated in the following of this paper that
examines an experimental situation where the initial
perturbations are conveyed by the liquid flow and are
therefore highly dependent on the liquid velocity.

Experimental investigation

This section summarizes the experimental setup and
diagnostic. More details are available in Grout et al. (2" ) )
and Dumouchel et al. (2" I'). A single injector, with a unique
discharge orifice, is used. The injector nozzle is a triple-disk
nozzle whose geometry is inspired from compound injector
encountered in low-pressure port-fuel injection engines.
Figures 3-a and 3-b provide a sketch of the triple-disk nozzle
whose orifice diameter is do, = 180 vtm. The liquid enters
disk 1, expands in the cavity disk (disk 2) and discharges
through disk 3 orifice. As soon as the liquid issues from the
nozzle, the flow expands and stretches as a 2-D liquid sheet.
Figure 3-c visualizes the liquid sheet at the nozzle exit. It can
be seen that perturbations of different characteristic
length-scale appear on the sheet edges in the near-nozzle
region. Then, the sheet more and more deforms and
reorganizes as a ligament network that eventually breaks into
droplets. Two main characteristics of the liquid flow at the
nozzle exit are responsible for this behaviour (Dumouchel et
al. 2005a). The nozzle internal geometry imposes drastic
flow deflections and favors the development at the exit
section of a counter-rotating double-swirl, characterized by a
non-axial kinetic energy, as well as a consistent turbulent
level. The double-swirl is responsible for the formation of the
liquid sheet at the nozzle exit and the turbulence initiates
perturbations. As going downstream, the double-swirl effect
weakens and liquid sheet contraction occurs due to surface
tension forces. The effect of this contraction is conditioned
by the deformation initially imposed by the turbulence and
results in the ligament network production. The rather good
efficiency of this atomization mechanism is due to the


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

shaping of the issuing liquid flow as a flat liquid sheet: the
characteristic length scale of a sheet (its thickness) being less
than the one of cylindrical jet (its diameter) that would
produce the injector in the absence of the double-swirl
structure.



Disk 2
D Eccentncty






-b- -c- -d-
Figure 3: a side view of the nozzle, b top view of the
nozzle, c visualization of the liquid sheet produced at the
nozzle exit (water, AP, = 0.4 MPa), d two gray-level image
of image c

Liquid L UL
Heptane 704 0.41 103 0.0206
Water 991 1.00 103 0.0720
Water-Ethanol 1% 986 1.14 10- 0.0659
Water-Ethanol 10% 972 1.43 10- 0.0461
Table 1: Liquid physical properties (percentages indicate
weight proportion)

Four different liquids are used. Their physical properties are
listed in Table 1. The injection pressure is kept low and
ranges from 0.15 to 0.5 MPa except for heptane, the lowest
surface tension fluid, for which it varies from 0.05 to
0.25 MPa. The gaseous Weber number of the issuing liquid
flows (based on the gaseous density, the orifice diameter and
the average liquid velocity) never exceeds 2.7 and indicates a
negligible contribution of the aerodynamic forces to the
atomization mechanism that is mainly controlled by the
surface tension forces. Experimental evidences of this very
point have been reported in a recent investigation
(Dumouchel and Grout, 2010).
As the atomization mechanism is orientated in a plane, all
liquid structures and droplets are inscribed in this plane. Thus,
a description of this mechanism can be satisfactorily
approached by 2-D visualizations. Backlight images of the
flow are taken with a light source that has a very short
pulse-duration (11 ns) in order to freeze the liquid system. A
camera with a high number of pixels (3040x2016 pixel) is
used to reach a good spatial resolution. The image covers a
field of 10.5x7 mm2 corresponding to a spatial resolution
equal to 3.47 utm/pixel. An example of image is shown in Fig.
3-c. The optical arrangement provides a depth of field equals
to 7 mm. This depth is much greater than the thickness of the
atomizing liquid flow. Thus, no liquid fragments and droplets
are out-of-focus and they are all visualized with a very good
contrast. Finally, 150 images are taken and analyzed for each
working condition. These images were treated in order to
provide two gray-level images as the one shown in Fig. 3-d
where the liquid appears in white on a black background. The
details of the image treatment protocol are described in Grout
et al. (2" )) and Dumouchel et al. (2" I').
The image analysis consists in measuring the cumulative
surface-based scale distribution S(D) of the liquid system






Paper No


during the atomization process. To achieve this, portion of
the liquid phase delimited by a rectangular Analyzing
Window (AW) is analyzed as a function of the position of the
window that is slid from the nozzle exit down to the spray
region. However, since the experimental protocol does not
provide a temporal resolution of the liquid system behaviour,
a representative spatial evolution of S(D) can be statistically
approached by averaging the information reported by the 150
images available for each operating conditions.
The AW height has been fixed to 200 pixels (= 697 vtm). The
influence of this parameter has been investigated in a
previous investigation (Dumouchel and Grout, 2009). It was
found that the cumulative scale distribution is independent of
the AW height when it is less or equal to 200 pixels. This is
due to the fact that the average cumulative scale distribution
is representative of a temporal averaging and not of a spatial
averaging. Thus, the choice of an AW height of 200 pixels
appears appropriate. The position of the AW is located by the
distance dAw from the nozzle exit and the AW middle line. In
the following, the position of the AW is replaced by the
equivalent time defined by:


t dA
V.


This equivalent time is assumed to be physically
representative because the Froude number is high enough
(> 30) to prevent any acceleration due to gravity and because
the surface fraction of the small liquid drops that could be
slowed down by air drag is less than 10% (Dumouchel and
Grout, 2009).


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

where the plateau is perceptible, the local fractal dimension is
almost independent of the scale denoting a fractal property of
the system in these scale and time intervals. This fractal
property characterizes the tortuosity of the interface and
ignores the system as a whole (Grout et al. 2007), and the
corresponding fractal dimension is a textural fractal
dimension as defined by Kaye (1989). It was reported (Grout
et al. 2007) that this fractal dimension correlates with the
liquid Reynolds number evidencing the impact of turbulence
on the liquid-sheet edge tortuosity in the near nozzle region.
For times greater than 200 vts, the plateau is not perceptible
anymore and 8D) monotonously increases. As for S(D),
AD) stabilizes For times greater than 300 vts.


Scale D (gm)
Figure 4: Temporal evolution of the cumulative
surface-based scale distribution S(D) (water, AP, =0.4 MPa)


Results


Figure 4 shows an example of measured cumulative scale
distributions (water, AP, = 0.4 MPa) as a function of the
equivalent time. The smallest time corresponds to the
position of the AW that is the closest to the nozzle exit
(dAw = 1.07 mm) and the greatest time corresponds to the
window localized in the spray region. The interesting result
shown in this figure is that the distribution S(D) continuously
evolves with time from the nozzle exit down to the spray
region. This temporal evolution describes the whole
atomization mechanism. In the small scale range
(D < 50 vtm) S(D) reports a 'parallel increase' with time that
illustrates the increase of surface fraction covered at these
scales during the atomization process, i.e., the emergence of
smaller and smaller liquid structures. When t> 300 vts, the
scale distribution is not time-dependent anymore and
characterizes the final spray. Being presented in a logxlog
plot, the cumulative scale distribution shown in Fig. 4 is
similar to the scale entropy (see Eq. (4)). Furthermore,
according to Eq. (2), the slopes of the curves shown on Fig. 4
allow the local fractal dimension AD) in the scale space to be
determined. This dimension is presented in Fig. 5.
Figure 5 illustrates clearly the fact that an atomization
mechanism is described by a scale- and time-dependent
fractal dimension. The local fractal dimension evolves from 1
to 2 when covering the liquid system scale range. For times
less than 200 vts, 8(D) first increases with D, reaches a
plateau whose inclination increases with time, and sharply
increases to reach 2 at D = Do. Within the scale interval


S(D) (-)


1 10 100 1000


Scale D (gm)
Figure 5: Temporal evolution of the scale-dependent fractal
dimension AD) (water, AP,= 0.4 MPa, to= 136.7 vts,
Dospray = 219 tm).

The cumulative scale distribution and the corresponding
local fractal dimension allow characteristic length and scale
times to be determined. Among the characteristic length
scales, the system reference scale Do (smallest scale for
which S(D)= 1 and 8(D) = 2) is interesting to consider.
Figure 6 shows the temporal evolution of the reference scale
during the atomization mechanism for water as a function of
the injection pressure.
At t = 0, the liquid is at the nozzle exit section where the
reference scale is equal to the diameter of the orifice, i.e.,
180 vtm. When time increases, Do increases, becomes a
maximum DOMa, and decreases to finally reach a constant
value Dospray. The increase of Do at the nozzle exit is a






Paper No


manifestation of the liquid system stretching in the plane of
observation due to the action of the issuing flow double-swirl
structure. This evolution is controlled by the dynamic of the
issuing flow and depends on the injection pressure. The
maximum reference scale DoM. is reached when the
stretching process ends because of modifications of the liquid
system shape imposed by surface tension forces. At this stage,
liquid gulfs and ligaments are created and prevent any further
increase of the reference scale. Then, the intensification of
the atomization mechanism leads to the formation of smaller
and smaller liquid structures, which is represented by a
continuously decreasing reference scale. The asymptotic
value Dospray, which corresponds to the reference scale of the
spray, is reached when the atomization process is completed.
The scale Dospray and the time at which it is reached are both
functions of the injection pressure. They are also dependent
on the liquid physical properties. Finally, it is interesting to
note in Fig. 6 that an increase of the scale DoMa is
accompanied by a decrease of the scale Dospray. This behavior
makes sense since an increase of DOMwa reveals a decrease of
the thickness of the liquid sheet and thinner liquid sheets
produce smaller drops.


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

The determination of the scale diffusivity makes use of a
characteristic feature of the local fractal dimension 8(D,t). As
shown in Fig. (5), this dimension reports a value independent
of the scale in a scale range around Dospray and in a time
interval around to. As noted above, this characteristic is
related to a fractal property of the liquid system within these
scale and time intervals. This also shows that at these scale
and time intervals, the exchange term in the diffusion
equation, i.e., 22'(x,t)/dx2 is very small compared to the
other terms and can be omitted. Thus, the simplified solution
provided by Eq. (9) should be applicable. This solution can
be rewritten as:


In(Y)


where the function Y is given by:


y_ (t)- So
s(t=0)- ,


Reference scale Do (gm)
1200
IV iP 0 25 MPa
1000 V*
7o v = o0 40 MPa
800 z P = o 50 MPa
800 *Ooos

600 \7 Water
a
400 *
*v
200 l u muS lni

0
0 100 200 300 400 500


Time t (gs)
Figure 6: Temporal evolution of the reference scale Do
(water, influence of the injection pressure)

As said above, characteristic times can be obtained from the
temporal evolution of the cumulative surface-based scale
distribution. For example, referring to Fig. 6, two
characteristic times can be defined, namely, the time to at
which Do = DoM.a and the time tl at which Do = Dospray. to
corresponds to the time at which the liquid sheet stops to
expand whereas time ti is the time at which the shape
evolution of the liquid system is almost completed.
The objective of the present work is to demonstrate the
possibility of describing the temporal evolution of the
cumulative scale distribution with the scale entropy diffusion
model established by Queiros-Conde 2 11 1') and introduced
in Section 2 of this article. First, as noticed above, the
cumulative scale distribution and the scale entropy are
equivalent functions (Eq. (4)). Second, the fractal dimension
associated to the evolution of the cumulative scale
distribution is scale- and time-dependent (see Fig. 5). The
scale entropy diffusion model has been established for such a
situation. This model suggests describing the evolution of the
scale entropy by the diffusion equation given by Eq. (8) and
that introduces the scale diffusivity. To demonstrate the
relevance of this model, we intend first to determine and to
investigate this parameter.


The function Y is calculated for the specific scale D = Dospray,
i.e., (t) in Eq. (18) is equal to 8Dospray,t). 8 is the fractal
dimension for this scale at infinite time. By definition of
Dospray, o = 2. Finally, (t = 0) is assumed to be equal to 1 to
be in agreement with the simplified model that lead to Eq. (9).
An example of the evolution of the function Y is shown in Fig.
7 for the case of water used at an injection pressure of
0.4 MPa.


Water
AP, = 0.4 MPa
D,,y 219 gm


Sto 137 gs


0 50 100 150 200 250


Time t (gs)
Figure 7: Temporal evolution of the function Y given by Eq.
(18) for water tested at P,= 0.4 MPa (Dospray= 219 vtm,
to= 137 vts)

It can be seen in Fig. 7 that around time to, the function Y
reports a linear dependence with time in agreement with the
solution obtained by the simplified case and given by Eq.
(17). Thus, the diffusion time z* introduced by this equation
can be determined. According to Queiros-Conde, this
characteristic time is associated to the scale range extending
from the scale for which Y has been calculated, namely
Dospray is the present case, to the reference scale of the system.
By definition, the reference scale of the liquid system around
time to where the simplified model applies is equal to DoM..
It is interesting to note in Fig. 6 that this reference scale is
almost constant in the vicinity of time to. This is an
hypothesis of the simplified model described by Eq. (9).






Paper No


Therefore, in agreement with the simplified model due to
Queiros-Conde, the scale diffusivity can be estimated by:


DoM
In' o? "
= n- (19)


For each operating condition, Eq. (20) was applied to
determine the scale diffusivity using the diffusion time
reported by Fig. 7. The results are presented in Fig. 8 which
shows the scale diffusivity for each fluid as a function of the
injection pressure.


30000

25000

20000

15000

10000

5000


Scale diffusivity Z(s-1)


0.1 0.2 0.3 0.4 0.5


Injection pressure AP, (MPa)
Figure 8: Measured scale diffusivity for each fluid as a
function of the injection pressure

The interesting result shown in Fig. 8 is the very well
organization of the scale diffusivity with the injection
pressure as well as with the liquid physical properties. The
scale diffusivity scale increases with the injection pressure or
when the liquid surface tension decreases. An increase of the
scale diffusivity can be associated to a more rapid diffusion
of scale entropy in the scale space and therefore to an
increase of the small scale representativity. This behavior is
in agreement with the production of smaller and smaller
droplets when the injection pressure increases or when the
surface tension coefficient decreases. These results are
relevant and plead to the favor of the use of the scale entropy
diffusion model to describe the evolution of the shape of an
atomizing liquid system. Figure 8 reports a linear relationship
between the scale diffusivity and the injection pressure. It has
been demonstrated in a recent investigation (Dumouchel and
Grout, 2010) that this behavior is valid when the liquid
Weber number of the liquid system defined by:


PLq do0
a


ranges from 300 to 2000. This interval corresponds to the
present experimental working conditions.
Using the experimental scale entropy (given by Eq. (4)) and
the scale entropy flux (given by Eq. (5)), the scale diffusivity
allows the scale entropy flux sink to be calculated. An
example of such a calculation is illustrated in Fig. 9.
In Fig. 9, x is calculated on the basis of the initial liquid
system reference scale, i.e., x = ln(D/do,). The smallest time
corresponds to the position of the AW that is the closest to the


-0.2

-0.4


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

nozzle exit (dAw= 1.07 mm) and the greatest time
corresponds to the window localized in the spray region.
When t increases, we clearly see in Fig. 9 that the reference
scale increases reaches a maximum and decreases as
evidenced in Fig. 6. The maximum reference scale DouM is
reached at to = 137 uts. For this time, we note that the scale
entropy scale flux is constant in the vicinity of the value of x
corresponding to the spray reference scale, i.e., x = 0.28.
According to Eq. (5), we see that this behavior is the
reproduction of the fractal property of the liquid system
evidenced in Fig. 5. The examination of the function ax,t) in
Fig. 9 is interesting. This function shows two parts: a
maximum in the large scale range that collapses as a function
of time, and a rather constant value in the small scale range.
Between these two parts, ax,t) is either negative or positive
according to the time.


"x,t) (-)


* 59ps
7 137 s


233ps


in ,, li,




q(x t) -.,
^,0(" ^^^s^pr-.
U -* I /


-0.6

-0.8 ** m


-1 0 1 2
x ln(D/d,) (-)
Figure 9: Evolution of the scale entropy (top), the scale
entropy flux (middle) and the scale entropy flux sink
(bottom) during the atomization process (water,
zP,= 0.4 MPa)

As explained in Section 2 of this paper, the function (x,t)
always exhibits a value independent of the scale in the small
scale range. This value, noted co(- ,t) is given by Eq. (12).
This value is visible in Fig. 9 for the three first time, i.e., 59,
137 and 233 vts. For the fourth time, the atomization process
is completed. Therefore, the diffusion equation reduces to Eq.
(6) which, in the small scale reports co(- ,t) = 0. This is
not visible in Fig. 9 because of the limitation of the spatial
resolution in the small scale range. To demonstrate this, we
can make use of the specific property of the scale entropy
flux sink given by Eq. (11) and that applies when the
atomization process is completed. In this work, we






7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


introduced the function Ot) by:


l W67 ) pL
0.435161 n' ( ) WeL F7 L O


c(t)= fcx,t)dx


where xm,, is imposed by the smallest scale that the spatial
resolution allows approaching, i.e., xm, =-3.26. Figure 10
presents examples of the temporal evolution of ot) for water
at several injection pressure. This figure shows that, as
expected, at) evolves towards an asymptotic value when
time increases but that this value is less that the theoretical
value which is 1 according to Eq. (11). This clearly
demonstrates a lack of spatial resolution in the small scale
range of the present investigation. However, we can see that
the limit reached by Ot) is rather close to 1.
In conclusion, we can say that the value of scale entropy flux
sink in the small scale range, i.e., co(-o,t), first increases
with time and then decreases to reach zero when the process
is completed.

a(t) (-)
5
0 25 MPa
4 0 30 MPa
S040 MPa
SVV9VW 0 50MPa
3 t) = 1
V
v*

1


where Ao = 10 3/02 2 being given in meter and a in
N/m. The comparison between the measured and the
calculated scale diffusivities is presented in Fig. 12 that
reports a very good agreement.




















Figure 11: Visualization of the liquid flow issuing from the
atomizer with a Water-Glycerol 20% mixture at
AP, = 0.35 MPa (V, = 15.4 m/s, WeL = 651)


20000


0 100 200 300 400 500 600


Time t (gs)
Figure 10: Evolution of at) defined by Eq. (21) as a
function of time (Water, influence of the injection pressure)

The negative values of the function Ox,t) evidenced in Fig. 9
for the smallest time and in the intermediate scale range is
interesting to consider. To illustrate the physical meaning of a
negative value of the scale entropy flux sink, we consider the
case of the liquid sheet shown in Fig. 11. This liquid sheet is
produced by using a Water-Glycerol 20% mixture at an
injection pressure of 0.35 MPa. The physical properties of
this fluid are: pL = 1052 kg/m3, a= 0.0685 N/m,
/L = 2.06 10-3 kg/ms. We see in Fig. 11, that for this working
condition, the atomizer produces an unperturbed and smooth
liquid sheet, that contracts under the action of the surface
tension forces to reorganize as a cylindrical jet. This denotes
that the presence of a sufficiently developed double swirl
structure at the nozzle exit to induce the production of a
liquid sheet, but an insufficient level of turbulence to trigger
the disintegration of this liquid sheet as observed with the
other fluids. Note here, that the absence of turbulence is
consistent with the increase of the liquid dynamic viscosity.
To analyze the situation presented in Fig. 11, we need first to
estimate the scale diffusivity. This is achieved by using the
results of a recent investigation (Dumouchel and Grout,
2010) that reported the following expression for the scale
diffusivity from the analysis of the results obtained for the
four fluids listed in Table 1:


16000

12000

8000 Heptane
SV Water
SWater-Eth 1%
4000 m Water-Eth 10%
Linear Reg
0
0 4000 8000 12000 16000 20000
Measured Z (s-1)
Figure 12: Comparison between the measured scale
diffusivities and those calculated with Eq. (22) (from
Dumouchel and Grout, 2010)

Equation (22) and Fig. 12 report no dependence between the
scale diffusivity and the liquid dynamic viscosity when this
parameter ranges from 0.41 10-3 to 1.43 10-3 kg/ms. As a first
approximation, we assume that Eq. (22) applies for the
Water-Glycerol 20% mixture even if the dynamic viscosity of
this fluid is greater than those for which Eq. (22) was
established. This assumption is not too penalizing if the
analysis is limited to the description of qualitative behavior.
The application of Eq. (22) for the water-glycerol 20% at
0.35 MPa reports a scale diffusivity equal to 9450 s-'. This
value allows the scale entropy flux sink to be calculated as a
function of time. Figure 13 shows the temporal evolution of
the scale entropy and of the scale entropy flux sink for the
liquid system shown in Fig. 11.


Paper No






Paper No


2(x)(-)


I i


70ps
2 92 ps
S v 160 ps
1 0o000 A 273 ps
386 ps
477 ps
567 ps


o0x,t) (-)


-2 -1


x ln(D/do) (-)
Figure 13: Temporal evolution of the scale entropy (top) and
of the scale entropy flux sink (bottom) for the liquid system
shown in Fig. 11 (Water-Glycerol 20%, AP, = 0.35 MPa)

Except in the vicinity of the reference scale, we see that, at all
times, the scale entropy reports a linear dependence with the
scale x with a slope equal to -1. This behavior is due to the
absence of perturbation on the liquid sheet edges. Thus, the
fractal dimension of the liquid contour in the small scale
range is equal to 1, which according to Eq. (5) is equivalent to
a scale entropy flux equal to -1. In this specific scale range,
the scale entropy increases when t e [70 pgs; 273 vts] and
decreases when t e [273 vts; 567 vts] and the reference scale
of the liquid system follows the same evolution. Thus,
Do = DoMax at to = 273 vts, time at which the liquid sheet is the
widest. The increase of the scale entropy indicates that the
small scales are less and less representative of the whole
system shape, and the decrease of the scale entropy after to
indicates the opposite. Since the scale entropy flux is
independent of the scale and of the time in the small scale
range, the scale entropy diffusion equation reduces to Eq.
(12). Thus, the variation of the scale entropy is controlled by
the value of c-co,t). When t e [70 vts; 273 vts], this value is
negative (see Fig. 13). A negative scale entropy flux sink
corresponds to a scale entropy flux source, and the locally
entering scale entropy flux imposed by (-c-,t) is locally
stored inducing an increase of the scale entropy all over the
scale range representing the liquid system and therefore an
increase of the reference scale. After to, c-co,t) is positive
and corresponds to a scale entropy flux sink that induces a
decrease of the scale entropy with time and therefore a
reduction of the reference scale.
Thus, the liquid sheet formation and its subsequent
contraction are both controlled by the values of a-c-,t).
Bearing in mind that the liquid sheet production is related to
the double-swirl intensity of the liquid flow at the nozzle exit
(Dumouchel et al., 2005a) the initial value of c(-o,t) is
clearly related to this characteristic feature. Furthermore,
since the sheet contraction results from the action of the
surface tension forces, the evolution of g-co,t) depends on


I IU


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

the physical phenomenon that controls the shape evolution of
the liquid system.
If we now go back to the function cx,t) shown in Fig. 9, we
may conclude that the negative values observed at the
smallest time (t = 59 pts) and for the intermediate scales are
related to the double-swirl structure of the issuing liquid flow.
However, as noticed above, for the small scale, gx,t) is
always positive: this positive scale entropy flux sink induces
a reduction of the scale entropy with time (see Fig. 9) which
indicates that these small scales are more and more
representative of the whole shape of the liquid system. The
positive values of Ox,t) in the small scale range controls the
tortuosity of the liquid sheet edges, tortuosity that correlates
with the turbulence of the liquid flow issuing from the nozzle
(Grout et al. 2007). This demonstrates that the initial function
cx,t) characterizes the initial liquid system. This makes
sense in the present situation for which the initial
perturbations of the atomization process are all imposed by
the liquid flow characteristics.
These considerations show that the scale entropy flux sink
#x,t) is a key parameter in the scale entropy diffusion model
and further investigations are required to propose a
modelization of this function whose form may be kept
general a priori.

Conclusions

This paper presents the first attempt in describing a liquid
atomization process on the basis of a multi-scale analysis that
consists in investigating the shape evolution of the liquid
system from the nozzle exit down to the spray region. The
shape of the liquid flow is characterized by the cumulative
surface-based scale distribution. The first interesting result is
that this distribution continuously evolves during the
atomization process. This result suggests modeling the
atomization process by describing the temporal evolution of
the cumulative surface-based scale distribution.
This work shows that this could be achieved by using the
scale entropy diffusion model that has been developed to
describe turbulent interface. The evolution of such interfaces
is associated to a scale- and time-dependent fractal
dimension. In atomization, the cumulative surface-based
scale distribution reports a similar characteristic feature.
Furthermore, the scale entropy and the cumulative
surface-based scale distribution are related to each other.
Thus, the scale entropy has been applied to describe the
shape evolution of the liquid system during the atomization.
The scale- and time-dependent diffusion equation of the
model introduces a scale diffusivity, parameter that
characterizes diffusion of scale entropy through scale space.
This parameter was measured in the present work. To our
knowledge, this was the first time such measurements were
performed. The results show a very organized dependence
between the scale diffusivity, the injection pressure and the
liquid physical properties, which gives credit to the use of the
scale entropy diffusion model.
This model also introduces the scale entropy flux sink
function which controls the scale entropy flux variation
throughout the scale space. The determination and analysis
of this function clearly evidences the relationship between
this function and the characteristics of the liquid flow issuing
from the nozzle in a situation where these characteristics
trigger the initial perturbations of the liquid system. This






Paper No


result also demonstrates the physical relevance of the model.
Thus, the scale entropy diffusion model appears as a new
promising lead to describe liquid atomization processes. It
introduces new characteristics such as the scale diffusivity
that could be used to identify, classify or understand
atomization processes. It must be added here that the
atomization process is a stage of paramount importance in
spray formation since it is the vital link between the nozzle
and the resulting spray. However, the analysis of this stage is
often disregarded due to a lack of appropriate tool and model.
It is believed that the scale entropy diffusion model could
advantageously fill this gap.

References

Chigier, N. The future of atomization and sprays. Proc. Of
ILASS-Europe 2005, Orleans, France, 5-7 Sept. 2005

Dumouchel, C., Cousin, J., Triballier, K. On the role of the
liquid flow characteristics on low-Weber-number
atomization processes. Exp. Fluids. Vol. 38, 637-647 (1" '. i)

Dumouchel, C., Cousin, J., Triballier, K. Experimental
analysis of liquid-gas interface at low-Weber-number:
interface length and fractal dimension. Exp. Fluids. Vol. 39,
651-666 (2005b)

Dumouchel, C., Cousin, J., Grout, S. Analysis of
two-dimensional liquid spray images: the surface-based scale
distribution. Journ. of Flow Visualization & Image
Processing. Vol. 15, 59-83 (2'" ",)

Dumouchel, C. On the experimental investigation on primary
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Dumouchel, C., Grout, S. Application of the scale entropy
diffusion model to describe a liquid atomization process. Int.
Journ. Multiphase Flow. Vol. 35, 952-962 (2'" '1)

Dumouchel, C., Grout, S. On the scale diffusivity of a 2-D
liquid atomization process analysis. Submitted to Physica A.
(2010)

Grout, S., Dumouchel, C., Cousin, J., Nuglisch, H. Fractal
analysis of atomizing liquid flows. Int. Journ. Multiphase
Flow. Vol. 33, 1023-1044 (2 '"-)

Hunt, J.C.R., Vassilicos, J.C. Kolmogorov's contribution to
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Kaye, B.H. A random walk through fractal dimensions, VCH,
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Mandelbrot, B. The fractal geometry of nature. WH Freeman
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Mansour, A., Chigier, N. Dynamic behavior of liquid sheets.
Phys. Fluids. Vol. 3, 2971-2980 (1991)

Queiros-Conde, D. A diffusion equation to describe scale-


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

and time-dependent dimensions of turbulent interfaces. Proc.
R. Soc. Lon. A. Vol. 459, 3043-3059. .2" i1)

Rayleigh, L. On the capillary phenomenon of jets. Proc. R.
Soc. Lon., Vol. 29, 71-97 (1878)

Shavit, U., Chigier, N. Fractal dimensions of liquid jet
interface under break-up. Atom. Sprays. Vol. 5, 525-543
(1995)

Sreenivasan, K.R., Meneveau, C. The fractal facets of
turbulence. JFM. Vol. 173, 357-386 (1986)




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