Air-blast atomization of a liquid film

Abstract

Air-blast atomizers are extensively used for a variety of purposes. Due to its complexity, the atomization mechanism has not been elucidated. In this study, a mechanistic model is proposed to predict the droplet diameter distribution based on the atomization process of a planar liquid film with co-current gas flows, and its validity is examined by comparing the estimated and measured droplet diameters using high-speed image analysis and laser measurement. As a result, using high-speed imaging, we clarified that the bag film rupture is caused not by the turbulence of the gas flow but by the impact of floating droplets on the liquid film of the expanding bag when the film is thin enough. The average thickness of the liquid film at the bag breakup is of the order of micrometres and varies greatly, resulting in a dispersed distribution of droplet diameters. After the film ruptures, the bag film shrinks towards its transversal and vertical rims due to surface tension, forming large-diameter ligaments. During the contraction process of the bag film, tiny droplets of the order of micrometers are formed at the edge of the perforation. Finally, the remaining ligaments with large diameters fragment into large droplets with submillimetre diameters. The good agreement between the measured and predicted droplet diameter distributions validated the mechanistic model.

Full text

J. Fluid Mech. (2024), vol.985, A36, doi:10.1017/jfm.2024.279
Air-blast atomization of a liquid film
Ippei Oshima1,†and Akira Sou2
1Institute of Fluid Science, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
2Graduate School of Maritime Sciences, Kobe University, 5-1-1 Fukaeminami-machi, Higashinada-ku,
Kobe, Hyogo 658-0022, Japan
(Received 26 August 2023; revised 11 March 2024; accepted 11 March 2024)
Air-blast atomizers are extensively used for a variety of purposes. Due to its complexity,
the atomization mechanism has not been elucidated. In this study, a mechanistic model is
proposed to predict the droplet diameter distribution based on the atomization process of
a planar liquid film with co-current gas flows, and its validity is examined by comparing
the estimated and measured droplet diameters using high-speed image analysis and laser
measurement. As a result, using high-speed imaging, we clarified that the bag film rupture
is caused not by the turbulence of the gas flow but by the impact of floating droplets on the
liquid film of the expanding bag when the film is thin enough. The average thickness of the
liquid film at the bag breakup is of the order of micrometres and varies greatly, resulting in
a dispersed distribution of droplet diameters. After the film ruptures, the bag film shrinks
towards its transversal and vertical rims due to surface tension, forming large-diameter
ligaments. During the contraction process of the bag film, tiny droplets of the order of
micrometers are formed at the edge of the perforation. Finally, the remaining ligaments
with large diameters fragment into large droplets with submillimetre diameters. The good
agreement between the measured and predicted droplet diameter distributions validated
the mechanistic model.
Key words: multiphase flow, gas/liquid flows, thin films
1. Introduction
Liquid film air-blast atomization with co-current high-speed gas flows is widely used to
produce tiny droplets in numerous applications, including gas turbines, spray painting and
spray coating. Figure 1 shows front and side views of a planar liquid sheet with co-current
gas flows at a relatively low gas velocity of 40 m s−1, which is smaller than the typical gas
†Email address for correspondence: i.oshima@tohoku.ac.jp
© The Author(s), 2024. Published by Cambridge University Press. This is an Open Access article,
distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/
licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original
article is properly cited. 985 A36-1
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I. Oshima and A. Sou
Bag Bag
0x
z
Ligament
10 mm 10 mm
(b)(a)
Figure 1. Liquid sheet atomization process with co-current gas flows. (a) Front view and (b) side view.
velocity in gas turbines. The liquid film oscillates longitudinally due to Kelvin–Helmholtz
(KH) instability, transversally due to Rayleigh–Taylor (RT) instability and finally breaks
up into droplets. The atomization process is multi-phase and multi-time scale phenomena
with various length scales, such as film thickness, atomizer lip thickness, which is the solid
wall between the gas and liquid inlet exits, wavelengths of the KH/RT instabilities, bag
breakup length, thickness of the boundary layer, turbulence scale and droplet diameter.
Due to its complexity, the basic principle of liquid film atomization has not yet been
clarified.
Numerous studies on the flapping and breakup processes of liquid films have been
conducted owing to the importance of spray diameter prediction and control for many
industrial devices. Squire (1953) visualized a sinusoidal fluctuating liquid sheet in stagnant
gas and proposed a model for the longitudinal wavelength λLon of the liquid sheet. Hagerty
&Shea(1955) performed a theoretical analysis of sinusoidal and dilational waves of
a liquid film. Fraser et al. (1962) and Dombrowski & Hooper (1962) proposed models
for the droplet diameter based on theoretical analysis, where a liquid film in this model
disintegrates into ligaments and then into droplets by the Rayleigh instability (Rayleigh
1878). The liquid film flapping process with co-current gas flow and with no lip was
investigated, and the vortex flow was observed around the film by the numerical simulation
(Odier et al. 2015). Two different conceptual numerical codes were used to simulate
the liquid flapping behaviour, and the usefulness of the Eulerian multi-fluid solver was
discussed (Zuzio et al. 2013). Lohsea and Villermaux reviewed the rupture of the liquid
film based on various influences, such as a laser pulse and heterogeneity of the surface
tension (Lohsea & Villermaux 2020). Tang et al. simulated the bag breakup of a droplet
using the algorithm, in which the liquid film was artificially punctured when it reached
a preliminary defined thickness (Tang, Adcock & Mostert 2023). The fragmentation
phenomena of films and ligaments are summarized by Villermaux (2006), and the droplet
diameter distribution formed by the ligament breakup was discussed by Villermaux et al.
(2004). They also showed that the droplet diameter distribution produced by the breakup of
the liquid film created by the swirl and fan spray nozzles can be expressed as a compound
gamma distribution with two parameters (Kooij et al. 2018).
Various empirical correlations for the droplet diameter of air-blast atomizers have been
proposed by many researchers and summarized by Lefebvre (1980,1992). However,
the previous empirical correlations are mostly available only under limited conditions
because these correlations do not take into account all of the complicated atomization
processes mentioned above. Therefore, these correlations require parameter tuning for each
application and condition.
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Air-blast atomization of a liquid film
To develop a mechanistic model for predicting the droplet diameter distribution under a
wide variety of conditions, it is necessary to clarify and model all fundamental processes.
The flapping phenomena of an air-blasted liquid sheet have been extensively investigated
by many researchers. Dumouchel (2008) reviewed atomization characteristics such as
the oscillation frequency and breakup length. Using a laser technique, Lozano et al.
(2005) measured flapping characteristics and proposed an empirical correlation for λLon.
Considering the effect of lip thickness on the length scale of the gas-phase boundary layer,
Oshima et al. (2017), Oshima & Sou (2019) proposed a correlation for λLon based on the lip
momentum ratio MRLip as a new dimensionless number using DLbecause the momentum
of the thin liquid film is exchanged downstream of the lip with that of the gas flow. Then,
they validated the correlation using their experimental results. Fernandez, Berthoumie &
Lavergne (2009) investigated the transversal oscillation phenomenon of a planar liquid film
flow and proposed correlations for the transversal wavelength λTra . Oshima & Sou (2021)
modelled the spanwise oscillation phenomenon based on the RT instability caused by the
acceleration of the liquid sheet. Matsuura et al. and Yoshida et al. studied the effect of the
discharged gas flow angle on the droplet diameter using annular and planar liquid sheet
atomizers (Matsuura et al. 2008; Yoshida et al. 2012). Inoue et al. investigated the spatial
spray flux to determine the local mass ratio of fuel to air in the combustor (Inoue et al.
2021). After the longitudinal and transversal oscillations, the liquid film is stretched by the
gas flow to form bags. It is well known that the breakup of bags efficiently produces tiny
droplets, while the remaining liquid becomes large droplets. There have been numerous
foundational studies on thin liquid films and ligaments. The bag breakup of a droplet in a
gas flow produces numerous tiny droplets, and a large liquid ring remains. Chou & Faeth
(1998) measured the amount of the remaining liquid ring using several liquids and reported
that approximately 52 % to 59% of the original droplet volume became the ring. Taylor
(1959) and Culick (1960) proposed a correlation between the film thickness and its velocity
based on the relationship between the shrinking velocity of the edge of a liquid film and
film thickness. McEntee & Mysels (1969) reconfirmed the validity of the Taylor–Culick
velocity model using soap films with a thickness slightly greater than 0.1 μm. When a thin
liquid film ruptures and contracts, the contracting rim velocity reaches the Taylor–Culick
velocity (Agbaglah, Josserand & Zaleski 2013) and RT instability causes the formation
of ligaments at the rim. The breakup of a ligament is based on the Rayleigh instability
(Rayleigh 1878) and Weber’s theory (Weber 1931; Dombrowski & Johns 1963), which are
often used to predict droplet size (Fraser et al. 1962; Dombrowski & Johns 1963; Inamura
et al. 2012). Finally, the remaining liquid that accumulates at the bag rim breaksup into
large droplets.
In the recent work, the bag formation process of a liquid film flow on the wall by the gas
flow was observed and the effect of the viscosity on the bag length was examined (Kant
et al. 2023). Jackiw and Asgriz examined the model for predicting droplet diameters in
the bag breakup process of a droplet. Based on their visualization and theoretical analysis,
they discussed each part of the rim and node. However, the tiny droplets caused by the
liquid film rupture were not discussed in detail (Jackiw & Ashgriz 2022). The numerical
simulation of the pre-filming air-blast atomizer was compared with their experimental
results such as breakup length and droplet diameters (Warncke et al. 2017). The effect of
the nozzle structure on the liquid film deformation from the pre-filming air-blast atomizer
was investigated by numerical simulation (An et al. 2023). Thus, there is interest in the
deformation and breakup processes of the liquid film from various perspectives.
In the present study, we developed a framework for predicting the droplet
distribution based on the phenomenological approach described above rather than
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I. Oshima and A. Sou
Liquid film A: Initially deformed
section of liquid film
C: Bag breakup D: Ligament breakup
B: Bag growth
DL
λtra
Gas
flow
Bag
A
B
Rim
Ligament
Lip
DLDLip
(1/2)λLon
Figure 2. Atomization model of an air-blasted liquid film.
empirical correlation. We first predicted the longitudinal and transversal wavelengths of
the oscillating liquid film by the KH–RT instability, which corresponds to the bag sizes.
Then, we calculated the diameter of the tiny droplets produced at the contracting film edge
after the rupture and the diameter of the large droplet transformed from the remaining
liquid. To validate the proposed model, high-speed visualizations of the atomization
process of a planar liquid film with co-current gas flows were performed, and the diameters
using a Phase Doppler Interferometer (PDI) system and image analysis were measured to
confirm the atomization model.
2. Development of a model to predict droplet size distribution
An outline of the proposed atomization model is illustrated in figure 2. The KH instability
causes longitudinal oscillation of a liquid film near the exit of an air-blast atomizer, and
the oscillating liquid sheet fluctuates in the transverse direction due to RT instability. The
rectangular element of the fluctuating liquid sheet is stretched downstream by the impact
of the co-current gas flow to form a bag. After the bag breaks up, tiny droplets form
at the edge of the perforation, and the remaining liquid becomes ligaments, eventually
becoming large drops. In the following sections, we explain the correlations used to predict
the diameters of the small and large droplets.
2.1. Longitudinal wavelength λLon
The liquid film oscillates longitudinally owing to KH instability, and the longitudinal
wavelength λLon is written as follows (Oshima & Sou 2019):
λLon
DLip =14.3
MRLip
,(2.1)
where DLip is the lip thickness. MRLip is the lip momentum ratio, which is defined as
follows:
MRLip =ρGV2
GDLip
ρLV2
LDL
,(2.2)
where ρGand ρLare the densities of the gas and liquid, respectively, VGand VLare the
gas and liquid velocities, respectively, and DLis the initial thickness of the liquid sheet.
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Air-blast atomization of a liquid film
2.2. Transversal wavelength λTra
After the longitudinal oscillation of the liquid film by the KH instability, the RT instability
induces the spanwise fluctuation of the liquid film, whose acceleration is caused first by
the KH instability and then by the drag force of the co-current gas flow. In this study, we
tentatively calculate the transversal wavelength λTra as (Oshima & Sou 2021)
λTra =cλKH ,(2.3)
where λKH is the transversal wavelength based on the acceleration caused by the KH
instability, and the coefficient cis given by
c=0.5,if λKH >2λDrag
1,otherwise.(2.4)
When λKH >2λDrag, the bag can be deformed into two horizontal bags. Here, λDrag is
the transversal wavelength due to the acceleration caused by the aerodynamic force of the
gas flow. These wavelengths are given by
λKH =2π
ωi_KH 3σ
ρLDLip
,(2.5)
λDrag =2π
VG−VL6σDL
ρGCD
,(2.6)
where σis the surface tension, and CDis the drag coefficient with a value of 2.0
(Varga, Lasheras & Hopfinger 2003). The growth rate ωi_KH of the horizontal liquid sheet
disturbance can be expressed as follows (Squire 1953):
ωi_KH =
kρGcoth(kDL)
ρL
1+ρGcoth(kDL)
ρL
(VG−VL)2−σk
ρG1+ρGcoth(kDL)
ρL,(2.7)
where kis the wavenumber of λLon, which is given by
k=2π
λLon
.(2.8)
2.3. Volumes of a bag and ligaments
A bag was formed between the two transverse rims and two vertical rims created by the
KH and RT instabilities, as illustrated in figure 2. The distance between the two transverse
rims is half that of λLon, and that of the vertical rims is λTra . Therefore, the volume vof
liquid forming the bag is expressed as follows:
v=1
2λLonλTr a DL.(2.9)
As soon as the bag is perforated, tiny droplets are formed at the hole edge. Finally, the
remaining liquid forms a rim, ligament and droplet. It was reported that 44 % of the droplet
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I. Oshima and A. Sou
Perforation edge
2r
Vcon
Vcon
λcap = (2π/3)r = 3.6
r
Figure 3. Cross-sectional structure of the moving edge of perforation on a bag.
volume became tiny droplets, and the remaining liquid became a ligament after the bag
breakup (Chou & Faeth 1998). Thus
vB=0.44v, (2.10)
vLi =0.56v, (2.11)
where vBand vLi are the total volumes of tiny droplets and ligaments, respectively.
2.4. Diameters of tiny droplets generated at the hole edge
A liquid element expands downstream by the co-current gas flow and breaks up. As
illustrated in figure 3, the edge of the liquid film with 2rthickness moves toward the rim
of the bag with Vcon at a constant velocity after a perforation due to the surface tension
force. In other words, the perforation expands, and the liquid film shrinks with Vcon at the
terminal velocity given by Culick (1960) as follows:
Vcon =σ
ρLr.(2.12)
The equation takes the time ttrans for the moving velocity of the perforation edge to reach
the Taylor–Culick velocity Vcon, which is given as follows:
ttrans ∼O⎛
⎝ρLr3
σ⎞
⎠.(2.13)
Due to the surface tension, the perforation edge becomes cylindrical, and the wavelength
λcap oftheneckisgivenby
λcap =2π
√3r.(2.14)
Hence, the acceleration aRim is roughly given as
aRim =Vcon
ttrans
.(2.15)
As shown in figure 4, in the spanwise direction of the perforation edge, periodic waves
are caused by the RT instability based on these accelerations, whose wavelength λRim is
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Air-blast atomization of a liquid film
Transversal rim
Perforation
Bag
aRim
λRim
Figure 4. Instability of the moving edge of a perforation.
expressed as
λRim =2π3σ
ρLain
,(2.16)
where ain denotes acceleration of the perforation edge.
A tiny droplet was formed by the growth of the two horizontal and vertical waves at the
perforation edge. The volume of each droplet is calculated as follows:
Vdrop_ba =2rλRimλcap =8π2r2σ
ρLaRim
.(2.17)
The arithmetic mean diameter DBag corresponding to Vdrop_ba, is expressed as follows:
DBag =6
π
Vdrop_ba1/3
=12
π
rλRimλcap 1/3
=26πr2σ
ρLaRim 1/3
.(2.18)
2.5. Diameter of large droplets caused by ligament breakup
Because gas flow does not affect ligament breakup, the Rayleigh or Weber theory
(Dombrowski & Johns 1963) is suitable for calculating the droplet diameter. After the
bag is ruptured, the remaining liquid, without bag breakup, collects at the transversal and
vertical rims and eventually becomes a ligament. The gas flow stretches the bag and the
vertical rim between two adjacent bags. The vertical rims are approximately two to four
times longer than λLon. Therefore, we estimate the length in this study as 3λLon. Assuming
that the ligament is a uniform cylinder, its radius Ris determined from vLi using the
following equation satisfying the conservation of the mass as follows:
R=vLi
π(3λLon +λtra).(2.19)
The droplet diameter DLi is obtained by the following Weber theory:
DLi =3.76R(1+3Oh)1/6,(2.20)
where Oh is the Ohnesorge number, which is defined by
Oh =μL
√2ρLσR.(2.21)
2.6. Summary of a mechanistic model for droplet diameter
The flowchart of the mechanistic model ofthe droplet diameter distribution is shown
in figure 5. The framework enables us to predict droplet diameters via the above
phenomenological modelling on the bag and ligament breakups, which can account for
the effects of velocity and physical properties of gas and liquid and injector geometries
without any tuning parameters.
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I. Oshima and A. Sou
Longitudinal wavelength: λLon = 14.3DLip
MRLip
1
Transversal wavelength: λTra = cλKH
Total volume of bag: v =
Volume of droplets by bag breakup:
vB = 0.44vVolume of Ligament
vLi = 0.56v
DBag = 23DLi = 3.76 π(3λLon + λtra)(1 + 3Oh)1/6
Vcon
vLi
ttrans
ttrans~aRim =
6πr2
Average droplet diameter by bag breakup: Average droplet diameter by ligament:
ρLDLip
=2cπ
0.5,
1,
if
othewise
λKH > 2λDrag
λTraDL
λLon
2
c =
{
3σ
ωi_KH
2
3
44
5
ρLaRim
ρLr3
ρLV2
con
,,
r =
σ
σ
σ
6
Figure 5. Flow chart of the mechanistic model of the air-blast liquid sheet atomization.
DG = 3.0 mm
DL = 0.2, 0.5 mm
DLip = 0.2 mm
DG = 3.0 mm
40 mm
Gas flow
Gas flow
Liquid
Lip
Air Liquid Air
DLip DL
x
z
(b)(a)
Figure 6. Planar air-blast atomizer for the liquid film.
3. Experimental set-up and conditions
A schematic diagram of the planar air-blast atomizer (Yoshida et al. 2012) is shown in
figure 6. In the present study, the thickness of a liquid film is 0.2 and 0.5 mm, the width
of the gas flow is 3.0 mm and the thickness of the lip is 0.2 mm. We assembled the
experimental apparatus shown in figure 7. Pure water at room temperature was injected
into the atmosphere using compressed gas. The liquid mass flow was controlled by a
needle valve and measured by a Coriolis flow sensor (KEYENCE, FD-SS20A). The blower
(HITACHI, VB-030-E3) ejected the gas flow, whose mass flow rate was controlled by
rotational control using an inverter, and was measured with a liquid column manometer.
The formation and breakup of bags and ligaments were recorded with a high-speed
camera (Vision Research, Phantom v211, and MIRO LAB310). Backlight images were
taken with a macro lens (Nikon, AI AF Micro-Nikkor 200 mm f/4D IF-ED), a close-up
ring and a metal halide lamp (Kyowa, MID-25FC, SIGMA KOKI, SHLA-150). The
acquisition rate was 7000–30 000 f.p.s., and the spatial resolution of the images was chosen
to be 20–100 μmpixel
−1to sufficiently capture the analysis object.
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Air-blast atomization of a liquid film
High-speed
camera
Compressor
Liquid
tank
Liquid
flow
Blower
Flow
meter
Regulator
Gas flow
Valve
Atmospheric
Rig
Light
Air-blast
atomizer
AirLip Liquid Air
DLip DLDG = 3.0 mmzx
Figure 7. Atmospheric experimental apparatus.
The droplet diameter and velocity were measured using a PDI system (Artium, PDI-200
MD). The collection angle was approximately 70°. The front and back focal lengths were
both 500 mm, and the slit aperture was set to 100 μm. The droplets generated by ligament
breakup were non-spherical and had a diameter greater than 150 μm. Because almost all
droplet diameters measured by the PDI system were less than 100 μm, we eliminated data
whose droplet diameters were greater than 100 μm using a filter. We collected 10000
droplets at each measurement point and examined the burst signals with an oscilloscope.
The diameters of the ligaments and droplets generated by ligament breakup were measured
using image analysis. Since the diameters of the ligaments were not uniform, we measured
the average diameters for each case. After the ligament breakup, the diameters of two
hundred large droplets were measured.
4. Results and discussion
4.1. Visualization of the bag breakup process
The high-speed images of a typical bag breakup process are shown in figure 8. It was found
that bag breakup is almost always triggered, not by turbulence or the Van der Waals force,
but by the collision of the rapidly expanding thin liquid film and floating droplets. The
ruptured liquid film contracted longitudinally and transversely toward the rims of the bag
due to surface tension to retain the ligaments. Many tiny droplets formed at the perforation
edge because the liquid film was very thin when the bag ruptured.
When a floating droplet impacts a thick liquid film with comparatively slow expansion
velocity prior to the bag’s large expansion, the film does not always rupture. The
impingement of a droplet on a thick liquid film is shown in figure 9. The preceding
observation indicates that the liquid film will only rupture when it is sufficiently thin and
the impact velocity is high. We consider the critical Weber number. As we presume that
the velocity of the bag expansion is 1 m s−1and liquid film thickness is 10 μm, the critical
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I. Oshima and A. Sou
Droplets
impingement
5 mm
Droplet
impingement
Bag rupture
Bag rupture Contraction
with
rim
and
ligament
Bag rupture
(a)(b)(c)(d)
(e)(f)(g)(h)
Figure 8. Bag breakup process initiated by a droplet impingement (DL=0.5 mm, VL=1m s
−1,VG=
30 m s−1); (a)t=0 ms, (b)t=0.09 ms, (c)t=0.18 ms, (d)t=0.27 ms, (e)t=0.36 ms, ( f)t=0.45 ms,
(g)t=0.55 ms and (h)t=0.64 ms.
Droplet
impingement Droplet
impingement
Droplet
impingement
Interface
disturbance Interface
disturbance
Interface
disturbance
3 mm
(a)(b)(c)(d)
Figure 9. Disturbance wave of a liquid film by the droplet impact (DL=0.2 mm, VL=1ms
−1,
VG=30 m s−1); (a) 0.12 ms, (b) 0.25 ms, (c)0.37msand(d) 0.49 ms.
Weber number is approximately 0.1. By taking into account the variations in local gas
velocity and local film thickness, the critical Weber number is estimated to be about one.
To estimate the diameters of the small droplets we must estimate the film thickness.
The measured contracting rim velocities VCon after film rupture are shown in figure 10.
Figure 10(a) shows an example of the measured VCon distribution, and figure 10(b)shows
the average VCon in each case. The maximum measurement error in the contraction
velocity VCon is estimated to be approximately at most 15% when we assume that the
bag takes the shape of a spheroid and remove the Vcon data that appear near the bottom
of the bags. Therefore, the depth of the field and the viewing angle were controlled in the
experiment. The measured VCon values ranged from 3to12ms
−1. The effects of VLand
VGon VCon were insignificant. In this study, we use VCon =5ms
−1at DL=0.5 mm and
VCon =7ms
−1at DL=0.2 mm as measured data.
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Air-blast atomization of a liquid film
15
12
9
6
3
0
DL = 0.5 mm
DL = 0.2 mm
VCon (m s–1)
VL = 0.7 m s–1 VL = 1.0 m s–1 VL = 1.0 m s–1 VL = 1.0 m s–1 VL = 1.5 m s–1 VL = 2.0 m s–1 VL = 2.5 m s–1 VL = 1.5 m s–1
VG = 21 m s–1 VG = 25 m s–1 VG = 30 m s–1 VG = 35 m s–1 VG = 30 m s–1 VG = 30 m s–1 VG = 30 m s–1 VG = 45 m s–1
0.8
0.7
0.6
0.5
Probability
0.4
0.3
0.2
0.1
0246
Vcon (m s–1)
810
(b)
(a)
Figure 10. Measured contracting rim velocity VCon.(a) Probability at VL=2ms
−1,VG=30 m s−1and
DL=0.2 mm and (b)averageVCon .
4.2. Visualization of the ligament formation process
Figure 11 shows the ligament formation process. The upper images show the original,
whereas the lower images highlight the perforations edge and longitudinal rims. At
t=0 ms, we can see the vertical rims between the bags and the perforation edge. The
perforation grows over time. As a result, the vertical rim collects the liquid and transforms
into a ligament, and the transverse rim becomes a liquid column with a larger diameter.
This process creates tiny droplets when the bag is ruptured, and several ligaments and large
liquid columns are formed.
4.3. Evaluation of the atomization model by the comparison between predicted and
measured droplet diameters
In this section, the estimated droplet diameters obtained with our atomization model are
compared with the experimental results to determine the validity of the proposed model.
First, the longitudinal and transversal wavelengths λLon and λTra are predicted using (2.1)
and (2.3). The validity of λLon and λTra correlations have been confirmed (Oshima & Sou
2019,2021). The bag volume, v, was calculated using λLon and λTra .Figure 12 shows the
estimated v.AsVLand DLincrease, vincreases, whereas an increase in VGdecreases v.
The discontinuity appears at low VGdue to the switching in the dominant acceleration for
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I. Oshima and A. Sou
Original imageWith highlight
Injector
Vertical rim
Perforation
edge
10 mm
(a)(b)(c)(d)(e)
Figure 11. Ligament formation process after the bag breakup (DL=0.2 mm, VL=1.5 m s−1,
VG=30 m s−1); (a)t=0 ms, (b)t=0.14 ms, (c)t=0.28 ms, (d)t=0.41 ms and (e)t=0.55 ms.
14
12
10
8
6
4
2
020 40 60
VL = 0.7 m s–1, DL = 0.5 mm
VL = 1.0 m s–1, DL = 0.5 mm
VL = 1.5 m s–1, DL = 0.5 mm
VL = 1.0 m s–1, DL = 0.2 mm
VL = 1.5 m s–1, DL = 0.2 mm
VL = 2.0 m s–1, DL = 0.2 mm
v (mm3)
80
VG (m s–1)
Figure 12. Predicted volume of a bag.
the RT instability, and its effect is involved in the coefficient cof (2.3). It was confirmed
that the nonlinear transition of λTra occurred from the measurement (Oshima & Sou 2021).
The volume of the ligament vLi was obtained from (2.11), which was used to calculate the
radius of the ligament Rusing (2.20). The predicted and measured Rare shown in figure 13.
The error bar indicates the standard deviations. As VGincreases or VLdecreases, Ralso
decreases. The Rdecreases slightly when DLdecreases. The mean error of the predictions
at DL=0.5 mm was 16 %, and that at DL=0.2 mm was 47%. The predicted and measured
results are in good agreement.
The predicted and measured large droplet diameters DLi obtained by ligament breakup
are shown in figure 14. The measured value is the mean diameter D10 obtained by
image analysis, and the error bars represent the standard deviations. Although a simple
comparison between the measured and predicted data is not possible due to the large
variations in the measured diameters, the predicted and measured data agree that the
droplet diameter decreases with increasing VGor decreasing VLand DL. The droplet
diameter derived from the transverse rim was strikingly large at a large VLor low VG. The
mean diameters D10 at large VLor low VGwere slightly larger than the predicted values.
The mean error of the predictions at DL=0.5 mm was 26 %, and that at DL=0.2 mm
985 A36-12
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Air-blast atomization of a liquid film
0.4
(a)(b)
0.3
0.2
0.1
R (mm)
020 40 60
VG (m s–1)
VL = 0.7 m s–1
VL = 1.0 m s–1
VL = 1.5 m s–1
Measured: VL = 0.7 m s–1
Measured: VL = 1.0 m s–1
VL = 1.0 m s–1
VL = 1.5 m s–1
VL = 2.0 m s–1
Measured: VL = 1.0 m s–1
Measured: VL = 1.5 m s–1
80
0.20
0.15
0.10
0.05
020 40 60
VG (m s–1)
80
Figure 13. Predicted and measured radius of ligament; (a)DL=0.5 mm and (b)DL=0.2 mm.
(b)(a)
20 30 40 50
VG (m s–1)
60 70 80
0
DLi (mm)
0.5
1.0
1.5
2.0
20 40
VG (m s–1)
60 80
0
0.5
1.0
D10: VL = 1.0 m s–1
D10: VL = 1.5 m s–1
D10: VL = 0.7 m s–1
D10: VL = 1.0 m s–1
D10: VL = 1.5 m s–1
Predicted D10: VL = 0.7 m s–1
Predicted D10: VL = 1.0 m s–1
Predicted D10: VL = 1.5 m s–1
Predicted D10: VL = 1.0 m s–1
Predicted D10: VL = 1.5 m s–1
Predicted D10: VL = 2.0 m s–1
Figure 14. Predicted and measured droplet diameter by ligament breakup; (a)DL=0.5 mm and
(b)DL=0.2 mm.
was 31 %. The predicted droplet diameters agreed with the measured values, except in the
preceding cases.
Finally, we address the predicted diameters of the small droplets. First, the thickness
of the liquid film was estimated. In the present study, we use the average value of the
measured VCon,i.e.VCon =5ms
−1at DL=0.5 mm and VCon =7ms
−1at DL=0.2 mm.
From (2.12), the liquid film thickness 2rcan be expressed as follows:
2r=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
2σ
ρLV2
con ∼2×0.072
998 ×52∼6μm,when DL=0.5mm
2σ
ρLV2
con ∼2×0.072
998 ×72∼3μm,when DL=0.2 mm.
(4.1)
Here, we examined the effect of the varying VCon on 2ras a sensitivity analysis. At
VCon =2and12ms
−1, the minimum and maximum film thicknesses were 2rmax ∼36 μm
and 2rmin ∼1μm, respectively. Therefore, 2rmay have a large deviation, and the
non-uniformity of the film thickness may cause a large variation in the droplet size
distribution.
Next, we estimated the acceleration at the perforation edge. If the film thickness of bag
2ris6or3μm, the delay time ttrans for the edge velocity to reach its terminal velocity
985 A36-13
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I. Oshima and A. Sou
VCon, calculated using (2.13), is as follows:
ttrans =0.58 μs,when DL=0.5mm
0.21 μs,when DL=0.2 mm.(4.2)
The acceleration aRim based on the initial accelerated motion is given as follows:
aRim =⎧
⎪
⎪
⎨
⎪
⎪
⎩
Vcon
ttrans ∼5
5.8×10−7∼8.7×106ms
−2,when DL=0.5mm
Vcon
ttrans ∼7
2.1×10−7∼3.3×107ms
−2,when DL=0.2 mm.
(4.3)
Therefore, the droplet diameter due to bag breakup can be obtained by solving (2.18),
using aRim
DBag =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
26πr2σ
ρLaRim 1/3
=26π(3×10−6)20.072
998 ×8.7×1061/3
∼15 μm,
when DL=0.5mm
26πr2σ
ρLaRim 1/3
=26π(1.5×10−6)20.072
998 ×3.3×1071/3
∼8μm,
when DL=0.2 mm.
(4.4)
The DBag was between 3 μm and 96 μm, since VCon was 2–12 m s−1. It is important to
consider the non-uniformity of the bag film thickness when predicting the droplet diameter
distribution. The mean droplet diameter decreased with increasing gas flow velocity.
Alternatively, it has been reported that the droplet diameter generated by the bag breakup
in a cross-flow is generally independent of the Weber number (Ng, Sankarakrishnan
& Sallam 2008). The bag breakup occurs when the liquid film is sufficiently thin,
below the critical Weber number, regardless of how the gas flow is injected. Therefore,
the fundamental characteristics of the bag breakup will be common. Measured droplet
distributions at x=0mm and z=15 mm are shown in figure 15. We can see a large
variation in the droplet diameter since the film thickness and ligament diameter show
large variations. We can confirm that the influence of the DLand VLon the diameter is
very small, and the modal diameter lies between 8 and 12 μm. The predicted results agree
well with the modal diameters.
Finally, predicted and measured droplet diameters were compared. Figure 16 shows the
relationship between measured D10 obtained by PDI optical measurements and predicted
DBag.AsVGincreases from 40 to 80 m s−1, measured values of D10 at DL=0.2 and
0.5 mm are almost constant. The order of predicted DBag agreed with measured D10,
indicating that the proposed model can capture the atomization phenomenon. However,
there is a gap between D10 and predicted results, which indicates that we have to compare
this with the other mean droplet diameter, e.g. modal diameter. The range of modal
diameter is in the range of 6–20 μm for all conditions. However, it is difficult to determine
at this stage which mean droplet diameter corresponds to the predicted droplet diameter.
We will solve the problem in the near future.
We conclude, based on the above discussion, that the framework of the mechanistic
model proposed in this study is plausible and provides an opportunity to predict the
droplet diameter distribution produced by each elementary process of the atomization
phenomenon.
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Air-blast atomization of a liquid film
0
0
200
400
600
800
1000
1200
20 40 60
Diameter (μm)
DL = 0.2 mm DL = 0.5 mm
Number of droplets (–)
80 100 0
0
200
400
600
800
1000
1200
20 40 60
Diameter (μm)
Diameter (μm)
DL = 0.2 mm DL = 0.5 mm
Diameter (μm)
80 100
0
0
200
400
600
800
1000
1200
20 40 60
Number of droplets (–)
80 100 0
0
200
400
600
800
1000
1200
20 40 60 80 100
(a)
(b)
Figure 15. Droplet size distribution measured by PDI; (a)VL=0.4 m s−1,VG=80 m s−1and
(b)VL=0.4 m s−1,VG=50 m s−1.
0
5
10
15
20
25
30
35
40
20 40 60 80
0
5
10
15
20
25
30
35
40
20 40 60 80
Diameter (μm)
VG (m s–1)VG (m s–1)
DBag
D10: VL = 0.4 m s–1
D10: VL = 1.0 m s–1
(a)(b)
Figure 16. Predicted and measured droplet diameter by bag breakup; (a)DL=0.2 mm and (b)DL=0.5 mm.
5. Conclusions
The atomization process of the liquid film induced by the gas flow was investigated. The
oscillation and breakup processes of the liquid film were initially discussed and modelled.
Second, the bag breakup process was clarified using high-speed imaging. We validated
our atomization model by measuring the droplet diameter using image analysis and PDI
measurements. Consequently, we reached the following conclusions:
(i) We clarified for the first time that the film rupture is almost always caused by the
impact of floating droplets on the expanding film of the bag, which is sufficiently
thin.
985 A36-15
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I. Oshima and A. Sou
(ii) The rim contraction velocity was in the range of 3–12 m s−1. The effects of gas and
liquid velocities on the contraction velocity were minimal in this study. The average
film thickness immediately after bag rupture was 3–6 μm whose value is calculated
by (2.12), and there was a considerable variation in film thickness, which may have
contributed to the droplet size distribution.
(iii) The liquid at the perforation edge contracted, forming ligaments that fragmented the
large droplets along the vertical and transverse rims.
(iv) We developed a mechanistic model for the droplet diameter distribution that provides
the first framework for the air-blast atomization process. Its validity was verified by
comparing the predicted and measured diameters using high-speed visualizations
and optical measurements.
(v) The breakup of the ligament generates droplets of the order of submillimetres, while
bag fragmentation generates tiny droplets with diameter in the micrometres range.
The proposed model can roughly predict the droplet diameters.
Supplementary movie. A supplementary movie is available at https://doi.org/10.1017/jfm.2024.279.
Acknowledgements. The author would like to thank Mr K. Oishi for his help in the measurement and image
analysis.
Funding. This study was supported by a JSPS KAKENHI Grant Numbers JP19K23489, 19KK0110 and
JP21K14084.
Declaration of interests. The author reports no conflict of interest.
Author ORCIDs.
Ippei Oshima https://orcid.org/0000-0003-3739-0578;
Akira Sou https://orcid.org/0000-0003-0638-1767.
REFERENCES
AGBAGLAH,G.,JOSSERAND,C.&ZALESKI, S. 2013 Longitudinal instability of a liquid rim. Phys. Fluids
25 (2), 022103.
AN,X.,DONG,B.,ZHANG,Y.,GENG,F.,ZHOU,X.,QIN,Y.&LI, W. 2023 Effects of structured prefilmer
lip on spray characteristics in planar prefilming airblast atomization. Aerosp. Sci. Technol. 140, 108426.
CHOU, W.-H. & FAETH, G.M. 1998 Temporal properties of secondary drop breakup in the bag breakup
regime. Intl J. Multiphase Flow 24 (6), 889–912.
CULICK, F.E.C. 1960 Comments on a ruptured soap film. J. Appl. Phys. 31 (6), 1128–1129.
DOMBROWSKI,N.&HOOPER, P.C. 1962 The effect of ambient density on drop formation in sprays. Chem.
Engng Sci. 17 (4), 291–305.
DOMBROWSKI,N.&JOHNS, W.R. 1963 The aerodynamic instability and disintegration of viscous liquid
sheets. Chem. Engng Sci. 18 (3), 203–214.
DUMOUCHEL, C. 2008 On the experimental investigation on primary atomization of liquid streams. Exp.
Fluids 45 (3), 371–422.
FERNANDEZ, V.G., BERTHOUMIE,P.&LAVERGNE, G. 2009 Liquid sheet disintegration at high pressure:
an experimental approach. C. R. Méc. 337 (6), 481–491.
FRASER, R.P., EISENKLAM,P.,DOMBROWSKI,N.&HASSON, D. 1962 Drop formation from rapidly
moving liquid sheets. AIChE J. 8(5), 672–680.
HAGERTY,W.W.&SHEA, J.F. 1955 A study of the stability of plane fluid sheets. J. Appl. Mech. 22 (4),
509–514.
INAMURA,T.,SHIROTA,M.,TSUSHIMA,M.,KATO,M.,HAMAJIMA,S.&SATO ,A.2012Spray
characteristics of prefilming type of airblast atomizer. In 12th Triennial International Conference on Liquid
Atomization and Spray Systems, Heidelberg.
INOUE,C.,YOSHIDA,H.,KOUWA,J.,IWAKI,Y.&ITOH, M. 2021 Measurement and modeling of planar
airblast spray flux distributions. Intl J. Multiphase Flow 137, 103580.
985 A36-16
https://doi.org/10.1017/jfm.2024.279 Published online by Cambridge University Press

Air-blast atomization of a liquid film
JACKI W,I.M.&ASHGRIZ, N. 2022 Prediction of the droplet size distribution in aerodynamic droplet breakup.
J. Fluid Mech. 940.
KANT,P.,PAIRETTI,C.,SAADE,Y.,POPINET,S.,ZALESKI,S.&LOHSE, D. 2023 Bag-mediated film
atomization in a cough machine. Phys. Rev. Fluids 8(7), 74802.
KOOIJ,S.,SIJS,R.,DENN, M.M., VILLERMAUX,E.&BONN, D. 2018 What determines the drop size in
sprays? Phys. Rev. X8(3), 031019.
LEFEBVRE, A.H . 1980 Airblast atomization. Prog. Energy Combust. Sci. 6(3), 233–261.
LEFEBVRE, A.H . 1992 Energy considerations in twin-fluid atomization. Trans. ASME: J. Engng Gas Turbines
Power 114 (1), 89–96.
LOHSE,D.&VILLERMAUX, E. 2020 Double threshold behavior for breakup of liquid sheets. Proc. Natl
Acad. Sci. 117 (32), 18912–18914.
LOZANO,A.,BARRERAS,F.,SIEGLER,C.&LÖW, D. 2005 The effects of sheet thickness on the oscillation
of an air-blasted liquid sheet. Exp. Fluids 39 (1), 127–139.
MATSUURA,K.,SUZUKI,S.,SUDA,M.,IINO,J.&HAYASHI, S. 2008 Effects of swirl on spray
characteristics of a counter-swirl airblast fuel injector. In 22nd European Conference on Liquid Atomization
and Spray Systems.
MCENTEE,W.R.&MYSEL S, K.J. 1969 Bursting of soap films. I. An experimental study. J. Phys. Chem.
73 (9), 3018–3028.
NG, C.-L., SANKARAKRISHNAN,R.&SALLAM, K.A. 2008 Bag breakup of nonturbulent liquid jets in
crossflow. Intl J. Multiphase Flow 34 (3), 241–259.
ODIER,N.,BALARAC,G.,CORRE,C.&MOUREAU, V. 2015 Numerical study of a flapping liquid sheet
sheared by a high-speed stream. Intl J. Multiphase Flow 77, 196–208.
OSHIMA,I.&SOU, A. 2019 Longitudinal oscillation of a liquid sheet by parallel air flows. Intl J. Multiphase
Flow 110, 179–188.
OSHIMA,I.&SOU, A. 2021 Transversal oscillation of a planar liquid sheet induced by co-current airflows.
Multiphase Sci. Technol. 33 (2), 53–67.
OSHIMA,I.,SOU,A.,KAWA BA TA ,R.&MATSUURA, K. 2017 Longitudinal wavelength of oscillating liquid
sheet with air flow. In 55th AIAA Aerospace Sciences Meeting.
RAYLEIGH,LORD 1878 On the instability of jets. Proc. Lond. Math. Soc. 1, 4–13.
SQUIRE, H.B. 1953 Investigation of the instability of a moving liquid film. Brit. J. Appl. Phys. 4(6), 167–169.
TANG,K.,ADCOCK, T.A.A. & MOSTERT, W. 2023 Bag film breakup of droplets in uniform airflows. J. Fluid
Mech. 970,A9.
TAYL OR, G. 1959 The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets. Proc. R. Soc. Lond.
Ser. A. Math. Phys. Sci. 253 (1274), 313–321.
VARGA, C.M., LASHERAS,J.C.&HOPFINGER, E.J. 2003 Initial breakup of a small-diameter liquid jet by a
high-speed gas stream. J. Fluid Mech. 497, 405–434.
VILLERMAUX, E. 2006 Fragmentation. Annu. Rev. Fluid Mech. 39 (1), 419–446.
VILLERMAUX, E., MARMOTTANT,PH.&DUPLAT, J. 2004 Ligament-mediated spray formation. Phys. Rev.
Lett. 92, 0745 01.
WARNCKE,K.,GEPPERTH,S.,SAUER,B.,SADIKI,A.,JANICKA,J.,KOCH,R.&BAUER,H.-J.2017
Experimental and numerical investigation of the primary breakup of an airblasted liquid sheet. Intl
J. Multiphase Flow 91, 208–224.
WEBER, C. 1931 Zum Zerfall Eines Flüssigkeitsstrahles. Z. Angew. Math. Mech. 11 (2), 136–154.
YOSHIDA,K.,IDE,K.,TAKAHASHI,S.,MATSUURA,K.,IINO,J.,KUROSAWA,Y.,HAYASHI,S.&
OHTA, Y. 2012 Airblast spray characteristics of planar liquid films in longitudinal gas-phase shear layers
at various ambient pressure conditions. In Proceedings of the 12th ICLASS,2012.
ZUZIO,D.,ESTIVALEZES, J.-L., VILLEDIEU,P.&BLANCHARD, G. 2013 Numerical simulation of primary
and secondary atomization. C. R. Méc. 341 (1), 15–25.
985 A36-17
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