Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Flapping instability of a liquid jet

Jean-Philippe Matas and Alain Cartellier
LEGI
CNRS Universite de Grenoble
BP 53, 38041 Grenoble Cedex 9
matas@hmg.inpg.fr


Keywords: Atomization, Spray, jet instability




Abstract

We present an experimental study of the flapping instability which appears when a coaxial liquid jet is
atomized by a cocurrent fast gas stream. When primary atomization does not lead to a total break-up of
the liquid jet, it undergoes a large-wavelength instability, characterized by very large amplitude
oscillations, and can break into large liquid fragments whose typical size is the jet diameter. These large
liquid fragments, and consequently the flapping instability, are to be avoided in applications related to
combustion where liquid droplets need to be as small as possible. We carried out experiments with air and
water coaxial jets, with a gas/liquid velocity ratio of order 50. We studied the consequence of the flapping
instability on the break-up of the liquid jet. Measurements of the frequency of the instability were carried
out. We suggest a mechanism where the flapping instability could be triggered by non axisymmetrical
KH modes.


Introduction

Airblast, or assisted atomization of a liquid jet is
fundamental to a large number of applications. In this
process the liquid is stripped from a cylindrical jet by a fast
co-current air-stream, and a spray is produced (Lasheras &
Hopfinger 2000). Applications range from injectors in
turboreactors, to cryotechnic rocket engines with LOX/H2.
This process is widely used, and has proven reliable, but
the mechanism by which the liquid bulk is broken into
droplets is still subject to controversy. A better
understanding of the different stages of the atomization
process could help improve the efficiency of combustion,
and decrease the amount of emissions.

Experiments carried out by coworkers on a plane
mixing layer (Raynal 1997, Ben Rayana 2007) and on a
coaxial injector (Hong 2003) have shown that the liquid
break up is the result of two successive instabilities
(Marmottant & Villermaux 2004). The first instability is
analogous to a Kelvin-Hemholtz instability, and leads to
the formation of waves at the interface between the liquid
and the fast gas stream. However, while Kelvin-Helmholtz
instability involves a discontinuity of the velocity profile
between the gas and liquid phases, the instability involved
here has been shown to rely on the smoothness of the
velocity profile, namely on the finite thickness of the gas
vorticity layer. Within an inviscid approximation, it
predicts that the wavelength X of the surging waves will be
given by (Raynal 1997, Marmottant & Villermaux 2004):


XKH = CKH (PL/PG)1/28


with 8G thickness of the gas boundary layer, PL and pG
the liquid and gas densities and CKH = 4 a dimensionless
coefficient. The axisymmetric waves resulting from this
instability are next accelerated by the fast gas stream, and
undergo a Rayleigh-Taylor transverse instability, leading to
the formation of liquid ligaments (Marmottant &
Villermaux 2004, Hong et al 2002, Varga et al 2003). These
ligaments grow and will eventually break into droplets,
whose size is therefore controlled by the thickness of
ligaments, i.e. the wavelength of the R-T instability.

If the liquid intact length is larger than the potential
cone, as is the case in our experiment, atomization of the
liquid jet is incomplete: while small droplets are still
produced in the potential cone region, far downstream the
liquid jet ends up breaking into large liquid lumps. Just
before its break-up, the liquid jet downstream the
potential cone exhibits a striking "oscillating" aspect, in
which the jet undergoes oscillations of a wavelength large
compared to the jet diameter (see figure 1). Our study is
devoted to this large scale instability, which we dub the
"flapping" instability.






Paper No


Figure 1: Instability of a liquid jet in airblast atomization
for: a) UG = 25 m/s and UL = 0.15 m/s ; b) UG = 30 m/s and
UL = 0.15 m/s.


We begin by presenting the experimental set-up used
for this study. The following part will be devoted to our
experimental results and the proposed mechanism for the
flapping instability.


Experimental Facility

The configuration used in our experiment is quite simple: a
round vertical liquid jet is entrained by a fast coaxial
annular gas stream (figure 2). Injection is made through
smooth convergent nozzles. The contraction is of a factor
6.9 for the liquid jet (exit diameter 8 mm), and 6 for the
gas jet (outer exit diameter 11.4 mm). The convergence in
the injector ensures that the intensity of perturbations is
reduced. The liquid is water. Gas flow is measured with a
mass flowmeter (Brooks Instruments), and liquid flow is
measured with a rotameter (Kobold).


Figure 2: Sketch of the injector.


The liquid velocity UL was varied in the range [0.11 m/s 1
m/s], and the gas velocity UG in the range [15 m/s 60 m/s].
For these ranges of velocities and the smooth convergent
injector we used, the flow is laminar in the liquid and gas
boundary layers. The three dimensional structure of the
flow was captured with a single camera and a vertical
mirror placed at an angle of 450 next to the nozzle (see
figure 3) : this device allows to capture on a same frame


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

the front view and side view of the atomized jet, and
consequently the three dimensional geometry of the flow.
The camera is a high speed camera (Phantom v12), and
frames are taken at a frequency in the range [500 im/s ;
2000 im/s] depending on the gas velocity and the expected
frequency of the instability.



Water






T 4Nozle


Iage i Camera
processmlg --


Figure 3: Sketch of the experimental set-up a) Front view ;
b) Top view


Results and Discussion

Figure 4a shows the aspect of the liquid jet for gas and
liquid velocities of UG = 40 m/s and UL = 0.3 m/s
respectively. For this regime, a large number of small
drops are produced near the injection, when liquid is
stripped from the bulk of the jet by the fast air stream.
However, it can be seen on figure 4 that large liquid
fragments remain intact on the axis of the jet: though the
jet undergoes a strong destabilization, these fragments
cause the formation of large drops far downstream. This
destabilization of the bulk of the jet is due to the flapping
instability studied in this work. When the gas velocity is
decreased, the growth-rate of the K-H and R-T instabilities
is strongly reduced and the flapping instability becomes
the dominant mechanism in the break-up of the jet: liquid
is not stripped from the jet anymore (see figure 1). This
instability of relatively large wavelength leads the jet to
twist and break into fragments whose size is of order R.

The flapping instability is therefore especially relevant in
situations when K-H and R-T instabilities fail to
completely atomize the jet: this occurs in particular when
the length of the gas potential cone (shown in dotted line
on the sketch of figure 4b) is shorter than the liquid intact
length. In our experiment this is generally true. This means
that the ultimate break-up of the jet is likely to be
controlled by the flapping instability.






Paper No


JI I
'S.




r.


Figure 4 : a) Airblast atomization for UG = 40 m/s and UL
= 0.3 m/s ; b) sketch emphasizing how the flapping
instability creates large liquid fragments.


In order to measure the intact length of the liquid jet, the
images taken with the device shown on figure 3 are
processed allowing to the following sequence: the images
are thresholded, we use a Matlab routine to find all paths on
the image, and the jet is taken to be the longer path on each
image. The intact length is taken to be the distance between
the injector and the point belonging to the jet which is
farthest from the injector. The intact length for a given set of
gas and liquid velocity is averaged over 300 images. This
process is only carried out for the front images of the jet (the
mirror images are not used for this measurement). The intact
length L is then nondimensionalized by the liquid jet
diameter D, results are shown on figure 5.


7.56

7

6.5 '-----'---...--'-'-
0.15 0.2 0.25
UL (m/


Figure 5: Liquid intact length
nondimensionalized by the liquid in
function of liquid and gas velocity.


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

The data of figure 5 show that as expected the liquid intact
length decreases with increasing gas velocity Ug, and
increases with increasing liquid velocity Ul. These
measurements were only carried out for conditions where
the flapping instability was present, i.e. the liquid jet
exhibits large scale oscillations. Raynal (1997) proposed a
scaling for the liquid intact length: L/D = 6/IM where M is
the momentum flux ratio M = pGUG2/PLUL2. This scaling
was verified experimentally in his plane shear layer
experiment (liquid sheet flowing on a solid wall and
atomized by a single gas stream).


Figure 6: Logarithmic plot of the ratio L/D as a function
of M. The equation of the power law fit shows a much
slower decrease than the M-1/2 scaling predicted by Raynal
(1997).


In figure 6 we plot the ratio L/D as a function of M: it
can be seen that though M appears to be a relevant
parameter (the data of figure 5 are collapsed), the scaling
law observed experimentally is quite different from the one
predicted Raynal (1997). The decrease of L/D with M is
much slower, and the values of L/D are significantly larger
(longer relative intact length). This could be due to the fact
that for our conditions the liquid intact length is set not
only by the amount of fluid stripped by the gas from the jet
(as argued by Raynal 1997), but also by the amplitude of
the flapping instability. As mentioned above, this is mainly
because the liquid intact length largely exceeds the length
of the potential cone.


A Ug = 16.3 m/s We next show measurements of the angle of aperture of
x Ug=S 19 mis the jet: these measurements are taken by superposing all
images obtained for a given set of conditions. Figure 7
S. ..... shows an example of result for this operation, for a liquid
0.3 0.35 0.4 velocity UL= 0.28 m/s and a gas velocity Ug= 14 m/s. The
) resulting image shows that there is a constant angle over a
relatively large distance. The angle is measured in the
region close to the injector, where the edges of the
L of the jet superposition are cleaner. We plot the measured angle
jector diameter D, as a when gas and liquid velocities are varied on figure 8. As
expected, the angle increases with the gas velocity and
decreases with increasing liquid velocity, i.e. when the
momentum of the liquid is increased. The measurements


-y=10.871 (-0.1616)


*


0
0
Sx Ug=6m


a Ug = 11.6 mis
a I






Paper No


for the lowest liquid velocity (Ul = 0.17 m/s, represented
by crosses on figure 8) stray a bit from this trend: this
could be due to the fact that for this very low liquid
velocity the jet is extremely thin a couple of diameters
downstream the injector (due to its acceleration), and
therefore easily deviated by the gas stream.


Figure 7 : Superposition of the successive positions of
the jet, for Ug = 14 m/s and Ul = 0.28 m/s.


25


x
20



C 15
A


10 x Ul=0.17mis
A Ul= 0.22 mis
So U= 0.28 mis
A Ul= 0.33 mis
a a U= 0.39 mis

10 12 14 16 18 20

Ug

Figure 8: Angle of aperture of the jet as a function of gas
velocity, for different liquid velocities.


Measurement of the wavelength of the instability has been
attempted, but results are inconclusive due to the strong
spatial variation of the wavelength as a function of the
downstream distance: this is evidenced on figure lb for
example, where it can be seen that the first wavelength is
significantly smaller than the second one (presumably due
to the increase in velocity of the liquid). We therefore
choose to focus instead on the frequency of the instability.

The main difficulty for this measurement is that
the flapping instability occurs roughly within a same plane,
but a plane of changing orientation. The orientation of the
plane of oscillation changes after a few periods, and
appears to be random. In order to get rid of the
tridimensionality of the instability, the images taken with
the device shown in figure 3 are processed according to the


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

following procedure: images are thresholded only the
dark region connected to the injector is retained (drops and
detached ligaments are removed) the center of this region
for a given line is computed on the mirror image (this is
position x) and on the front image (position y) the
distance of the jet center from the axis of the injector is
then defined as r=(x2+y2)1 the variations of r as a
function of time are calculated for several heights
(distances from the injector) a Fourier transform of this
signal yields the frequency of the oscillation.

We find that the maximum frequency is independent of the
distance from the injector, provided that the measurement
is made far enough from the injector. Measurements close
to the injector might be affected by the Kelvin-Helmholtz
instability: the latter instability is expected to produce
axisymmetric waves close to the injector, but any alteration
to the symmetry of the waves could induce a variation of
the distance r (distance of the jet center from the axis) at
the KH frequency. This is why we take care to measure the
frequency of the flapping instability farther downstream, in
regions where the flapping is visually predominant
(criterion: the axis of the jet is displaced from the axis of
the injector of a distance larger than R).


100 -


70 -


10 15 20 25 30 35
U


Figure 9: Frequency of the flapping instability as a
function of gas velocity, for different liquid velocities.

The frequency of the flapping instability, shown on figure 9,
increases with gas velocity. Only the points where a
maximum frequency was clearly apparent in the spectrum
were kept for the results of figure 9. We can compare these
data points with the measurements of the Kelvin-Helmholtz
instability carried out in a previous experiment (see figure
10). The KH instability was measured just below the
injector, by aiming a laser at the edge of the jet, and
analyzing the signal received by a photodiode located
opposite the laser: the signal is periodically interrupted by
the KH waves, and the frequency can be measured precisely
up to much larger velocities than with image processing
methods. It can be seen on figure 10 that the frequency of
the KH instability is slightly larger than the flapping
instability, but that both frequencies are relatively close. We


* Ul=0.17 ms
o UI=0.22 mis
A UI=0.28 ms


0
A A




o

*


IIIIIIIIIIIIIIIIIII IIIII






Paper No


emphasize that both instabilitites were measured at different
locations: very close to the injector for the KH instability,
and far downstream for the flapping: measurements of both
series were made as independent as possible, and the
closeness of the results is probably a clue to the physical
link between both instabilities in our experimental
conditions.

120
SUl= 0.22mis A
110 A UI= 0.28 mIs *
0 KH UI= 0.22 mfs
A KH UI= 0.28 ms o
100
o *
90

80
A 0o
70 k.
0,
060

50. . .
10 15 20 25 30 35
LJ
g
Figure 10: Frequency of the flapping instability (filled
symbols) and of the Kelvin-Helmholtz intability (empty
symbols) as a function of gas velocity Ug.


Though the range of gas velocity Ug is not large
enough to predict a scaling of the flapping frequency with
gas velocity, the experimental points are consistent with a
frequency proportional to Ug. This is the scaling law
experimentally measured by Lozano et al (2005) on their
liquid sheet experiment (a rectangular liquid sheet atomized
by two parallel gas streams). This is also the scaling found
by Couderc (" )1) in his numerical simulation of the same
liquid sheet configuration. The analogy between their
configuration and ours resides in the limit of the jet
instability when the radius is decreased: for a large radius
KH instability is merely a surface instability, but when R
becomes of the order of magnitude of the KH instability
wavelength, it can be expected that sinuous modes
analogous to the flapping modes of a liquid sheet may start
to overcome the axisymmetric instability in our
experiment. In a flapping sheet it is known (Lin 2003) that
the sinuous mode overcomes the varicose mode: it is
relatively easier for waves on opposing sides to propagate
with opposing phases (sinuous mode) than with the same
phase (varicose mode).

In our configuration this has been verified by
solving the inviscid stability analysis in axisymmetric
geometry for helical modes of the form e(k-0t+nO) with n/0.
We solve the resulting dispersion relation for spatial modes,
i.e. complex wavenumber and real frequency. We find that
modes for n=l are unstable. Modes for n>l are all stable.
The focus of this paper being on the experimental study, we
just present on figure 10 the comparison of the resulting
growth rate for the axisymmetric mode (n=0) and the helical
mode (n=l): it can be seen that when the radius of the jet is
decreased (in figure 11, R is nondimensionalized by the
vorticity thickness 6), the growthrate of the helical mode
overcomes that of the varicose mode. In our experimental


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

conditions we have R/6 -20 (the value increases when the
gas velocity is increased), which corresponds on figure 11 to
the region where the helical mode is significantly more
unstable than the n=0 mode. This predicts that the KH
instability for our conditions should produce non
axiymmetric surface waves.

This might be at the origin of, or at least
enhance, the flapping instability: figure 12 shows a PIV
visualization of the gas velociy field around the liquid jet for
experimental conditions for which the flapping instability is
observed farther downstream. The PIV is carried out with a
laser slice of the coaxial jet, and a seeding of the gas phase
with oil droplets. Figure 12 shows how the gas jet is
detached from the liquid jet downstream the first KH wave,
causing large gas recirculations scaling with the KH
wavelength. In particular, it can be noticed how the slight
dissymmetry in the KH wave (the left side of the wave is a
bit ahead the right side) induces a dissymmetry in the way
the gas jet impacts the liquid jet downstream the large
recirculations: we suspect this is the mechanism responsible
for the flapping instability. This scenario has been observed
on recent 2D numerical simulations of our experimental
conditions (Matas et al 2008).

80 1


ki 60
50

40

30

20

10


- ki n=0 (m-1)
Ski n=1 (m-1)


0 20 40 60 80 100 120 140


Figure 11: Result of the inviscid stability
analysis in cylindrical geometry: growthrate of the unstable
axisymmetric and helical modes as a function of the ratio
R/6. The helical mode has a larger growthrate.


Figure 12: PIV visualization of the gas velocity
field around the liquid jet: the flow is non axisymmetric. UG
= 10 m/s and UL = 0.4 m/s.






Paper No




Conclusions

We have studied the flapping instability of a liquid jet, an
instability occurring downstream the Kelvin-Helmholtz
instability when the liquid jet has not been fully atomized.
We have presented measurements of the frequency of this
instability, which is close to but smaller than the frequency
of the KH instability. We offer a scenario for the
development of the instability, based on the dissymmetry
induced in the gas flow when non axisymmetric modes
overcome the varicose modes. Further experiments need be
carried out for different conditions (in particular different
radii of the liquid jet, different gas injectors) in order to
investigate more precisely this hypothesis, and to precise the
scaling of the flapping frequency.

Acknowledgements

We acknowledge the contribution of several undergraduate
students on this experimental set-up: Angelique Sage,
Antoine Delon and Aleix Poch Parera.

References

Lasheras, J.C., and Hopfinger, E. J. Liquid Jet instability
and atomization in a coaxial gas stream. Annu. Rev. Fluid
Mech., 32, 275-308, (2 '""' ).

Raynal, L. Instability et entrainement a l'interface d'une
couche de melange liquide-gaz.: PhD Thesis, UJF
Grenoble France (1997).

Marmottant, P and E. Villermaux. On spray formation. J.
Fluid Mech., 498, (21 ".4).

Hong, M. Atomisation et melange dans les jets coaxiaux
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Hong, M., Cartellier A. & Hopfinger E.J., Atomization and
mixing in coaxial injection, Proc. 4th Int. Conf. on
Launcher Technology, Liege Belgique (2i "12).

Varga, C.M., Lasheras, J.C. and Hopfinger, E.J., Initial
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stream, J. Fluid Mech., 497, (21 1'").

Ben Rayana F., Contnribution a l'etude des instabilites
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gouttes, PhD Thesis, INP Grenoble France.

Lozano, A., Barreras., F., Siegler, C. and LOw, D., The
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(2005).

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

Lin S.P., Breakup of liquid sheets and jets, Cambridge
University Press, 2003.

Matas J.-P., Estivalezes J.-L. & Cartellier A., Atomisation
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