A model is currently under development to predict the occurrence and outcome of spray droplet breakup induced by aerodynamic forces and droplet collisions. It is speculated that these phenomena may be significant in determining the droplet size distribution in a spray subjected to acoustic velocity fluctuations. The goal is to integrate this breakup model into a larger spray model in order to examine the effects of combustion instabilities on liquid rocket motor fuel sprays. The model is composed of three fundamental components: a dynamic equation governing the deformation of the droplet, a criterion for breakage based on the amount of deformation energy stored in the droplet and an energy balance based equation to predict the Sauter mean diameter of the fragments resulting from breakup. Comparison with published data for aerodynamic breakup indicates good agreement in terms of predicting the occurrence of breakup. However, the model significantly over predicts the size of the resulting fragments. This portion of the model is still under development

Jacobs, H. R.

Wert, K. L.

English

NASA Technical Reports Server

DEVELOPMENT OF A DROPLET BREAKUP MODEL CONSIDERING
AERODYNAMIC AND DROPLET COLLISION EFFECTS
K. L. Wert and Dr. H. R. Jacobs
Propulsion Engineering Research Center
and
Department of Mechanical Engineering
The Pennsylvania State University
University Park, Pa 16802
SUMMARY:
A model is currently under development to predict the occurrence and outcome of spray droplet breakup
induced by aerodynamic forces and droplet collisions. It is speculated that these phenomena may be significant
in determining the droplet size distribution in a spray subjected to acoustic velocity fluctuations. The goal is to
integrate this breakup model into a larger spray model in order to examine the effects of combustion instabilities
on liquid rocket motor fuel sprays. The model is composed of three fundamental components: a dynamic
equation governing the deformation of the droplet, a criterion for breakage based on the amount of deformation
energy stored in the droplet and an energy balance based equation to predict the Sauter mean diameter of the
fragments resulting from breakup. Comparison with published data for aerodynamic breakup indicates good
agreement in terms of predicting the occurrence of breakup. However, the model significantly overpredicts the
size of the resulting fragments. This portion of the model is still under development.
TECHNICAL DISCUSSION:
The work to be discussed here is part of an ongoing numerical study of several aspects of the
interaction between transverse acoustic fluctuations and atomized liquid sprays. Therefore it is relevant to the
study of combustion instabilities in liquid propellent rocket motors since it is known that these instabilities result
from a coupling between the combustion and fluid dynamic processes of the motor and the chamber acoustic
resonance modes. It has been conjectured that for a spray subjected to acoustic waves, the displacements of the
droplets due to the acoustic velocity fluctuations may have a significant impact on the spray pattern and droplet
size distribution downstream of the injector.
In a previous study [Wen (1992)], numerical solutions were obtained for a model of a nonevaporating,
pressure-atomized spray subjected to a transverse, one-dimensional acoustic field. This model only examined
potential droplet agglomeration. Examination of the results showed an increase in mean droplet size downslream
of the injector compared to the same spray injected into a quiescent medium. An issue not addressed in this
earlier work, however, is that of possible drop breakup downstream of the near-injector primary atomization
zone. Just as acoustic velocity fluctuations were shown to enhance droplet coalescence, therefore increasing
droplet size, so too may these fluctuations lead to enhanced droplet fragmentation.
Downstream drop breakup, or secondary atomization, can occur through two processes, both of which
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areof potential importance for acoustically perturbed sprays:
1. Aerodynamic breakup where the relative velocity between the drop and the continuous phase is
sufficient to fragment the drop.
2. Collision-induced breakup where the energy of the colliding drops is sufficient to fragment the drops.
To examine the importance of these two fragmentation modes, a breakup model is currently under development.
When completed, this submodel will be integrated into the overall spray model. Previous efforts to model
droplet breakup have focused on only the aerodynamic breakup mode [O'Rourke and Amsden (1987) and
lbrahim et al. (1990).] However, what is clearly needed is a model able to treat both aerodynamic and collision-
induced modes. The remainder of this discussion will describe the droplet breakup model in its cm'rent state of
development and provide some preliminary comparisons with experimental data.
The droplet breakup phenomenon was approached from the standpoint of energy conservation; thus,
the analysis began by integrating the differential mechanical energy conservation equation over the volume of a
drop of arbitrary shape. The velocity of a fluid element within the drop was then decomposed into two
components: a mean velocity, equal to the velocity of the droplet mass center, and a fluctuating component that
is nonzero for a deforming droplet. Substituting this decomposition into the energy equation and subtracting out
the terms for the mean droplet energy (much like developing the turbulent kinetic energy equation), yielded an
integro-differential equation governing the deformation velocity field of the droplet. This equation contains
terms related to the temporal variation of the deformation kinetic energy, the surface tension energy generation
rate, the dissipated energy due to viscous effects and an energy source term that must be constituted to account
for aerodynamic surface forces and droplet collision.
While the energy equation gives a necessary condition that the deformation velocity components must
satisfy in order to satisfy mechanical energy conservation, it is not possible to use this equation to solve for the
deformation velocity field within the drop. To do this analytically would require the solution of the Navier-
Stokes equations subject to the boundary conditions at the drop surface. Clearly a generalized analytical solution
is not possible and a full numerical solution of the flow field within each spray droplet is not practical for
implementation into an overall spray model. Thus it was necessary to specify an appropriate deformation
velocity field. To do this, the droplet was viewed prior to breakup as deforming in one of two fundamental
modes:
1. From a sphere to an oblate spheroid. This approximates the flattening of the droplet experienced
initially in the aerodynamic breakup mode [Nigmatulin (1991) and Cliff (1978).]
2. From a sphere to a prolate spheroid. When fragmentation occurs after the temporary coalescence of
two colliding droplets, the droplet initially deforms into a shape much like a prolate spheroid before
further deforming into a dumbbell shape and fragmenting [Ashgriz and Givi (1987) and (1989).]
By including both prolate and oblate deformation modes, the model can account for the initial stages of both
aerodynamic and collision-induced breakup effects.
A velocity field was subsequently developed which not only satisfies both of the above mode shapes,
83
butwhichsatisfiesincompressiblecontinuityaswell. Substitutingthisfield into the deformation mechanical
energy equation yielded a second-order, nonlinear ordinary differential equation. This equation governs the
temporal variation of the streamwise axis of the spheroid, 2 b:
pdv_rl 2+[ a_21 db d2b
i0 [_ kb] ] dt dt 2 2 b3_dt} J + 4Eub -_ -_ + -_ -_ = Ei
(1)
where Pd is the drop density, Va is the drop volume, 2b is the streamwise axis length, 2a is the cross-stream drop
diameter, ¢y is the surface tension, S(b/a) is a function of spheroid geometry, _ is the drop viscosity and E i is the
energy input source term. From left to right, the terms are: the deformation kinetic energy term, the surface
tension energy term, the energy dissipation term and the energy input source term.
The development of a deformation velocity field allowed the explicit evaluation of all the terms of the
deformation mechanical energy equation except one, the energy input source term, Ei. This term must
incorporate both aerodynamic and collision effects. To evaluate the aerodynamic energy source term required
knowledge of the pressure distribution on the droplet surface as a function of spheroid shape and relative
velocity. In the paper of Masliyah and Epstein (1970), the authors reported numerically-determined surface
pressure distributions at Re = 1 and Re = 100 for spheroids of various major to minor axis ratios. It should be
noted that Reynolds numbers of the order of 100 are typical for sprays. Integration of the vector dot product
between the surface pressure force and the surface velocity over the drop surface area yielded the aerodynamic
energy input rate. At present, this has been done for the oblate data only (b/a < 1) as this is of most concern for
aerodynamic-induced breakup. For simplicity, the derived points were correlated by the expression
_] w tlOZO
/ ! _-o.n
(2)
where Et_o is the aerodynamic energy input term, pc is the continuous phase density and U is the relative
velocity while f, and f2 are unit step functions such that f,=l for b/a > 0.5 and f2--I for b/a < 0.5. The Reynolds
number is based on the cross-stream diameter, 2_ Equation 2 correlates the data to within 2 % for the points at
Re ffi 100. Note the weak Reynolds number dependence. The strongest dependence is on the spheroid geometry
manifested through the axis ratio, b/a.
Equation 1 governs only the lowest-order deformation mode of a droplet prior to breakup. The final
stages of droplet fragmentation are dominated by the development of higher-order modes (e.g. the dumbbell-
shaped breakup of a droplet formed by two colliding droplets.) The reader is referred to Nigmatulin (1991) for a
discussion of the various aerodynamic breakup modes. Since analytical treatment of these higher order modes
was not desired for the sake of simplicity, the conditions under which a droplet fragments were specified in
terms of a critical deformation energy level of the modeled fundamental modes. It is postulated that a single
84
criticaldeformatione ergylevelforbreakupexistsforbothcollisionandaerodynamic-inducedbr akup.
Toestimatethiscriticalenergylevel,thebmaryfueldropletcollisiondataof AshgrizandGivi (1989)
wasused.In theirwork,theauthorsobservedcollisionsof pairsof fueldropletshavingvariousrelative
velocitiesandrelativesizes.Thecriticalenergylevelforbreakup was derived by applying an energy balance to
the test case in which the droplets just had sufficient relative velocity such that the droplet formed from the
coalesced pair fragmented. From this energy balance, it was possible to derive the critical dimensionless energy:
Ea'crl_ - 1.48 (3)
Here E_,_t is the critical deformation energy, which comprises the deformed kinetic and surface tension energies.
This energy is nondimensionalized by the product of the drop surface tension and the square of the spherical
diameter of the fragmenting drop, D o.
Together, equations 1, 2 and 3 form the basis for predicting whether aerodynamic-induced breakup will
occur. As a test, the model predictions were compared with droplet breakup data available in the literature. In
the recent work of Hsiang and Faeth (1992), the authors conducted experiments on the properties of drop
deformation and secondary breakup for shock wave initiated disturbances. To recreate the shock condition in the
model, a step velocity change was specified. Noting that equation 1 is a second-order equation, the two
specified initial conditions were that the drop was initially spherical and that it possessed no deformation kinetic
energy ( b = 0 and db/dt = 0, respectively.) Equations 1 and 2 were solved numerically using Heun's method.
Care was taken to ensure a time step independent solution.
The model was used to determine what step change in velocity was necessary to bring about droplet
breakup. This was done for several of the fluids
considered by Hsiang and Faeth. Figure 1 shows the
critical Weber number, pcUo2Ddc, necessary to
bring about breakup of the drop as a function of the
Ohnesorge number, _/(pdDoO) °_ where Uo is the
imposexi step velocity change. The Ohnesorge
number is a measure of the ratio of liquid viscous
forces to surface tension forces. The points are the
predictions of the model while the solid line is from
the data presented by Hsiang and Faeth (1992),
which includes data from past studies. For We above
the line, the droplet is predicted to fragment;
however, for We below the line, the drop merely
undergoes deformation and oscillation. The model
reasonably predicts the variation in the critical Weber
Drop Deformation and Breakup Regime Map
We
10'
101
0 0001
-;--;;.;£;................1......
n later ]
A Olycerol: 217 [
v Glycerol: 637 ]
• Glycerol: ?57 _]a
• Glycerol: 927 '7
-- Hslang & Faet_ J
/
Drop Breakup • • /
/
1f7 V
_scillatJon
• ,,,,,,,I • ,,,,,,,I • ,..,,--I • ,.,,-,,J i ii|,-
0 001 0.01 0.1 1 10
Oh
Figure 1. Deformation and breakup regime map.
85
number with increasing drop viscosity effects. The worst agreement is around Oh = 0.2, where the model
predicts a critical We that is about 40 % high. It should be noted, however, that for sprays Oh < 0.01 is
expected. For this range the agreement between the model and experimental data is excellent. Since the critical
breakup energy was derived from considerations of drop collision data, its success in predicting aerodynamic
breakup lends support to the hypothesis that, at least for small effects of viscosity, the critical breakup energy
applies equally well to both collision and aerodynamic-induced fragmentation.
Having found the model able to adequately predict the occurrence of aerodynamic breakup, it was
necessary to prescribe a method for predicting the outcome of breakup. For this, the simple energy balance
method recommended by O'Rourke and Amsden (1987) and Ibrahim et al. (1990) was used. In this method, the
deformation energy of the droplet (both kinetic and surface tension energy) is equated to the surface energy of a
monodisperse group of spherical droplets. Thus the
deformation energy is converted into surface energy
of the fragments. Performing this energy balance
allows the prediction of the size of the fragments,
which is taken to be the Sauter mean diameter, SMD,
of the fragments.
The results of using this method are shown
in figure 2. The points are the model predictions, the
solid line is the best-fit line provided by Hsiang and
Faeth (1992) and the dashed lines represent the
spread in their data. Examining figure 2 it is seen
that the model predicts the trend of the data quite
well, but significantly overpredicts the size of the
SMD resulting from breakup. This was not
unexpected. Recall that the oblate spheroid
Correlation of the SMD after Secondary Breakup
100 ...... , ......
-- Cor"reJaUon ol HIlang & Fael.h
o 'llater
O n-Heptane _ _/
v Glycerol: 63X _
°go / _ /
o 0 0P 4)
/ /
OA 1 10
O, /p,_)-*"_lud/O,.,v.u )]uO,,v.tJ.'3/o
Figure 2. Correlation of the SMD after breakup.
deformation model treats only the initial stages of
droplet deformation. The higher-order deformation modes that ultimately lead to breakup are neglected in favor
of the critical energy criterion of equation 3. Thus the model neglecls the energy that enters the droplet through
the aerodynamic forces acting on the higher-order deformation modes. It is believed that the neglect of this
energy, which would be available to generate additional droplet surface area thus producing smaller fragments, is
the source of the overprediction of the SMD. An attempt to account for the energy that enters the droplet as a
result of these higher-order modes is under development.
Work that remains to be done on the model includes the derivation of Et for droplet collision, further
comparison of model predictions with available experimental data and integration of the breakup model into the
larger spray acoustics model.
86
CITED REFERENCES:
Ashgriz, N. and Givi, P., 1987. "Binary Collision Dynamics of Fuel Droplets," Int. J. of Heat and Fluid Flow,
Vol. 8, No. 3, pp. 205-210.
Ashgriz, N. and Givi, P.,1989. "Coalescence Probabilities of Fuel Droplets in Binary Collisions," Int. Comm.
Heat Mass Transfer, Vol. 16, No. 1, pp. 11-20.
Cliff, R., 1978. Bubbles Drops and Particles, Academic Press.
Hsiang, L.-P. and Faeth, G. M., 1992. "Near-Limit Drop Deformation and Secondary Breakup," Int. J.
Multiphase Flow, Vol. 18, No. 5, pp. 635-652.
Ibrahim, E. A., Yang H. Q. and Przekwas, A. J., 1990. "Modeling of Spray Droplets Deformation and Breakup,"
ASME Fluids Engineering Division, Vol. 99, pp. 475-479.
Masliyah, J. H. and Epstein, N., 1970. "Numerical Study of Steady Flow Past Spheroids," J. Fluid Mech., Vol.
44, part 3, pp. 493-512.
Nigmatulin R. I., 1991. Dynamics of Multiphase Media, Hemisphere Publishing Corporation.
O'Rourke, P. J. and Amsden, A. A., 1987. "The Tab Method for Numerical Calculation of Spray Droplet
Breakup," SAE Paper 872089.
Weft, K. L., 1992. Drot) A221omeration in an Atomized Spray Subiected to a Transverse Acoustic Field,
Applications to Combustion Instabilities in Liquid Propellant Rocket Motors, M.S. Thesis, The Pennsylvania
State University.
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Development of a droplet breakup model considering aerodynamic and droplet collision effects

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