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Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.
A97-36448
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AIAA-97-3298
Modeling the Effect of Unsteady Chamber
Conditions on Atomization Processes
K. M. Rump & S. D. Heister
School of Aeronautics and Astronautics
Purdue University
West Lafayette, IN 47907-1282
33rd AIAA/ASME/SAE/ASEE Joint
Propulsion Conference & Exhibit
July 6-9, 1997 / Seattle, WA
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
1801 Alexander Bell Drive, Suite 500, Reston, VA 22091
Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.
MODELING THE EFFECT OF UNSTEADY
CHAMBER CONDTIONS ON ATOMIZATION
PROCESSES
K. M. Rump *and S. D. Heister f
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April 24, 1997
Abstract
AP
e
pressure drop across the orifice
ratio of gas to liquid density
surface curvature
density
surface tension
velocity potential
phase lag
The response of fluid flow through an orifice to chamber pressure oscillations and the evolution of the corresponding liquid jet has been investigated through
the use of a two-dimensional model based on the
Boundary Element Method. A series of parametric
simulations have been performed using this tool to
evaluate the effect of orifice length, injection conditions, and the amplitude and frequency of chamber
pressure oscillations on the dynamic orifice massflow,
phase lag, and jet surface shape. Results indicate
substantial differences between the 2-D simulations
and 1-D theory, particularly in the case where the
orifice passage is short. Simulations have also been
performed to address the velocity profile at the exit
plane under these unsteady injection conditions.
p
&
<£
$
Subscripts
£
g
j
osc
ss
oo
rnin
max
Nomenclature
Introduction
K
liquid phase properties
gas phase properties
nodal location
oscillation amplitude
steady state value
far field condition
minimum value
maximum value
An area of interest in the study combustion stability in liquid-propellant rocket engines is the dy• orifice radius
a
namic behavior of fluid flow through an orifice (injec; orifice discharge coefficient
CD
k
: disturbance frequency or wave number tor) brought about by unsteady chamber conditions.
This phenomena has been theorized to be a possible
I
: orifice length
explanation for high-frequency combustion instabil: mass flow rate
m
ities in liquid engines1'2. In addition, the injector
n
: normal distance
phase-amplitude response may be used to determine
P
: pressure
: surface velocity in the normal direction the type or even to suppress instability mechanisms
g
which may occur during combustion3. Atomization
r
: radial distance
or breakup behavior of the liquid jet or droplets flow: time
t
ing from the injector also plays an important role in
: mean liquid velocity
U0
combustion stability. Droplet size, distribution, and
We
: Weber number, We = piU^a/cr
jet breakup length all effect the combustion process
z
• jet axial location
and in turn are influenced strongly by injector dy: surface slope
ft
namic response characteristics.
"Research Assistant, Member, AIAA
In spite of its potential importance in explain^ Associate Professor, Member, AIAA
ing
instabilities, dynamic orifice flows have not been
Purdue University, West Lafayette, Indiana, 47907
Copyright © 1997 by the American Institute of Aeronautics studied in great detail. While there have been sevand Astronautics, Inc. All rights reserved.
eral studies investigating combustion behavior un-
1
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Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.
der forced oscillations, few have focused on the injection/atomization response. One exception is the
early work of Reba and Brosilow4 who studied the
effects of large amplitude axial acoustic disturbances
in the injection chamber on jet breakup length and
periodicity of droplet formation for liquids injected
into the chamber. They concluded, based on their
observations, that the primary factor in determining
distance between drops was the oscillation in the orifice mass flow rate and developed an analytical model
which predicted the amplitude and phase shift of the
orifice flow response to a pressure oscillation in the
injection chamber.
Recently, there have been a number of advancements in the numerical modeling of jet breakup and
atomization processes resulting in boundary element
method (BEM) models capable of simulating unsteady, nonlinear coupled gas/liquid flows5"8. Most
relevant to the work described in this paper is a finite jet BEM model of flow from an orifice developed
by Hilbing et. al.5. While model did not include
gas effects, Hilbing did complete preliminary work
on a coupled gas/liquid finite jet model which was
the starting point for the present investigation. This
paper will describe the development of a this fully
coupled model to examine the phase-amplitude response to forced oscillations in the downstream pressure. The effects of orifice length, oscillation frequency and amplitude, and jet injection velocity are
addressed through a series of parametric simulations
using the model.
The axisymmetric computational domain and
boundary conditions used in this model are given in
Figure 1, where q and qg are velocities normal to the
boundary in the liquid and and gas domains respectively. Gas nodes are denoted with an "x", liquid
boundary nodes with a "o", and liquid interior nodes
with a "+". Nodes have been placed on the interface between the domains, along the radial axis in
both domains, in the interior of the liquid domain
at the orifice exit, and on the outer boundary in the
gas domain. The interior nodes permit the calculation of axial velocity within the liquid at the orifice
exit plane. By modifying the discretization procedure
used for nodes on the boundary, fluid velocity at locations in the interior of either domain can be calculated
using only information at the boundary nodes. This
permits determination of accurate boundary conditions at any location if it is desired to split off a part
of the domain.
Under the assumptions discussed above, the dynamics of both liquid and gas flows are governed by
Laplace's equation:
= 0=
(1)
where <j> and <j>g and are the liquid and gas velocity
potentials. The unsteady Bernoulli equation provides
the boundary conditions at the gas/liquid interface.
The coupled, nondimensional equations for the liquid
and gas phases at the free surface are:
(2)
2
Model Development
The model developed for this study uses Boundary
Element Methods to simultaneously solve for conditions in gaseous and liquid phases with the use of
a computational mesh containing nodes only on the
boundaries of the domain. A capability to calculate
conditions within the interior of either fluid was implemented in order to investigate velocities within
the orifice passage. As in previous modeling5"8, the
free surface is resolved with full fourth-order accuracy. Both liquid and gas phases are assumed to
be inviscid and incompressible, and gravity (or other
body forces) are neglected. In addition, the range of
acoustic wavelengths studied was assumed to be much
larger than the length of the jet such that spatial variations within the chamber gas are negligible and the
incompressible assumption is prudent. This assumption is also justified because the injector is typically
located near a velocity node and pressure antinode
such that minimal spatial variations are present.
+ P, = 0
(3)
Here, we choose the orifice radius (a), liquid density
(pi), and average orifice exit velocity (U0) as dimensions. In Eqs. 2 and 3 K is the local surface curvature,
e is the density ratio (pg/pt), and We is the Weber
number (piU%a/cr). These two equations are coupled
through the gas pressure (Pg) at the gas/liquid interface. This pressure, in turn is dependent on the
shape and velocity of the interface, and on the value
of the farfield gas pressure which is periodically varying with time due to the acoustic disturbance. In the
liquid phase the Bernoulli equation is used to specify <j> at the inlet, while the liquid velocity normal to
the boundary, q = d<f>/dn (where n is the outward
normal to the boundary), is specified along the inlet
walls (5 = 0). In the gas phase, the gas velocity normal to the boundary, qg, is set to zero along the wall,
4>g is set on the top surface via integration of Eq.
3, and qg = — q along the gas/liquid such that gas
and liquid nodes remain coincident on this boundary
throughout the simulation.
Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.
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The far-field gas pressure, Pgas, is assumed to be
represented by a simple sinusoidal function with frequency k. Note that under the nondimensionalization
employed, this dimensionless frequency is equivalent
to a dimensionless wavenumber since the jet travels
one radius over a dimensionless time of unity. The
resultant form for the orifice pressure drop, (AP), is
given by:
derivative d<j>/ds is obtained via 4th-order centered
differencing5.
Equations 5-7 (and Eqs. 2,3 on inflow/outflow
boundaries) are advanced in time using a 4th-order
Runge-Kutta time integration. Reference 7 provides
a description of the treatment of the gas-pressure
which couples the two Bernoulli equations on free
surface nodes. In addition, the treatment for internal
nodes is discussed in Ref. 9. As solutions proceed,
AP =
sin(kt)
(4) the free surface is regrid at each timestep using cubic
splines so as to keep the spacing between nodes conwhere A.PS3 is the steady state pressure drop and stant. Nodes are added as the jet emanates from the
APOJ!C is the oscillation amplitude. APS., in turn is orifice exit plane in order to meet this requirement.
given by Pgas — P(«,, where P^ is the liquid pressure
at the orifice inlet. Under the assumed nondimensionalization, Pgas — Pix = 0.5 + I/We; a result obtained
Model Validation
from the steady form of Bernoulli's equation.
At each time interval, Eq. 1 is solved using a
second-order accurate BEM as described in Ref. 5.
In this methodology, the quantities <j>, <j>g, q, and qg
are presumed to vary linearly between adjacent surface nodes. This solution process provides unknown
boundary velocities or velocity potential values at
all nodes. Because nodes on the gas/liquid interface
are permitted to move with time, an Eulerian - Lagrangian transformation is required7. Assuming that
nodes are "tracked" with the local velocity in the liquid, the proper form of the Bernoulli equation for free
surface nodes becomes:
-Ps-——
(5)
(6)
Dt
for the liquid and gas domains, respectively.
The position of nodes on the free surface is determined through basic flow kinematics considerations:
Dz_
Dt
_
~d~z
DT
~Dt
_
dr
(7)
where D()/Dt indicates a Lagrangian derivative for
points on the surface moving with same velocity as
the local liquid. Since the BEM solver returns velocities normal to the free surface, velocities must be
transformed to the inertial (r, z) coordinate system:
A /
f)/k
'dr ~ 'fos
(8)
where /? is the local wave slope and d<j>/ds is the
velocity tangential to the local surface. Surface
slope is determined by fitting a parabola through the
point in question and its nearest neighbors, and the
A standard test case (steady inflow, We=l7.Q, e =
0.01) was utilized in determining model sensitivity to
numerical parameters. While this Weber number is
far below that of actual liquid engine injection conditions, computational limitations restricted us to consider low values of this parameter. However, the influence of Weber number on results is addressed in the
following section for a limited range of values. The
assumed initial jet geometry included a small amount
of fluid outside the orifice. This fluid was assumed to
be cylindrical in shape with a hemispherical nosecap.
Calculations were terminated an instant before the
first droplet pinched from the surface. Grid function
convergence studies were undertaken by comparing
the jet shape at this event for different grid spacings.
Figure 2 presents results of these simulations for
three node spacing convergence test runs. These runs
were undertaken for node spacings of 0.5, 0.25, and
0.1. Although the pinch location is significantly further downstream, the actual shape of the jet with a
node spacing of 0.1 is quite similar to that of the jet
with a spacing of 0.25. Therefore, a spacing of 0.2 was
actually used for follow on runs. Similar convergence
tests for time step and gas domain radius indicated
that values of 0.002 for time step was adequate to insure temporal accuracy. In addition, a series of simulations using different sizes of gas domain indicated
that results were insensitive to this parameter if the
outer portion of the domain were placed at least 10
jet radii from the centerline. Of the three parameters tested, grid spacing appeared to have the largest
effect on jet shape, while time step had the smallest.
Additional tests were undertaken to determine the
minimum distance which interior nodes could be
placed from the boundary and to verify that predictions at interior nodes were consistent with flow
conditions on the boundary of the liquid domain. It
Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.
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is well known in BEM literature that singularities in
integrations lead to reduced accuracy if interior nodes
are placed too close to boundaries. Numerous studies were conducted9 to insure accuracy of velocities
obtained in interior nodes. Results of some of these
studies are shown in Figure 3 which highlights the
error in axial velocity (d<f>/dz) as a function of the
dimensionless distance from the wall. Based on the
results of both tests, the interior node closest to the
wall was placed a distance 50% greater than the node
spacing used along the boundary.
Results
A series of 24 simulations were performed to assess
the influence of oscillation wavenumber and amplitude (k, A.POSC), orifice length to radius ratio (//a),
We on the orifice mass flow rate response, jet shape,
and the interior velocity profiles. Additional simulations examined the effect of density ratio, (e), on
jet breakup in a non-oscillating gas. A typical calculation started with 126 nodes on the boundary of
the liquid, 89 nodes on the boundary of the gas, and
8 nodes in the interior of the jet. The number of
nodes on the liquid and gas boundaries increased as
the calculation progressed, due to the growth of the
jet. Each run required approximately 50,000 cpu seconds to complete.
The effect of gas liquid density ratio on the formation of a jet issuing from an orifice into a gas was
investigated by performing simulations to the point
of droplet pinch for four different values of e, from
e = 0.001 to 0.05, at We = 17.6. The results of
these runs are shown in Figure 4. It is seen that increases in e have the effect of decreasing jet breakup
length, and flattening the drop produced when the
jet pinches. This flattening becomes increasingly violent such that at e = 0.05, the droplet has flattened
into a mushroom cap. It is interesting to note that
e must be increased by an entire order of magnitude
(relative to the air / water value of e « 0.001) before
significant differences in jet shape are observed.
The effects of k, I/a, APOSC, and We on orifice
flow rate, surface evolution of the resultant jet, and
interior velocity profiles were studied by performing
a series runs which varied the parameter of interest
while all other parameters were held constant. In
each of the runs, the far field gas pressure (Pgas) was
varied periodically while liquid pressure at the orifice inlet, Piao, was held constant. This resulted in
periodic oscillations in the pressure drop across the
orifice, AP as specified by Eq. 4. Orifice flow rate
was calculated by integrating the values of velocity
in the z direction at nodes on the inlet boundary and
at interior nodes at the orifice exit across the appropriate area. The flow rate at the two locations was
then averaged to provide the results shown in the following figures. Profiles of velocity as a function of
radial location within the jet were calculated using
the interior velocity calculation procedure developed
for this study. A density ratio of e = 0.01 was used
in all these simulations.
The effect of oscillating orifice pressure drop is
shown in Figure 5 which compares a jet injected into
a constant pressure gas and a jet injected into a gas
with oscillating pressure. It is observed that the jet
injected into the constant pressure gas exhibits no
perturbations in surface shape until slightly before
the pinch point. On the other hand, periodic surface
perturbations are evident along the entire length of
the jet injected into the oscillating gas.
In their investigation of liquid jet behavior in
the presence of gas pressure oscillations, Reba and
Brosilow4 derive a 1-D, analytical model which can
be used to predict the mass flow rate frequency response to an oscillating chamber pressure. Effects
of I/a on frequency response are also described by
this model. Using this model, the dimensional orifice
mass flow rate is given by,
m =
&P'oscsin(k't' - a)
(9)
where,
(10)
and primes denote dimensional quantities. Here,
APg5C is the amplitude of the pressure drop oscillation, nig,' is the steady state mass flow rate through
the orifice, A'0 is the orifice area, and CD is the orifice
discharge coefficient. Since the BEM model assumes
inviscid flow, CD is equal to unity. Nondimensionalizing Eq. 9 gives:
K&.Poscsin(kt — a)
TO = ————.
—————— + TT
(11)
where,
w
1
a = — — *tan -it
(jr)\
(12)
As for APOSC, a rnosc can be defined such that m(t) =
ihss + rnoscsin(kt — a) where where mss is the steady
state mass flow rate through the orifice and —a is the
phase lag of the response.
As part of this series of investigations, predictions
from the BEM model were compared with analytical
predictions from Reba and Brosilow's model. Figure
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Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.
6 indicates the orifice mass flow rate behavior predicted by the BEM through the course of a typical
simulation. The predictions of the analytical model
and AP are are also shown. Mass flow rate data
is presented normalized with respect to the steadystate orifice flow rate (tn(t)/rhss) and the orifice pressure drop is normalized by the steady-state pressure
drop (AP(t)/APss). For the conditions noted in Fig.
6 caption, the 2-D BEM results predict a lower amplitude, but a greater phase lag than the 1-D analytic
results. Therefore, the main implication of 2-D flow
effects is to increase the capacitance of the nozzle. In
Fig. 6, the BEM massflow grows slightly with time
due to the varying capillary pressure drop across the
free surface. At larger Weber numbers, this effect
becomes vanishingly small. As mentioned previously,
we were unable to obtain results for large Weber numbers due to computational limitations.
Figures 7 and 8 address the effect of orifice design
(I/a) on amplitude and phase lag of the orifice massflow, and on the resultant jet surface shapes. In Fig.
7, orifice massflow characteristics are presented for a
range of frequencies, k. The overall region in which
dynamic response is important is roughly limited to a
frequency (wavenumber) range 0.1 < k < 10. Above
this frequency range, the jet is unresponsive to the
imposed oscillation; below this range a quasi-steady
behavior is observed. In Fig. 7, the 2-D BEM results
predict more compliance within the injection process
than the simple 1-D result. As one might expect, the
discrepancy is greatest for the short nozzles; at the
larger I/a = 5 values, the two results are reasonably
close. The BEM results also indicate that 2-D effects
tend to increase the phase lag for the shorter nozzle
(Fig. 7, I/a = 1) as compared to the analytic result.
For the longer nozzles, the phase angles for the two
approaches agree reasonably well. Both results predict an asymptotic approach to —90° as suggested by
basic control theory.
The effect of orifice length on jet surface profiles
near the droplet pinch event is detailed in Fig. 8.
Here, the amplitude and wavenumber (frequency) of
the disturbance are held fixed at the values shown in
the figure caption. The increased capacitance of the
longer nozzles leads to a reduction in the amplitude of
disturbances generated by the pressure perturbation.
This factor results in an increase in jet breakup length
with increasing orifice length as shown in Fig. 8.
Figures 9 and 10 address the effect of disturbance
magnitude (AP05C/AP»S) on amplitude and phase lag
of the orifice massflow, and on the resultant jet surface shapes. In Fig. 9, orifice massflow characteristics
are presented for a range of disturbance frequencies
for a fixed orifice design (I/a = 3). In comparing
these results with Fig. 7, we see that the amplitude
and phase angle response is a much weaker function
of disturbance amplitude than of orifice design. In
addition, the BEM results suggest a decreased sensitivity to amplitude variation as compared to the analytic predictions. However, both results predict that
the phase angle is insensitive to the amplitude of the
disturbance. The effect of disturbance magnitude on
jet surface shapes is shown in Fig. 10 for the fixed
orifice design and injection conditions noted in the
figure caption. Even though the lower disturbance
has an amplitude three times that of the upper disturbance, the overall jet breakup length is nearly the
same. This conclusion is definitely dependent on the
wavenumber of the disturbance, since there are some
obvious nonlinear wave interactions in the lower plot
in Fig. 10.
The effect of Weber number on orifice flow rate oscillation parameters is displayed for We = 5.0, 10.0,
and 20.0 in Table 1. Reba and Brosilow's analytical
model does not predict any Weber number effects on
flow rate oscillation response amplitude or phase lag
since capillary forces are neglected in their derivation.
For the low Weber number regime investigate in our
simulations, results do indicate a strong Weber number dependence on both mosc/ms, and the phase lag
angle <3>. Physically, the Weber number influence is
caused by variations in capillary forces with changes
in jet shape. The nonlinear coupling between the
jet shape and gas pressure fields leads to the behavior noted in Table 1, i.e. increasing surface tension
(decreasing We) tends to "stiffen" the system, but
reduces the frequency response.
Since increasing We results in smaller response amplitude and phase lag some of the conclusions regarding phase lag in the above discussion of Figs. 7 and 9
may be altered for high Weber number conditions. In
other words, phase lags for high velocity jets would
be closer to the analytic predictions than those indicated in Figs. 7 and 9. However, Table 1 indicates
that phase angle corrections are asymptotically vanishing in the limit of high Weber numbers, so the
results in Figs. 7 and 9 certainly predict qualitative
behavior for the high Weber number case.
Figure 11 shows surface profiles of the jet at droplet
pinch for two Weber numbers. Increases in Weber
number are seen to increase the size of disturbances
on the surface of the jet, and the length of the jet at
droplet pinch. This would seem to support the contention that a coupling of oscillating gas pressure and
curvature effect orifice flow rate response. Presumably the Weber number dependence vanishes as We
approaches infinity, since the jet shape would have
negligible influence on orifice flow in this limit.
Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.
driving solutions to the slug-flow profile expected in
Table 1: Comparison of mosc/mss and Flow Rate the limit as I/a —> oo. Results from other simulaOscillation Phase Angle, $, for k = 0.5, I/a = 3.0, tions discussed above indicate that variations in wave
APOSC/APSS = 0.5
number had little effect on axial velocities, but had a
more pronounced effect on radial velocities at the exit
We mosc/mss $ (deg)
plane. Increases in Weber number produced substan0.211
-63.0
5.0
tial changes in radial velocities; interested readers are
-51.4
10.0
0.189
referred to Ref. 9 for more detail on these results.
0.168
-48.7
20.0
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Conclusions
As mentioned in the model development section,
we included a capability to investigate the velocity
profile at the orifice exit plane. This development
was undertaken in order to assess flow angularity at
the exit plane during various portions of the imposed
periodic excitation cycle. In addition, this capability
provides a mechanism to generate exit flow conditions
for more conventional CFD models aimed at investigating flows within orifice passages. To illuminate the
internal velocity profiles, a large simulation was conducted for the case of chamber pressure oscillations
causing severe disturbances to orifice flow rate and
jet shape. This was achieved by setting APOSC/AP55
= 0.75 and I/a = 0.1. A much finer interior grid was
used in this run, as was a finer grid on the boundary;
this allowed interior nodes to be placed closer to the
boundary and velocity behavior in this region to be
examined more closely. A case with severe surface
disturbances was selected in order to examine the effect of large pressure oscillations on radial velocities,
d(j>/dr, at the orifice exit; large surface disturbances
insured that significant interior velocities in the r direction were present for examination.
Figure 12 presents both axial and radial velocity
distributions at four distinct times during the periodic process; the time at which both flowrates and
pressure drops are both minimum and maximum values. Axial velocities, 5</>/<9z, show radial variations
of the order of 5-10% with the velocity at the center
of the jet being lower than that at the edge. The
variation in axial velocity is also seen to be greater
when flowrates and pressure drops are at their maximum values. The radial velocities in Fig. 12 show
substantial radial variations indicating a tendency for
the exit flow to be focused slightly inward toward the
center of the jet. As in the case of the axial velocities,
the greatest radial velocities are developed at times
when flowrates or pressure drops reach maximum values. Exit flow angles (measured with respect to the z
axis) consistent with the results in Fig. 12 vary over
a range from 0-5°.
Increasing the orifice length reduced the magnatude of both axial and radial velocity distributions
A coupled liquid/gas, finite length jet model has been
developed utilizing the Boundary Element Method in
order to assess the dynamic response of a single orifice
to an imposed periodic downstream pressure. Twodimensional results from this tool have been compared with the 1-D model of Reba and Brosilow4 in
many cases. Results from both models indicate a
dynamic orifice response in a range of dimensionless
frequencies/wave numbers which lie roughly between
0.1 and 10. Below this lower bound, quasi-steady
behavior is present, while the orifice is unresponsive
above the upper bound.
In general, the 2-D effects addressed in the BEM
lead to a reduction in the amplitude of the massflow response as compared to the 1-D analytic results.
Results are quite sensitive to the orifice length and
the increased capacitance effect is most pronounced
for shorter nozzles. The 2-D results also predict increased phase lags for shorter nozzles; results for long
nozzles agree reasonably well with the 1-D theory.
The 2-D results of response of the liquid massflow
show surprising insensitivity to the amplitude of the
imposed oscillation in chamber pressure, predicting
much less variation than that of the 1-D theory. Both
1-D theory and 2-D simulations show the phase lag
to be unaffected by the magnatude of the imposed
oscillation. Variations in Weber number indicate a
decrease in both the amplitude and phase lag of the
response as We is increased. Velocity profiles at the
orifice exit plane show a tendency for the jet to "neck
down" at this location; response in this regard is most
pronounced when the imposed pressure perturbation
or orifice massflow reach maximum values.
6
Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.
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References
1. Bazarov, V. G., "Influence of Propellant Injector Stationary and Dynamic Parameters
on High Frequency Combustion Stability",
AIAA Paper 96-3119, July 1996.
q=0
/
+
Liquid
2. Jensen, R., (Ed.), "JANNAF Subcommit- 60
tee on Combustion Stability - Annual Report," 27th JANNAF Combustion Meeting,
Figure 1: Schematic of Computational Domain and
Cheyenne, Wyoming, 1990.
Boundary
Conditions
3. D. T. Harrje and F. H. Reardon, "Liquid
Propellant Rocket Combustion Instability,"
NASA SP-194 (1972).
4. Reba, I. and Brosilow, C., "Combustion Instability: Liquid Stream and Droplet Behavior", Wright Air Development Center,
Technical Report 59-720, 1960.
5. Hilbing, J. H., Heister, S. D., and Spangler,
C. A., "A Boundary Element Method for
Atomization of a Finite Liquid Jet", Atomization and Sprays, V 5, No. 6, pp 621-638,
= o.io
1995.
ds = 0.25
6. Spangler, C. A., Hilbing, J. H., and Heister,
ds = 0.50
S. D., "Nonlinear Modeling of Jet Atomization in the Wind-Induced Regime", Physics
Figure 2: Effect of Grid Spacing (ds) on Jet Profile,
of Fluids, V 7, No. 5, pp 964-971, 1995.
We = 17.6, e = 0.02
7. Heister, S. D., "Boundary Element Methods for Two-Fluid Free Surface Flows", To
Appear, Engineering Analysis with Boundary Elements, 1996.
8. Heister, S. D., Rutz, M., and Hilbing, J.,
"Effect of Acoustic Perturbations on Liquid Jet Atomization", Journal of Propulsion and Power, V13, No. 1, pp. 82-88,
1997.
9. Rump, K, "Modeling the Effect of Unsteady Chamber Conditions on Atomization Processes", MS Thesis, Purdue University, 1996.
1.0-r
As
Figure 3: Errors in axial velocity (ff) for Interior
Nodes as a Function of Radial Distance from Centerline
Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.
(a) - Const. AP
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t =8
(a)
£= 0.001
8= 0.02
Y7A
(b) - Osc. AP
t=16
(a)
V/A
(b)
e= 0.03
WX
t = 24
(a)
(b)
YZA
e= 0.05
Figure 4: Effect of Density Ratio on the Evolution of
a Jet Under Steady Inflow Conditions, We = 17.6
Figure 5: Comparison of Jet Formation During Oscillating and Steady Flow, AP0iC/APSi = 0.75, k =
2.0, We= 17.6
*-• o* o
o a !-*>
• i £\ (
CD
p
> O 3.
p
Normalized Magnitude: -?2i & AECI
Af
mss
ss
p p p
_ * _ * _ . _ » _ »
b l G ) - ^ J C D C O -
^ P Ch
1
^ * K D C O 4 > . C n
Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-3298
3 O 05
(X ™
^8
S±>
|
a
t' 3
iv>
o
S.
W «•
J=S
7
•<: 3
R
1
a<
5*0
Si S?
HrJ O
§ <n
l§-^
?lf
Mass Flow Rate Response Phase Angle, (Deg.)
Normalized Mass Flow Rate Response Amplitude, -
Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-3298
Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.
10
10
10
Nondimensional Pressure Oscillation Wave Number, k
Figure 8: Comparison of Jet Profiles for Various Orifice Lengths: k - 1.0, APOSC/&PS, - 0.5, We = 10
10'
10"
10"
10'
Nondimensional Pressure Oscillation Wave Number, k
Figure 9: Orifice Mass Flow Rate Amplitude, Phase
Response for Various Imposed Disturbance Amplitudes: BEM Comparison with Analytical Predictions
for k - 0.5, I/a = 3.0, We = 10
10
Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.
s.o •
0.0
I———————— ^
——————— _—^J
-
-5.0 •
—>
-t—
1.2
!
Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-3298
0.0 -
-
*
*
*
*
*
*
"*
. . " • " " " " *
'
*
+
+
+
S.O -
-5.0
*
m *
£-AP» =°-re
——•/>
00
10.0
20.0
1.1
+
-<HN
^)( )
rolro
1
+ +
:
*
+
+
+
+
+
•f
+
* m = riW
x
*
m = n \T*
+ AP-AI^
o AP=AI^
0.9
, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
30.0
0.8
Figu re 10: Effect of Pressure Oscillation Magnitude
on J st Profiles for k - 0.5, I/a - 3.0, We - 10
X
0.7
*
X
3
X
X
X
X
0.2
X
X
X
X
0.4
X
X
X
X
0.6
*
X
0.8
1
Radial Node Location
+
Oi
+
8
-t-
+
-0.01
8
+
+
x
8 x
-0.02
-e-l •-
S x
io\ro
-0.03
o
x
X
-0.04
X
0
. " x
o
-0.05
I——————————I
» rt"=riW
-0.06
x m = mMa
-0.07 •
+ AP = AP
o AP=AP^,
-0.08
)
Oi
_
^
x
x
x x x
*
»
o _
S
0.4
0.6
°
» 8 8 "
0.8
1
Radial Node Location
Figure 12: Orifice Exit Plane Velocity Profiles Highlighting Axial (d(j>ldz) and Radial (d<j>/dr) Velocities for I/a = 0.1, APOSC/AP,,5 = 0.75, k = 0.5,
We = 10.0
Figure 11: Effect of Weber Number on Jet Profiles
for AP05C/AP,S = 0.5, k = 0.5, I/a = 3.0
11
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