Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. A97-36448 Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-3298 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 Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-3298 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 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. 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. Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-3298 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. Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-3298 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 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. 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 Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-3298 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. Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-3298 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 Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1997-3298 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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