ILASS Americas, 19th Annual Conference on Liquid Atomization and Spray Systems, Toronto, Canada, May 2006

A Model for the Penetration of a Liquid Jet in Crossflow

A. Mashayek, A. Jafari and N. Ashgriz* Department of Mechanical and Industrial Engineering

University of Toronto Toronto, ON, M5S 3G8, Canada

Abstract

A theoretical model for the penetration of a liquid jet in subsonic gaseous cross flow (JICF) is developed. The model allows for the deformation of the jet cross section from circular to elliptical shapes along its path. A force balance analysis of an elliptical liquid element is performed. Aerodynamic, viscous and surface tension forces are considered, while counting for the nonlinear terms at large deformations. Effect of the mass shedding is also included in the model. Mass shedding has a considerable effect on the jet trajectory and deformation at higher Weber numbers. In addition, the drag coefficients of the elliptical cylinders of different cross sections are calculated numerically for Reynolds numbers between 200 and 8000 and for different ellipse aspect ratios. It is observed that the drag coefficient of the cylinder changes considerably with Reynolds number and the jet deformation. The change in the drag force affects the jet path in the gas stream noticeably. It was observed that the liquid-to-gas momentum ratio is not the only governing parameter in predicting the jet trajectory. Air Weber number, mass shedding from the jet, jet cross sectional deformation, variation in the drag coefficient and variation in the liquid and gas properties all affect the jet penetration.

1

* Corresponding Author

Introduction Liquid jet injection into gaseous crossflows has

applications in fuel injection systems such as gas turbines, afterburners, augmenters and ramjet/scramjet combustors. This type of radial fuel injection into a crossflow improves fuel atomization and vaporization characteristics and is commonly used in LLP (Lean Premixed Prevaporized) combustion systems. Over the past years the environmental issues concerned with the emission levels of the combustors have increased and new tighter regulations have been set. Previous studies have concluded that better homogeneity in the air-fuel mixture is necessary for reducing NOx emissions. On the other hand, tendency to lower the fuel consumption requires higher fuel-to-air ratios which in return cause undesirable consequences in terms of emission. These issues along with the instabilities occurred in the systems due to higher pressures and temperatures have invoked the need for more detailed understanding of this type of atomization

Jet in cross flow atomization problem involves very complex flow physics such as strong vortical structures, small scale wave formation, stripping of small droplets from the jet surface, and formation of differently sized ligaments and droplets. Therefore, the complete numerical simulation of such a problem, resolving most important flow scales on the Eulerian frame, is not still feasible, especially for industrial applications. These issues signal the demand for some simpler, yet reliable, models that can be used for industrial design purposes and can take into account important parameters such as flow conditions and physical properties of the liquid and gas phases.

In this study, we will focus on the jet trajectory by developing a model for the jet deformation and penetration. Details of the theoretical model are provided and results for different cases at various test conditions are presented and discussed.

There are numerous theoretical and experimental investigations of the JICF problem in the literature. Some correlations have been offered for the jet in cross flows based on experiments. Chen et al. [1] studied the effect of momentum ratio ( ) on the jet penetration. They changed the momentum ratio from 3 to 45 at room temperature and at pressures between 1 and 2 bars at the Mach number of 0.4 and offered a correlation for the trajectory. Wu et al. [2] studied the breakup of liquid jets in crossflows experimentally at normal temperature and pressure. They studied the effects of air Weber number and momentum ratio on the jet trajectory. The breakup regimes of the liquid column were characterized with respect to the Weber number in their work. Also, a phenomenological model for the jet penetration was presented that led a trajectory correlation.

2( ) /(l l g g

q u uU U

Generally, in the correlations proposed, the jet trajectory is given as a function of the liquid-to-gas momentum ratio and nozzle diameter. Some recent works have included the effects of temperature [5], pressure [6] and air Weber number [6] in their correlations.

Heister et al. [7], Li and Karagozaian [8], Nguyen and Karagozian [9] and Inamura [10] have modeled the jet cross section as an ellipse with its major axis perpendicular to the crossflow in order to better simulate the crossflow-induced deformation of the jet. Inamura [10] modeled the liquid jet trajectory by considering the liquid column’s deformation. His model predicted the jet penetration very close to the nozzle, within horizontal distances of 1.5-2 diameters, which is way before the jet’s final disintegration. He used the Clark’s [11] model for a two-dimensional (2-D) drop to predict the deformation of the jet cross section from circle to ellipse. However, the Clark model was derived for the small deformation of 2-D drops at low altitudes of oscillations and neglected the nonlinear terms. Ibrahim et al. [12] modeled the drop deformation and breakup taking into account the nonlinear terms and concluded that Clark’s model overestimated the drop deformations at larger altitudes of oscillations.

Mazallon et al. [13] experimentally investigated the primary breakup of nonturbulent round liquid jets in gaseous crossflows. They focused on the details of the jet deformation properties, time of onset of primary breakup, liquid column wavelengths and liquid surface wavelengths. Their measurements were carried out in normal temperature and pressure (NTP) conditions for momentum ratios of 100-8000 and suggested qualitative similarities between the primary breakup of nonturbulent round liquid jets in crossflows and the secondary breakup of drops. This similarity has been reported in a number of other articles. Model

The model presented here can be generally divided into four sections. First, the basic equation of deformation of the jet cross section is derived. Second, the equation for the jet trajectory is obtained. Third, the mass reduction from the jet is discussed. Finally, time varying drag coefficient is calculated and implemented in the equations.

2 )

Jet Deformation

A cylindrical element of infinitesimal thickness h from the liquid column is shown in Fig. 1. At any point, the element is perpendicular to the path its center of mass travels. The spatial location of this element as it moves along the jet trajectory is illustrated in Fig. 1.

The element is deformed and moved along the jet trajectory due to the interplay between aerodynamic, surface tension and viscous forces. Here, we make the

assumption that the jet cross section changes from a circle to an ellipse. The aspect ratio e=a/b of the ellipse, which is defined as the ratio between ellipse’s semimajor a and semiminor b axes, increases with time till the breakup location. It should be noted that this assumption is an approximation of the liquid jet’s behavior and its precision is being discussed in this article. In reality, the jet cross section changes slightly into a kidney shape [8]. ]. Clark [11] proposed that the cross section deformation is dependant on the viscous (dissipative) force, an interfacial tension (restoring) force, and an inertial force. His model was based on the analogy between an oscillating two-dimensional drop and a forced mass-spring system, as shown in Fig. 2. Using the same analogy, We will perform a force balance in the x2 direction (Fig. 2) in the cross sectional plane to obtain the equation for the oscillation of the element. Viscous Force

Formulating the element deformation in terms of the motion of the center of mass of the half-element, Clark [11] obtained the viscous force by dividing the energy dissipation per unit thickness of the half-element by 2dy where dy is the y increment. Modifying the linearized 2-D viscous force offered by Clark for a circular cross section to a non-linearized force for an elliptical cross section, we obtain:

2

2

/2

v l eq

dy dtF r h

yS P �

§ ·¨© ¹

¸ (1)

where req is the radius of a circle with equal area to that of the instantaneous elliptical cross section. As will be explained in the mass reduction section, the initial cross sectional area of ʌr

20 will not remain constant as the

element moves along the jet trajectory. Surface Tension Force

The surface tension force acting through the center of mass of the half element would be :

1

2s

dAF

dyV � § ·

¨ ¸© ¹

(2)

The lateral surface area of the element is the

instantaneous perimeter of the ellipse times the element thickness. Using Cantrell’s proposed formula for an ellipse [14], the lateral surface area will be:

> @4( ) 2(4 ) /A h a b ab HS u � � � (3)

where, 1

[( ) / 2]m m mH a b � . In practice, an exponent m=0.825 (33/40) yields an error of less than 0.0085% for any ellipse [14]. By differentiating equation (3) and substituting into (2) we get the surface tension force acting through the center of mass of the half element becomes:

2 234(1 )

8s

cF h r a

d

SV � � u u � �ª º

« »¬ ¼ (4)

where c and d are:

2 1 22 (4 )( )m m m

c r a r aS � � � � 1�

12

2( )2

mm m m

ma r a

d

���

External Force

The work done on the whole element due to external pressure is:

1

2 pdW pA dy � . (5)

Here, Ap is the frontal area of the element and the pressure term is the gas stagnation pressure:

21

2 g reluU

(6)

where urel is the velocity of the airflow normal to the element. Since the air flow is not always normal to the element’s surface, the normal component of the air velocity should be considered in calculating the aerodynamic force as shown in Fig. 1. The relative velocity can be written as:

cos ( )rel air

u u T (7)

where ș is the angle between the vertical direction and the normal vector to the element. Velocity projection is vital since assuming a constant free stream velocity leads to overestimating the total force exerted on the element. This large force, as previously used in some analytical models, will make the jet cross section deform faster and to a larger extent. It will be shown in the next section that as the aspect ratio increases, the drag coefficient changes noticeably which has considerable effects on the jet trajectory. By substitution (7) and (6) into (5) and dividing both sides by dy we get:

21( cos( ))

2p g air

dWF bh u

dyU T (8)

Equation of Deformation Writing the force balance in the x2 direction, we

obtain the following relation:

p v s elementmF F F y � � �� , (9)

Using equations (1), (4) and (9) for the forces, assuming melement = 0.5ȡl ʌabh, and dividing all the terms by the element thickness h yield to the nonlinear equation of oscillation in the form of:

2

1 2 32

d y dyc c c

dt dt� �

§ · § ·¨ ¸¨ ¸ © ¹© ¹

4c (10)

1

1

2 lc abU S

2 2

2l

abc

y

S P § ·¨ ¸© ¹

2 2

3

34(1 )

8

cc r a

d

SV � u � �ª º

« »¬ ¼

2

4

1( cos( ))

2 g airc b uU T

Jet Trajectory

In this section, we will perform a force balance in the X-Z plane to obtain the jet trajectory. The coordinate system is chosen as shown in the Fig. 1 with the origin located at the left corner of the injection point. This choice has been made to write the equation of trajectory for the liquid jet’s windward boundary rather than the center line. Faero is the aerodynamic force due to the air flow. F1 is the shear force from the lower element and F2 is the shear force from the upper element. The aerodynamic force is:

21

2aero D air relF C uU A (11)

The projected relative velocity is defined in

equation (7) and the frontal area of 2A a h u is used. Thus, equation (11) becomes:

2( cos ( ))

aero D air airF C a h uU T . (12)

Making the same assumption as Inamura [10] made, we assume the jet velocity along the trajectory to be constant and equal to the initial injection velocity. This assumption is reasonably valid up to the breakup location and we shall use it since this study is aimed at calculation of the jet trajectory up to the breakup point. Using this assumption, the velocity of the center of mass of the element can be written in the form of:

sin ( )

x jU v T (13)

cos ( )y j

U v T (14)

Now, balancing the forces in the X direction for the full element, we have:

1 2cos( ) ( ) cos( )element aero

m x F F FT T � �<<

(15)

Differentiating equation (13) with respect to time and Equating it with equation (15) and solving for ș, we get the equation for jet curvature angle with respect to time:

( ) c

aero

l j

d F Shear

dt ab h v

os( )T T

U S

� (16)

where:

1 2 sin( )l j

Shear F F ab v dS P T � . (17)

Mass Reduction As the jet moves along its trajectory, droplets and

fragments strip from the liquid column. The rate of mass stripping at different heights of the liquid column strongly depends on the air flow conditions and the breakup regime. Wu et al. [2] and Becker and Hassa [3] found the breakup regime to be dependent on the aerodynamic air Weber number. They concluded that at low Weber numbers, the breakup type would be bag or bag-shear breakup. However, as the Weber number increases, the shear breakup regime will be dominant. Mazallon et al. [13] studied the breakup regimes of liquid jet injections in crossflow in detail over a wide range of test conditions and liquid properties. Since the current model studies the deformation and trajectory of the liquid column, we limit the scope of the present study to bag/shear and shear breakup regimes. Thus, according to [13], the mass stripping in our model starts after meeting the following criteria:

(We =60<We )

Critical Local (18)

where :

2 2l air

Local

u aWe

U

V (19)

The equation for the mass shedding from the

cylindrical element of our model would be:

3 / 2

shed M s3

M ( ) R4 *

l air

st

d G H u

t

S U tu (20)

This equation is taken from [16,17,15] with two

modifications. First, the term ts/t* has been added to control the shedding rate to increase essentially linearly with distance away from the stripping start point. Ts is the time elapsed from the start of the mass stripping. Second, since the original equation was derived for mass shedding from a liquid drop, we need to multiply the mass shedding formula by the ratio of the element’s mass to that of a spherical drop with the radius req. The mass ratio in equation (20) is:

2

3

34 43

l eq

M

eq

l eq

h r hMass ratio R

rr

U S

U S . (21)

Drag Coefficient

Drag coefficient CD plays an important role in the prediction of the jet trajectory. As discussed earlier, it appears in equations (12) and (16) which predict the rotation angle of the liquid column. Various drag coefficients have been proposed for the liquid jets in crossflow. Inamura [10] assumed the value of unity for the drag coefficient which is usually the value for circular cylinder at low Reynolds numbers. Wu et al. [2] obtained a value of 1.696 for the drag coefficient. Sallam et al [18] offered the value of 3 for non-turbulent jets at the shear breakup regime. In this study, drag coefficients were calculated for circular and elliptical cylinders of different aspect ratios at different Reynolds numbers. Drag coefficients were calculated numerically for e=1, 2, 4 at air Reynolds numbers of 150-8000 where air Reynolds number is defined as:

Re /

air air air airu dU P . (22)

The calculations were carried out using FLUENT

Computational Fluid Dynamics (CFD) code [19]. Unsteady incompressible Navier-Stokes equations are solved. The equations are discretized using second order spatial and temporal schemes. Laminar equations were solved for Reynolds numbers up to 2000 and for

higher Reynolds numbers, the turbulent equations are solved using Spalart-Allmaras (SA) [20] turbulence model. The calculated results for e=1 cases are plotted and compared with the experimental results of Achenbach [21] in Fig. (3). As the plot shows, the results are in good agreement with the experimental results. The details of the numerical simulations can be found in [22].

In order to apply the results of the calculated test cases into our model, interpolation is needed between the different Reynolds numbers and different aspect ratios. Since our calculations are limited to maximum aspect ratio of e=4, an upper limit for interpolation is needed. The shape of the cross section of the cylinder becomes nearly a flat plate as the aspect ratio increases and thus, the drag coefficient of the flow past a normal infinite flat plate is set to be the upper limit. For a 2D normal flat plate of infinite aspect ratio, CD is 1.98 [23], [24]. Figure (4) plots the calculated data points for the three different aspect ratios and the curves fitted to the data points.

The value of CD has been set to 2 for (e � 10) as shown in the figure. Followings are the equations for the curves fitted to the data points:

3 2-12 -8DC = (-3*10 )Re +(5*10 )Re - 0.0002Re + 1

e=1: (23)

4 3 2-16 -12 -8DC = (-4*10 )Re +9*10 Re - (9*10 )Re + 0.0004Re + 1.96

e=2: (24)

-0.2166

DC = 7.3979Re

e=4: (25)

The value of the drag coefficient at each time step

in the model will be computed by first calculating the Reynolds number and then interpolating linearly between the above curves according to the values of the instantaneous aspect ratio, e. It should be noted that the above values are only valid for flows with Reynolds numbers in the approximate range of 200 and 8000 and the fitted curves would deviate drastically from the reality when the Reynolds number exceeds the mentioned range.

Summary of the Equations

Equations (10), (13), (14), and (16) are integrated simultaneously using 4th order Runge-Kutta method with time steps smaller than 10-6 sec. Equation (10) solves for the deformation of the cross section and equation (16) solves for the ș, the trajectory angle. Finally, equations (13) and (14) are integrated to determine the X and Z coordinates of the center of mass

of the elliptical element versus time. The following transformations should be applied to transfer the coordinates from the center of mass of the element to the upper boundary of the spray plume (Fig. 5). These transfers help plotting the jet trajectory rather than the path of the center of mass of the element

. . 0 . . cos( )

u b c mX r X b T � � (26)

. . . . sin( )

u b c mZ Z b T � . (27)

The subscripts “u.b.” and “c.m.” correspond to upper

boundary and center of mass respectively. Results and Discussion

In this section, the results of the trajectory calculations will be presented and compared with the available data in the literature. The effects of different parameters such as nozzle diameter, momentum ratio, air Weber number, mass reduction and the drag coefficient on the jet penetration and deformation will be discussed.

Jet Deformation

Figure (6) shows the calculated jet deformation from the side view for a case with momentum ratio of 60 and global air Weber number of 73. The curve is compared with the numerical 3-D Volume of Fluid (VOF) simulations of Madabushi et al. [25] and are in good agreement. Since the Weber number is 73, the mass reduction model is active. Figure (7) shows the jet cross section for the same test case at different heights and compares them with the results of Madabushi et al. [25]. As the figures show, the calculated frontal area of the cross section is almost the same as the results of Madabushi. This confirms that the deformation equation applied along with the drag coefficient curves work reasonably well. Figure (8) plots the variation of non-dimensional deformation of the cross stream axis of the liquid column, a/r0, versus the Weber number at the onset of breakup and compares the results with those of Mazallon et al. [13]. The onset of breakup is defined the same way as [13] here.

Calculations were done for ethanol and water within the ranges of (9.3<q <150) for ethanol and (3.9<q<72.5) for water. The nozzle diameter is 0.5mm for all the calculations and the jet velocity is fixed at 11.8 m/s while changing the Weber number with increasing the air velocity.

Previous studies have discussed the analogy between the deformation and breakup of the liquid column to those of a liquid drop. Thus far, several characteristics of the liquid drop deformation and breakup have been extended and used in the JICF

studies. Figure (9) compares the calculated deformation of the liquid jet with those of the 3-D drop calculated using the DDB model of Ibrahim et al. [12] , 2-D drop of Clark [11] and the jet model used by Inamura [11]. The calculations are for water and air at NTP with jet velocity of 11.8 m/s and air velocity of 60 m/s. The jet (drop) diameter is 0.5mm. The calculated curve for the same case but without the mass stripping model is also included in the figure to show the effect of mass reduction on the deformation. Effect of momentum ratio on the trajectory

Figure (10) demonstrates the jet trajectories calculated for three cases with constant Weber number and various liquid-to-air momentum ratios. Figure 10(a) shows the 3-D view of the jets. At lower momentum ratios, the jet bends and deforms faster while with the increase in the q, the jet penetrates faster and its cross section undergoes more deformation at larger heights allowing the jet to further penetrate into the gas stream. Figure 10(b) compares the calculated trajectories with the experimental results of Wu et al. [2]. As discussed in the earlier studies, the momentum ratio is the most important parameter in determining the trajectory of liquid jet in the gas stream. However, it is not the only factor that controls the jet path. Figure (11) compares the calculated curve with some available correlations. As observed in the figure, the present model gives the best trajectory prediction compared to other theoretical models and correlations.

Figure (12) compares the trajectory for almost constant momentum ratio and variable Weber numbers. The Weber number grows with increase in the nozzle diameter leading to a decrease in the jet penetration. The figure also compares the curve for the present calculation and the empirical correlation of Wu et al. [2]. Although the penetration is almost the same, the trajectories are slightly different as expected due to the factors discussed so far and in following sections.

Effect of the Drag Coefficient

Figure (13) plots the jet trajectories calculated for two cases of the same momentum ratio of 72 but different air Weber numbers. The jet trajectory is calculated using the CD curves of equations 23-25. The results are computed for constant CD of 1.0 for circular cylinders and the empirical value of 1.7 by Wu et al. [2] as well. The constant CD of unity obviously leads to over estimation of the jet trajectory since it does not take into account the deformation of the liquid jet and variation in Reynolds numbers. The results of trajectory from the present model using numerical CD curves are in good agreement (less than 5.5% of difference) with the trajectory results using the average CD value of 1.7 given by Wu et al. [2]. This shows that the average CD from our calculations should be close to 1.7. This is

confirmed by the data of Table (2) which contains the time averaged drag coefficients obtained for different test cases. The average value is in the range of 1.6 to 1.7 which is in good agreement with the empirical value of CD =1.7 suggested by Wu et al. [2]. Khosla et al. [15] have used a similar value of CD =1.7 and reported satisfactory results. Effect of Mass Stripping

As mentioned in the previous section, the mass stripping decreases the column cross section and drag force. Figure (14a) shows the column cross section’s deformation with and without the shedding model for a case with Weber number of 67, water jet velocity of 20 m/sec, air density of 1.2 kg/m3 and nozzle diameter of 0.5mm. Since the initial Weber number is above the stripping criteria, the mass stripping has a considerable effect on the penetration. Figure (14b) shows the side view of the jets and Fig. (15) compares the deformed cross section for both cases from a top view.

Summary and Conclusion

Deformation and trajectory of a liquid jet in the gaseous crossflow was modeled and compared with some experimental and theoretical data available in the literature. The results suggest that: x Jet cross sectional deformation plays a great role

in predicting the trajectory since it has substantial effects on the drag force exerted on the column. A good approximation of the deformation evolution is inevitable for accurate prediction of other characteristics of the liquid jet in crossflows.

x Assuming a constant drag coefficient in calculation of the jet penetration may lead to good results for the maximum penetration but, an adaptive drag scheme is needed to capture the correct trend of the liquid trajectory. The CD curves offered in this study have been calculated using a CFD code for the Reynolds number of 200 to 8000 and aspect ratios, e, of 1 to 4. This scheme is believed to predict the drag force satisfactorily for the mentioned ranges.

x In addition to the liquid-to-gas momentum ratio, q, the gas Weber number also plays an important role in the penetration and atomization of the liquid jets in crossflows. It determines the start of mass stripping from the liquid column, the rate of shedding and the breakup regime. A local Weber number based on the instantaneous cross stream axis of the jet’s cross section has been defined. This method enables us to have control over the start of the mass stripping from the column after the value of the local Weber exceeds a critical value. This means that if the mass stripping occurs for a jet, it may start at different heights of the jet depending on the flow conditions and not

necessarily from the injection point. This idea is approved by experimental pictures of various jets[2,3,13].

x Mass stripping plays an important role in predicting the jet penetration since it changes the jet’s cross section and reduces the drag force on the jet column. However, as the mass of the liquid element decreases, the ratio of its momentum to the momentum of the gas stream decreases and tends to bend and deform more. Thus, the rate of mass stripping plays an important role on the jet trajectory. As the Weber number grows, these effects become more important especially in the transition regime that the Weber number passes the criteria.

x Unlike previous theoretical models and correlations for JICF which are limited to a specific range of flow conditions, this theoretical model has no constant which needs a correlation and it creates satisfactory results for different conditions. The significance of the present model is that it does not need any tuning in order to properly work for a wide range of parameters.

Nomenclature r radius a semimajor b semiminor e aspect ratio y deformation h thickness s location along the trajectory t time A area ș angle between trajectory and vertical plane W work p pressure F force m mass u velocity U density ȝ viscosity ı surface tension coefficient CD drag coefficient q Liquid/gas momentum ratio We Weber number Re Reynolds number Subscripts g gas

l liquid rel relative j jet u.b. upper boundary c.m. center of mass v viscous

s surface eq equal References

1. Chen, T.H., Smith, C.R., Schommer, D.G. and Nejad, A.S., 31st AIAA Aerospace Sciences

Meeting & Exhibit, Reno, NV, January 1993. 2. Wu, P.K., Kirkendall, K.A., Fuller, R.P. and

Nejad, A.S., Journal of Propulsion and Power, 13(1):64-73 (1997).

3. Becker, J. and Hassa, C., Automization and

Spray, 12:49-67 (2002). 4. Wotel, G.J., Gallagher, K.E., Caron, S.D.,

Rosfjord, T.J., Hautman, D.J., and Spadaccini, L. J., Wright Lab., TR-9102043, Wright-Patterson AFB, OH, 1991.

5. Lakhamraju, R.R. and Jeng, S.M., 18th

Annual

Conference on Liquid and Atomization and

spray Systems, Irvine, CA, May 2005. 6. Elshamy, O.M. and Jeng, S.M., 18

th Annual

Conference on Liquid and Atomization and

spray Systems, Irvine, CA, May 2005. 7. Heister, S.D., Nguyen, T.T. and Karagozian,

A.R., AIAA Journal, 27(12):1727-1734 (1989). 8. Li, H.S. and Karagozian, A.R. AIAA Journal,

30(7):1919-1921 (1992). 9. Nguyen, T.T. and Karagozian, A.R., Journal of

Propulsion and Power, 8(1): pp. 21-29 (1992). 10. Inamura, T., Journal of Propulsion and Power,

16(1): 155-157 (2000). 11. Clark, M.M., Chemical Engineering Science

43(3):671-679 (1988). 12. Ibrahim, E.A., Yang, H.Q., and Przekwas, A.J.,

Joural of Propulsion and Power, 9(4): 651-654 (1993).

13. Mazallon, J., Dai, Z., and Faeth, G.M., Atomization and Sprays, 9(3): 291-311 (1999).

14. http://home.att.net/~numericana/answer/ellipse.htm

15. Khosla, S., and Crocker, D.S., 17th

Annual

Conference on Liquid Atomization and Spray

Systems, Arlington, VA, May 2004. 16. Chryssakis C.A. and Assanis D.N., 18

th Annual

Conference on Liquid and Atomization and

spray Systems, Irvine, CA, May 2005. 17. Rangers A.A. and Nicholls J.A., AIAA Journal,

7(2):285-290 (1969). 18. Sallam, K.A., Aalburg, C., and Faeth, G.M.,

AIAA J., 42:2529-2540 (2004). 19. FLUENT User Manual, Fluent Inc. 20. Spalart, P.R., and Allmaras, S.R., AIAA Paper

92-0439 (1992). 21. Achenbach, E.,Journal of Fluid Mechanics 34,

625-639 (1968) .

22. Mash, A. Jafari, A. and Ashgriz, N.,“On the drag of elliptical cylinders with normal attack angles”, in preparation.

23. Ringuette, M.R., “Vortex Formation and Drag on Low Aspect Ratio, Normal Flat Plates”. Ph.D. Thesis, California Institute of Technology, 2004.

24. McCormick, B.W.,. Aerodynamics, Aeronautics,

and Flight Mechanics, 2nd Ed., John Wiley & Sons, 1995.

25. Arienti M., Madabhushi R.K., Van Slooten P.R. and Soteriou M.M., 18

th Annual Conference on

Liquid and Atomization and spray Systems, Irvine, CA, May 2005.

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Table 1. Test condition for high pressure cases from Rachner et al. [26]. Test Case

name Air pressure

[bar] Air Temp.

[K] Air Density

kg/m3 Air Velocity

[m/s] q

q18 5.8 280 7.19 100 18 p9 8.7 280 10.78 100 6

u75 5.9 285 7.18 75 6 u75q2 5.9 285 7.18 75 2

Table 2. Time averaged drag coefficients calculated for different test cases

We q Time Averaged CD

34 72 1.58 140 72 1.6 71 9.9 1.61 139 9.4 1.65 281 10 1.68

Figure 1. Schematic of the jet element movement along the trajectory and the aerodynamic force.

Figure 2. Analogy between an oscillating two-dimensional drop and a forced mass-spring system

Figure 3. Comparison of time-averaged numerically calculated drag coefficient values with the experimental results for circular cylinders (e=1).

Figure 4. Calculated drag coefficients for circular and elliptical cylinders (e=1, 2, 4) and the curves fitted to the data.

Figure 5. Center of mass coordinates versus upper boundary coordinates.

(a) (b)

Figure 6. Comparison of the calculated jet trajectory (a) with the results of Madabushi et al. [25] (b) (reproduced from [25] by permission).

(a)

(b)

(c)

Figure 7. Comparison of the calculated jet cross sections left, with the results of Madabushi et al. [25] right(reproduced from [25] by permission). (a) y=0, (b) y=2d, (c) y=6d

Figure 8. Variation of Non-dimensional deformation of the jet cross section in the cross stream axis direction (a/r0) at the onset of breakup. Theoretical and experimental results are compared.

Figure 9. Comparison of the calculated deformation of the liquid jet using present model with those for the 3-D drop of the DDB model of Ibrahim et al. [12] , 2-D drop of Clark [11] and the jet model of Inamura [11]. water and air at NTP, Vj=11.8 [m/s], Uair=60 [m/s].

(a) (b)

Figure 10. Effect of momentum ratio on the trajectory at constant Weber number (a) 3-D jet view (b) calculated trajectories compared to the experiments of Wu et al. [2]

Figure 11. Comparison of the jet trajectory from the present model with experiment and different correlations.

Figure 12. Effect of Weber number on the trajectory at constant momentum ratio .

(a)

(b)

Figure 13. Jet trajectories calculated for two cases of the same momentum ratio q=72 but different air Weber numbers (a) We=34.3 (b) We=140. Various drag coefficients are used.

(a) (b)

Figure 14. Deformation of jet cross section with and without mass shedding model for a case with Weber number of 67, water and air at NTP, Vj=20 m/sec, Uair=90 m/sec and nozzle diameter of 0.5 mm. (a) 3-D view (b) side view

Figure 15. Comparison of the deformed cross section with and without mass reduction from the top view.