1

28th Symposium on Naval Hydrodynamics Pasadena, California, 12-17 September 2010

Bubble Augmented Waterjet Propulsion: Numerical and Experimental Studies

Xiongjun Wu, Jin-Keun Choi, Chao-Tsung Hsiao, Georges L. Chahine (DYNAFLOW, INC., U.S.A.)

ABSTRACT

In this paper, we present our efforts to test experimentally and numerically the concept of bubble augmented jet propulsion. The numerical model is based on a two-way coupling between an Eulerian description of the flow field and a Lagrangian tracking of bubbles using the Surface Averaged Pressure (SAP) model. The numerical results compares favorably with nozzle experiments. A half 3-D nozzle for bubble augmented propulsion was designed and optimized using the developed numerical code, which in the end yielded a design that produces net thrust increase with increased bubble injection rate. The numerical code can be a robust tool for the general two-phase flow nozzle design as demonstrated in the present numerical and experimental studies. INTRODUCTION

Bubble injection into a waterjet to augment thrust has received a revived interest after studies of Albagli and Gany (2003) and Mor and Gany (2004) showed interesting results indicating that bubble injection can significantly improve the net thrust and overall propulsion efficiency of a water jet. Several prior studies (Mottard and Shoemaker, 1961, Muir and Eichhorn, 1963, Witte, 1969, Pierson, 1965, Schell et al., 1965) explored similar ideas, and some of these studies adopted the naming Water Ramjet to such a system by analogy to ramjet aerodynamic propulsion systems.

Analytical, numerical and experimental evidences to the augmentation of the jet thrust due to bubble growth in the jet stream, even though preliminary, make the idea of bubble augmented thrust attractive. Unlike traditional propulsion devices which are typically limited to less than 50 knots, this propulsion concept promises thrust augmentation even at very high vehicle speeds (Mor and Gany, 2004)).

Various prototypes have been developed and tested, e.g., Hydroduct by Mottard and Shoemaker

(1961), MARJET by Schell et al. (1965), Underwater Ramjet by Mor and Gany (2004). Figure 1 shows how the concept works: fluid enters a diffuser and is first compressed (ram effect), then it mixes with high pressure gas injected via mixing ports, which constitutes the energy source for the ramjet, and then the multiphase mixture is accelerated by passing through a converging nozzle. Such prototypes have shown a net thrust due to the injection of the bubbles. However, the performance may be strongly related to the efficiency and proper operation of the bubble injector and the overall propulsion efficiency was typically less than that anticipated from the mathematical models used. This may be a result of a poor mixing efficiency at the injection, possible flow chocking or of weaknesses in the modeling (Varshay, 1994, 1996).

Previous empirical and numerical studies of bubbly flow through a nozzle (Tangren et al., 1949, Muir and Eichhorn, 1963, Ishii et al., 1993, Kameda and Matsumoto, 1995, Wang and Brennen, 1998, Preston et al., 2000) have modeled mixture passing through a nozzle. With these models, it is found that expansion of a compressed gas bubble-liquid mixture is an efficient way to generate the momentum necessary for additional thrust (Albagli and Gany, 2003, Mor and Gany, 2004)). However approximations and simplifications in these models dictates a numerical tool which can provide a more detailed analysis of the two-phase flow, and which correctly includes the dynamic behavior of bubbles to simulate water ramjet propulsion (Chahine et al., 2008).

At DYNAFLOW, our approach is to consider the bubbly mixture flow inside the nozzle from the following two perspectives: • Microscopic level: Individual bubbles are tracked

in a Lagrangian fashion, and their dynamics are followed by solving the surface averaged pressure Rayleigh-Plesset equation. The bubble responds to its surrounding medium described by its mixture density, pressure, velocity, etc.

2

• Macroscopic level: Bubbles are considered collectively and define a void fraction space distribution. The mixture medium has a time and space dependant local density which is related to the local void fraction. The mixture density is provided by the microscale tracking of the bubbles and the determination of their local volume fraction.

The two levels are fully coupled: the bubble

dynamics are in response to the variations of the mixture flow field characteristics, and the flow field depends directly on a function of the bubble density variations. This is achieved through two-way coupling between the unsteady Navier Stokes solver 3DYNAFS-VIS© and the bubble dynamics code 3DYNAFS-DSM©. A quasi-steady version of the code was also developed, and was extensively used to conduct parametric studies useful for fast estimation of the overall performance of selected geometric design.

Figure 1. Concept sketch of bubble augmented jet propulsion. GOVERNING EQUATIONS

The mixture medium satisfies the following general continuity and momentum equations:

( ) 0mm mt

ρ ρ∂+∇ ⋅ =

∂u , (1)

( )223

mm m m ij m m

D pDt

ρ µ δ µ⎧ ⎫= −∇ +∇ − ∇⋅⎨ ⎬⎩ ⎭

u u , (2)

where, the subscript m represents the mixture medium, and ijδ is the Kronecker delta. The mixture density and

the mixture viscosity for a void volume fraction α can be expressed as:

( )1m gρ α ρ αρ= − + , (3)

( )1m gµ α µ αµ= − + , (4)

where the subscript represents the liquid and the subscript g represents the gas bubbles. The flow field has a variable density because the void fraction varies in space and in time. This makes the overall flow field problem similar to a compressible flow problem.

LAGRANGIAN BUBBLE TRACKING

Lagrangian bubble tracking is accomplished by 3DYNAFS-DSM© (or a corresponding FLUENT User Defined Function called Discrete Bubble Model (DBM)). It is a multi-bubble dynamics code for tracking and describing the dynamics of bubble nuclei released in a flow field. The user can select a bubble dynamics model; either the incompressible Rayleigh-Plesset equation (Plesset, 1948) or the compressible Keller-Herring equation (Gilmore, 1952, Knapp et al., 1970, Vorkurka, 1986). In the first option, the bubble dynamics is solved by using a modified Rayleigh-Plesset equation improved with a Surface-Averaged Pressure (SAP) scheme:

32 0

0

2

3 1 2 42

| |,

4

km

v g encm

enc b

R RRR R p p PR R R

γ µρ

⎛ ⎞⎛ ⎞+ = + − − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠−

+u u

(5) where R is the bubble radius at time t, R0 is the initial or reference bubble radius, γ is the surface tension parameter, µm is the medium viscosity, ρm is the density, pv is the liquid vapor pressure, pg0 is the initial gas pressure inside the bubble, k is the polytropic compression law constant, uenc is the liquid convection velocity vector and ub is the bubble travel velocity vector, and Penc is the ambient pressure “seen” by the bubble during its travel. With the Surface Averaged Pressure (SAP) model, Penc and uenc are the average of the pressure over the surface of the bubble. If the second option is adopted, the effect of liquid compressibility is accounted for by using the following Keller-Herring equation.

2

3 20

0

3 11 1 12 3

4 | |2 ,4

m m m m m

km enc b

v g enc

R R R R dRR Rc c c c dt

R Rp p PR R R

ρ

µγ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + − = + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ −⎛ ⎞+ − − − +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

u u

(6)

where cm is the speed of sound in the mixture medium. The bubble trajectory is obtained from the

bubble motion equation (Johnson and Hsieh, 1966):

( ) ( )

( )1/2

1/4

32

( )1 .2 ( )

b b

b

b b b

b m mD b b b

m m ijm bb

lk kl

d RFdt R x

Kv ddd gdt dt R d d

ρ ρρ ρ

ρ ρ ρρρ ρ ρ

∂= − + − +

∂

−⎛ ⎞+ − + + −⎜ ⎟⎝ ⎠

u uu u u u u

uu u u

(7) The first term in Equation (7) accounts for the drag force effect on the bubble trajectory. The drag coefficient FD is determined by the empirical equation

3

of Haberman and Morton (1953). The second and third term in Equation (7) account for the effect of change in added mass on the bubble trajectory. The fourth term accounts for the effect of pressure gradient, and the fifth term accounts for the effect of gravity. The last term in Equation (7) is the Saffman (1965) lift force due to shear as generalized by Li & Ahmadi (1992). The coefficient K is 2.594, ν is the kinematic viscosity, and dij is the deformation tensor. 1-D MODELING

To study conditions where the flow can be considered 1-D with average cross section quantities, the governing equations for unsteady 1-D flow through a nozzle of varying cross-section, A, can be written as

1 0,

1 0,

m m m

m m m m m

u At A x

u u Au pt A x x

∂ρ ∂ρ∂ ∂∂ρ ∂ρ ∂∂ ∂ ∂

+ =

+ + =

(8)

where mρ , mu , p are the mixture density, velocity and pressure.

A one dimensional code, 1-D BAP, was developed for this study (Chahine et al., 2008). The liquid was assumed to be incompressible so that all compressibility effects of the mixture arose from the disperse gas phase only. The void fraction, α , is determined by the volume occupied by the bubbles per unit mixture volume, additionally, it is assumed that other than at the injection locations, no bubbles are created or destroyed. For steady conditions, the time dependant term can be ignored. In addition, if spherical, the local bubble dynamics in the nozzle is governed by either the Rayleigh-Plesset equation or by assuming that bubbles readily equilibrate with the local pressure, this quasi-steady approximation can be written by removing all dynamics components from the Rayleigh-Plesset equation. 3-D UNSTEADY FULLY COUPLED MODELING

The 3D coupling between the mixture flow field and the bubble dynamics and tracking is realized by coupling the 3DYNAFS_VIS© and 3DYNAFS_DSM©. The unsteady two-way interaction can be described as follows:

• The bubbles in the flow field are influenced by the local densities, velocities, pressures, and pressure gradients of the mixture medium. The dynamics of individual bubbles and Lagrangian tracking of them are based on these local flow variables as described in Equations (5) to (7).

• The mixture flow field is influenced by the presence of the bubbles. The local void fraction, and accordingly the local mixture density, is modified by the migration and size change of the bubbles, i.e., the bubble population and size. The flow field is adjusted according to the modified mixture density distribution in such a way that the continuity and momentum are conserved through Equations (1) and (2).

The two-way interaction described above is

very strong as the void fraction can change significantly (from near zero in the water inlet to as high as 70% at the nozzle exit) in Bubble Augmented jet Propulsion (BAP) applications. Void fractions based on �-cell concept were introduced in the 3-D space to compute the void fraction (Chahine et al., 2008). THRUST DEFINITIONS

There are two definitions for thrust calculations based on the application type. For ramjet type propulsion, the thrust of the nozzle can be computed by integrating the pressure and the momentum flux over the surface of a control volume that contains both the inlet and outlet of the nozzle.

( )2

m mAT p u dAρ= +∫∫ , (9)

where mu is the axial component of the mixture velocity.

For waterjet type applications, what counts is the exit water jet momentum flux, the thrust of the nozzle can be computed by integrating only the momentum flux over the exit surface of the nozzle.

2

om mA

T u dAρ= ∫∫ , (10)

where oA is the exit surface area of the nozzle.

With the 1-D assumption, the thrust for the ramjet is defined as follows:

( ) ( )2 2, , , ,R o o i i m o o m o m i i m iT p A p A A u Auρ ρ= − + − , (11)

where the subscripts i and o represent the inlet and the outlet of the ramjet respectively. The component of the thrust described in the first parenthesis in (11) is the contribution from the pressure variation between the inlet and the outlet. The second parenthesis represents the thrust due to the momentum change between the inlet and the outlet.

The thrust for a waterjet of the nozzle with the 1-D assumption can be simply written as:

4

2, ,W m o o m oT A uρ= . (12)

EXPERIMENTAL SET-UP

The test setup used in this study is shown in Figure 2. The tests were conducted in DYNAFLOW’s 72 ft wind-wave tank. The flow was driven by two 15 HP pumps (Goulds Model 3656), each of which is capable of producing a maximum flow rate of 550 gpm at a pressure drop of 25 psi. The two pumps can work in parallel to boost the flow rate and a bypass line is used to adjust the flow rate. The nozzle test section is placed below the free surface. The wind wave tank is used as a very large water reservoir, so that the possible accumulation of air bubbles generated from the testing is minimized. A flow adaptor is used to convert the flow from a circular cross section to a matching cross section with the shape of the test section geometry. A flow straightening section is inserted between the flow adaptor and the nozzle inlet.

Reluctance type pressure transducer arrays are arranged in the nozzle walls to measure the pressures at different locations along the test section. Pressure signals are sent to the data acquisition system for data logging.

Various methods were used for velocity measurement based on experimental conditions:

• A planar PIV system was used to characterize the flow field without air injection or in the region before air injection.

• An optical bubble tracking method was used for velocity measurement for flows with low void fraction when individual bubbles could be tracked to characterize the flow field.

• For flows with high void fraction which pose difficulties for optical method, Pitot tubes and Kiel probes were used for velocity measurement. Void fraction effects are accounted for using the following formula proposed by Bosio and Malnes (1968):

(13)

The void fraction was measured or estimated

using several methods: • The nominal void fraction was obtained by

dividing the air flow rate by the liquid flow rate. • A photographic method was used to measure

the void fraction with image analysis. This was applicable only to low void fraction medium where the bubbles did not overlap too much.

• The attenuation of laser light through the bubbly medium due to light scattering on the bubble surfaces.

• An acoustic method that utilizes DYNAFLOW’s ABS ACOUSTIC BUBBLE SPECTROMETER® was used to measure bubble size distribution and integrate it to obtain the void fraction (Prabhukumar et al., 1996, Duraiswami et al., 1998, Chahine et al., 2001, 2008, Chahine, 2008b).

• Conductivity probes were used to measure conductivity and convert this to void fraction information. This method can be applied to a wide range of void fraction, but appears more accurate for the higher void fraction.

Figure 2: Sketch of the test setup for the Bubble Augmented Jet Propulsion experiment. 2-D EXPERIMENT

The test section for two dimensional experiments consisted of two acrylic side-walls and two inserts that were sandwiched between the two side-walls. The inserts were machined to form the desired profile of the test section. This approach enables one to test different test section profiles by replacing the inserts. Figure 3 shows the dimensions of the two dimensional nozzle with an expansion exit, which was reported previously (Chahine et al., 2008).

Figure 3: Dimensions of a 2-D nozzle with an expansion exit.

Figure 4 shows a picture of the nozzle in the

experimental setup. The intention of the addition of the exit expansion was to make the bubbles grow more at the throat and to enhance the bubble effects. Using the

2

1 2 .(1 / 2)

LL

PVρα∆

=−

5

pressure ports on the side wall, the pressure was measured at four locations along the length of the nozzle and at a location upstream of the nozzle. These pressure measurements were compared with the 1D model and the 3DYNAFS_VIS© predictions. Figure 5 shows the comparison of the pressure along the nozzle at 200 gpm inlet flow. The 1D model slightly over-predicted the pressure drop at the throat (x=105 cm) because the model does not consider the friction loss. However, the model captures the overall pressure behavior well and is useful for design calculations. The 3DYNAFS_VIS© prediction agrees quite well with the measurements. A similar good agreement is observed at other flow rate conditions.

Figure 4: Photographs of the experimental setup of the nozzle with an expansion exit. The flow is from right to left.

70

80

90

100

110

120

130

-20 0 20 40 60 80 100 120 140

X (cm)

Pres

sure

(kPa

)

1D_BAP 0.0%

3DynaFS_VIS 0.0%

EXPERIMENTS

Figure 5: Measured and predicted of pressures along the centerline of the nozzle (200 gpm, no bubble injection).

2

4

6

8

10

12

14

0 5 10 15 20Nominal Void Fraction (%)

Velo

city

(m/s

)

1D_BAP 100gpm1D_BAP 200gpm1D_BAP 300gpmEXP 100gpmEXP 200gpmEXP 300gpm

Figure 6: Predicted and measured section averaged velocities at the exit vs. nominal injection void fraction.

Increased velocity at the outlet with bubble injection is a key factor for thrust augmentation. Figure

6 compares the measured velocity at the exit with the 1D model predictions for various flow conditions: different void fractions and different flow rates. The measured velocities were averaged outlet velocities obtained by a Pitot tube at several locations in the outlet plane. The nominal void fraction was defined based on the flow rate of the upstream water inflow and the air flow rate at the injector. The overall agreement between the prediction and measurement is satisfactory. This comparison indicates that the variation of the outlet velocity with respect to the change of bubble injection amount is well captured by the 1D model.

Figure 7: Variation of the total thrust and contributions from the pressure component and the velocity component with void fraction.

Figure 7 shows the variation of the thrust with void fraction for the 200 gpm water flow. With this nozzle geometry, the baseline thrust (Ramjet thrust) is already negative at zero void faction due to the small exit area. However, there is a net thrust gain as the void fraction increases. There was about 10 N thrust increase at 10% void fraction. 3-D GEOMETRY DESIGN

In our continued tests, the two dimensional set up was extended to what we dubbed half 3-dimensional set up, which is half of the full three dimensional axisymmetric nozzle with a vertical cut through a center plane. This set up represents the flow of the full three dimensional nozzle more closely and enables good flow visualization through the flat transparent center-plane. The initial design of the half 3-D nozzle maintained the same geometry of the 2-D nozzle with an expansion exit as shown in Figure 3 in the cut-through plane. Figure 8 shows the CAD drawings of the half 3-D nozzle with the expansion exit.

1D-BAP was used to evaluate the effects of the throat on the performance of the half 3-D nozzle

-100.00

-80.00

-60.00

-40.00

-20.00

0.00

20.00

40.00

60.00

80.00

0 2 4 6 8 10 12

Nominal Void Fraction (%)

Thru

st (N

)

Pressure ContributionVelocity ContributionTotal Thrust

6

with an expansion exit. Figure 9 shows, for a parametric study, variations of the cross section outline along the nozzles for different throat geometries. Notice that the only varying parameter in these profiles is the diameter of the throat.

V

V

Figure 8: CAD rendered drawing of the built half 3-D nozzle with an expansion without the side plate (top); CAD drawing of the complete nozzle with a flow adaptor and a flow straightener sections (bottom)

Figure 10 shows the predicted normalized thrusts for different nominal void fractions at injection for the different geometry profiles. The color of the thrust curve corresponds to the color of the nozzle geometry in Figure 9. The thrust is normalized by the inlet momentum flux defined as

(14)

Figure 9: Cross section area changes along the nozzles for different parametrically examined throat areas.

As indicated in Figure 10, the expansion exit actually causes a thrust decrease with bubble injection; this prompted the modification of the base nozzle design to a nozzle without the expansion exit section. Figure 11 shows the dimensions of the half 3-D nozzle,

which has the same dimensions as those of the 2-D nozzle shown in Figure 4 without the expansion exit.

Figure 10: The change of thrust normalized with inlet momentum flux with nominal void fraction at injection. The different colors correspond to the colors of the examined profiles shown in Figure 9.

Figure 11: Dimensions of the half 3-D nozzle selected for testing. AIR INJECTION OF HALF 3-D NOZZLE In order to achieve bubble injection as uniformly as possible, an air injector covering the outer half circular boundary of the nozzle was used. Figure 12 shows a CAD drawing of the outer air injector. A half circular air chamber was covered by a flexible porous plate that was curved to match the nozzle profile and was used as the nozzle boundary wall. Compressed air was forced through the porous plate to inject bubbles in the nozzle. As shown on the right picture in Figure 12, the air chamber was partitioned into six separate small chambers and each chamber had its own air supply line such that more uniform air injection could be achieved by adjusting the pressured applied to each chamber. The flexible porous plate was 1/8 inch thick and has pore sizes of 7 µm. To characterize the bubble sizes generated from the flexible porous plate, an experiment as illustrated in Figure 13, was used to measure the bubble size distribution. A 1 inch square porous plate was glued at one end of square Plexiglas tube and compressed air was applied to the other end

2/ l in inT T A Vρ=

7

of the tube and forced through the porous plate to generate bubbles.

Figure 12: CAD drawing of the air injector positioned in the outer boundary of the nozzle. On the left is the injector assembly and on the right is an exploded view of the air chamber and porous membrane.

Figure 14 shows bubbles generated at different air flow rate, both the number of bubbles and the bubble sizes increase with increased air flow rate. The mean bubble radius was 860 µm, 1100 µm, and 1300 µm for air flow rates at 1 L/min, 2 L/min, and 3 L/min respectively. This was however in absence of shear or liquid flow.

Figure 15 shows a picture of bubbles generated from the outer air injector. Most of the bubbles were concentrated on the outer boundary, and the bubble concentration was significantly lower in the central region. In order to achieve more uniform distribution, an inner air injection scheme, as show in Figure 16, was also designed to fit in the nozzle. The inner air injection consists of a flat injector and a half circular injector with total three air injection faces, one on the flat injector and two on both sides of the circular injector.

Figure 13: Experiment setup for characterizing the bubbles generated from the porous plate.

To study how the flow field in the nozzle was affected by the inner injector, we conducted a fast CFD simulation. Figure 17 shows the simulation domain of the nozzle with the inner injector. A symmetry plane was used to speed up the computation. Figure 19 and

Figure 20 shows comparisons of the pressure and velocity distributions with and without the inner air injector, the inlet velocity was 4.67 m/s (300 gpm). As shown in the figures, the effects of the inner injector on the flow field are not significant from both the pressures and the velocities. The differences of thrust in this case with and without the inner air injector were 0.3 and 1.9 N for waterjet and ramjet thrusts respectively.

Figure 14: Variation of bubble generation with different air injection rate, from left to right 1 L/min, 2 L/min and 3 L/min.

Figure 15: Air injection from the outer air injector in the ½ 3D BAP nozzle.

Figure 16: A sketch of the inner air injector.

To achieve better control over the injected bubbles distribution, the inner air injector also has independent air chambers such that air supply to each air chamber can be adjusted to achieve overall uniform bubble injection. Figure 21 shows a picture of air injection testing of the inner injector alone set in standstill water. Combined with the outer air injector, a more uniform bubble injection can be achieved compared to using just a single air injector. Figure 22 shows a picture of the half 3-D nozzle set up in the

Porous membrane

Air chamber

Chamber partitions

8

tank. As shown in the figure, multiple air hoses are connected each air chamber of the air injectors to control the air supply independently.

Figure 17: The simulation domain of the nozzle used to study the effects of the inner injector on the flow field of the nozzle.

Figure 18: Close up of the inner injector grid.

Figure 19: Comparison of the pressure distributions along the nozzle with and without the inner injector.

Velo

city

(m/s

)

Figure 20: Comparison of the velocity distributions along the nozzle with and without the inner injector.

Figure 21: Air injection from the inner air injector.

Figure 22: A snap shot of the half 3-D nozzle set up in the tank.

THRUST MEASUREMENTS ON HALF 3-D NOZZLE

To estimate the nozzle thrust using (11), the

pressure and velocity at the inlet and the outlet of the nozzle were measured. PIV was used to measure the inlet velocity and a Pitot tube traversing vertically through the outlet section of the nozzle was used for outlet velocity measurement. The Pitot tube setup can also be seen in Figure 22.

Pitot tubeFlow

9

Figure 23: Measured inlet velocity profiles at different bubble injection rates. Inlet velocities are 2.4 m/s and 3.57 m/s, corresponding to water flow rates of 154 gpm and 230 gpm respectively.

Figure 24: Measured outlet velocity profiles at different bubble injection rates. Nominal inlet velocities are 2.4 m/s and 3.57 m/s, corresponding to water flow rates of 154 gpm and 230 gpm respectively.

Figure 23 and Figure 24 show the measured velocity profiles with and without bubble injection (the void fraction varied from 0% to 27% for the inlet velocity of 2.4 m/s and from 0% to 16% for the inlet velocity of 3.57 m/s). As shown in the figures, the inlet velocity profiles are little affected by the air injection downstream and are symmetric in the cross section from the top to the bottom. However, we see from the outlet velocity profiles that, as expected, the velocities increase with increased rates of bubble injection, and the velocities become skewed to a higher velocity at the top when there is bubble injection. This imbalance

is more prominent with higher void fraction of bubble injection. This is caused by the fact that bubbles rise to the top due to buoyancy as they move downstream in the nozzle. This creates an asymmetric void fraction distribution at the exit. The larger the void fraction, the more significant is the difference. Figure 25 shows an example view of bubble distribution along the nozzle in an experiment. The nominal inlet velocity was 3.57 m/s with the nominal void fraction of air injection of 16%. The void fraction at the top is seen higher as the flow moves towards the exit. This effect will be less prominent at higher liquid flow.

Figure 25: View of the contraction section of the ½ 3D BAP nozzle. Higher void fraction at the top section as bubbles keep rise under gravity. Nominal inlet velocity is 3.57 m/s and nominal void fraction at injection is 16%.

Figure 26 and Figure 27 show comparisons of the inlet and outlet velocity profiles between experiments and 3-D simulations with 3DYNAFS_VIS©. As we can see the BAP inlet velocity profiles in the numerical simulations match with the measurements, differences are probably due to the fact that the actual inlet velocity profile at the upstream section after the flow straightener is not a uniform velocity profile as assumed in simulation. For the exit velocity profiles, however, the simulations under predicted the velocity for the 3.57 m/s case. We are looking further into the reason for this discrepancy.

Velocity (m/s)

z(m

m)

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-60

-40

-20

0

20

40

60

V= 2.4 m/s, α = 0.0V= 2.4 m/s, α = 0.11V= 2.4 m/s, α = 0.27V= 3.57 m/s, α = 0.0V= 3.57 m/s, α = 0.057V= 3.57 m/s, α = 0.16

Velocity (m/s)

Z(m

m)

6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

V= 2.4 m/s, α = 0.0V= 2.4 m/s, α = 0.11V= 2.4 m/s, α = 0.27V= 3.57 m/s, α = 0.0V= 3.57 m/s, α = 0.057V= 3.57 m/s, α = 0.16

Flow

10

Figure 26: Comparison of inlet velocity profile BAP between experiments and simulations. The dots are from the experiments and solid lines are from the 3DYNAFS_VIS© simulations.

Figure 27: Comparison of exit velocity profiles between experiments and simulations. The dots are from experiments and solid lines are from 3DYNAFS_VIS© simulations. Figure 28 and Figure 29 show measured pressure distributions at the nozzle inlet and outlet for different air injection rates. Both the inlet and outlet pressures increase with increased void fraction, and the pressure increase is more prominent at higher nominal inlet velocity. The outlet pressure distributions exhibit a skewed distribution similar to the outlet velocity profile, i.e. the pressure in the top section is higher than the pressure at the lower section and the difference is more prominent with increase bubble injection and velocity. This phenomenon is also attributed to the asymmetric void fraction distribution in the vertical direction that causes the skewed outlet velocity profile.

Figure 28: Measured pressure distributions at the inlet for different air injection rates with nominal inlet velocities of 2.4 m/s and 3.57 m/s.

Figure 29: Measured pressure distribution at the outlet for different air injection rates with nominal inlet velocities of 2.4 m/s and 3.57 m/s.

With the velocity profiles and pressure distributions available at both inlet and exit, the ramjet thrust as defined in Equation (11) is computed from the experiment data and is compared with 1-D BAP simulation, as shown in Figure 30. As shown in the figure, the net thrust increases for both nominal velocities are negative, i.e. the thrust becomes more negative with air injection; this effect increases with increased air injection, and with higher inlet velocities. The comparison between the measurement and 1-D BAP simulation indicates that the simulation predicts larger thrust decrease compared to measurement. However, the simulation predicts the same trend with air injection increase.

If we define a normalized thrust augmentation, ξ , as following:

P (PSI)

z(c

m)

0 1 2 3 4-4

-2

0

2

4

V= 2.4 m/s, α = 0.0V= 2.4 m/s, α = 0.11V= 2.4 m/s, α = 0.27V= 3.57 m/s, α = 0.0V= 3.57 m/s, α = 0.057V= 3.57 m/s, α = 0.16

P (PSI)

z(c

m)

2 4 6 8 10 12 14 16-6

-4

-2

0

2

4

6

V= 2.4 m/s, α = 0.0V= 2.4 m/s, α = 0.11V= 2.4 m/s, α = 0.27V= 3.57 m/s, α = 0.0V= 3.57 m/s, α = 0.057V= 3.57 m/s, α = 0.16

11

, ,R R

R

T TTαξ−

= (15)

in which ,RT α and RT are ramjet thrusts with and

without bubble injection. As shown in Figure 31, ξ increases with increased void fraction, i.e. the increased void fraction augments what RT is.

α

T R,α

-TR,N

-0.1 0 0.1 0.2 0.3 0.4-150

-100

-50

0

50Exp. Vi = 2.4 m/sExp. Vi = 3.57 m/s1D-BAP Vi = 2.4 m/s1D-BAP Vi = 3.57 m/s

Figure 30: Comparison between measurement and 1-D BAP simulation for the net ram jet thrust increase with void fraction at nominal velocity of 2.4 m/s and 3.57 m/s.

α

ξ

-0.1 0 0.1 0.2 0.3 0.4-0.1

0

0.1

0.2

0.3

0.4Exp. Vi = 2.4 m/sExp. Vi = 3.57 m/s1D-BAP Vi = 2.4 m/s1D-BAP Vi = 3.57 m/s

Figure 31: Comparison between measurements and 1-D BAP simulations for the normalized net ram jet thrust increase with void fraction at nominal velocity of 2.4 m/s and 3.57 m/s.

Figure 32 compares the waterjet thrust as defined in Equation (12) between the measurement and 1D-BAP simulation, since only outlet momentum flux is considered, the thrust increases with increased bubble injection. Both experimental measurements and numerical simulations show similar trend. The simulations again overpredicted the thrust increase than the measurements. The discrepancies between the numerical and experimental results that we did not observe in earlier studies are under further investigations.

α

T W,α

-TW

,N

-0.1 0 0.1 0.2 0.3 0.4-50

0

50

100

150Exp. Vi = 2.4 m/sExp. Vi = 3.57 m/s1D-BAP Vi = 2.4 m/s1D-BAP Vi = 3.57 m/s

Figure 32: Comparison between measurement and 1-D BAP simulation for the waterjet thrust increase with void fraction at nominal velocity of 2.4 m/s and 3.57 m/s. 3-D NOZZLE GEOMETRY OPTIMIZATION

Thrust measurement from both experiments and numerical simulations show that the half 3-D nozzle geometry used in the previous section could not achieve the desired thrust augmentation effects. To study the effects of nozzle geometry on the thrust augmentation, a series of numerical studies were conducted to provide guideline for nozzle geometry optimization.

Figure 33 show the variation of the normalized thrust augmentation, mξ , with the normalized nozzle outlet area, C for inlet velocity 2.4 m/s. C is defined as the ratio of the exit area to the inlet area. mξ is defined as net thrust increase with bubble injection normalized by the inlet momentum flux as following:

,R Rm

m inlet

T TTαξ−

−= . (16)

Where m inletT − is inlet momentum flux. As indicated in the figure, both 0-D BAP (a simplified analytic approach (Sing et al, 2010)) and 1-D BAP results show that in order to obtain a positive thrust augmentation gain, i.e. 0mξ > , the exit area has to be large enough such that C is greater than about 0.6. Figure 33 also shows that there is an optimal C value for different bubble injection void fractions, and the optimal C value is around 1.0. Notice that the base half 3-D nozzle described before has a C value of 0.27 that is well below the minimum C value for positive net thrust increase. This is consistent with the measurements.

12

Cotraction Ratio, C: Aexit/Ainlet

ξ m

0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.02

0

0.02

0.04

0.06

0.08 α0 = 0.01α0 = 0.05α0 = 0.1α0 = 0.01α0 = 0.05α0 = 0.1

Numerical (1-D BAP) : Dashed Lines

Analytical (0-D BAP) : Solid Lines

Vinlet = 2.4 m/sPexit = 101325 Pa

Figure 33: Variation of the normalized thrust augmentation (Equation (16)) with normalized nozzle exit area at 2.4 m/s.

Figure 34. Variation of the normalized thrust augmentation (Equation (15)) with normalized nozzle exit area at 2.4 m/s.

Additionally, Figure 33 indicates that for a given nozzle inlet area, the net thrust increase is determined only by the exit area and is not controlled in a major way by nozzle length and cross section variation in between. Therefore the easiest modification to our experimental base nozzle was to cut short the exit contraction section such that C become close to the optimal value for thrust augmentation. Figure 34 shows the corresponding variation of ξ as defined in Equation (15) with C. ξ indicates that there is a region with 0.75 < C < 1.0, where the bubble injection is detrimental.

Figure 35 and Figure 36 shows 1-D BAP simulations of the net thrust increase if the nozzle contraction section was shortened differently for nominal flows of 2.4 m/s and 3.57 m/s. The resulting total BAP length is measured from the nozzle inlet. As shown in the figures, the best nozzle length for

maximum net thrust increase falls between nozzle length of 0.9 m and 1.02 m, which correspond to C values of 1.46 and 0.78. Therefore, for the optimized nozzle, the converging section of the nozzle was cut down to 0.3153 m which gives a C value of 1.0 and a nozzle length of 0.9775 m. Figure 37 shows the dimension of the modified nozzle with a shortened converging section.

BAP Length, m

T R,α

-TR,N

0.6 0.7 0.8 0.9 1 1.1 1.2-3

-2

-1

0

1

2

3

α = 0.05α = 0.10α = 0.15α = 0.20

Figure 35: Variation of net thrust increase with nominal void fraction for different lengths of the half 3-D nozzle, and a nominal inlet velocity of 2.4 m/s.

BAP Length, m

T R,α

-TR,N

0.6 0.7 0.8 0.9 1 1.1 1.2-10

-5

0

5

10

α = 0.05α = 0.10α = 0.15α = 0.20

Figure 36: Variation of net thrust increase with nominal void fraction for different lengths of the half 3-D nozzle and a nominal inlet velocity of 3.57 m/s.

C

ξ

0 0.5 1 1.5 2 2.5-0.1

-0.05

0

0.05

0.1 α0 = 0.01α0 = 0.05α0 = 0.1α0 = 0.01α0 = 0.05α0 = 0.1

1-D BAP

Vinlet = 2.4 m/sPexit = 101325 Pa

0-D BAP

13

Figure 37: Dimensions of the modified half 3-D nozzle cut short from the base nozzle for optimal thrust augmentation. THRUST PREDICTION OF THE OPTIMIZED 3-D NOZZLE

Based on the newly selected half 3-D nozzle geometry, both 1-D and 3-D two-way coupling numerical simulations were conducted to evaluate the performance.

Figure 38: Pressure contour from 3 D computations at a nominal inlet velocity of 3.11 m/s without air injection.

Figure 39: Axial velocity contour from 3 D computations at anominal inlet velocity of 3.11 m/s without air injection.

Bubble radius, m

Num

bero

fbub

bles

inin

ject

or

0.0001 0.0002 0.0003 0.0004 0.0005 0.00060

100

200

300

400

Figure 40: Bubble size distribution for the bubble injection used in the 3D two-way coupling simulation.

Figure 41: Exit velocity profile from 3DYNAFS_VIS© predictions for different void fractions of bubble injection at a nominal inlet velocity of 3.11 m/s.

Figure 38 and Figure 39 show the pressures and flow velocities along the axial direction at a nominal inlet flow velocity of 3.11 m/s (200 gpm) without air injection. The pressure contours indicate that the pressure varies significantly along the axial direction but is quite uniform vertically. The velocity contours show that there are re-circulating zones near the air injection area.

14

Figure 42: 3DYNAFS_VIS© results showing average pressure distributions along the nozzle for different nominal void fractions at a nominal inlet velocity of 3.11 m/s.

Figure 43: 3DYNAFS_VIS© results showing average velocity variations along the nozzle for different nominal void fractions at a nominal inlet velocity of 3.11 m/s.

In order to estimate the nozzle performance, 3DYNAFS© 3D two–way coupled simulations were conducted for different bubble injection rates at nominal inlet flow velocity of 3.11 m/s. Figure 40 shows the bubble size distribution of the bubbles injected at a base line void fraction of 4.43%. Bubble distributions for higher void fractions have similar distribution shape but the number of bubbles is multiplied by the ratio to the base line void fraction, the bubbles are injected into the flow through the inner and outer injectors.

Figure 41 shows the exit velocity profile for different void fractions. As the curves show, the profiles have similar shapes but the velocity magnitudes increase significantly with increased bubble injection. The average pressure and velocity distribution along the nozzle are shown in Figure 42 and Figure 43, as expected, higher bubble injection rates cause a pressure increase at the inlet, and significantly increase in the exit velocity.

Nominal injection void fraction, αn

Nor

mal

ized

thru

stga

in,ξ

m

0 20 40 60 800

0.2

0.4

0.6

0.8

1

1.2

3DynaFS-VIS©

1D BAP0D BAP

Figure 44: Comparison of the normalized thrust augmentation computed from the 1 D and 3 D models for different nominal void fractions at a nominal inlet velocity of 3.11 m/s.

Figure 45: Snap shots of the modified half 3-D nozzle with direct force measurement at low and high bubble injection rate, flow rate at 200 GPM.

Figure 44 shows the predictions of the normalized net thrust increase, mξ , with the void fraction from the 0D and 1D-BAP as well as the 3D simulations. All simulations show positive net thrust for the optimized half 3-D nozzle. 0D and 1D results match closely. The predictions from the 1D and 3D codes match pretty well at low void fractions and start to deviate at void fractions higher than 20%, 3DYNAFS_VIS© predicts higher thrust increase than what 1D BAP predicts, the difference is due to the fact that 1D BAP does not fully consider bubble dynamics as 3DYNAFS_VIS© does. These effects become stronger as the void fraction increases. Experiments are presently setup and will be conducted in the near future to compare with the simulations.

15

Figure 45 shows two snap shots of the set up of the optimized nozzle with low and high void fraction of bubble injection, the study of the modified half 3-D nozzle with direct force measurement is still on-going. CONCLUSIONS

We summarized in this paper our efforts to study both experimentally and numerically the interaction between purposely injected bubbles and the flow in a waterjet nozzle. Numerical modeling was based on two-way coupling between a model of the mixture flow field and Lagrangian tracking of the injected bubbles. The experimental studies were performed on a laboratory large scale nozzle to validate and compare with the numerical results.

A 2-D nozzle with expansion exit was first studied. The numerical and experimental results agreed well, and a net thrust increase could be measured. To better study the physics of a real BAP while maintaining good accessibilities for experimental measurements, half 3D nozzles were built and studied. Numerical studies were first used to design a base nozzle and then compared with experiment results. Numerical predictions showed the same trend of thrust changes as measured for the baseline nozzle.

A one-dimensional numerical tool was used to systematically study the performance of different nozzle geometries, and an optimal nozzle geometry to produce the best net thrust increase was selected. Comprehensive numerical studies based on the optimized nozzle with different models show net thrust increase which grows with increasing void fraction. An improved experimental set up with the optimized nozzle and with direct force measurement is under way.

ACKNOWLEDGEMENT

This work was supported by the Office of Naval Research under the contract N00014-09-C-0676, monitored by Dr. Ki-Han Kim. We gratefully acknowledge this support.

REFERENCES Albagli, D. and Gany, A., “High Speed Bubbly Nozzle Flow with Heat, Mass, and Momentum Interactions”, International Journal of Heat and Mass Transfer, Vol. 46, pp. 1993-2003, 2003.

Bosio, J. and Malnes, D. "Water Velocity Measurements in Air-Water Mixture," Euromech Colloquiurn No. 7, Grenoble, France, 1968. Chahine, G.L., “Cloud Cavitation Theory”, Proc. 14th Symposium on Naval Hydrodynamics, Ann Arbor, Michigan, National Academy Press, 1983. Chahine, G.L., “Dynamics of the Interaction of Non-Spherical Cavities”, in Mathematical Approaches in Hydrodynamics, ed. Miloh, T., SIAM, Philadelphia, 1991. Chahine, G.L., “Nuclei Effects on Cavitation Inception and Noise”, Keynote Presentation, Proc. 25th Symposium on Naval Hydrodynamics, St. John’s, Canada, Aug., 2004. Chahine, G.L., “Numerical Simulation of Bubble Flow Interactions”, Proc. 2nd International Cavitation Forum, Invited Lecture, pp.72-87, 2008a. Chahine, G.L., “Development of an acoustics-based instrument for bubble measurement in liquids”, Paris ASA meeting, Acoustics 08, 2008b. Chahine, G.L. and Kalumuck, K., “BEM Software for Free Surface Flow Simulation Including Fluid Structure Interaction Effects”, International Journal of Computer Applications for Technology, Vol. 3/4/5, 1998. Chahine, G.L., Kalumuck, K.M., Cheng, J.-Y., and Frederick, G.S., “Validation of Bubble Distribution Measurements of the ABS Acoustic Bubble Spectrometer® with High Speed Video Photography,” Proc. Fourth International Symposium on Cavitation CAV2001, Pasadena, CA, 2001. Chahine, G.L., Perdue, T.O., and Tucker, C.B., “Interaction between an Underwater Explosion Bubble and a Solid Submerged Structure”, DYNAFLOW, INC., Technical Report 89001-1, 1988. Chahine, G.L., Hsiao, C.-T., Choi, J.-K., Wu, X., “Bubble Augmented Waterjet Propulsion: Two-Phase Model Development and Experimental Validation”, 27th Symposium on Naval Hydrodynamics, Seoul, Korea, October, 2008. Choi, J.-K and Chahine, G.L., “Non-Spherical Bubble Behavior in Vortex Flow Fields”, Computational Mechanics, Vol. 32, No. 4-6, pp. 281-290, 2003. Choi, J.-K. and Chahine, G.L., “Noise due to Extreme Bubble Deformation near Inception of Tip Vortex

16

Cavitation”, Physics of Fluids, Vol. 16, No. 7, pp. 2411-2418, 2004. Choi, J.-K. and Chahine, G.L, “Modeling of Bubble Generated Noise in Tip Vortex Cavitation Inception”, ACTA Acustica United with Acustica, The Journal of the European Acoustics Association, Vol. 93, pp. 555-565, 2007. Duraiswami, R., Prabhukumar, S., and Chahine, G.L., “Bubble counting using an inverse acoustic scattering method”, J. of Acoustic Society of America, Vol. 104, pp. 2699-2717, 1998. Gilmore, F. R., California Institute of Technology, Div. Rep. 26-4, Pasadena, CA, 1952. Gumerov, N. and Chahine, G.L., “An Inverse Method for the Acoustic Detection, Localization, and Determination of the Shape Evolution of a Bubble”, IOP Publishing Ltd. on Inverse Problems, Vol. 16, pp. 1741-1760, 2000. Haberman, W.L. and Morton, R.K., “An Experimental Investigation of the Drag and Shape of Air Bubbles Rising in Various Liquids”, Report 802, DTMB, 1953. Hsiao, C.-T. and Chahine, G.L., “Numerical Simulation of Bubble Dynamics in a Vortex Flow Using Navier-Strokes Computation”, Proc. Fourth International Symposium on Cavitation CAV2001, Pasadena, CA, 2001. Hsiao, C.-T. and Chahine, G.L., “Prediction of Vortex Cavitation Inception Using Coupled Spherical and Non-Spherical Models and Navier-Stokes Computations”, Proc. 24th Symposium on Naval Hydrodynamics, Fukuoka, Japan, 2002. Hsiao, C.-T. and Chahine, G.L., “Prediction of Vortex Cavitation Inception Using Coupled Spherical and Non-Spherical Models and UnRANS Computations”, Journal of Marine Science and Technology, Vol. 8, No. 3, pp. 99-108, 2004a. Hsiao, C.-T. and Chahine, G.L., “Numerical Study of Cavitation Inception due to Vortex/Vortex Interaction in a Ducted Propulsor”, Proc. 25th Symposium on Naval Hydrodynamics, St. John’s, Canada, Aug., 2004b. Hsiao, C.-T. and Chahine, G.L., “Scaling of Tip Vortex Cavitation Inception Noise with a Bubble Dynamics Model Accounting for Nuclei Size Distribution”, Journal of Fluid Engineering, Vol. 127, pp. 55-65, 2005.

Hsiao, C.-T., Chahine, G.L., and Liu, H.L., “Scaling Effects on Prediction of Cavitation Inception in a Line Vortex Flow”, Journal of Fluids Engineering, Vol. 125, pp. 53-60, 2003. Ishii, R., Umeda, Y., Murata, S., and Shishido, N., “Bubbly Flows through a Converging-Diverging Nozzle”, Physics of Fluids, Vol. 5, pp. 1630-1643, 1993. Johnson, V.E. and Hsieh, T., “The Influence of the Trajectories of Gas Nuclei on Cavitation Inception”, Proc. Sixth Symposium on Naval Hydrodynamics, pp. 163-179, 1966. Kalumuck, K.M., Duraiswami, R., and Chahine, G.L., “Bubble dynamics fluid-structure interaction simulation by coupling fluid BEM and structural FEM codes”, Journal of Fluids and Structures, Vol. 9, pp. 861-883, 1995. Kameda, M. and Matsumoto, Y., “Structure of Shock Waves in a Liquid Containing Gas Bubbles”, Proc. IUTAM Symposium on Waves in Liquid/Gas and Liquid/Vapor Two Phase Systems, pp. 117-126, 1995. Knapp, R.T., Daily, J.W., Hammit, F.G., Cavitation, ed. Hill, M., New York, 1970. Li, A. and Ahmadi, G., “Dispersion and Deposition of Spherical Particles from Point Sources in a Turbulent Channel Flow”, Aerosol Science and Technology, Vol. 16, pp. 209-226, 1992. Mor, M. and Gany, A. “Analysis of Two-Phase Homogeneous Bubbly Flows Including Friction and Mass Addition”, J. of Fluids Engineering, Vol. 126, pp. 102-109, 2004. Mottard E.J. and Shoemaker, C.J. “Preliminary Investigation of an Underwater Ramjet Powered by Compressed Air”, NASA Technical Note D-991, 1961. Muir, T.F. and Eichhorn, R., “Compressible Flow of an Air-Water Mixture through a Vertical Two-Dimensional Converging-Diverging Nozzle”, Proc. Heat Transfer and Fluid Mechanics Institute, Stanford Univ. Press, pp. 183-204, 1963. Noordzij, L. and van Wijngaarden, L., “Relaxation Effects Caused by Relative Motion on Shock Waves in Gas-Bubble/Liquid Mixtures”, Journal of Fluid Mechanics, Vol. 66, pp. 15-143, 1974. Ovadia, Y. and Strauss, J., “The Nautical Ramjet – an Alternative Approach”, SciTech 2005 project report,

17

Technion Israel Institute of Technology, Israel, <http://noar.technion.ac.il/scitech2005/mechanical3.pdf>, 2005. Pierson, J.D., “An Application of Hydropneumatic Propulsion to Hydrofoil craft”, Journal of Aircraft, Vol. 2, pp. 250-254, 1965. Plesset, M.S., “Dynamics of Cavitation Bubbles”, Journal of Applied Mechanics, Vol. 16, pp. 228-231, 1948. Prabhukumar, S., Duraiswami, R., and Chahine, G.L., “Acoustic measurement of bubble size distributions: theory and experiments”, Proc. ASME Cavitation and Multiphase Flow Forum, Vol. 1, pp. 509-514, 1996. Preston, A., Colonius, T., and Brennen, C.E., “A Numerical Investigation of Unsteady Bubbly Cavitating Nozzle Flows”, Proc. of the ASME Fluid Engineering Division Summer Meeting, Boston, 2000. Rebow, M., Choi, J., Choi, J.-K., Chahine, G.L., and Ceccio, S.L., “Experimental Validation of BEM Code Analysis of Bubble Splitting in a Tip Vortex Flow”, Proc. 11th International Symposium on Flow Visualization, Notre Dame, Indiana, 2004. Saffman, P. G., “The Lift on a Small Sphere in a Slow Shear Flow”, Journal of Fluid Mechanics, Vol. 22, pp. 385-400, 1965. Singh, S., Choi, J.-K., Chahine, G.L., “Optimum Configuration of an Expanding-Contracting-Nozzle for Thrust Enhancement by Bubble Injection”, Proceedings of ASME International Mechanical Engineering Congress and Exposition, Vancouver, British Columbia, Canada, 2010. Schell Jr., et al., “The Hydro-Pneumatic Ram-Jet”, US Patent 3,171,379, 1965. Tangren, R.F., Dodge, C.H., and Seifert, H.S., “Compressibility Effects in Two-Phase Flow”, Journal of Applied Physics, Vol. 20 No. 7, pp. 637-645, 1949. Valenci, S., “Parametric Sea Trials of Marine Ramjet Engine Performance”, M.Sc. Thesis, Technion Institute of Technology, 2006. van Wijngaarden, L., “Linear and Non-linear Dispersion of Pressure Pulses in Liquid Bubble Mixtures”, Proc. 6th Symposium on Naval Hydrodynamics, ONR, 1966.

van Wijngaarden, L., “On the Equations of Motion for Mixtures of Liquid and Gas Bubbles”, Journal of Fluid Mechanics, Vol. 33, pp. 465-474, 1968. van Wijngaarden, L., “One-Dimensional Flow of Liquids Containing Small Gas Bubbles”, Annual Review of Fluid Mechanics, Vol. 4, pp. 369-396, 1972. Varshay, H. and Gany A., “Underwater two phase ramjet engine”, US Patent 5,598,700, 1994. Varshay, H. and Gany A., “Underwater two phase ramjet engine”, US Patent 5,692,371, 1996. Vokurka, K., “Comparison of Rayleigh’s, Herring’s, and Gilmore’s Models of Gas Bubbles”, Acustica, Vol. 59, No. 3, pp. 214-219, 1986. Wang, Y.-C. and Brennen, C.E., “One-Dimensional Bubbly Cavitating Flows through a Converging-Diverging Nozzle”, Journal of Fluids Engineering, Vol. 120, pp. 166-170, 1998. Witte, J.H., “Predicted Performance of Large Water Ramjets”, AIAA 2nd Advanced Marine Vehicles and Propulsion Meeting, AIAA Paper No. 69-406, 1969. Young S., Hsiao, C.-T., and Chahine, G. L., “Effect of Model Size and Free Stream Nuclei on Tip Vortex Cavitation Inception Scaling”, Proc. Fourth International Symposium on Cavitation CAV2001, Pasadena, CA, June 20-23, 2001.