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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.

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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:

( ) 0m m mt + =

u , (1)

( )223

mm m m ij m m

D pDt

= +

u u , (2)

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

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

( )1m g = + , (3) ( )1m g = + , (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

+ = +

+u u

(5) where R is the bubble radius at time t, R0 is the initial or reference bubble radius, is the surface tension parameter, m is the medium viscosity, m 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

+ = + +

+ +

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

= + +

+ + +

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

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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, 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

+ =

+ + =

(8)

where m , 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, , 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 approximati