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2009,21(3): 316-332 DOI: 10.1016/S1001-6058(08)60152-3

NUMERICAL SIMULATION OF BUBBLE FLOW INTERACTIONS*

CHAHINE Georges L. DYNAFLOW INC., 10621-J Iron Bridge Road, Jessup, Maryland 20794, USA, E- mail: glchahine@dynaflow-inc.com (Received October 18, 2008, Revised December 15, 2008) Abstract: Bubble flow interaction can be important in many practical engineering applications. For instance, cavitation is a problem of interaction between nuclei and local pressure field variations including turbulent oscillations and large scale pressure variations. Various types of behaviours fundamentally depend on the relative sizes of the nuclei and the length scales of the pressure variations as well as the relative importance of bubble natural periods of oscillation and the characteristic time of the field pressure variations. Similarly, bubbles can significantly affect the performance of lifting devices or propulsors. We present here some fundamental numerical studies of bubble dynamics and deformation, then a practical method using a multi-bubble Surface Averaged Pressure (DF-Multi-SAP) to simulate cavitation inception and scaling, and connect this with more precise 3-D simulations. This same method is then extended to the study of two-way coupling between a viscous compressible flow and a bubble population in the flow field. Key words: cavitation, bubble flow interaction, numerical simulation 1. Introduction

Study of cavitation inception teaches us that liquids rarely exist under a pure monophasic form and that bubble nuclei are omnipresent. These nuclei are very difficult to eliminate and are always in any industrial liquid at least in very dilute concentrations. However, most engineering studies, and rightfully so, ignore the presence of this very dilute often invisible bubble phase, and consider only the liquid phase to conduct analytical or numerical simulations. This applies to benign situations where pressure variations are not significant and where cavitation, dynamic effects, gas diffusion, and heat transfer do not result in dramatic modification of the flow to warrant inclusion in the modelling of both phases. In this contribution, we are concerned with those conditions where it is important to either evaluate whether cavitation may occur and/or to model the flow in presence of bubbles in significant local concentrations, sizes, or numbers to play a significant role in the flow dynamics. Under these conditions, interaction between the bubbles and

*Biography: CHAHINE Georges L. (1947-), Male, Ph. D.

the flow can be significant and need to be properly addressed. Three families of situations can be distinguished:

(1) Cavitation Inception in uniform or acoustic flow fields.

(2) Cavitation Inception in vortical flow fields. (3) Developed cavitation and bubbly flows. In the first two cases, especially if cavitation

inception is determined acoustically (i.e., prior to bubbles becoming visible), the interaction between bubbles and liquid flow is 1-way, i.e., the liquid dynamics affects the bubble dynamics, while bubble effects on the hydrodynamic flow field are negligible. In the third case, bubbles presence and behaviour is influential enough to affect significantly the basic flow field and implementation and use of 2-way interaction models is warranted.

In this contribution, we will describe the work we have conducted at DYNAFLOW over the recent years in order to address the above aspects. This is not intended to be a review of the research field, however, references to other contributions will be done as appropriate. The subdivision of the article follows the

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three families of situations listed above. 2. Cavitation inception

Cavitation and bubble dynamics have been the subject of extensive research since the early works of Besant in 1859[1] and Rayleigh in 1917[2]. A host of papers and articles and several books[3-10] have been devoted to the subject. Various aspects of the bubble dynamics have been considered and included one or several of the physical phenomena at play such as inertia, interface dynamics, gas diffusion, heat transfer, bubble deformation, bubble-bubble interaction, electrical charge effects, magnetic field effects, etc.. Unfortunately, very little of the resulting knowledge has succeeded in crossing from the fundamental research world to the applications world, and it is uncommon to see bubble dynamics analysis made or bubble dynamics computations conducted for example by propeller or pump designers. This is due to the perceived impracticality of using the methods developed but for experts. However, with the tremendous advances in computing resources, these communities now commonly use Navier Stokes solvers and CFD codes to seek better solutions[11-15]. The challenge is thus presently for the cavitation/bubble community to bring its techniques to par with the single phase CFD progress. Consideration of bubble dynamics within a CFD computation should also become common practice. Indeed, the tools already developed are at the reach of all users and should be adopted as more CFD tools to use for advanced design[12,13]. 2.1 Presence of cavitation nuclei

A traditional cavitation inception engineering definition is: a liquid flow experiences cavitation if the local pressure drops locally below the liquid vapour pressure, vP . This definition of cavitation inception is only true in static conditions when the liquid is in contact with its vapour through the presence of a large free surface and is in practice applicable for liquids saturated with bubbles. For the more common condition of a liquid in a flow, or in a rotating machine, liquid vaporization can only occur through the presence of microscopic free surfaces or microbubbles, also called cavitation nuclei. These are micron and sub-micron sized bubbles in suspension in the liquid or trapped in particles crevices. Several techniques have been used to measure these nuclei distributions both in the ocean and in laboratory cavitation channels. These include Coulter counter, holography, light scattering methods, cavitation susceptibility meters, and acoustic bubble spectrometers[16-22].

Therefore, any fundamental analysis of cavitation inception has to start from the observation that, any

real liquid contains nuclei which when subjected to variations in the local ambient pressure will respond dynamically by oscillating and eventually growing explosively (i.e., cavitating). Cavitation inception cannot be defined accurately independent of this liquid bubble population (sometimes characterized by liquid strength[23]) and independent of the means of cavitation detection. The cavitation inception is in fact a complex dynamic interaction between the nuclei and their surrounding pressure and velocity fields, interaction that can be different between the small and large scale of the same configuration. In addition, the experimental means to detect and decide on calling cavitation inception (practical threshold used by the experimentalists) will affect the results and could be different between laboratory experiments and full scale tests. 2.2 Inception in quasi uniform flow: spherical model

When the pressure variations to which a bubble is subjected are not slow compared to the bubble response time, the nuclei cannot instantaneously adapt, inertia effects become important, and one needs to consider bubble dynamics effects. This is the case for nuclei travelling through rotating machinery. The nuclei /bubbles then act as resonators excited by the flow field temporal and spatial variations. In the case of a vortical flow field the strong spatial pressure gradients (in addition to the temporal) strongly couple with the actual bubble motion (i.e., position vs. time) to result in a driving force that depends on the resonator reaction. This makes such a case, considered in the next section, much more complex than what occurs for a travelling bubble about a foil where, relatively speaking, the position of the bubble is less coupled to its dynamics.

The flow field pressure fluctuations have various time scales: e.g., relatively long for travelling cavitation bubbles over a blade or captured in a vortical region flow, or very short for cavitation in turbulent strongly sheared flow regions. The amplitudes of these fluctuations and the relationship between the various characteristic times determine the potential for cavitation inception.

For cavitation inception in uniform flow fields it is common to use spherical bubble models to examine the dynamics. This is justified as long as the local velocity and pressure gradients around the bubble are weak compared to the radial gradients, and as long as proper account is made of the bubble time dependent location in the flow field i.e., account for any slip velocity relative to the liquid flow. Spherical models have been extensively studied following the original works of Rayleigh[1] and Plesset[10]. For instance, if we limit the phenomena to be modelled to inertia, small compressibility of the liquid, compressibility of the bubble content, we obtain the Gilmore[24]

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differential equation for the bubble radius . We modified this equation to account for a slip velocity between the bubble and the host liquid, and for the non-uniform pressure field along the bubble surface

( )R t

[25]. (This is further discussed later.) The resulting Surface-Averaged Pressure (SAP) equation applied to Gilmores equation[25,26] becomes:

3 11 + 1 = 1+ +2 3 d

R R R RRR Rc c cU

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(2) When the bubble reaches the axis just upstream of the minimum pressure, it develops an axial jet on its downstream side which shoots through the bubble moving in the upstream direction. Even at this stage, the spherical model provides a very good approximation because the bubble is more or less spherical until a thin jet develops on the axis.

(3) The bubble behaviour becomes highly non-spherical once it passes the minimum pressure location. It elongates significantly and can reach a length to radius ratio that can exceed 10. The bubble then splits into two or more daughter bubbles emitting a strong pressure spike followed later by other strong pressure signals when daughter bubbles collapse. Two axial jets originating from the split and a strong pressure signal during the formation of the jets are observed. Fig.1 Illustration of the acoustic pressure emitted by a bubble

in a vortex field as a function of the cavitation number. Note that the bubble behaviour near and above the cavitation inception is quasi-spherical[32]

Fig.2 High speed photos of spark-generated bubble collapsing

between two solid walls, and resulting acoustic signal. Peak signals at splitting and sub-bubbles collapses[32]

2.4 Experimental verification

This behaviour supports the hypothesis that the noise at the inception of the vortex cavitation may originate from bubble splitting and/or from the jets formed after the splitting. This is an important

conclusion that has been preliminarily confirmed experimentally[32,36,38]. Three types of tests were conducted: spark generated bubbles, laser generated bubbles, and electrolysis bubbles injected in vortex lines. Figure 2 shows high speed photography and acoustic signals of bubble splitting between two rigid walls. A small but distinct pressure spike is formed at splitting followed but a more significant spike during the collapse of the sub-bubbles. The second set of experiments was conducted in a vortex tube, where bubbles generated by electrolysis were injected and observed once captured by the vortex line. Figure 3 shows the elongated bubble dynamics and the corresponding pressure signals[35]. The third set of experiments was conducted at the University of Michigan[36,38] using laser induced bubbles (in the vortex and far upstream) and the flow field of a tip vortex behind a foil. (Shown in Fig.4) Comparisons between the observations and the 2DYNAFS simulations showed very good correspondence as shown in Fig.5. Detailed correspondence was harder to establish because of difficulty of measurements of all experimental microscale conditions. Fig.3 High speed photos of an electrolysis bubble captured in

a line vortex, and resulting acoustic signal indicating peak signals at splitting and collapse[32]

Fig.4 Bubble behaviour in a vortex flow field:

, , = 4.51mmcR2=0.2123 m /s* =1.72V ,

, =10m/sUf 0 = 750 mR P . 3-D view of the bubble just before splitting predicted by 2DYNAFS and observed at University of Michigan using laser-induced bubbles at the center of the vortex[36]

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Fig.5 Bubble behaviour in a vortex flow field[26] 2.5 Bubble splitting criteria

The numerical simulations resulted in the following observations on the splitting of bubbles captured in a vortex (see Fig.6), which we use in our predictive models[31-35]:

(1) An explosively growing bubble splits into two sub-bubbles after it reaches its maximum volume, (equivalent radius, ) and then drops to 0.95

. maxR

maxR(2) The two resulting sub-bubbles have the

following equivalent radii: 0.90 and 0.52 . maxR maxR(3) The locations of the two sub-bubbles after the

splitting are at 0.95 and 4.18 on the vortex axis.

maxR maxR

(4) The pressure generated by the subsequent formation of reentrant jets in the sub-bubbles can be approximated by a function of V [36].

Since the noise associated with the jet formation appears to be much higher than the pressure signal from the collapse of a spherical bubble, it is desirable to include the splitting and the associated jet noise in simulations with multiple bubble nuclei. Fig.6 Sketch of the successive phases of a bubble behavior in a

vortex flow field

2.6 Inception in vortical flow fields: Fully non-spherical In order to study the full 3-D interaction between

a bubble and a complex flow field, two methods were developed. The first, using our boundary element code, 3DYNAFS, enables study of full bubble deformations during capture but neglects the effects that the bubble may have on the underlying flow field. The second method accounts for full 2-way bubble/flow field interaction, and considers viscous interaction. This model is embedded in an unsteady Reynolds- Averaged Navier-Stokes (RANS) code, DF-UNCLE1, with appropriate free surface boundary conditions and a moving Chimera grid scheme[26]. This full 2-way interaction non-spherical bubble dynamics model has been successfully validated in simple cases by comparing the results with base case results obtained from the Rayleigh-Plesset equation and 3DYNAFS for bubble dynamics in an infinite medium both with and without gravity. Fig.7 Comparison of the bubble radius versus time for the

spherical models (the conventional and the SAP model) and the 3-D 1-way and 2-way UnRANS computations[26]

As an illustration, Fig.7 shows results of a bubble

interacting with the tip vortex of an elliptical foil. The bubble elongates once it is captured, and depending on the cavitation number, either forms a re-entrant jet directed upstream or splits into two sub-bubbles. When 2-way interaction is taken into account, further smoothing of the bubble surface is exerted by viscosity resulting in a more distorted but overall more rounded bubble. Figure 8 illustrates the various stages of the interaction between a bubble and a tip vortex flow.

1DF_UNCLE, new dubbed 3DYNAFS-VIS is based on UNCLE developed by Mississippi State University that we extended to 3-D cavity and free surface dynamics.

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2.7 Validation of the SAP model Here we compare the SAP spherical model and

the 3-D interaction non-spherical models to predict tip vortex cavitation inception for a tip vortex flow generated by a finite-span elliptic hydrofoil[26]. The flow field was obtained by a RANS computation, which provided the velocity and pressure fields for all compared models: the classical spherical model, the SAP model, the one-way interaction model where the bubble deformed and evolved in the vortex field but did not modify it, and finally the fully coupled 3-D model in which unsteady viscous computations included modification of the flow field by the presence of the bubbles. Fig.8 Comparison of the maximum bubble radius vs.

cavitation number between the spherical models (the conventional and the SAP model) and the 3-D 1-way and 2-way UnRANS computations[26]

Comparisons between the various models of the

resulting bubble dynamics history, and of the cavitation inception values obtained from many tested conditions, reveal the following conclusions, illustrated in Fig.7 and Fig.8:

(1) The bubble volume variations obtained from the full 2-way interaction model deviate significantly from the classical spherical model due to the interaction between the bubble and the vortex flow field.

(2) Differences between the 1-way and 2-way interaction models exist but are not major.

(3) Using the SAP scheme significantly improves the prediction of bubble volume variations and cavitation inception. SAP appears to offer a very good approximation of the full interaction model. 2.8 Modelling of a real liquid nuclei field

We have developed the following general approach to address bubble flow interactions in practical conditions. In order to simulate the water conditions with a known distribution of nuclei of various sizes, as illustrated in Fig.9, the nuclei are

considered to be distributed randomly in a fictitious supply volume feeding the inlet surface or release area of the computational domain. The fictitious volume size is determined by the sought physical duration of the simulation and the characteristic velocity in the release area as illustrated in Fig.9. The liquid considered has a known nuclei size density distribution function, , which can be obtained from experimental measurements

( )n R[17-22] and can be

expressed as a discrete distribution of M selected nuclei sizes. Thus, the total void fraction, D , in the liquid can be obtained by

3

=1

4=3

Mi

ii

RND S (5) where is the discrete number of nuclei of radius iN

iR used in the computations. The position and thus timing of nuclei released in the flow field are obtained using random distribution functions, always ensuring that the local and overall void fraction satisfy the nuclei size distribution function. Fig.9 Illustration of the fictitious volume feeding the inlet to

the nuclei tracking computational domain (or release area) and example of resulting nuclei size distribution satisfying a given nuclei density distribution function

2.9 Effect of a propeller on a nuclei field

In order to simulate the effect of a propeller on a nuclei flow field, the methods described above were used. Gas diffusion into the bubbles was also

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included, since it is very important in this case. For the nuclei field, the fictitious volume size was determined as the product of the release area, the sought physical duration of the simulation, and the characteristic inlet velocity as illustrated in Fig.10.

For the results shown below we have selected characteristic values: size range of 10 to 200 m and void fraction, . Figure 11(a) shows the selected nuclei size number density and Fig.11(b) shows the numbers of discrete bubble sizes and band widths selected.

5= 3.23 10D u

Fig.10 Illustration of the fictitious volume feeding the inlet to

the nuclei tracking computational domain (or release area)

Fig.11 (a) The nuclei size distribution selected to resemble

the field measurement by Medwin[19] and (b) the resultant number of nuclei released

In order to minimize CPU time, a time of release,

, during which actual bubble computations are conducted, was selected and corresponded to a meaningful bubble population distribution in the fictitious bubble supply volume. The resulting bubble behavior was then assumed to be repeatable for the next . This was repeated as many times as

necessary and the effects on the results were analyzed. Figure 12 shows the time variation of the total number of bubbles within the computational domain for different number of repeats. The results clearly indicate that a steady state is achieved after

with a minimum of 10 repeats effectuated. Otherwise, the two-phase medium is in a transient phase. Repeating the number of bubbles more than 10 times increases the period of quasi-steady state in the computational domain.

t'

t'

> 0.03mst

Fig.12 Effect of the bubble release duration on the total

number of bubbles within the computational domain. Computations were conducted for s and results were then repeated to cover much longer release times

-3= 3.7 10t' u

Fig.13 Front and side view of the bubbles in the compu-

tational domain at a given time during the computation. Bubble sizes are to scale in Section b, They are enhanced by a factor of 5 in Sections a and c

Figure 13 illustrates the sizes and locations of bubbles that became visible. Both front and side

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views are shown at a given instant for . It is noted that for the side view the bubbles in Section b were illustrated with the actual sizes, while they were amplified 5 times to make them more visible in print in Sections a and c. Figure 13 illustrates the marked increase in visible bubbles downstream of the propeller, traveling bubble cavitation, tip vortex cavitation and a resemblance of sheet cavitation.

= 1.75

In order to illustrate the results in a simplified manner, a time dependent void fraction by section,

( , )x tD , is defined by integrating bubble volumes in between x and + dx x :

( , )

3

=12 2

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( , ) = 14

N x ti

iR

x tD d x

DS

S '

(6)

where is the shaft diameter, and is the number of bubble within

d ( , )N x tx' at a given time, . The

time-averaged void fraction of t

( , )x tD at , over a time period can also be defined as:

1tdt

1+d

1

1= ( ,d

tt

t)dx t t

tD D (7)

Figure 14 shows the distribution of D along x .

It is seen that the void fraction increases significantly as a result of cavitation on the blades (first hump) and in the tip vortices (second hump). The void fraction decreases as the encountered pressure by the bubble increases. However, the downstream void fraction is 1.4 times that of upstream. Fig.14 The time-averaged void fraction, D , distribution along

the axial direction for and 5= 3.23 10D u = 1.75V The effect of propeller cavitation on the number

of observable bubbles is much more important than on D . Figure 15 shows the bubble size distribution at

. It is seen that a large number of bubbles of 400 m radius are now present in the field, while all bubbles upstream were smaller than 200 m.

= 0.2mx

The effect of the cavitation number, the initial nuclei size distribution, and the initial dissolved gas concentration, on the bubble nuclei entrainment and dynamics were shown below.

Fig.15 Comparison of the initial nuclei size distribution and

that resulting at = 0.2mx Cavitation number: The time-averaged void fraction and downstream

nuclei size distribution obtained at three different cavitation numbers, =V 1.5, 1.65 and 1.75, are compared in Fig.16 and Fig.17. In the three cases the void faction increases significantly near the propeller then plateaus. As the cavitation number decreases, the void fractions within the cavitation areas increase. The relative importance of the tip vortices diminishes relative to the propeller blade cavitation. With smaller values of V , the bubbles retain a much larger size downstream of the propeller due to enhanced net gain of gas mass diffusing into the bubbles. The downstream void fraction D at is 50 times the upstream value for

= 0.2mx=V 1.5, 2.35 times for

=V 1.65 and 1.4 times for = 1.75V . The largest downstream bubble radii observed are 800 m for

= 1.5V , 600 m for = 1.65V and 400 m for = 1.75V , as compared to 200 m at release.

Fig.16 Comparison of the void fraction variations along x

for different cavitation numbers Nuclei distribution:

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The effect of the initial nuclei size distribution on the results is shown in Fig.18. A bubble size distribution ranging from 10-200 m is compared to that of a uniform distribution of 100m bubbles with the same initial void fraction. Higher bubble void fractions are seen near the propeller blade region for the initial uniform nuclei size case. However, this does not result in any significant difference in the void fraction downstream. Figure 19 compares the resulting downstream nuclei size distributions at . It is seen that the largest bubble radius is 400 m for the non-uniform nuclei distribution, while it is 300 m for the uniform case.

= 0.2mx

Fig.17 Bubble size distribution at for three

different cavitation numbers = 0.2mx

Fig.18 Comparison of the void fraction variation along the

streamwise direction for two initial nuclei size distributions

Fig.19 Bubble size distribution at for two initial

nuclei size distributions = 0.2mx

Dissolved gas concentration: The effect of the dissolved gas concentration on

the bubble dynamics is illustrated in Fig.20 by doubling the dissolved gas concentration. This results

in further increase in the void fraction downstream. However, although the dissolved gas concentration was doubled, the void fraction increased only 11%.

Fig.20 Comparison of the void fraction variation along the

streamwise direction for two initial gas concentrations 3. Modeling of bubbly flows 3.1 Bubbly flow and propeller/hydrofoil

In order to address problems where bubbles significantly modify the flow, we have developed a two-way coupled method between the viscous solver: 3DYNAFS-VIS and multi-bubble dynamics solver, DF_MULTI_SAPP . The coupling between the two-phase continuum flow model and the bubble discrete dynamics/tracking model is an essential part of this method and can be described as follows:

(1) The bubbles in the flow field are influenced by the local densities, velocities, pressures, and their gradients in the medium. The dynamics of the individual bubbles and their tracking are based on these local flow variables.

The local properties of the mixture continuum are determined by the bubble presence. The void fractions, and accordingly the mixture local densities, are determined by the bubble population and size, and evolve as the bubbles move and oscillate. The flow field is adjusted according to the modified mixture density distribution while satisfying mass and momentum conservation.

(2) The void faction is obtained by subdividing the computational domain in D -cell where D is computed and measuring the volumes of all bubbles in such cells. An D -cell is large enough for the statistics to be meaningful, i.e., it should contain a large enough number of bubbles, and small enough, compared to the flow field local characteristic length, such that bubbles of equal size in the cell will behave practically the same way.

This approach was applied to a NACA4412 foil at zero angle of attack with various void fractions. Figure 21 shows the void fraction distribution for a hydrofoil operating in a bubbly flow with 5 % free stream void fraction. It is seen that the void fraction level decreases near the leading edge due to screening effects. High concentrations of bubbles are observed near the suction side surface, while the concentration

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is very low in the wake region. Fig.21 Void fraction distribution for a hydrofoil operating in a

bubbly flow Figure 22 shows the effect of the void fraction on

the pressure coefficients on the foil surface. As D increases, the pressure on the suction side increases, which results in loss of lift compared to the single phase flow. The drag has contributions from both skin friction and pressure drag. Skin friction decreases when bubbles are present near the hydrofoil surface, the drag due to the pressure can increase. To illustrate this, we plot the pressure coefficients with respect to the vertical coordinate in Fig.23. The BC and CD portions of the hydrofoil surface contributes to drag, while the AB and AD portions contribute to thrust. The total pressure drag is related to the area between these curves, which increases as the void fraction increases. Fig.22 Pressure coefficient vs. x curves on the surface of the

hydrofoil for various void fractions The changes of lift and drag from the single

phase flow are shown in Fig.24. It is seen that the percentage of lift reduction is greater than the percentage of void fraction increase. The total drag resulting from both pressure drag and skin friction is shown to increase as the void fraction is increased because the increase of the drag due to the pressure is greater that the reduction of drag due to the skin friction.

Fig.23 Pressure coefficient vs. z curves on the surface of the

hydrofoil for various void fractions Fig.24 Hydrofoil performance, the lift and drag changes as

functions of void fraction 3.2 Bubble augmented propulsion

Several publications[39-46] and patents[47-49] claim that bubble injection in a propulsor or a waterjet can significantly augment the thrust even at high speeds. Experimental work has been conducted where fluid enters the ramjet and is compressed by passing through a diffuser (ram effect). High pressure gas is then injected into the fluid via mixing ports and constitute the energy source for the ramjet. The multiphase mixture is then accelerated by passing through a converging nozzle. Various prototypes have been developed and tested in the past and have shown a net thrust due to the injection of bubbles. However, the overall propulsion efficiency was typically less than that anticipated from mathematical models. These results require further modelling and controlled experimentation as we are presently undertaking at DYNAFLOW.

First approximation of the multiphase flow field inside the air augmented nozzle were obtained by neglecting all non-uniformities in the cross-section and considering only a quasi-one dimensional flow field[50-54]. The fluid medium is considered as the mixture of water and bubbles which satisfies the

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following continuity and momentum equations:

+ =m m mt 0U Uw w < u (8)

D 2= + 2D 3

mm m m ij mptU P G P