Bubble Augmented Waterjet Propulsion: Two-Phase Model Development and Experimental Validation

27 th Symposium on Naval Hydrodynamics Seoul, Korea, 5-10 October 2008 Bubble Augmented Waterjet Propulsion: Two-Phase Model Development and Experimental Validation Georges L. Chahine, Chao-Tsung Hsiao, Jin-Keun Choi, Xiongjun Wu (DYNAFLOW, INC., U.S.A.) ABSTRACT The concept of thrust augmentation through bubble injection into a waterjet has been the subject of many patents and publications over the past several decades. Computational and experimental evidences of the augmentation of the jet thrust through bubble growth in the jet stream explain recent rise in interest in the concept. To generalize the idea to practical designs, an adequately validated modeling tool is needed especially since published models are based on significant simplifications. In this paper, we present numerical and experimental studies which aim at developing and validating a numerical code to simulate and predict the performance of a two-phase flow water jet propulsion system. The numerical model is based on the two-way coupling between the flow field and the Lagrangian tracking of large number of bubbles using the Surface Averaged Pressure (SAP) model. Validation studies are conducted using experiments with a two-dimensional flow bubble-augmented propulsion test setup. Bubbles were introduced into the flow with various injection methods, and the bubble behaviors were observed. The velocity and pressure fields were measured, and thrust augmentation was deduced from these measurements. Results show measurable thrust augmentation. INTRODUCTION Two-Phase bubbly flows have been the subject of a large number of research publications due to the profound effect that the presence of the bubbles has on the flow. Depending on the application, these effects can be either deleterious or beneficial. Recently, Albagli and Gany (2003) and Mor and Gany (2004) presented interesting results indicating that bubble injection in a waterjet can significantly improve the net thrust and overall propulsion efficiency. A key aspect of this concept is that the thrust augmentation can be achieved even at very high vehicle speeds unlike traditional propulsion devices which are typically limited to less than 50 knots. 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. In fact, this idea had been proposed earlier, and several prior studies exist in the open literature (Mottard and Shoemaker, 1961, Muir and Eichhorn, 1963, Witte, 1969, Pierson, 1965, Schell et al., 1965). By analogy to ramjet aerodynamic propulsion systems, several of these studies have adopted the naming Water Ramjet to such a system. Experimental work was conducted with fluid that enters the ramjet and is compressed first by passing through a diffuser (ram effect). High pressure gas is then injected into the fluid via mixing ports and constitutes the energy source for the ramjet. The multiphase mixture is then accelerated by a converging nozzle. 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). The Hydroduct and the Underwater Ramjet were tested by towing the device at a given speed and comparing the measured total thrust with the energy stored in the injected gas. Some open sea trials were also conducted by Valenci (2006). Such prototypes have shown a net thrust due to the injection of bubbles. However, 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 or of weaknesses in the modeling (Varshay, 1994, 1996). Flow chocking due the expanding bubble could also occur under certain operating conditions, which would limit the maximum efficiency; however, this has not been reported. Based on the results from these early studies, performance of the ramjet may be strongly related to the efficiency and proper operation of the bubble injector. The more general problem of bubbly flow through a nozzle (not necessarily for propulsion) has been 1

extensively studied empirically and numerically (Tangren et al., 1949, Muir and Eichhorn, 1963, Ishii et al., 1993, Kameda and Matsumoto, 1995, Wang and Brennen, 1998, Preston et al., 2000). Although the analysis of such flow did not consider the process of gas injection, the mathematical equations regarding the state of the mixture as it passes through the nozzle should be the same. Based on these models, the expansion of a compressed gas bubble-liquid mixture was found to be an efficient way to generate the momentum necessary for additional thrust (Albagli and Gany, 2003, Mor and Gany, 2004)). In these studies, the inviscid governing equations were averaged in a given cross section and one-dimensional analysis was conducted. Spherical bubbles of the same size or of given distribution were considered, and the bubble dynamics was deduced from the Rayleigh-Plesset equation. Even though earlier experiments (Muir and Eichhorn, 1963) revealed that the velocity of the gas phase is faster than the mixture velocity due to the negative pressure gradient, most of the theories assumed that the bubble and the liquid have the same velocity. This approximation was also used in more recent studies (Wang and Brennen, 1998, Preston et al., 2000). In some analyses, additional simplifications were introduced where the bubble dynamic equations was substituted with a simpler quasi-static equilibrium equation (Mor and Gany, 2004). With the increasing interest in the water ramjet propulsion, there is a need for a numerical tool which can provide a more detailed analysis of the two-phase flow, and which correctly includes the dynamic behavior of bubbles. At DYNAFLOW, we have been developing numerical methods to predict the dynamics of bubbles (Chahine et al., 1988, Chahine, 1991, Kalumuck et al., 1995, Chahine and Kalumuck, 1998) and the behavior of small cavitation nuclei (Chahine, 1983, 2004, 2008a, Hsiao and Chahine, 2001). These methods were well validated in the past against experiments as well as against other numerical methods. One of the achievements of these efforts is the development of the Surface Averaged Pressure (SAP) model (Hsiao and Chahine, 2001, 2002, 2003, 2004a, 2004b, 2005, Young et al., 2001, Hsiao et al., 2003, Choi and Chahine, 2003, 2004, 2007, Chahine 2004, 2008a, Rebow et al., 2004), which is very suitable for the simulation of many bubbles/nuclei in the flow field. The resulting code, DF_MULTI_SAP, has been coupled with steady and unsteady Navier-Stokes solvers such as DF_UNCLE TM (DYNAFLOW s version of UNCLE), CFDShip-Iowa (work done at PennState), and INS3D. Very recently, we have developed a FLUENT User Defined Function (UDF) version of DF_MULTI_SAP allowing this bubble dynamics and tracking module to be coupled with the commercial flow solver FLUENT. We will use DF_MULTI_SAP here with CFD solvers to model the bubbly mixture in the waterjet. 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 closely followed by solving the surface averaged pressure Rayleigh-Plesset equation. The bubble responds to its surrounding medium that is described by its mixture density, pressure, velocity, etc. Macroscopic level: Bubbles are considered collectively and define the local void fraction. The mixture medium has a time and space dependant average density which is related to the local void fraction. This is similar to a compressible flow in that the mixture density depends on the local pressure. However, the relation between the density and the pressure is different from that of the classical equation of states. The mixture density is provided by the microscale tracking of the bubbles and the determination of their local volume fraction. The full coupling between the two levels is twoway: the bubble dynamics are in response to the variations of the mixture flow field characteristics, and the flow field is directly a function of the bubble density variations. The Navier Stokes solver DF_UNCLE TM was extended to allow a varying density medium, and a 3D, fully unsteady, two-way coupling between DF_UNCLE TM and DF_MULTI_SAP was implemented to describe the two-phase bubbly flow. 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 any geometric design. We are also conducting small scale experiments to visualize and measure the flow characteristics of the considered bubbly flow inside the nozzle in an effort to understand the physics better and to develop a more detailed modeling for bubble injection schemes. ONE-DIMENSIONAL APPROACH 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 (Wang and Brennen, 1998, van Wijngaarden, 1966, 1968, 1972, Noordzij and van Wijngaarden, 1974). The use of a quasi one-dimensional model for the study of two-phase flow in a nozzle goes back to the early work of Tangren et al. (1949) in which the bubbles were assumed to be very small and well dispersed in 2

the fluid and are in equilibrium with the locally prevailing flow conditions. Although this simple model neglects inertial effects on the bubble dynamics, it can capture the compressible nature of the two-phase mixture and can be used to determine, for example, the choking flow conditions. This quasi-steady model was further improved by modeling the relative motion of the bubbles with respect to the liquid. Such a model was used for the analysis of bubbly flows in a converging diverging nozzle by Ishii et al. (1993). A more complex model able to characterize the bubble dynamics using the Rayleigh-Plesset equation was proposed by van Wijngaarden (1966, 1968, 1972). By capturing inertial effects, this model is able to represent transient shock waves in a bubbly mixture. In particular, analyses of bubbly flows in converging diverging nozzle using this type of model were presented in the work of Wang and Brennen (1998) and Preston et al. (2000). The unsteady 1-D flow through a nozzle of varying cross-section can be described by where ρ, m ρm 1 ρmuma + = 0, t A x ρmum 1 ρmumaum p + + = 0, t A x x (1) u, m p are the mixture density, velocity and pressure, and A is the cross-sectional area of the nozzle. In the above equations, the liquid phase can be taken to be either compressible or incompressible. In this study, 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 which can be written using the number of bubbles of radius R per unit volume, N(R): 4 = ( ). (2) 3 3 α πrn R For the purpose of the present calculations, we have assumed that other than at the injection locations, no bubbles are created or destroyed. The conservation of the number of bubbles can be written as: N 1 um AN + = 0. (3) t A x Concerning the description of the local bubble dynamics in the nozzle, we assume that the bubble behavior is governed by the Rayleigh-Plesset equation. Alternatively, a simplified model for the bubble dynamics could be written if we assume that bubbles readily equilibrate with the local pressure. This quasisteady approximation can be written by removing all dynamics components from the Rayleigh-Plesset equation. The above set of equations was integrated in space using an adaptive embedded explicit Runge-Kutta integration. Using this type of integration, the solution was computed using both 4 th and 5 th order accurate scheme for each marching step. The difference between the two is taken as a measure of the integration error and is used to control the forward step size. In addition to the above explicit scheme, an implicit scheme was used for the numerical integration, which made the convergence more robust. LAGRANGIAN BUBBLE TRACKING Lagrangian bubble tracking is used in DF_MULTI_SAP and the corresponding FLUENT User Defined Function (UDF) version which is called Discrete Bubble Model (DMB). 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 for 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 Rayleigh-Plesset equation improved with a Surface-Averaged Pressure (SAP) scheme: 3k 3 2 1 R0 2γ 4μR RR + R = pv + pg0 Penc 2 ρ R R R 2 enc b + u u, 4 (4) where R is the bubble radius at time t, R 0 is the initial or reference bubble radius, γ is the surface tension parameter, µ is the viscosity, ρ is the density, p v is the liquid vapor pressure, p g0 is the initial gas pressure inside the bubble, k is the polytropic compression law constant, u enc is the liquid convection velocity vector and u b is the bubble travel velocity vector, and P enc is the ambient pressure seen by the bubble during its travel. With the Surface Averaged Pressure (SAP) model, P enc and u enc 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. 3

R 3 R 2 1 R R d 1 RR + 1 R = 1+ + c 2 3c ρ c c dt 3k R0 2γ 4μR (5) pv + pg0 Penc R R R 2 enc b + u u, 4 where c is the speed of sound in the liquid. The bubble trajectory is obtained from the bubble motion equation derived by Johnson and Hsieh (1966): dub ρ 3R ρ u = FD( u ub) + ( u ub) + ub dt ρ 2R ρ x b b 1 ρ du du ( ρ ρ) b b + g 2 ρ dt dt + ρ b b 1/2 Kv ρd + ( u ub ). ρ Rd ( d ) b ij 1/4 lk kl (6) The first term in Equation (6) accounts for the drag force effect on the bubble trajectory. The drag coefficient F D is determined by the empirical equation of Haberman and Morton (1953). The second and third term in Equation (6) 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 (6) 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 d ij is the deformation tensor. FLOW FIELD SOLUTION The mixture medium should satisfy the following general continuity and momentum equations: ρm + ( ρmu m) = 0, (7) t Dum 2 ρm = pm + 2μmδij μm( u m), (8) Dt 3 where, the subscript m represents the mixture medium, and δ is the Kronecker delta. The mixture density and ij the mixture viscosity for a void volume fraction α can be expressed as m ( 1 ) ρ = α ρ + αρ m ( 1 ) μ = α μ + αμ, (9) g, (10) g where, the subscript represents the liquid and the subscript g represents the gas in the 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. COUPLING BETWEEN THE FLOW MEDIUM AND THE BUBBLES The coupling between the mixture flow field and the bubble dynamics/tracking is the essential part of the present research. The two-way interaction can be described as follows: The bubbles in the flow field are influenced by the local density, velocity, pressure, and pressure gradient 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 (4) to (6). The mixture flow field is influenced by the presence of the bubbles. The void fraction, and accordingly the 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 (7) and (8). The two-way interaction described above is very strong because the void fraction can change significantly in Bubble Augmented jet Propulsion (BAP) applications, from near zero in the water inlet to as high as 70% at the nozzle exit. While the first component of the coupling (flow to bubble) is straightforward through the Lagrangian bubble tracking, the second component of the coupling (bubble to flow) can be handled in many different ways. The essence of this bubble-to-flow coupling is to determine the medium density through computation of the void fraction from the bubbles. We hare considered three options to calculate this void faction: 1. Sum the volumes of all the bubbles in a well defined elementary volume of the mixture and divide this by this elementary volume. We call this finite volume the α-cell. The advantage of this approach is that the dynamics of each individual bubble are included in the computation. However, this requires a large enough number of bubbles so that the α-cell contains a statistically meaningful number of bubbles. If there are N spherical 4

bubbles of radius R in the α-cell of a volume V, the void fraction α is calculated by α = i= N i= 1 4 π R 3 V 3 i. (11) 2. Based on the tracking of a small number of bubbles, construct the field of void fraction using initial (or injection) void fraction. The advantage of this method is the substantially reduced computational time due to the smaller number of bubbles to track. This approach is appropriate for preliminary computations in design stages when a quick turn-around time is required. A well defined α-cell volume V, which would be much larger than the first approach described above, is also needed in this approach to obtain a representative bubble radius in the volume and to assign a calculated void fraction to the α-cell. One simple way of constructing the field of void fraction is to scale the initial void fraction, α, using the radius o of the tracked bubble, R, and the initial radius of the bubble, R ; o TEST CASE STUDY The tow pool experiment of Ovadia and Strauss (2005) was chosen as a test case since measured data is available. The nozzle is shown in Figure 1. The water comes through the inlet, flows through a diffuser followed by an expansion with a bubble injector, and finally exits through a converging nozzle. The tests were performed at 8 m/s, and thrust, air flow, and pressure were measured at four locations inside the nozzle. The flow was modeled both in DF_UNCLE and FLUENT. Boundary conditions were set so that the inlet velocity was 8 m/s and the exit pressure was 1 atm. Figure 2 shows the velocity distribution on the center plane of the nozzle. As expected, the velocity inside the nozzle decreases as the cross sectional area increases up to the bubble injector, and then increases toward the exit as the cross section narrows. The corresponding pressure, as shown in Figure 3, is the highest just downstream of the bubble injector. The inlet pressure, which is predicted to be 84 kpa, matches with the experimental observation reported in Ovadia and Strauss (2005). 3 R α = α o. (12) Ro 3. The third approach is to use the concept of clone bubbles. In this case an α-cell would be composed of a small set of different bubble radii of with a number of identical bubbles assigned to each size. The computations are then reduced to only computing the nominal bubbles and to expanding the results to the clone bubbles. This approach optimizes speed of computation vs. completeness. In all three approaches, the selection of the α-cell volume V is important. This volume cannot be too small so that either there are no bubbles in the volume or the bubble is larger than the volume, i.e. the volume should be much larger than a characteristic bubble volume. On the other hand, the characteristic length of the α-cell should be smaller than the local characteristic flow length such that the flow characteristics vary little over the α-cell volume to enable assumption of uniformity of the medium quantities in this cell. We select the α-cell volume to satisfy the above two conditions and introduce appropriate smoothing of the void fraction over neighboring cells.. Figure 1: Geometry of the nozzle inside wall as modeled seen from the downstream of the nozzle exit. Figure 2: The axial velocity distribution on the center plane of the nozzle. The unit is m/s. 5

Figure 3: The pressure distribution on the center plane of the nozzle in Pascals. Figure 4: Convergence of the computed bubble radius of the tracked bubble. STEADY STATE 1-D APPROACH OF THE TWO-WAY INTERACTION METHOD A steady 1-D approach can be used within the DF_UNCLE TM / DF_MULTI_SAP coupled approach. All flow and bubble quantities are assumed to be the same in a cross section of the nozzle. One then obtains the void fraction from tracking bubbles through the 1- D flow field. We can then conduct an iteration procedure for the two-way interaction between the bubbles and the flow. The procedure is as follows: a) Track the bubble in the flow field using the liquid flow as an initial guess at the first step. From there on, the previous updated flow field is used at each iteration. b) Find the tracked bubble radius as a function of the axial coordinate along the nozzle. c) Deduce the medium density from the void fraction using Equation (9). d) Solve the medium flow field equations using the updated medium density. e) Re-compute bubble motion and size as in step 1 and iterate until convergence. The iteration converges well for the simulated nozzle. In Figure 4, the convergence of the bubble radius is shown indicating good convergence as early as in the 3 rd iteration. Figure 5 shows a similar fast convergence of the corresponding flow field solution, illustrated by the pressure distribution along the nozzle. Figure 5: Convergence of the pressure along the centerline of the nozzle. The unit of the pressure is Pa. For illustration, the behavior of a 5 mm radius bubble injected at a pressure of 2 atm and at 1.5 m downstream of the inlet is shown in Figure 6. Since the pressure inside the nozzle at the injection location is about 1 atm, the bubble expands after the injection then oscillates as it flows down and reaches an equilibrium radius. Figure 7 shows the converged void fraction distribution inside the nozzle if a set of such bubbles were injected. It is zero from the inlet to the injection location. Then it suddenly rises at the injection point and overshoots due to the bubble oscillations. It then rises gradually due to the pressure drop along the nozzle till the outlet. The pressure distributions along the nozzle axis are shown in Figure 8. The axial velocity (Figure 9) increases toward the outlet, and reaching the value of 15 m/s which is much higher than the exit velocity of the water only flow. 6

Figure 6: Bubble radius of an initial 5mm bubble as it flows downstream inside the nozzle. Figure 9: Converged axial velocity distribution on the center plane (top), and the axial velocity along the centerline (bottom) in m/s. With the 1-D assumption, the thrust for the inside of the nozzle is defined as follows: α Figure 7: Void fraction distribution inside the nozzle. The initial void fraction at the injection is 0.5 (the dashed line). 2 2 ( o o i i) + ( ρm, o o m, o ρm, i i m, i ) T p A p A Au Au, (14) where the subscripts i and o represent the inlet and the outlet respectively. The component of the thrust described in the first parenthesis in (14) 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 predicted from the steady 1-D model over a range of void fractions is shown in Figure 10. As the void fraction increases, there is a large gain in the momentum component of the thrust, which is countered by a loss in the pressure component. As a result, the total thrust increases as the void fraction increases. At the initial void fraction of 50%, the total thrust gain above the water only flow is about 1,100 N, that is, an enhancement of about 1.75. Figure 8: Converged pressure distribution along the centerline of the nozzle (Units are Pa). 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 the nozzle. A 2 ( m m) T = p+ρ u da, (13) where u m is the axial component of the mixture velocity. Figure 10: Predicted thrust for various initial void fractions. 7

EFFECT OF THE BUBBLE DYNAMICS Figure 11 compares for the same configuration the quasi-steady and the Rayleigh-Plesset bubble based models. The operating conditions were determined such that the exit pressure was atmospheric and the inlet velocity was held constant at 8 m/s. The bubbles were injected near the end of the injection section of the nozzle. For the cases presented here, the injection was implemented as a discontinuity in the void fraction. The pressure was kept constant across it while the velocity was suddenly increased in order to preserve the mass flow rate. The pressures for a void fraction injection of 20% and 50% are compared in Figure 11. It is important to note that for a 20% injection void fraction, the two models provided nearly identical answers. However, at 50% injection void fraction, the pressure field exhibited a marked difference between the two models. It is important to note that the quasi-steady approach predicts larger bubbles at the nozzle exit. This is expected since the inertial effects modeled by the Rayleigh-Plesset equations will retard bubble growth. 1.5 m). After the injection, α increases mildly for the injection α =20%, while it increases more rapidly for the α=50%. The large difference between DBM+ Fluent approach and 1-D quasi-steady model is observed near the outlet. This is due to the bubble dynamics that includes the slip of the bubble relative to the fluid. Figure 12: Comparison of the total thrust obtained by the DBM+Fluent approach and the two 1-D models. Figure 11: Comparison of the pressure field for the quasisteady and Rayleigh-Plesset models for the 1-D problem. COMPARISON OF 1-D MODELS WITH DISCRETE BUBBLE MODEL The predictions using the Discrete Bubble Model (DBM) and FLUENT solver are compared to the two other 1-D models described earlier. The total thrust is compared in Figure 12 and the pressure distributions along the nozzle are compared in Figure 13. They match well for low void fractions, but the DBM deviates from the others for higher void fractions. For instance, the pressure distributions obtained by the DBM approach coupled with FLUENT and by the 1-D quasi-steady model match well for α=20%, but are different for α=50%. In the case of α=50%, the difference of the pressure gradient is prominent at the nozzle exit. Figure 14 shows the void fraction along the nozzle. It is zero from the inlet to the injection (at Figure 13: Comparison of the pressure distribution along the length of the nozzle predicted by the DBM+Fluent approach and the 1-D quasi-steady model. The injection void fractions of 20% and 50% are shown. Figure 14: Comparison of the void fraction along the length of the nozzle predicted by the DBM+Fluent approach and the 1-D quasi-steady model. The injection void fractions of 20% and 50% are shown. 8

One essential aspect of the BDM approach is that the bubbles are tracked according to the correct bubble motion Equation (6). This is an important improvement over the other methods which assume that the bubble follows the flow field (Albagli and Gany, 2003, Mor and Gany, 2004, Preston et al. 2000, Wang and Brennen, 1998). In order to illustrate the difference, the slip velocity, which is defined here as the mixture flow velocity minus the bubble velocity, is shown in Figure 15. The slip velocity is obtained by the coupling of DF_UNCLE TM and DF_MULTI_SAP and by our 1D model including slip velocity effects for the case of a converging-diverging nozzle. Our two different numerical methods produced satisfactorily close results. As can be observed in the figure, the bubble velocity is a higher than the mixture flow velocity in the converging section. After passing the throat, the bubble slows down in the diverging section and eventually becomes slower than the mixture flow at the exit. This slip velocity has an influence on the bubble dynamics as can be seen in Equations (4) and (5). DF_MULTI_SAP coupled approach can run unsteady two-way coupled 3D simulations. The procedure for these unsteady computations is as follows: 1. Initialize the flow field, e.g., from liquid only steady state solution or from a known two-phase initial solution. 2. Advect the bubbles one time step in the flow field following injection. 3. Based on the resulting bubble size and location, compute the void fraction α ( x, yzt,, ), using the α-cells. ρ x, yzt,, from 4. Deduce the medium density m ( ) α ( x, yzt,, ). 5. Solve the flow field using the updated medium density, and proceed to the next time step. 6. Repeat from step 2 to step 5 until the desired simulation time is reached. Figure 16 shows an example simulation using unsteady two-way coupling between DF_UNCLE TM and DF_MULTI_SAP. Three different bubble sizes were injected from a bottom port near the inlet. It is observed that due to buoyancy the larger bubbles rise faster than the smaller bubbles, and contribute predominantly to the medium density. Eventually, bubble stratification depending on size is observed over the length of the nozzle (Figure 17). Figure 18 shows an instantaneous snapshot of bubbles, medium velocity vectors, and density contours for another case of the bubble injection from a vertical strip at the widest section of the propulsor. Figure 15: Comparison of the slip velocities (mixture velocity minus bubble velocity) predicted by DF_UNCLE TM + DF_MULTI_SAP approach and our 1D model for a converging-diverging nozzle. The nozzle converges between 0.58 m and 1.08 m and diverges from 1.08 m to 1.3 m (exit). Bubbles are injected at 0.58 m. 1 mm 3-D UNSTEADY BUBBLE/FLOW COUPLING 500 μm 100 μm To study full 3-D two-way interactions, the void fraction is defined in the 3-D space using the α-cell concept introduced earlier. The bubble number and size in each α-cell are used to compute the void fraction. Improvement in the distribution of the void fraction by introducing a smoothing over the neighboring α-cells was necessary to improve convergence of the solution. The DF_UNCLE TM and Figure 16: DF_UNCLE TM / DF_MULTI_SAP coupled simulation for bubbles of different sizes injected upstream from a bottom port. 9

Supporting rigs 4 flang es 4 pipe 4 Te st se c tio n 4 Flow adaptor Supporting beam Wind wave tank 4 nip ples Pu m p 3x 3 4 bulkhead fitting Ba ll Va lve 4 flow m e ter 4 c o up lin g 4 flang es 4 pipe Figure 19: Sketch of the test setup for the Bubble Augmented Jet Propulsion experiment. Figure 17: Bubble stratification at a later time than shown in Figure 16. The bubbles on top of the mixture density contours (top) and the same bubbles colored by their size (bottom). Measurement Stations Air Injection Flow Straightening Section Computational domain Figure 18: Snapshot of computed bubble distribution and mixture medium density distribution on the center plane of the bubble augmented propulsion test setup. Only the portion of the nozzle between the injection and the exit is shown. EXPERIMENTAL SET-UP The test setup used in this study is shown in Figure 19. The tests were conducted in DYNAFLOW s 72 ft wind-wave tank. The flow is driven by a 15 hp pump (Goulds Model 3656) capable of a maximum flow rate of 550 gpm at a pressure drop of 25 psi and 4 inch diameter piping. 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 not an issue. A flow adaptor is used to convert the flow from a circular cross section to the rectangular cross section of the test section geometry. A flow straightening section is inserted between the flow adaptor and the nozzle inlet. A photograph of the test section placed inside the wave tank is shown in Figure 20. The test section consists of two acrylic size-walls and two inserts that are sandwiched between the two side-walls. The inserts are machined to form the profile of the test section. This setup enables one to change the test section easily by replacing only the inserts to form the geometry of different profiles. Figure 20: Experimental setup of the observation and measurement section used in the test. The flow is from right to left. In computational studies, only the nozzle section indicated in the figure is modeled. Several designs of bubble injection were studied. Figure 21 shows the bubble injection from top and bottom ports. This shows that, if an axisymmetric propulsor employs injection from the cylindrical wall, the bubbles could remain only around the periphery of the flow. Figure 22 shows a series of snapshots from a high speed video taken near a bottom injection port. The injected bubble plume is seen interacting with the boundary layer of the liquid flow, and the bubbles are observed to break up into smaller sizes due to the shear. Figure 23 shows a case of bubble injection from side ports which here distribute the bubbles more uniformly inside the propulsor. The void fraction was measured or estimated using several methods; Nominal void fraction can be obtained by dividing the air flow rate by the liquid flow rate. Laser light attenuation is used. A laser light attenuates through the bubbly medium due to light scattering on the bubble surfaces. Figure 24 shows a test setup for the calibration, which illustrates the operation of the laser light attenuation method. Figure 25 shows the void fraction along the propulsor measured by the laser light attenuation method. Another way of measuring the void fraction is to use an acoustic method. Figure 26 shows a test setup to measure bubble size distribution (and thus the void fraction) using DYNAFLOW s ABS ACOUSTIC BUBBLE 10

SPECTROMETER (Prabhukumar et al., 1996, Duraiswami et al., 1998, Chahine et al., 2001, 2008, Chahine, 2008b). A direct way to measure the void fraction is image analysis. This applicable only to low void fraction medium where the bubbles do not overlap too much. Bubble Column Laser Photo Detector Figure 24: Void fraction measurement by laser light attenuation is calibrated using a bubble column. Figure 21: Bubble injection from the top and bottom ports. The main flow is from right to left. flow Figure 25: Void fraction variation along the length of propulsor measured by laser light attenuation method. time Figure 22: Bubble injection from a bottom port. 2080 ml/min air, 200 gpm water flow, bottom port, 1/500 sec. between pictures Figure 26: Void fraction measurement setup using ABS ACOUSTIC BUBBLE SPECTROMETER. Figure 23: Bubble injection from four side ports near the inlet. The flow direction is from the right to the left. Figure 27 shows the bubbles injected by a sintered plate. Under the sintered plate, four air chambers were installed so that the pressure in each chamber can be adjusted. With this arrangement, the bubble injection can be adjusted more uniformly in the vertical direction regardless of the hydrostatic pressure difference between the top and bottom of the sintered plate. As demonstrated by comparing the two photos in Figure 27, a strobe light can enhance the image quality for this purpose. Figure 28 shows an image with backlighting which enhances the contrast and helps the image analysis. The void fraction analyzed from this image is 3.2%. Figure 29 shows the bubble size 11

distribution right downstream of each of the four sections of the sintered plate injector. Figure 27: Bubble injection using sintered plate. Pictures taken with a continuous light (left) and with a strobe light (right). Figure 28: Photographs of bubbles injected into the BAP test section. Void fraction is 3.2%. THRUST MEASUREMENTS In order to estimate the nozzle thrust using Equation (14), the pressure and velocity at the inlet and the outlet of the nozzle were measured. A Pitot tube traversed vertically through the inlet and outlet section of the nozzle for this measurement. Figure 30 shows the measured pressure and velocity profiles without bubble injection (0% void fraction), and Figure 31 shows similar profiles with a bubble injection of 7.2% nominal void fraction. Compared to the liquid only flow, the 7.2% void fraction case shows an increase of both the outlet velocity and the inlet pressure. Increase of the outlet velocity contributes to an increase in thrust due to momentum increase, while an increase of the inlet pressure reduces the thrust. However, the momentum gain is larger than the pressure loss and there is a net gain in the thrust as shown in Figure 32. The curve marked with sintered plate inj. (1/2 exit) corresponds to the case discussed above. This nozzle has a modification of the outlet height which is a half of that of the original nozzle. A sudden rise of thrust around 1% void fraction is not yet fully understood, but the chocking due to large bubble growth is suspected as responsible for the plateau at higher void fraction. This is being further investigated. The three other curves (port injection 1, port injection 2, and sintered plate injection) in the figure are obtained with two different combinations of injection ports and sintered plate injection in the original diverging-converging nozzle (Figure 17, Figure 23). Bubble Size Distribution (bubbles/ cm 3 ) 8 7 6 5 4 3 2 1 Bottom - sec. 1 Midle lower- sec.2 Middle upper - sec. 3 Top - sec. 4 0 200 400 600 800 Bubble Radius (µm) Figure 29: Bubble size distribution right after the sintered plate injector. The distributions are compared for the four sections of the injector from top to bottom. The average void fraction is 1.6%. Figure 30: Measured pressure and velocity profiles at the inlet and the exit of the nozzle without bubble injection. Water flow rate is 200 gpm. 12

Figure 33: Photographs of the experimental setup of the nozzle with an expansion exit. The flow is from right to left. 115 110 Figure 31: Measured pressure and velocity profiles at the inlet and the exit with bubble injection of 7.2% void fraction. Water flow rate is 200 gpm. Thrust Increase (N) 25 20 15 10 5 0-5 Port Injection 1 Port Injection 2 Sintered Plate Injection Sintered Plate Inj. (1/2 Exit) 0 2 4 6 8 Void Fraction (%) Figure 32: Thrust augmentation measured experimentally. Four curves are for two different nozzle outlet heights and three different injection methods of the first nozzle geometry. NOZZLE WITH AN EXIT EXPANSION A nozzle with an expansion exit (Figure 33) was also studied. The intention of this nozzle geometry modification is to make the bubbles grow more at the throat and to enhance the bubble effects. Using the 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. This pressure is compared with the 1D model and the DF_UNCLE TM predictions in Figure 34 for 100 gpm inlet flow. The 1D model overpredicts the pressure drop at the throat (x=105 cm) because the model does not consider the friction drag. However, the model captures the overall pressure behavior well and is useful, for example, in design calculations. The DF_UNCLE TM solution agrees well with the measurement. A similar good agreement is observed at higher flow rate conditions; 200 gpm (Figure 35) and 300 gpm (Figure 36). Pressure (kpa) 105 100 95 90 85 80 1D_BAP 0.00% DF_UNCLE 0.00% EXPERIMENTAL -20 0 20 40 60 80 100 120 140 X (cm) Figure 34: Measured and predicted pressures along the centerline of the nozzle (100 gpm, no bubble injection). The axial coordinate x is 0 at the inlet and 130 cm at the exit. Pressure (kpa) 130 120 110 100 90 80 70 60 1D_BAP 0.00% DF_UNCLE 0.00% EXPERIMENTAL -20 0 20 40 60 80 100 120 140 X (cm) Figure 35: Measured and predicted of pressures along the centerline of the nozzle (200 gpm, no bubble injection). Pressure (kpa) 160 140 120 100 80 60 40 20 1D_BAP 0.00% DF_UNCLE 0.00% EXPERIMENTAL -20 0 20 40 60 80 100 120 140 X (cm) Figure 36: Measured and predicted pressures along the centerline of the nozzle (300 gpm, no bubble injection). 13

The change of the velocity at the outlet with bubble injection is the key factor of the thrust augmentation. The outlet velocity distribution is measured by a Pitot tube at several locations in the outlet plane to obtain a flux averaged outlet velocity. It is then compared with the 1D model prediction for various flow conditions and void fractions as shown in Figure 37. The nominal void fraction is 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. The thrust augmentation estimated for this nozzle geometry for the range of void fraction studied, however, is not too large because the upstream pressure at the inlet increases as much countering the gain in the outlet momentum. An extension of this study to a much higher void fraction is under preparation. Figure 38 shows the axial velocity of the mixture and the bubble for the case of 150 gpm water flow and the bubble injection of 2.37% nominal void fraction. In the 1D model a 200 μm bubble is assumed while in the experiments a distribution of bubble sizes were observed. The flow velocity is measured by a Pitot tube, and the bubble velocity is measured by particle image velocimetry (PIV). Without seeding, the PIV correlates the velocity of bubble between two consecutive images. The trend of bubble moving faster in the contraction and slower in the expansion is well captured in the experiment and in the prediction. This part of the study is still on-going. Velocity (m/s) 14 12 10 8 6 4 2 1D_BAP 100gpm 1D_BAP 200gpm 1D_BAP 300gpm EXP 100gpm EXP 200gpm EXP 300gpm 0 5 10 15 20 Nominal Void Fraction (%) Figure 37: Predicted and measured flux averaged velocity at the exit vs. nominal injection void fraction. Averaged Velocity (m/s) 10 8 6 4 2 0 Flow velocity-exp Flow velocity-1d-bap Bubble Velocity-Exp Bubble Velocity-1D-BAP 0 20 40 60 80 100 120 140 X (cm) Figure 38: Comparison of measured and predicted bubble velocity and mixture velocity (150 gpm, 2.4% nominal void fraction). CONCLUSIONS We have presented in this paper our efforts to model 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 using our Surface Averaged Pressure (SAP) model. The SAP model is based on studies of single bubble dynamics and deformation in complex flow fields and uses the acquired knowledge in a simplified Rayleigh-Plesset like model using bubble surface averaged quantities to compute the bubble behavior in the flow field. This SAP model enabled reasonable computation times of a real nuclei field distribution. The coupling can be as simple as a 1D steady solution model or can be quite involved to describe full 3D unsteady flows. The thrusts predicted by the different methods agreed well for low void fractions, but deviates at high void factions. The prediction indicated a substantial thrust augmentation at high void fractions. We have also conducted validation studies of the numerical predictions using experiments with a twodimensional bubble-augmented propulsion test setup at DYNAFLOW. Nozzle geometry was varied and bubbles were injected into the flow with injectors at various locations including a flush mounted sintered metal injector. The bubble injection from wall ports indicated a potential issue of bubble mixing in the full scale system. The bubble size distribution and void fraction were measured by laser light attenuation method, by ABS ACOUSTIC BUBBLE SPECTROMETER, and by image analysis for low void fraction cases. The most probable bubble size generated through the sintered plate injector was 100~200 μm. The void fraction we tested was up to 20%, but we will cover higher void fraction ranges in the near future. The two dimension 14

flow test setup turned out to be an excellent visualization tool for this study. The velocity and pressure fields were measured, and thrust augmentation was deduced from these measurements. The trend of increased exit momentum at higher void fraction was obtained both through predictions and measurements. Bubble velocities were measured by PIV and the mixture velocity was measured by a Pitot tube. These two velocities agreed well with their corresponding predictions. Actual thrust augmentations were measured, but the thrust gain appeared to saturate in the experiments probably due to bubble coalescence as the bubble becomes too large. In the future, we will improve the instrumentation for better accuracy, and move to a 3D axisymmetric nozzle geometry with higher void fraction ranges both in the experiment and in the computation. In the framework of this project, such experiments would provide a key contribution to our understanding of the successes and failures of previous prototypes, in addition to providing critical support for the design of next generation waterjet devices. ACKNOWLEDGEMENT This work was supported by the Office of Naval Research under the contract N00014-07-C-0427, monitored by Dr. Ki-Han Kim. 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. 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. 2 nd 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., Tanguay, M., Lorraine, G., Acoustic Measurements of Bubbles in Biological Tissue, WIMRC Cavitation: Turbo-machinery and medical applications, Warwick University, 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 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. 15

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