Proceedigs of IMECE008 008 ASME Iteratioal Mechaical Egieerig Cogress ad Expositio October 31-November 6, 008, Bosto, Massachusetts, USA IMECE008-68609 BUBBLE RISE AND DEPARTURE FROM A VISCOUS LIQUID FREE SURFACE Mehra Mohammadi Farhagi Mohammad Passadideh-Fard Graduate Studet Assistat Professor Departmet of Mechaical Egieerig, Departmet of Mechaical Egieerig, Ferdowsi Uiversity of Mashhad, Ira Ferdowsi Uiversity of Mashhad, Ira farhagi.mehra@gmail.com mpfard@um.ac.ir Behtash Bagheria Graduate Studet Departmet of Mechaical Egieerig, Ferdowsi Uiversity of Mashhad, Ira Behtash60@gmail.com ABSTRACT I this paper, the risig of a sigle bubble i a viscous liquid ad its departure from the free-surface are simulated usig a trasiet D/axisymmetric model. To predict the shape of the bubble deformatio, the Navier-Stokes equatios i additio to a advectio equatio for liquid volume fractio are solved. A modified Volume-of-Fluid (VOF techique based o Yougs algorithm is used to track the bubble deformatio. To validate the model, the results of simulatios for termial rise velocity ad bubble shape are compared with those of the experimets. Next, the effect of differet parameters such as iitial bubble radius, chael height, ad liquid viscosity ad surface tesio o the shape ad rise velocity of the bubble is ivestigated. Fially, the iteractio of the bubble with the free surface durig its departure from the liquid is simulated, ad the results are compared qualitatively with experimetal photographs. INTRODUCTION The dyamic behavior of two-phase flows is of great importace i various processes ragig from egieerig applicatios to evirometal pheomea. The presece of air bubbles i hydrodyamic systems ofte reveals may udesirable effects such as early erosio, loss of efficiecy or flow irregularities. May idustrial applicatios ivolve two-phase flows with or without mass (ad/or heat trasfer. Examples i Chemical Egieerig iclude bubble colums, loop reactors, agitated stirred reactors, flotatio, or fermetatio reactors. For the desig of efficiet two-phase reactors detailed kowledge of, say, Bubble sizes ad shapes, slip velocities, iteral circulatio, Swarm behaviors are of fudametal importace. Numerical studies of two-phase flows are carried out to aalyze the iterface behavior of oe air bubble movig oisig i a liquid. I the past decade a umber of techiques, each with their ow particular advatages ad disadvatages, have bee developed to simulate complex multi-fluid flow problems. Level set methods [1-5] are desiged to miimize the umerical diffusio hamperig shock-capturig methods ad typically defie the iterface as the zero level set of a distace fuctio from the iterface. The advectio of this distace fuctio evolves with the local fluid velocity. Level set methods are coceptually simple ad relatively easy to implemet. Whe the iterface is sigificatly deformed, level set methods suffer from loss of mass (volume ad hece loss of accuracy. A well-kow method for trackig the free surface of a liquid is Volume-of-Fluid (VOF techique [6] where the computatioal domai is characterized by a scalar color fuctio f whose value is oe for a cell full of liquid ad zero for a empty cell. A cell with a value betwee zero ad oe idicates a free-surface cell. I additio to the value of the color fuctio the iterface orietatio eeds to be determied, which follows from the gradiet of the f fuctio. Roughly two importat classes of VOF methods ca be distiguished with respect to the represetatio of the iterface, amely simple lie iterface calculatio (SLIC ad piecewise liear iterface calculatio (PLIC. Earlier works with VOF were geerally based o the SLIC algorithm itroduced by Noh ad 1 Copyright 0xx by ASME
Woodward [7] ad the door-acceptor algorithm published by Hirt ad Nichols [8]. Moder VOF techiques iclude the PLIC method of Yougs [9]. The accuracy ad capabilities of the older VOF algorithm such as the Hirt ad Nicholas VOF method were studied by Rudma [10]. A drawback of these VOF methods for advectig gas bubbles is the so-called artificial (or umerical coalescece of gas bubbles which occurs whe their mutual distaces is less tha the size of the computatioal cell. Frot trackig methods [11-13] make use of markers (for istace triagles, coected to a set of poits, to track the iterface whereas a fixed or Euleria grid is used to solve the Navier-stokes equatios. This method is extremely accurate but also rather complex to implemet due to the fact that dyamic re-meshig of the Lagragia iterface mesh is required ad mappig of the Lagragia data oto the Euleria mesh has to be carried out. Difficulties arise whe multiple iterfaces iteract where all require a proper sub-grid model. Cotrary to most other methods, the automatic mergig of iterfaces does ot occur i frot trackig techiques due to the fact that a separate mesh is used to track the iterface. This property is advatageous i case swarm effects i dispersed flows eed to be studied. A 3D frot trackig method was used by Va Sit Aalad et al. [14] to simulate a sigle bubble risig i water. The frot trackig algorithm predicted reasoably well the rise velocity ad aspect ratio of a sigle air bubble risig i water for diameters i the rage of 1 mm to 7 mm. Computatioal studies cocerig the motio of a bubble ear a ifiite free surface are more abudat, though agai ot so pletiful as with the rigid boudary problem. Early studies i this area datig back to the Secod World War are cocered with the motio of uderwater explosio bubbles: for example, the work by Herrig [15] ad subsequet umerical study by Taylor [16]. These ecessarily model the bubble as spherical ad so the effects of buoyacy ad earby boudaries are to displace the bubble rather tha deform its shape. Oe early cosideratio of the o-spherical motio of the bubble was performed by Leoir [17], who employed a simple boudary itegral techique to model the motio of the bubble. Blake ad Gibso [18] employed a approximate itegral equatio techique to model the motio of the bubble ad free surface. Durig expasio ad early collapse, the calculated motio was show to be i good agreemet with experimets. However, upo formatio of the liquid jet withi the bubble, this model fails ad calculatios are ceased. Later work by Kucera ad Blake [19] has show such approximate methods to compare well with boudary itegral studies for bubbles ot too close to a boudary, with a reasoable agreemet possible dow to stadoff distaces of about two maximum bubble radii up util the time of jet formatio. More recet calculatios usig these techiques may be foud i Referece [0]. I this study, the iterface of a bubble i a liquid durig its rise ad departure from the free-surface are simulated usig a trasiet D/axisymmetric model. A modified Volume-of-Fluid (VOF techique based o Yougs algorithm is used to track the bubble deformatio. To validate the model, umerical results are compared with those of the experimets for termial rise velocity ad bubble shape. The effect of differet parameters such as iitial bubble radius, chael height, ad liquid viscosity ad surface tesio o the shape ad rise velocity of the bubble is ivestigated. NOMENCLATURE de D f g V p R t V Δρ Δp equivalet bubble diameter bubble diameter fractioal amout of liquid gravitatioal acceleratio Velocity pressure bubble radius time termial bubble rise velocity desity differece pressure differece Greek letters µ Viscosity ρ desity surface tesio τ viscous stress tesor Dimesioless 4 M (= Δρ g μl Morto umber 3 ρl Δρ gd Eo (= e Eotvos umber Re (= ρlutde Reyolds umber μ l μv Ca ( = Capillary umber ΔρgR Bo ( = t Bod umber Subscripts ad superscripts L liquid phase G gas phase NUMERICAL METHOD The mai issue regardig the developed model is the advectio of the bubble iterface usig VOF method. I this sectio, we preset a brief accout of the umerical method. I modelig we ca reasoably assume that the multi-fluid system studied i this paper is a isothermal system of two Newtoia, Copyright 0xx by ASME
icompressible ad immiscible fluids. The flow goverig equatios are: r V r = 0 (1 r V r 1 r 1 r t 1 r ( + (VV = p + τ + Fb ρ ρ ρ where V is the velocity vector, p is the pressure, ρ is the desity ad F b represets body forces actig o the fluid. The bubble iterface is advected usig VOF method by meas of a scalar field f whose value is uity i the liquid phase ad zero i the gas. Whe a cell is partially filled with liquid, f will have a value betwee zero ad oe. 1 i liquid f = > 0, < 1 at the liquid-gas iterface (3 0 i gas The discotiuity i f is propagatig through the computatioal domai accordig to: df f f = + V f = 0 ( exact = (V. f (4 dt Although the velocity field is divergece free, the term ( r V r has a order of O(ε i umerical solutio. Therefore, i order to icrease the accuracy of the umerical solutio, Eq. 4 is used i the coservative form as where f ( umerical = (V. f ( V f = (Vf f f ( exact = ( umerical + ( V f (6 For the advectio of volume fractio f based o Eq. 4, differet methods have bee developed such as SLIC, Hirt- Nichols ad Yougs PLIC [8]. The reported literature o the simulatio of free-surface flows reveals that Hirt-Nichols method has bee used by may researchers. I this study, however, we used Yougs method [6, 8, 9], which is a more accurate techique. Assumig the iitial distributio of f to be give, velocity ad pressure are calculated i each time step by the followig procedure. The f advectio begis by defiig a itermediate value of f, f = f δ t ( Vf ~ The it is completed with a divergece correctio ~ = + + 1 f f δ t( V f (5 (7 (8 A sigle set of equatios is solved for both phases, therefore, desity ad viscosity of the mixture are calculated accordig to: ρ = fρl + (1 f ρg, μ = fμl + (1 f μg where subscripts L ad G deote the liquid ad gas, respectively, μ is the viscosity. New velocity field is calculated accordig to the two-step time projectio method as follows. First, a itermediate velocity is obtaied, V ~ V r 1 r t r 1 r = (VV + τ + g + Fb δ t ρ ρ (9 (10 The cotiuum surface force (CSF method [6, 1] is used to model surface tesio as a body force ( F b that acts oly o iterfacial cells. Pressure Poisso equatio is the solved to obtai the pressure field, r V ~ r r 1 r [ p 1 + ] = ρ (11 δ t Next, ew time velocities are calculated by cosiderig the pressure field implicitly, r V ~ r V + 1 1 r = p + 1 δ t ρ (1 RESULTS As a first step, the model was subjected to several tests i order to validate its results. The first case cosidered was that of a sigle bubble durig its rise i a liquid; a case for which experimetal results are available i terms of termial bubble rise velocity agaist its diameter. The measured data performed by Grace [] for air bubbles i water is give as a diagram show i Fig. 1. The default material properties used i the simulatios are give i Table 1.. properties water air desity ρ l =998. kg/m 3 ρ a =1.1 kg/m 3 viscosity μ l =100 10-6 kg/(m.s μ a =18.4 10-6 kg/(m.s surface tesio =0.073 N/m Tab. 1: Material properties A axisymmetric coordiate system was used i the model to simulate the deformatio of the bubbles risig i a vertical tube. The tube diameter was assumed to be aroud four times as that of the bubble diameter i order to reduce the wall ifluece o bubble movemet. Bubbles with diameteaged from 0.8 mm to 10 mm were simulated; larger oes ted to break up before they reach their termial velocity. 3 Copyright 0xx by ASME
uder cosideratio. U t represets the termial rise velocity of the bubble. Fig. 1: A compariso betwee the results of simulatios with those of the experimets [] for termial rise velocity agaist iitial bubble diameter. Based o the experimets, the rise velocity should be located i the regio surrouded by the solid lies. The results of the model, preseted i Fig. 1, are located i the same regio where observed by experimets. The upper boudary of this regio correspods to pure systems, while the lower curve belogs to cotamiated systems. As see from the figure, icreasig the bubble diameter icreases the rise velocity up to a certai limit after which the bubble starts to oscillate. I this regime, the rise velocity remais early costat. Addig further to the bubble diameter chages the deformatio behavior to the spherical cap regime where the rise velocity agai icreases with diameter. For the bubbles smaller tha the 0.5 mm there is a icreasig deviatio of simulated to measured velocities, which occurs maily because of the so-called parasitic currets. These currets are due to iaccuracies i the calculatio of surface tesio forces, i particular because of errors i the iterfacial ormal ad curvature. MODEL VALIDATION Grace [3] has aalyzed a large body of experimetal data o shapes ad rise velocities of bubbles i quiescet viscous liquids ad has show that this data ca be codesed ito oe diagram, provided that a appropriate set of dimesioless umbers is used. A represetatio of the Grace diagram [3] is show i Fig. where dimesioless umbers Morto (M, Eötvös (Eo, ad Reyolds (Re are give by 4 M= Δρ g μl Δρ gd, Eo= e, Re= ρlutde (13 3 ρl μl where the equivalet diameter d e is defied as the diameter of a spherical bubble with the same volume as that of the bubble Fig. : Grace bubble diagram [3] for the shape ad termial rise velocity of gas bubbles i quiescet viscous liquids. Poits A, B, ad C show the cases for which the simulatio results were compared with measuremets as give i Table. I Table, the values of the selected Morto ad Eötvös umbers are give for simulatios of bubbles i differet regimes of bubble deformatio. I this table, Re exp ad Re model represet the bubble Reyolds umbers obtaied from the Grace diagram ad calculated from the model, respectively. Bubble regime M Eo Re Exp. Re Model Poit (Fig. spherical 1.4 0.01 1 1. A wobblig 14.5 1 10-9 100 00 B skirted 14.56 1 15 18 C Tab. : Morto (M ad Eötvös (Eo umbers for simulatios of bubbles i differet regimes accordig to Grace bubble diagram [3]. Effect of Importat Parameters I this sectio, the effect of differet parameters such as tube size, surface tesio ad viscosity o the shape ad rise velocity of the bubble is ivestigated. I Fig. 3, the effect of tube diameter o the bubble shape ad termial rise velocity is show. Velocity distributios alog with flow streamlies at a time istat are also displayed i the figure. Free-slip boudary coditios were applied at all cofiig walls. As see from the figure, icreasig the tube diameter icreases the rise velocity 4 Copyright 0xx by ASME
characterized by Reyolds umber. The data used for this simulatio is give i Table 3. The desity ad viscosity ratio i this simulatio is believed to be sufficietly high to mimic gas liquid systems with sufficiet accuracy ad much higher tha the ratio used by Sabisch et al. [4]. capillary wall begis to overcome the icrease i the buoyacy force with icreasig the bubble size. 00 Experimet Simulatio Dimesioless Termial Velocity 150 100 50 0 0 0. 0.4 0.6 0.8 1 1. Fig. 3: The effect of tube diameter o the bubble shape ad rise velocity parameters liquid gas desity ρ l =1000 kg/m 3 ρ g =10 kg/m 3 viscosity μ l =0.1 kg/(m.s μ a =0.001 kg/(m.s surface tesio =0.0673 N/m, - bubble diameter - 0.01 m Tab. 3: Data used for the simulatio to study the effect of tube diameter. The effect of the bubble size o termial velocity is show i Fig. 4. The rise velocity is made dimesioless usig Δ ρgr t / μl, where R t is the tube radius; the bubble radius is also made dimesioless usig the tube radius as R b /R t. This figure shows the compariso betwee umerical results ad measuremets [5]. The experimets were performed for systems with small values of Ca/Bo, where Ca μv ad Bo are Capillary umber ( Ca = ad Bod umber ΔρgR ( Bo = t, respectively. As see from the figure, the termial velocity icreases with bubble radius upto a maximum at a itermediate bubble radius of R b /R t =0.5. After this poit, the termial velocity decreases with further icrease of the bubble radius. This is because the largeetardig effect of the Dimesioless Radius ( R b /R t Fig. 4: Dimesioless rise velocity agaist dimesioless bubble radius. It is well kow that surface tesio causes a excess pressure iside a bubble give by the Laplace equatio as Δ p = R b for a spherical shape, where is the surface tesio coefficiet ad R b the bubble radius. Figure 5 shows the effect of surface tesio o bubble rise velocity. The data used i this simulatio are those give i Table 3 except for surface tesio which varied from 0.03 N/m to 0.15 N/m. From the figure, it ca be clearly see that the bubble rise velocity icreases with liquid surface tesio. Bubble rise velocity (m/s 0.3 0.5 0. 0.15 0.1 0.05 0 0 0.05 0.1 0.15 Surface tesio (N/m Fig. 5: The variatio of bubble rise velocity versus surface tesio for a bubble of 0.01 m i diameter. 5 Copyright 0xx by ASME
Figure 6 displays the effect of viscosity o termial rise velocity for a bubble with a iitial diameter of 0.01 m. For a give bubble size, the bubble rise velocity is reduced as viscosity ratio is icreased. This result was expected because icreasig the viscosity ratio decreases the iterfacial motio due to viscous forces. The relative reductio i termial rise velocity from the maximum to the miimum value also becomes less proouced as the viscosity ratio icreases. 0.3 0.5 Bubble rise velocity (m/s 0. 0.15 0.1 0.05 0 0 1 3 4 5 6 Viscosity ratio ( μ l / μ g Fig. 6: The variatio of bubble rise velocity versus viscosity ratio for a bubble of 0.01 m i diameter. Bubble Departure from the Free-Surface I this sectio, the effect of the bubble departure from the free surface is ivestigated. Whe gas bubbles rise through a pool of liquid ad approach the free surface, the various violet motios associated with the free-surface breakup geerate droplets that may persist i the surroudig gas to costitute a spray. The spray formatio by this pheomeo (bubblig at the free surface is show i Fig. 7 where the free-surface dyamic behavior after a sigle bubble departure from the liquid is show usig both umerical model ad experimets [6]. A schematic of the importat processes durig this pheomeo is show i Fig. 8. A thi film of liquid is formed o top of the bubble before it leaves the free surface. The breakup ad disitegratio of this film create a spray of droplets with sizes i the order of oe teth of the bubble diameter. The surface waves formed durig the free-surface rise propagate iward (toward the ceter ad outward away from the bubble. A upward jet is formed at the ceter of the disruptio; the disitegratio of this jet also creates small droplets. I geeral, the droplets formed from the disitegratio of this jet are bigger i size tha those formed from the breakup of the thi film. The material parameters used for the simulatios i this sectio are give i Table 1. Fig. 7: Qualitative compariso betwee calculated images ad experimetal photographs [6] for the deformatio of the free surface ad formatio of a spray durig a sigle bubble departure. Fig. 8: Stages of a bubble breakig through a free surface The importat effects of the bubble departure o the free surface deformatio appear after the bubble left the liquid. As see from Fig. 7, the liquid rises at the departure poit; this pheomeo causes a low speed liquid spray risig from the free surface. The height of the spray is affected by differet parameters amely the tube diameter, the iitial bubble radius ad the distace betwee the iitial bubble locatio ad the free-surface level. Figure 9 shows the effect of the tube diameter (made dimesioless by the bubble diameter D o the height of the free surface spray. As see from the figure, icreasig the tube diameter icreases the height of the liquid 6 Copyright 0xx by ASME
risig above the free surface. This result was expected because icreasig the tube diameter icreases the bubble rise velocity which i tur causes a higher spike. The Height of the Liquid Spray (mm 16 15 14 13 1 11 10 9 8 1 3 4 5 6 Dimesioless Tube Diameter Fig. 9: The variatio of the height of the liquid spike versus dimesioless tube diameter. Figure 10 shows the effect of bubble radius o the height of the spray. From this figure, it ca be see that the height is a liearly icreasig fuctio of the bubble radius. The iitial bubble ceter for the simulatios show i this figure was located at a distace beeath the free surface equal to four times as much as the bubble radius. 17 CONCLUSION I this paper, a axisymmetric VOF method was used to simulate the rise ad iteractio of gas bubbles i a viscous liquid. The model was validated by a compariso betwee umerical results with available measuremets for the bubble deformatio ad velocity durig its rise i a liquid. The effect of icreasig diameter o bubble rise velocity was also ivestigated ad compared well with that of the experimet. Next, we ivestigated the effect of importat parameters o the bubble rise velocity. A bubble movig i a arrower tube (smaller diameter was foud to reach a smalleise velocity. Surface tesio ad viscosity had adverse effects o the bubble movemet. While icreasig surface tesio raised the bubble rise velocity, icreasig liquid viscosity had a opposite effect. Fially, the effect of the bubble departure o the liquid free surface was ivestigated. The three parameters cotrollig the height of the liquid above the free surface (spray formatio were foud to be the tube diameter; the bubble radius; ad the distace betwee the iitial bubble ceter ad the free-surface level. Icreasig the tube diameter ad the bubble radius icreased the height of the spray. REFERENCES [1] Sussma, M. Smereka, P. ad Osher, S., A level set approach for computig solutios to icompressible two-phase flow, J. Comput. Phys, Vol. 114, pp. 146-159, 1994. [] Sethia, J.A., Level Set Methods, Cambridge Uiversity Press, Cambridge, UK, 1996. [3] Sussma, M., ad Smereka, P., Axisymmetric free boudary problems, J. Fluid Mech., Vol. 341, pp. 69 94, 1997. The Height of the Liquid Spray (mm 16 15 14 13 1 11 10 [4] Sussma, M, ad Fatemi, E., A efficiet iterface-preservig level set redistacig algorithm ad its applicatio to iterfacial icompressible fluid flow, SIAM J. Sci. Comput., Vol.0, pp.1165-1191, 1999. [5] Fedkiw, R., ad Osher, S., Level-set methods: a overview ad some recet results, J. Comput. Phys, Vol. 169, p. 463, 001. [6] Pasadideh-Fard, M. ad Roohi, E. A Computatioal Model for Cavitatio Usig Volume-of-Fluid Method, 14 th Aual Cof. of CFD Caada, Kigsto, 006. 9 3 4 5 6 7 8 Bubble Radius (mm [7] Noh, WF, ad Woodward, PR., SLIC (Simple Lie Iterface Calculatio method, I: Va Voore AI, Zadberge PJ, eds. Lecture Notes iphysics. Vol. 59. Berli: Spriger-Verlag; 330, 1976. Fig. 10: The variatio of the height of the liquid spike versus bubble radius. [8] Hirt, C.W., ad Nichols, B.D., Volume of fluid (VOF method for the dyamics of free boudaries, J. Comput. Phys, Vol. 39, p. 01, 1981. 7 Copyright 0xx by ASME
[9] Yougs, DL., Time-depedet multi-material flow with large fluid distortio, I: Morto KW, Biaes MJ, eds. Numerical methods for fluid dyamics. New York, NY: Academic Press, pp. 73-85, 198. [10] Rudma, M., Volume-trackig methods for iterfacial flow calculatios, Iteratioal joural of umerical methods i fluids, Vol. 4, pp. 671 691, 1997. [11] Uverdi, SO, ad Tryggvaso G. A frot-trackig method for viscous, icompressible multi-fluid flows, J. Comput. Phys, Vol. 100, pp. 5-37, 199. [1] Esmaeeli, A., ad Tryggvaso, G., Direct umerical simulatio of bubble flows, Part I. Low Reyolds umber arrays. J. Fluid Mech., Vol. 377, pp. 313 345, 1998. [13] Esmaeeli, A., ad Tryggvaso, G., Direct umerical simulatio of bubble flows, Part II. Moderate Reyolds umber arrays. J. Fluid Mech., Vol. 385, pp. 35 358, 1998b. [14] Va Sit Aalad, M., Dijkhuize, W., Dee, N. G., ad Kuipers, J. A. M., Numerical simulatio of behavior of gas bubbles usig a 3- D frot-trackig method, AIChE J, Vol. 5, pp. 99 110, 006. [1] Bussma M., Mosthghimi J., ad Chadra S., "O a Three- Dimesioal Volume Trackig Model of Droplet Impact", Phys. Fluid, Vol. 11, p. 1406, 1999. [] Clift, R, ad Grace, J.R., ad Weber, M., 1978, Bubbles, drops ad particles, Academic Press, New York. [3]Grace, JR., Shapes ad velocities of bubbles risig i ifiite liquids, Tras ICHemE.:51:116-10, 1973. [4] Sabisch, W., Worer, M., Grotzbach, G., Cacuci, D.G. Driedimesioale umerische Simulatio vo aufsteigede Eizelblase ud Blaseschwarme mit eier Volume-of Fluid- Methode. Chemie Igeieur Techik 73, 368 373, 001. [5] Matroushi, Eisa A., Ieteractio ad coalescece of bubbles ad drops movig through a tube, Ph.D. Pesylvaia State Uiversity, 000. [6] Blachard, D.C., The electrificatio of the atmosphere by particles from bubbles i the sea, Progr. i Oceaography, Vol. 1, p. 7, 1963. [15] Herrig, C. Theory of the pulsatios of the gas bubble produced by a uderwater explosio, I: Hartma GK, Hill EG, editors. Uderwater explosio research, vol. II. Washigto, DC: Office of Naval Research; 1949. [16] Taylor, GI. Vertical motio of a spherical bubble ad the pressure surroudig it, I: Hartma GK, Hill EG, editors. Uderwater explosio research, vol. II. Washigto, DC: Office of Naval Research; 194. [17] Leoir, M. Calcul ume rique de l implosio d ue bulle de cavitatio au voisiage d ue paroi ou d ue surface libre. J Me c; 15(5:75 51, 1976. [18] Blake, JR, ad Gibso DC. Growth ad collapse of a vapour cavity ear a free surface, J. Fluid Mech., Vol. 111, pp. 13 40, 1981. [19] Kucera, A. ad Blake, JR. Approximate methods for modelig cavitatio bubbles ear boudaries, Bull Aust Math Soc, Vol.41, pp.1 44, 1990. [0] Cox, E. The source sigature due to the close iteractio of marie seismic airgus, PhD Thesis, The Uiversity of Birmigham, 003. 8 Copyright 0xx by ASME