INVESTIGATION OF VOID FRACTIONS AND BUBBLE PLUME PROPERTIES UNDER BREAKING WAVES

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Proceedigs of he 6 h Ieraioal Coferece o he Applicaio of Physical Modellig i Coasal ad Por Egieerig ad Sciece (Coaslab16) Oawa Caada May 10-1 016 Copyrigh : Creaive Commos CC BY-NC-ND 4.

H WAI PENGZHI LIN 1 Deparme of Civil ad Eviromeal Egieerig The Hog Kog Polyechic Uiversiy Hog Kog aglia.scu@16.com Deparme of Civil ad Eviromeal Egieerig The Hog Kog Polyechic Uiversiy Hog Kog oyx.

1 Proceedigs of he 6 h Ieraioal Coferece o he Applicaio of Physical Modellig i Coasal ad Por Egieerig ad Sciece (Coaslab16) Oawa Caada May Copyrigh : Creaive Commos CC BY-NC-ND 4.0 INVESTIGATION OF VOID FRACTIONS AND BUBBLE PLUME PROPERTIES UNDER BREAKING WAVES LIAN TANG 1 ONYX W.H WAI PENGZHI LIN 1 Deparme of Civil ad Eviromeal Egieerig The Hog Kog Polyechic Uiversiy Hog Kog aglia.scu@16.com Deparme of Civil ad Eviromeal Egieerig The Hog Kog Polyechic Uiversiy Hog Kog oyx.wai@polyu.edu.hk Sae Key Laboraory of Hydraulics ad Mouai River Egieerig Sichua Uiversiy Chegdu Chia cvelipz@scu.edu.c ABSTRACT Numerical ivesigaio of urbule flow field uder breakig waves is sill of grea challeges especially i he high iesiy air bubble regio maily due o he poor kowledge of bubble plume properies durig wave breakig. I his sudy he air eraime model based o laboraory measured average air bubble size specrum is icorporaed io a wodimesioal umerical model (NEWFLUME) o ivesigae bubble plume evoluios uder breakig waves i he surf zoe. The model solves he Reyolds equaios for urbule flows ad employs he volume-of-fluid (VOF) mehod o rack free surfaces. The bubble eraime process is simulaed by usig umerically compued isaaeous flow field ad he experimeally deermied model of bubble size specrum. The simulaed resuls of breakig wave heigh disribuio velociy profiles ad he urbule kieic eergy are verified by he experimeal daa. The void fracios ad iegral properies of bubble plume are examied ad discussed. KEWORDS: breakig wave; bubble plume; void fracios 1 INTRODUCTION I he surf zoe wave breakig geeraes high iesiy of urbulece ad erais large volumes of air bubbles ha ca exed 0.5m or more below he surface ad coribue sigificaly o he dyamics urbule flow field (Deae ad Sokes 00). The eraied air bubbles play a impora role i wave eergy dissipaio (Rapp ad Melville 1990; Iafrai 011) ad alerig he urbulece ad vorex field (Waaabe e al. 005). To sudy hese effecs a beer udersadig of he void fracios ad bubble plume properies uder breakig waves is impora. Several laboraory experimes sudies o bubble plume uder breakig waves i he surf zoe have bee coduced. Deae ad Sokes (00) measured he bubble size disribuio uder he laboraory-scale iside breakig waves. They foud ha he bubbles larger ha abou 1 mm have a iiially bubble size desiy proporioal o he bubble radius wih powerlaw of -10/ while he smaller bubbles wih -/. Cox ad Shi (00) carried ou laboraory sudies o void fracio flucuaios i he regio close o breakig wave fro. They showed ha he maximum esemble-averaged void fracios were bewee 15 ad 0%. Mori ad Kakuo (008) measured he void fracio ad flow velociy uder laboraory-scale breakig waves. They foud ha he void fracio shows liear depedece o he urbulece iesiy. Blekisopp ad Chapli (007) measured he ime-depede disribuio of void fracio i laboraory. The resuls suggesed ha air eraime ad splash geeraio accou for a leas 6.5%~14% of he oal eergy dissipaio ad he perceage icreases wih he iesiy of he breakig wave. Lim e al. (015) measured he velociies ad void fracio uder a useady plugig breakig i laboraory. They foud ha he void fracio shows liear growh ad expoeial decay rae. Besides laboraory experimes umerical modellig of bubble dyamics uder breakig waves is sill limied. Terrill e al. (001) proposed a umerical model based o a advecio-diffusio equaio o describe populaios i he surf zoe wihou cosiderig bubble eraime processes. Moraga e al. (008) developed a sub-grid model o deermie he bubble source due o air eraime uder breakig waves. Shi e al. (010) preseed a D model based o mixure heory. I heir model hey relaed he air eraime o urbulece producio a he air waer ierface. Ma e al. (011) proposed a air eraime model based o he assumpio ha he oal eergy required for bubble formaio is liearly proporioal o he urbulece dissipaio rae ad he icorporaed i i o a VOF model TRUCHAS (Rider ad Kohe 1998) o ivesigae 1

2 he bubble plume dyamics uder breakig waves. Their model could predic he void fracios ad bubble plume evoluio well uder spillig breakig wave bu showed overesimaio of he sream wise velociies above he sill waer level. I his sudy he experimeally based air eraime model developed by Ma e al. (011) is icorporaed io he model NEWFLUME which is developed by Li ad Xu (006) for geeral urbule surface flows simulaios o ivesigae he void fracios ad bubble plume characerisics uder breakig waves i he surf zoe. The spillig regular wave ru-up a 1/5 slope is simulaed by he prese model ad he umerical resuls are verified by he measureme daa of Cox ad Shi (00). MODEL DESCRIPTION I he prese work we cosider he air-waer mixure as a mixure phase ad solve he mixure equaios derived by Buscaglia e al (00) for urbule flow field bu add he addiioal shear sress erm i he momeum equaio o represe he momeum dissipaio which caused by bubble-liquid ieracios. The goverig equaios are give as: ui 0 x ui ui 1 p 1 i TDi u gi x x x i 0 i 0 0 where u i is he mixure phase velociy of i-h compoe i 1 is he ime is he mixure desiy ad p is mixure phase pressure respecively. 0 is he referece desiy for icompressible liquid phase i is he Reyolds shear sress TDi is he addiioal shear sress gi The mixure desiy is calculaed by is he i-h compoe of he graviaioal acceleraio. 1 a a w a where a is he air desiy ad w is he waer desiy a is he void fracio of he air bubbles. The urbulece closure is accomplished by he oliear k model (Li ad Liu 1998a) he equaios are give by: k k k ui u u iu x x k x x ui u C1 i C x x x k x k where k is urbule kieic eergy ad is he urbulece dissipaio rae is he kiemaic molecular viscosiy ad is he eddy viscosiy. Cd k C1 ad C are empirical coefficies (Rodi 1980) Cd 0.09 k C C 1.9 (6) The bubble plume populaio is calculaed by solvig he bubble umber desiy equaio which is give by The void fracio ca be calculaed by N N N u E b b b b b x x b x 4 r N N a b b 1 where N is he bubble specrum for he -h bubble group N is he oal bubble group umber b b is he Schmid umber of gas wih b 1.0 (Carrica e al. 1998). ub is he bubble velociy for he -h bubble group which wih characerisic radius r b E accous for he bubble eraime source from he free surface. The bubble velociy is calculaed by b u u u b m r i which ur is he bubble rise velociy for he -h bubble group which is relaed o he bubble diameer d b : (1) () () (4) (5) (7) (8) (9)

u r db g a 1 db 0.068mm Falvery 1980 9 w 1 db g a 1 0.068mm db 0.8mm Comole 1979 18 w 0.5gdb.14 0.8mm db 10.0mm Comole 1979 wdb uar gdb db 10.

3 u r db g a 1 db 0.068mm Falvery w 1 db g a mm db 0.8mm Comole w 0.5gdb mm db 10.0mm Comole 1979 wdb uar gdb db 10. 0mm Falvery 1980 The bubble eraime source E is calculaed by he formulaio developed by Ma e al. (011) which relaed he bubble b eraime wih urbulece dissipaio rae : E b 1 c b f rb rb 1 4 l r f r r N b b b i1 where c b is bubble eraime coefficie is surface esio f rb is he bubble size specrum which is deermied by experime measured daa of Deae ad Sokes (00). For bubbles larger ha 1 mm urbule fragmeaio deermies bubble size disribuio resulig i a iiially bubble desiy proporioal o he bubble radius o he power of -10/. Smaller bubbles are creaed by e ad drop impac o he wave face wih a -/ powerlaw scalig which gives he bubble size specrum as: RESULTS AND DISCUSSIONS f r 10 b b b b b r b is he widh of each bubble group ad r whe r 1.0mm r whe r 1.0mm A spillig breakig wave ru-up o a 1/5 slope (Cox ad Shi 00) is simulaed by he preseed model. The simulaio seup is show i Figure 1. The sill waer deph i he wave flume is h=0.51m he wave rai has a wave heigh H=0.11m wih wave period T=.0s. The measured breakig poi is locaed a x b =18.07m. Three measureme cross-secios are locaed a (a): x=18.81m (b): x=18.94m ad (c): x=19.17m respecively. (10) (11) (1) Figure.1 Skech of he experime seup ad he compuaio domai. I he umerical simulaios he compuaioal domai is 1m log by 0.7m high wih he slopig beach oe 6.0m from he ile boudary. The compuaioal domai is discreized io a grids sysem wih uiform grid x0.0mi he x direcio ad o-uiform grid i he z direcio wih he miimum grid z0.006 mear he free surface. The ime sep is auomaically adused durig he compuaio o saisfy he compuaio sabiliy ad o save he compuaio ime..1 Compariso of experimeal daa ad umerical resuls I his secio comparisos bewee he experimeal daa ad umerical resuls are preseed for he wave heigh mea sream wise velociies ad urbulece kieic iesiy. Figure shows he compariso of wave heigh disribuio he simulaed wave heighs are i good agreeme wih he experimeal daa alhough he prediced maximum wave heigh is a lile smaller ha measureme. Figure shows he comparisos of mea velociies a hree verical cross-secios i he surf zoe. A each locaio comparisos are made a four differe elevaios z=0.5cm -0.5cm -1.5cm -4.5cm from he sill

4 waer level. Though he simulaed mea velociies are larger ha he measureme i he lower regio z=-4.5cm he overall agreeme bewee he umerical resuls ad experimeal daa is accepable especially he maximum velociies above he sill waer level which show good agreeme wih he measured daa. Figure. Comparisos of measured (do) ad simulaed (solid lie) wave heighs alog he flume. (a) (b) (c) Figure.. Comparisos of horizoal mea velociies bewee experimeal daa (lef paels) ad umerical resuls (righ paels) a secios (a) x=18.81m (b) x=18.94m ad (c) x=19.17m across he waer deph z=0.5cm ( ) z=-0.5cm (-.-.) z=-1.5cm (---) z=-4.5cm ( ). Due o lack of measured urbulece kieic iesiy he simulaed urbulece iesiy is compared wih measured 4

5 horizoal velociy variace which is show i Figure 4. The simulaed urbulece kieic iesiy shows muliple relaio wih he average value of he horizoal velociy variace. Besides he decay reds bewee he urbulece iesiy ad he horizoal velociy variace are cosise. The comparisos wih he measured daa idicae ha he prese model is capable of simulaig he urbule flow field i he aeraio regio uder breakig waves. Figure.4. Comparisos of measured horizoal velociy variace ad simulaed urbulece kieic iesiy (solid lie) a secio b (x=18.94m) a differe verical locaios.. Iegral properies of bubble plume uder breakig waves The capabiliy of he proposed model for predicig urbule flow field uder breakig waves has bee esed ad verified. I his followig he model is applied o ivesigae he iegral properies of bubble plume. The characerisics of bubble plume ca be described by he oal volume of air eraied V variaio of he bubble plume area A ad he poeial eergy of he eraied bubble plume E. They ca be esimaed by V E where ' is void fracios wih boudaries of % ad 97% z is he isaaeous verical disace from grid ceer o he free surface. The iegraio regio is coduced i 17.0m x 1.0m ad 0.8m z 0.66m which covers he whole bubble plume evoluios area. The grids i his area wih void fracio ' bewee % ad 97% will be recorded o calculae bubble plume area A ad o iegrae he above equaios. Comparisos of void fracio disribuios a hree verical cross-secios are show i Figure 5. The overall umerical resuls agree well wih he measured daa wih he prediced void fracio wih maximum value 5% is abou 10% higher ha he measured daa a secio (a) z=1.5cm; hough he model shows uderesimaio of he void fracio a secio (b) z=.5cm ad secio (c) z=1.5cm. I ca be foud ha he void fracio decreased rapidly ad chaged o early zero wihi 0.1T afer wave breaks which idicaes ha almos all he eraied bubbles degassed i 0.1T. This fidig is cosise wih he experimeal resuls of Lamarre ad Melville (1991) which showed ha he degassig rae afer wave breakig was very quickly ad less ha 5% of he iiial eraied bubbles remaied i he waer colum oe period afer breakig. ' A ' A da z da (1) (14) 5

6 (a) (b) (c) Figure.5. Comparisos of measured (do) ad simulaed (solid lie) void fracios a differe verical elevaios a cross-secios (a) x=18.81m (b) x=18.94m ad (c) x=19.17m. Figure 6 shows he evoluios of bubble plume volume ad bubble plume area which are ormalized by he peak value i a wave cycle V max ad A max respecively. The volume icrease rapidly ad reaches he peak value a 0.T afer wave breaks. The he volume decays similarly i a expoeial way as he bubble plume degasses. A secod peak is observed aroud 0.5T which could be coeced wih he air eraime iduced by he bore impigeme o he slope. Compared wih he volume he bubble plume area shows a mild decay red ad is i o hurry o decay afer reachig he maximum value bu maiai a cosa value for abou half a wave period. This is maily coribued o he diffusio processes of bubbles i he waer especially he small bubbles which have smaller rise velociies (buoyacy forces) ad ca say for loger ime i he waer. This also idicaes ha he void fracios uder breakig waves are domiaed by he eraied larger bubbles. The poeial eergy of he bubble plume which also ca be described as he eergy ha are required o maiai he eraied 6

7 bubbles i he waer is show i Figure 7. Similar o he bubble plume volume wo peaks are observed for he bubble plume poeial eergy. The decay red of poeial eergy is cosise wih he decay of bubble plume volume which idicaes ha he air eraime processes is closely relaed o he eergy dissipaio durig wave breakig. Figure.6. Volume per ui widh of bubble plume VV max ad he bubble plume area AA max. 4 CONCLUSIONS Figure.7. Poeial eergy of he eraied bubble plume EE max. I his sudy he air eraime model based o experimeally deermied iiial bubble size specrum disribuio uder he laboraory-scale breakig waves is icorporaed io he umerical model NEWFLUME o ivesigae he breakig waves i he surf zoe. The bubble eraime process is simulaed by usig umerically compued isaaeous flow field ad he laboraory based bubble size specrum. The reliabiliy of he model is validaed i erms of wave heigh evoluio mea velociy ad urbulece iesiy. The he model is applied o sudies he void fracios ad bubble plumes properies. The void fracio disribuios wih maximum value up o 5% i he regio close o he breakig poi are well capured by he model hough i shows slighly overesimaio ad uderesimaio a some locaios he overall comparisos are good. The evoluios of bubble plume volume ad poeial eergy show similar icrease ad decay red hey reach he maximum value aroud 0.T rapidly ad decay similarly i a expoeial way as he degassig of he eraied bubbles. The bubble plume area maiais he maximum value as log as half a wave cycle ad shows mild decay rae due o he small bubbles diffusio effecs. 7

8 ACKNOWLEDGEMENT The sudy was suppored i par by he Hog Kog Research Gras Coucil (Gra No: PolyU 57/1E)Naioal Naural Sciece Foudaio of Chia (Gra No: ). REFERENCES Buscaglia G. C. Bombardelli F. A. Garcıá M. H. 00. Numerical modelig of large-scale bubble plumes accouig for mass rasfer effecs. Ieraioal oural of muliphase flow 8(11) Blekisopp C. E. ad J. R. Chapli Void fracio measuremes i breakig waves Proc. R. Soc. A Carrica P. Boeo F. Drew D. Lahey R The ieracio of backgroud ocea air bubbles wih a surface ship. Ieraioal oural for umerical mehods i fluids 8(4) Cox D. T. Shi S. 00. Laboraory measuremes of void fracio ad urbulece i he bore regio of surf zoe waves. J. egieerig mechaics 19(10) Deae G. B. Sokes M. D. 00. Scale depedece of bubble creaio mechaisms i breakig waves. Naure 418(6900) Comole R. (1979). Termial velociy of a gas bubble i a liquid very low viscosiy. Joural of mechaical applicaio () Falvey H. T. (1980). Air-waer flow i hydraulic srucures. NASA STI/Reco Techical Repor N Iafrai A Eergy dissipaio mechaisms i wave breakig processes: Spillig ad highly aeraed plugig breakig eves. Joural of Geophysical Research: Oceas ( ) 116(C7). Lamarre E. Melville W. (1991). Air eraime ad dissipaio i breakig waves. Naure 51(66) LimH.-J. K.-A. Chag Z.-C.. Huag. B. Na 015. Experime sudy i plugig breakig waves i deep waer J. Geophys.Res. Oceas Li P. Liu P. L.-F.1998a. A umerical sudy of breakig waves i he surf zoe. Joural of Fluid Mechaics Li P. Xu W NEWFLUME: a umerical waer flume for wo-dimesioal urbule free surface flows. J. Hydraulic Res 44(1) Ma G. Shi F. Kirby J.T A polydisperse wo fluid model for surf zoe bubble simulaio. J. Geophys Res: Oceas ( )116(C5) doi:10.109/010jc Moraga F. J. P. M. Carrica D. A. Drew ad R. T. Lahey Jr A sub-grid air eraime model for breakig bow waves ad aval surface ships Compu. Fluids Mori N. T. Suzuki ad S. Kakuo Experimeal sudy of air bubbles ad urbulece characerisics i he surf zoe J. Geophys. Res. 11 C05014 doi:10.109/006jc Rapp R. Melville W Laboraory measuremes of deep-waer breakig waves. Philosophical Trasacios of he Royal Sociey of Lodo. Series A Mahemaical ad Physical Scieces 1(16) Rider W.J. Kohe D.B Recosrucig volume rackig J. Compu. Phys Rodi Turbulece Models ad Their Applicaio i Hydraulics. Ieraioal Associaio for Hydraulic Research Publicaio 9-0. Terrill E. J. W. K. Melville ad D. Sramski Bubble eraime by breakig waves ad heir ifluece o opical scaerig i he upper ocea J. Geophys. Res

F P = A. PRESSURE In general terms, pressure conveys the idea of a force. Pressure, P, is the force, F that acts on a given area, A:

F P = A. PRESSURE In general terms, pressure conveys the idea of a force. Pressure, P, is the force, F that acts on a given area, A: GASES Gases differ sigificaly from solids ad liquids. A gas expads spoaeously o fill is coaier. Gases are highly compressible. Gases form homogeeous mixures wih each oher regardless of he ideiies or relaive