TWO-PHASE FLOW BALANCE EQUATIONS

Transcription

1 WO-PHASE FOW BAANCE EQAONS Chsohe Moe EN/ER/SSH/ CEA eobe 7, ue es Mays, 8054 RENOBE CEEX 9 e : () hsohemoe@eaf ouo oa saaeous baae equaos oooga equaos Mass baaes 5 Momeum baaes (Newo s aw) 6 4 oa eegy baaes (fs e) 7 5 Seoay baae equaos 8 5 wo-fu fomuao 9 6 Eame of aao : he Rayegh equao fo a shea aou bubbe 0 6 Mass baae equaos 6 Momeum baae equaos 6 Eegy baae equaos 64 Rayegh equao 7 Eame of aao : Haama souo fo he asao of a shea uso a ey sous fu 7 Smfyg assumos, baae equaos a bouay oos (obem seg) 7 eemao of he eoy a essue fes a aou he gobue 4 7 Foe eee o he gobue 7 wo-fu aeage equaos 9 Aeagg oeao 9 Pmay aeage baae equaos 0 Mass baae equao 0 Momeum baae equao oa eegy baae equao 5 Aeage oooga equaos 0 Vo fao oooga equao 0 efaa aea oooga equao 4 ubuee equaos 4 Hyb aoah fo sese wo-hase fows 6 4 eso of a ouao moo-sese sze bu mu-sese eoy 7 4 efo of aabes a eao of he ma baae equaos 7 4 Cosue of he efaa foe fo eeg fows 40 4 eso of a ouao mu-sese sze (a eoy) 4 4 Mass a momeum baae equaos 44 4 eomea momes baae equaos 46 5 Vaous sea ases 47 5 ema eoy 47 6 he eame of he NEPNE_CF oe 49 6 he mass baae equaos he NEPNE_CF oe 50

2 6 he momeum baae equaos he NEPNE_CF oe 50 6 he eegy baae equaos he NEPNE_CF oe he ubuee baae equaos he NEPNE_CF oe he geomea momes baae equaos he NEPNE_CF oe 59 Refeees 6 ouo hs ouse s a fs ouo o wo-hase fow moeg s eequses ae a goo owege of esoa auus a a bas owege of he sbuo heoy, wh of ouse a goo owege of assa fu mehas he am of hs ouse s o ese a eae mae he baae equaos goeg wo-hase fows, whh ae usefu o he umea eo of suh fows Suh a ouse ao be ehause ue o he hess of he sube, bu we hae e o ge o he sues he ma oos whh ae eessay o mase befoe eeg he umea oos eoe o wo-hase fow sues hese oos a be assfe wo ma aegoes he fs oe whh has emege he as eaes s base o he RANS aoah (RANS meas Reyos Aeage Nae-Soes) hs aegoy s auay usefu fo egeeg aaos whee oy he age saes (o eees) of he fow fes ae ese he seo aegoy s goue o he aoym NS (fo e Numea Smuao) s omeey ffee fom he fs se a he fow eas ( sae as we as me) ae soe umeay he fs aegoy of oos s base o aeage baae equaos, ooso o he seo aegoy whh s base o oa saaeous (uaeage) oes he mao ffuy of he RANS aoah s he osue obem ose by he aeage equaos ug he aeagg oess, a o of fomao has bee os (se he sma fow eas ae o aessbe he souo) Howee, he mea (aeage) effe of he os eas o he aeage quaes s o eggbe a a As a osequee, he hyss mus oe osue aws fo a ea umbe of uow ems aeag he aeage equaos heefoe, he ma ffuy of he RANS aoah s esseay of mahemaa a hysa aue he ffues of he NS aoah ae omeey ffee Hee, he osue ssue s ese se he oa saaeous baae equaos ae soe ey, whou ay of aeagg he ffuy aeas esseay he age amou of fow eas whh ae eessay o auae Vey owefu omues ae eessay, wh age quaes of memoy Ee wh suh owefu omues, he fows whh ae ameabe o smuae ae que ese ems of he Reyos umbe, umbe of mobe efaes (e bubbes o oes efaes) a so o ao, sea ag agohms ae ofe eessay o smuae auaey hese mobe efaes So we a say ha he mao ffues eouee he NS aoah ae esseay umea a of aa oessg aue he a of hs ouse s he foowg oe he oa saaeous baae equaos ae esee seo hese equaos ae usefu a wo-fo mae Fs, hey ae he bas equaos soe by he NS oos Seo, hey aso fom he heoea bass o eeo he aeage equaos whh ae soe by he RANS oos he oa saaeous (uaeage) equaos ae somemes ae he moso equaos (o equaos a a he moso ee of eso) a he aeage equaos ae somemes ae he maoso oes he aeage baae equaos ae ee he seos a 4 Seo eses he so-ae wo-fu moe he wo-fu moe osues a ey geea mahemaa fame whee he equaos ae we fo he wo hases a symmea way, whou assumg ay aua efaa ofguao (o fow egme)

3 Seo 4 s eoe o sese wo-hase fows, whh gous bubby, oe a auae fows Oe of he wo hases s assume o be sese a age umbe of usos (o aes o gobues) o he ohe hase whh s ae he ouous (o ae) hase ue o he obous ssymmey of suh s of fows, he aeage equaos a be we s a ssymmea mae, efeg he fow ssymmey he sese hase s esbe a mae aaogous o he moeues he oe of he e heoy of gases Seea aaages a be gae by usg hs of eso: he equaos fo he sese hase ae ease o ee a o ee se hey esembe o he equaos goeg a sge ae he e o ay s a os of geeay a he ffuy o mae he oeo wh he ohe (ouous) hase whh s aways eae he oe of he wo-fu moe ue o he fa ha he equaos fo he sese hase ae eae a ffee mae ha he equaos fo he ouous hase, hs aoah s ae hyb he seo 5 s eoe o he smfe suy of aous sea ases A hs me, oy he ase of he ema eoy of a sg bubbe a qu has bee esee he as seo 6 ges a omaso of he equaos use he NEPNE_CF oe o he ea equaos ee he eous seos hs ges a eame of umea aao o he eae he oe of he RANS aoah he mao assumos mae by he NEPNE-CF eam ae se a he smfe equaos ae esee oa saaeous baae equaos oooga equaos e: (, ) 0 F () be he geomea equao efg he ffee efaes he fow e F be ose hase a F be egae hase he Phase ao Fuo (PF) s a bay fuo whh a be efe as: (, ) (, ) Y( F(, ) ) () whee Y s he Hease sbuo he u eo oma o he efae a ee ouwa fom hase (,) a be efe assay as (As, 96): F/ F () e w be he eoy fe assoae o he efaa sufae As F s eay zeo fo a os oae o he efae, s oee me eae a he eoy w s : F w F 0 (4) Fom ()-(4), oe a eue he oma saeme see of he efae (ehaye, 98):

4 F / w w (5) F Oe moa ema s ha wo ffee eoy fes w ffeg oy hough he agea omoe w w (w) ge se o he same efae moo aog o Eq (4) heefoe, he oma eoy omoe s he oy oe o be eae uambguousy o he sufae moo Fom he efos (), oe a eue he foowg eessos fo he saa a me eaes: ( F) ( F) F F (6) whee s he a sbuo, whh s he eae of he Hease sbuo Y Fom (4) a (6), oe a eue he foowg oooga equao: w 0, (7) Fom eaos () a (6), o a aso eue: ( F) F (8) whee s a a sbuo hag he ffee efaes as a suo s ae a oa saaeous efaa aea oeao by Kaaoa (986) e us fsh hs seo by emag some eesg oees of he PF As hey ae bay fuos, hey efy ha: K K 0 (9) As w be see ae, he aeage faos of esee of he wo hases ae efe as he aeages of he oesog PF ( K < K > whaee he of aeagg oeao eoe by < >) Howee, he wo aeage faos of esee K ae o bay fuos; hee o o efy eaos e (9) hs s a moa ffeee bewee oa-saaeous a aeage quaes a has some osequees o he oesog baae equaos he aaage of he eaos (9) efe by he PF s ha we a we, fo ay wo quaes A a B haaezg hase : ( KA K )( KBK ) KA KBK (0) 4

5 Mass baaes e a beg he esy a eoy fes fo hase A so-ae sge fu esy a a sge fu eoy a be efe as:, () ue o he oey (0), we a we: () Whe hee s o e mass geeao he wo-hase meum as a whoe, he sge fu efes he we ow mass baae equao: ( ) 0 () Hee, seg () a () o (): 0 (4) Sg eaes, Eq (4) a be ewe: ( ) 0 (5) se oe of he wo hases (e ouse he efaa sufae), he eaes of he PF ae, aog o he eessos (6), a oe s ef wh he usua mass baae equao fo hase ( o ): ( ) 0 (6) Combg (6) wh (5) ges he foowg equao a o he efaes: w ( w ) 0 (7) ag he oooga equao (7) o aou, he fs wo ems saea, a oe s ef wh: ( w ) 0 (8) 5

6 whh s he mass baae fo he efaes a ges he omeme o he mass baae se hases (6) sg (8) a efg: ( w ) ( w ) m& (9) whh s he mass ga ue o hase hage (eaoao o oesao) e u oume, he quay m& beg he mass ga e u sufae he efaa mass baae equao (8) s heefoe: m & 0 (0) showg ha hee s o mass aumuao a he efaes, a osequee of he m assumo ha he efaes ae mmaea sufaes, ayg o mass Momeum baaes (Newo s aw) Poeeg he same mae as fo he mass baaes, he mue momeum baae eas (Kaaoa, 986): g Fs () whee a eoe he essue a he sous sess eso hase he eos g a F s eoe he gay aeeao a he sufae eso foe eseey Aog o ehaye (974), he sufae eso foe e u efaa sufae has he foowg eesso: F σ σ () s s whee s o usefu o ese he sese of he u oma eo se aeas we he fs em of he RHS (Rgh Ha Se) of () he egee of he u oma eo ges he oa uaue, equa o we he mea uaue (As, 96), a he as em σ s s he sufae gae of he sufae eso oeffe: he so-ae Maago effe efg a mue essue a a mue sous sess eso as (): (), s easy o see ha he momeum equao he sge fu fomuao eas: ( ) ( ) g ( σ sσ) (4) 6

7 Ee fo he as em, he equao (4) s he same ha he momeum baae fo a sge fu, hee he ame of he fomuao he as em, sef o wo-hase fows, eeses he sufae eso foe ag o a u oume of he wo-hase meum oag efaes Poeeg as fo he mass baaes, he momeum baaes fo hase a fo he efae a be seaae fom he mue baae () he momeum baae fo hase s he assa oe: ( ) g (5) a he momeum baae fo he efae eas: σ σ s s F m& (6) 4 oa eegy baaes (fs e) Aog o Kaaoa (986), he oa eegy fo he wo-hase mue s efe by: s u e (7) whee e s he sef ea eegy fo hase a u s s he efaa eegy e u sufae he mue oa eegy obeys o he foowg baae equao (Kaaoa, 986): s s s s s u w F Q g q w u e u e (8) whee q eoes he hea fu ue o ouo se hase, Q eoes a ossbe hea soue hase a s eoes a soue em of Aog o Kaaoa (986), he oa saaeous AC (efaa Aea Coeao) obeys o he foowg baae: [ ] s w (9) Moe (007) ges he ea eesso of he soue em s ue o sehg of he efaes (e he absee of bea-u a oaesee): 7

8 s s w (0) Fom (8), s ossbe o seaae he oa eegy baaes fo hase a fo he efae he same mae as he eeg aagahs he oa eegy baae equao hase eas: ( ) ( ) Q g q e e () sg (9) a (), he emag of (8) ges he oa eegy baae fo he efae: [ ] s s s e m q w F u w u & () Now, efg he foowg aoa mue quaes: Q Q, q q, e e () he equao (8) a be ewe he sge fu fomuao: ( ) ( ) s s s s s u w F Q g q w u e u e (4) 5 Seoay baae equaos ag he o ou of he momeum equao (5) by he eoy ges he e eegy baae equao: ( ) ( ) g : (5) Subag he e eegy baae (5) fom he oa eegy baae () ges he foowg ea eegy baae equao: [ ] [ ] : Q q e e (6) Subsag e mue by he mass baae equao (6) fom (6), he o oseae fom of (6) s obae: 8

9 e q Q e e ˆ e : wh : (7) he oao / sas fo he maea (o oee) eae foowg he hase s moo sg e C, * whee C, a eoe eseey he sef hea a osa oume a he hase emeaue, Eq (7) a be ewe fo he emeaue as he ma aabe: C, q λ ( λ ) Q : wh : (8) whee he Foue s aw has bee assume o eess he oue hea fu q efg he ehay by he sum of he ea eegy a he eegy assoae o essue foe: h e (9) he ea eegy (6) o (7) a be ewe he fom of a ehay baae equao: [ h ] [ h ] q Q : (40) efg he oa ehay by he sum of he ehay a of he e eegy: H h (4) he equao fo H a be ee smy by ag he equaos (40) a (5) fo he wo foms of eegy oae he oa ehay: [ H ] [ H ] q Q ( ) g (4) 5 wo-fu fomuao he equaos (), (4) a (4) osue he sge fu o oe fu fomuao fo he wo-hase fow Aohe usefu fomuao s he wo-fu fomuao whee he 9

10 equaos fo he wo hases ae eae (o soe) eeey hese equaos a be obae fom he sge hase baae equaos fo mass (6), momeum (5) a oa eegy () by muyg hem by he PF a eoug he eaes he mass baae of he wo fu fomuao eas: ( ) ( w ) m & (4) he momeum baae of he wo fu fomuao eas: ( ) m ( ) ( ) g & (44) he oa eegy baae of he wo fu fomuao eas: e e m& e q ( q ) ( ) ( ) g Q (45) s easy o efy ha, summg he equaos (4), (44) a (45) o he wo hases ( a ) a ag o aou he efaa baaes (0), (6) a () wh he efos (), () a (), he baae equaos of he sge fu fomuao a be eee 6 Eame of aao : he Rayegh equao fo a shea aou bubbe We ose a sge aou bubbe mmese a qu ue he foowg hyoheses (ehaye, 98): (H) o gay (H) shea symmey (H) sge omoe qu (H4) Newoa qu (H5) Cosa qu sosy µ (H6) qu obeyg Foue s aw (H7) Cosa qu hema ouy λ (H8) Sge omoe aou (H9) Newoa aou (H0) Cosa aou sosy µ V (H) aou obeyg Foue s aw (H) Cosa aou hema ouy λ V (H) Cosa sufae eso σ 0

11 6 Mass baae equaos ue o he assumo of obem shea symmey (H), he equaos ae we shea ooaes e he assumos (H) a (H8), he mass baae equaos fo he qu a aou hases (6) beome: ( ) ( ) 0 w 0 w V V V (46) Whee s he aa sae o he bubbe ee, of aus R, a w V, ae he aa omoe of he aou a qu eoes he efaa mass baae equao (0) eas: ( ) ( ) R o R w R w V V & & (47) Whee R & s he me ae of hage of he bubbe aus R a s equa o he oma saeme see of he efae w o hs sme obem he seo e aes ha he aues ae ae a he efae 6 Momeum baae equaos e he assumos (H), (H), (H), (H4), (H5), (H8), (H9) a (H0), he qu a aou momeum baae equaos (5) ea, shea ooaes: µ µ V V V V V V V V V w w w 4 w w w w w w 4 w w w (48) he efaa momeum baae equao (6) eas, ue he aoa assumo (H): ( ) ( ) µ µ σ R w w 4 R w w 4 w R w w R w R V V V V V V V & & (49) 6 Eegy baae equaos he hose fom of he eegy baae equaos s he emeaue equao (8) usg assumos (H6) a (H) sg aso (H), (H7) a (H), he equao (8) we fo he wo hases shea ooaes ea:

12 V C C,,V V w w V V λ λ V V V ( w ) : ( w V ) : V V (50) whee we mae he aoa assumo: (H4): o hea soue Q he bu of he hases he efaa eegy baae s o usefu a w o be we hee 64 Rayegh equao he sues of aou bubbe yams, a sea fom of he qu momeum equao, he Rayegh equao, s ofe use We mae wo aoa assumos: (H5) he qu s omessbe (H6) he aou esy s eggbe wh ese o he qu esy he assumo (H5) aows o egae ey he qu mass baae equao (46) whh ges: () A w (5) As a esu, he qu momeum equao (48) aes he foowg sme fom, whaee he qu sosy: w w w (5) ag he esu (5) o aou a egag (5) fom R o fy ges he foowg esu: w & R Rw & (5) w whee he seo e aes quaes whh ae ae o he bubbe efae ( R()) a he oeo eoes a me eae Assumo (H6) eabes o smfy he efaa mass baae (47) as R&, heefoe Eq (5) beomes: w RR& R& (54) whh s ae he Rayegh equao

13 7 Eame of aao : Haama souo fo he asao of a shea uso a ey sous fu 7 Smfyg assumos, baae equaos a bouay oos (obem seg) he souo esee heeafe has bee ee by Haama (9) a summaze by Caee (008) osss he suy of he asao of a ey sous oe a ey sous fu (eeg fow) he foowg assumos ae eessay: (H) he fow s saoay (H) he wo hases ae omessbe (H) eeg fow, e Re << (H4) Newoa fus wh osa soses (H5) he shea oe s asag whou ay aeeao (H6) o hase hage (e ehe eaoao o oesao) (H7) he fow s assume o be asyme (H8) Cosa sufae eso σ e he fou assumos (H)-(H4), he mass a momeum baae equaos eah hase (6) a (5) eue o: m, 0 µ (55) Whee m, s a mofe essue fo hase, ug he gay em oug he oy ω o( ), he seo equao (55) a be ewe: m, µ µ ω ω (56) ag suessey he oaoa a he egee of (56), he essue a he oy ae hamo quaes: ω m, 0 0 { ω ω 0 ω 0 (57) he foowg, he sese hase (he uso) w be eoe by a he ouous fu suoug hs uso w be eoe by e he oo of o hase hage (H6) a f we am o s of oe hase o he ohe a he efae, he emaa oos o he shea gobue sufae ea:

14 (58) e he assumos (H6) a (H8), he efaa momeum baae equao (6) egeeaes o: ( ) σ (59) Poeg (59) o he oma (o he efae) a agea eos ges:,, H,, σ (60) whee H s he mea uaue, equa o /R fo a shea sufae of aus R he bouay oos (BC) a fy ea: e 0 z (6) whee s he asao eoy of he gobue ee We hoose o wo a efeee fame e o he gobue e z be he symmey as (H7) a θ be he age measue fom he z as 7 eemao of he eoy a essue fes a aou he gobue We ae seahg he souo ue he foowg fom: ψ ψ, θ (6) s θ θ s θ Whee ψ s ae he Soes seam fuo he oy eo ω has oy oe o zeo omoe ω aog he base eo e φ (H7) whh eas : ω ω E ω e ϕ wh : θ s θ ˆ θ s θ θ θ ( ) E ( ψ ) θ s wh : θ o gθ θ (6) Hee, he fs equao (57) beomes: E 4 ( ) 0 ψ (64) 4

15 whee he oeao E 4 meas he oeao E ae wo mes he BCs (6), (58) a (60) beome: ψ s θ θ ψ s θ 0 { osθ { s θ ψ ψ 0 e R θ θ ψ ψ e R ψ ψ µ µ s θ θ s θ θ ψ ψ µ µ e R σ R e R (65) he souo s seahe he foowg fom: ψ (, θ) F ( ) s θ (66) We oba: E E 4 e ( ψ ) s θ( F F / ) ( ψ ) s θ( f f / ) F 4 F () (4) f A F 8F 8F 4 B osθ A s θ 4A B B 0 C 0 C θ C / (67) Fo he eea fe, he eoy emas fe whe es o fy, mes A 0 Ohewse, he fs wo BCs (65) ge C -/ Fo he ea fe, he eoy mus ema fe a he og (e 0) so we hae eessay: B 0 Whee : 5

16 ( ) ( ) ( ) ( ) ( ) θ θ θ θ θ θ ψ θ θ ψ θ θ B s B os C 4A s C A os s / B, s C A, 4 (68) he fouh e (65) ges he wo oos 0 a he efae R : 0 R R B 0 C R A (69) he ffh e (65) ges he uque oo θ θ a he efae R : R R B C A R 4 (70) Fay, he as BC (65) (equay of agea sesses a he efae) ges : µ µ 5 R 4 R R B R C A 4 (7) oug he foowg oao of he soses ao: ˆ µ µ κ (7) he souo of he sysem of ageba equaos (69)-(7) s: κ κ κ κ κ κ 4 R / R B 4 C 4R A (7) he seam fuos a he eoes eah hase ae he eue: 6

17 ψ ψ (, θ) θ θ (, θ) 4R osθ s θ R R κ R κ / R κ 4 osθ 4R κ 4 κ s θ R κ κ 4 4 κ κ / R κ 4 κ / R κ 4 κ κ κ κ s θ κ κ s θ (74) he momeum equao (55) he aows o auae he mofe essue fe eah hase: m, m, osθ 5 0 µ R κ R osθ κ 0 µ ( κ) (75) he essue aue a he uso ee s eue fom he oma baae a he efae (see (65) 6 ) : σ 0 (76) R 0 he sufae eso σ obues o he essue ee se he uso 7 Foe eee o he gobue he foe eee o he uso by he suoug fu s ge by he foowg eesso: F σ S wh σ e e e e,, θ θ, { ϕ ϕ 0 (77) By symmey, hs foe has oy oe o zeo omoe F z he eo of he eae eoy : ( os θ os θ s θ) S wh S R s θθ ϕ Fz,,θ (78) hs foe, ofe ae he ag foe, ues hee obuos: 7

18 Pessue foe (fom ag): / κ z, ( os θ) S πµ R (79) κ F Vsous foe oma o he efae (s ag): 8 z, (, os θ) S πµ R (80) κ F Vsous foe age o he efae (s ag): κ z,θ (, θ s θ) S 4πµ R (8) κ F he sum of he hee obuos (79)-(8) ges he oa ag foe eee o he gobue: F z κ / 4 κ / R 6πµ R C ae Re ˆ (8) κ Re κ ν Whee C s he ag oeffe whh s eae o he ag foe by he foowg efo: C z (8) πr F f we mae he ao κ efe by (7) e o fy (he sosy of he fu uso s fey geae ha he oe of he suoug fu), we eee he eesso of he Soes ag foe o a so ae (eg Oeseé, 006): 4 C (so ae) (84) Re he ohe mg ase s he oe obae by mag κ 0 (he sosy of he fu uso s ) whh oesos aomaey o he ase of a ea bubbe: F z 6 R 4πµ R C wh Re ˆ (ea bubbe) (85) Re ν Wha s eesg s o see how he eae esus (79)-(8) egeeae hese wo mg ases he omaso of he ffee foe obuos fo he so ae o oe ha, a fo he ea bubbe o he ohe ha, ae ge he foowg abe: uso µ κ ˆ µ F z, F z, F z,θ z F 8

19 so πµ R 0 4πµ R 6πµ R Cea bubbe πµ R πµ R πµ R hs omaso usaes he fuee of he of bouay oos o he uso sufae shou be oe ha he essue obuo s o ea he wo ases he agea sesses ge o obuo fo a ea bubbe bu ge he wo-h of he oa ag foe fo a so ae he sous sesses he oma eo ge a obuo oy fo he ea bubbe ase, whh aso equas he wo-h of he oa ag wo-fu aeage equaos hs seo, he aeage equaos of he wo-fu moe ae ee hese equaos ae he aeage foms of he oa saaeous wo-fu equaos ha hae bee esee seo 5 he eesso wo-fu sgfes ha he wo hases ae eae seaaey, ooso o some sme moes ha use baae equaos fo he mue osee as a whoe Neeheess, he wo hases ae o eee, se hey ae eae hough efaa eao ems whh ae he aeage foms of he ems og he oa saaeous equaos Aeagg oeao Hee we efe he oees of he aeagg oeao ha w be use o he oa saaeous equaos o he aeage oes he oa saaeous equaos ae somemes ae moso equaos, o equaos a he moso ee, ooso o he aeage equaos ae maoso oes he aeagg oeao heefoe efes a bge bewee he moso eso of he fow heomea (ug a he saa a emoa eas) o a smfe maoso eso, haaeze by a smae umbe of feeom egees, whh s oseaby heae o auae by umea meas, a whh s ofe suffe o he egee Fom a umea o of ew, he moso ee s he oma of NS (e Numea Smuao) a he maoso oe s he oma of RANS (Reyos Aeage Nae-Soes) smuaos, o ea assa eessos use sge-hase fow auaos A emeae smuao oma s ES (age Ey Smuao) whh s ofe use sge-hase fow auaos, bu seems ffu o use fo wo-hase fows he ese sae of he a hee es a o of aeagg oeaos use by moees a eemeaos hese aeagg oeaos a be assfe o hee ma aegoes: saa, emoa a sasa (esembe) aeagg oeaos Some omose aeagg oeaos, esug fom he suemoso of seea bas aeagg oeaos, ae somemes use A hese oeaos ae of ffee aue, eah oe beg aae o a ea ass of obems fo whh hey wee ee Fo eame, he me aeagg oeao s moe ofe use he oe of sasay saoay fows, ee f s aaby oma s o omeey ese o hs of fows he moe fuamea of aeagg s he sasa aeagg, whh a be eae aaageousy by a me o a saa aeagg oeao some aua suaos (saoay fows fo me aeage, homogeeous fow fo saa aeage), og he egoy assumo shou be e m ha a he aeagg 9

20 oeaos o o hae he same oees Cea aeagg oeaos, ae Reyos oeaos (Sagau, 998) ae ofe use beause hey ge he smes fom of he aeage equaos wha foows, we ea he bas oees of suh a Reyos oeao: φ ψ φ ψ () a φ a φ (a e) () φ ψ φ ψ () φ s φ s s, (4) φ φ (5) φ 0 φ ˆ φ φ (6) whee φ a ψ eoe wo fe quaes (e quaes eeg o me a sae ooaes) he aeagg oeao s eoe by baes < > he eaos () a () eess he eay of he aeagg oeao, a oey whh s ommo o a s of aeagg oeao he eao (4) eesses ha me a saa eaes a be emue wh he aeagg oeao whou oug ay aoa em wo-hase fows, whee souous fes ae eouee a he efaes, hs oey s a oy by eeg he fes he sese of geeaze fuos (o sbuos) whh has bee oe seo he ohe hee eaos (), (5) a (6) eess ha a aeay aeage quay s uaffee by a seo aao of he aeagg oeao As a osequee, f he fuuag quay s efe as he ffeee bewee he quay sef a s aeage (Eq 6), he aeagg of hs fuuag quay ges zeo hs ey moa oey s sef o a Reyos aeagg oeao, a s o ue fo eame, he ase of a saa aeagg oeao use ES he esembe (o sasa) aeagg oeao has hs oey, a we w assume ou subseque eeomes, ha he hose aeagg oeao s a Reyos oe Pmay aeage baae equaos By may, we mea he mass, momeum a oa eegy baae equaos he oa saaeous fom has bee summaze seo 5, he fom of he wo-fu fomuao Hee, s suffe hee o ay ey he aeagg oeao o oba he may aeage baae equaos Mass baae equao he aeage of he mass baae equao (4) ges: 0

21 m& (7) Seea maoso (aeage) quaes ae he efe he aeage fao of esee of hase, somemes ae he o fao o he hase hou, s efe as he aeage of he PF, hee: (8) he (s) hase aeage esy s he efe by: ψ ψ ψ (9) eg Fae aeagg s assay efe fo ems o quaes weghe by mass o oume mass: φ φ φ eg (0) A he e, we aso efe he aeage mass ga e u oume e u me ue o hase hage by: m& () As a osequee of he eeg efos, Eq (7) a equaey be ewe as: () he equao () s he assa fom fo he mass baae equao of he wo-fu moe (eg sh, 975) Eq () shou be suemee by he aeage fom of he efaa mass baae (0) Muyg (0) by a ag he aeage, he foowg mass um oo s obae: 0 () Momeum baae equao Aeagg Eq (44), we oba:

22 m& g (4) whee he seo e egous he momeum efaa asfes he fs efaa asfe m& s he ehage of momeum assoae o he mass asfe by hase hage s ofe ae he eo foe he wo ohe efaa ems ae he mea efaa foes ue o essue a sous sesses Eq (4) s a ea equao a efes ey he oa saaeous fom eeoe seo 5 f he mass baae equao () seems o be ommo o a wo-fu esos eeoe by he ffee auhos, he suao s o so sme fo he momeum equao May aas a be eeoe fom Eq (4) aog o he hoes mae by he ffee auhos o efe maoso aabes hese maoso aabes shou be fu of hysa sgfae a hs hysa sgfae ees o he obem beg sue Fo eame, safe fows ae omeey ffee fom sese fows (e bubbes, oes o auae fows) A safe fow s a fow whee he gas a qu hases ae suemose oua, seaae by a uque ouous efae a safe fow, he wo hases ay heefoe a symme oe a he equaos shou efe hs symmey O he oay, fo bubby o oe fows, oe hase (ae he sese hase) s osue fom sma usos embee he ohe hase (ae he ouous hase) he wo hases ae heefoe o symme, a hs asymmey s aso efee by some wo-fu moes eseay eoe o sese fows (eg Zhag & Posee, 994) hs seo, we mae o aua assumo o he fow ofguao, a we ese some efos a mauaos whh ae oose que geea wo-hase fow e boos (sh, 975; ew & Passma, 999; Oeseé, 006) efg hase aeage essue a sous sess eso a sma mae ha Eq (9), Eq (4) a be ewe: m& ( ) ( ) g (5) efg he fuuag eoy aou s Fae aeage: (6) a be show by usg Eqs (), (5) a (6) ha: (7)

23 he Reyos sess eso fo hase s efe a mae aaogous o sge-hase fow by: (8) Now we mus eame he efaa asfe ems he eo foe s ofe eesse by oug a ew aeage eoy weghe by hase hage: m& (9) he efaa essue foe s uey oma o he efae hs s o he ase of he sous foe whh has oma a agea omoes aog o: ( ) 44 (0) A fs fom of he aeage momeum equao a be ge: ( ) ( ( ) g M () whee M s efe as he aeage efaa momeum asfe: M m& () Now we w foow he boo fom sh & Hb (006) o eomose he efaa asfe em of momeum M hey aso efe he mue momeum soue ue o sufae eso: M m s ( σ s σ) F () he auhos oue he sufae mea aues: ψ ψ ψ (4) a fo a abay fuo ψ efe a he efae, whee a s efe as he aeage efaa aea oeao efg efaa-aeage essue a sous sesses, he efaa asfe of momeum s eomose he foowg mae:

24 M M M M M ( ) ( ) M wh : (5) M he fs hee ems he RHS of Eq (5) ae he oma omoes wheeas he as wo ems ae agea omoes he o fao gae aeas ue o he aeage fom of he eao (8) : (6) he oma foe M eeses he fom ag a f foe asg fom he essue mbaae a he efaes he agea foe M eeses he s ag ue o he mbaae of shea foes he wo foes ae ombe o efe he oa geeaze ag foe: M M M (7) oug he aeage mea uaue of he efaes H as we as he aeage sufae eso σ, sh & Hb (006) ewe he mue momeum soue he foowg mae: M m ( ( H σ H σ ) s σ) H σ (8) he seo em aes o aou he effe of hages of he mea uaue Negeg he Maago effe (as em he RHS of Eq (8)), he mue momeum soue s aomae by: M M m H m H H m ( σ H σ ) H σ M wh : (9) he eo soue H M m s he effe of he hages of he mea uaue o he mue momeum he aeage fom of he oa saaeous efaa baae of momeum (6) s: M M m (0) oug he eomoso (5) wh (7) o (), he foowg fom of he momeum baae s obae: 4

25 ( ) ( ) ( ) M g () Subag fom () he mass baae equao () eousy mue by he mea eoy a eaagg, he foowg o oseae fom of he momeum baae equao s obae: ( ) ( ) ( ) M g () whee he maea eae foowg hase s mea moo: () oa eegy baae equao Aeagg Eq (45) ges: q Q g q e m e e & (4) Now e us efe he foowg maoso quaes (sh & Hb, 006): he aeage ubue e eegy: K (5) 5

26 he aae ea eegy whh s he sum of he mea ea eegy a he ubue e eegy: K e (6) he ubue hea fu whh aes o aou he ubue eegy oeo as we as he ubue wo: e q (7) he efaa suy of oa eegy o he h hase, whh gous a he efaa asfe ems Eq (4): q e m E & (8) sg hese efos, Eq (4) beomes: ( ) [ ] E Q g q q (9) Muyg Eq () by a aeagg, he foowg um oo fo he oa eegy s obae: s s s s s u w u w F E wh E E (40) whee E s s he sufae eegy soue fo he mue hs meas ha eegy a be soe a o eease fom efaes Now, foowg he saa meho of seo 5, we w ee seoay foms of he eegy baae equao og he momeum equao () by he mea eoy, he foowg mehaa eegy equao s obae: 6

27 ( ) ( ) ( ) M g (4) Subag Eq (4) fom Eq (9), he mea aae ea eegy equao s obae: [ ] ( ) M E Q : q q (4) he efaa asfe he hema eegy equao (4) has a sea fom whh ombes he mass, momeum a oa eegy asfe ems: M E Λ (4) Now, oug he aae ehay as he sum of he mea ehay a he ubue e eegy: K h Η (44) he hema eegy equao (4) a be ewe fo he aae ehay: [ ] ( ) ( ) Q : q q Λ Η Η (45) Now we w eame eas he oe of he efaa hema eegy asfe Λ Sag fom he efos (), () a (8), we oba he eesso of Λ as a fuo of he moso fes: 7

28 ( ) q e m Λ & (46) he, oug he foowg maoso quaes: q a q e m & (47) whee s he efaa aeage of he aae ea eegy weghe by hase hage a s he hea u e u efaa aea, a beg he efaa aea e u oume oug he efos (47) o (46), hs beomes: q ( ) a q Λ (48) he mehaa em Eq (48) s a e b moe ffu o eess ems of maoso quaes he essue a sous sess eso mus be eomose o efaa aeage aues (Eq 4) a fuuag as A he e, he foowg esu s obae (sh & Hb, 006): ( ) ( ) W M (49) whee s he ubue wo of he efaa foes Subsug Eq (49) o Eq (48), he maoso efaa hema eegy asfe beomes: W ( ) W M a q Λ (50) aaogy wh (44), he aae mea ehay a efaes weghe by hase hage a be efe: Η (5) 8

29 he we hae: ( ) W M a q Η Λ (5) s saghfowa o oba E fom he eaos fo Λ, M a, heefoe we hae fom (5), (7) a (5) he foowg esu: ( ) W M a q E Η (5) Subsug hese esus o he oa eegy equao (9) a o he aae hema eegy equao (45), hese equaos beome: oa eegy equao: ( ) [ ] ( ) W M a q Q g q q Η (54) Aae hema eegy equao: [ ] ( ) ( ) [ ] ( ) ( ) W M a q Q : q q Η Η Η (55) efg he ubue eegy soue by a he sous ssao em ue o he mea eoy gae by, hus: Φ Φ ( ) : W Φ Φ (56) he equao (55) a be ewe equaey o he foowg o oseae fom: 9

30 Η [ ( q q )] Φ Φ ( ) ( ) ( ) Η Η q a M Q (57) Aeage oooga equaos Vo fao oooga equao We fs aeage he oooga equao (7) fo he PF As he fao of esee of hase s efe as he aeage of he PF (Eq 8), he aeage of (7) ges a eouo equao fo Afe some sme mauaos, a usg Eq (9), he foowg equao s obae: m& ( ) (58) a be see ha hs equao esembes, bu s o ea o, he mass baae equao () We a mae use of he oa saaeous mass baae (6) ewe ue he foowg fom: (59) whee we use he foowg efo of he moso maea eae: (60) base o he oa saaeous eoy, whh s ffee fom he maoso maea eae () base o he aeage eoy he equao (58) beomes: m& ( ) (6) hs equao a be omae o he mass baae equao () whh a be ewe: (6) a be see ha he wo equaos (6) a (6) ae equae f, a oy f, he esy of hase emas osa, e hase s omessbe f he esy of hase aes o he moso sae, he a (6) a (6) ae o equae he 0

31 fs oe s a equao fo he o fao whh s s fom he mass baae equao, he ffeees omg fom oa omessby aaos efaa aea oooga equao A seo oooga equao s ge by he oa saaeous efaa aea oeao (AC) baae equao (9) s aeage ges he foowg AC baae equao: a [ a w ] a s (6) whee a s he aeage AC efe by he foowg eao: a (64) he quaes w a s ae he efaa aeage efae eoy a soue em e u sufae hs as quay s ue o umeous hysa heomea e oaesee a uues, hase hage, efaa sabes a so o Neeheess, s o easy o oue souous heomea, e oaesee a bea-u, he geea oe esee hee hey w be oue ae, a seo sef o sese wo-hase fows, e bubby o oe fows 4 ubuee equaos ese he fa ha s que ffu o efe ue ubuee wo-hase fows, beause he fuuaos obsee he bu hases ae aso sogy oue o he ea moemes of efaes, auhos ofe foow he same e of easog o ee baae equaos fo ubuee wo-hase fows ha sge-hase fows We ae oee hee wh RANS aoahes (RANS meas Reyos Aeage Nae-Soes) whh hae ohg o see wh ES (age Ey Smuao) eae o moe fuamea sues he ffee RANS aoahes a be assfe aog o he umbe of baae equaos use o auae he aeage effe of ubuee ya aoahes use zeo, oe, wo o ee moe baae equaos o esbe ubuee, he moe hag he geae umbe of equaos beg obaby he Reyos Sess Moe (RSM) whh uses see ubuee equaos (eg Shese, 99) Hee we foow he assa (sge-hase) aoah o ee baae equaos fo ubuee (Shese, 99), a smy ee hese equaos o wo-hase fows by oseg he fuos he sese of geeaze fuos, a usg he oos eeoe seo he moe moa mea quay assoae o ubuee s obaby he Reyos sess eso whh has bee efe (8) Aog he oaos of Shese (99), we efe o eefe he Reyos sess eso as he oube eoy oeao: R (65)

32 he oga sess eso (8) a be eee by smy mae R he Reyos sess eso baae equaos ae obae by he foowg meho Fs of a, we ee he equao fo he fuuag eoy whh s efe by Eq (6) hs a be oe by subag he equao fo he mea eoy () fom he equao fo he oa saaeous eoy (5) oe o o hs, we mae he smfyg assumo of a omessbe hase, hee hee: Smfy g assumo : e (66) Hee s o usefu, hs aagah, o mae he so bewee he Fae aeage a he hase aeage e he assumo of a osa esy, Eq (5) a be ewe: ( ) g (67) e he same assumo, Eq () a be ewe: R R M g (68) Subag (68) fom (67), he foowg equao s obae: R R M (69) whee we hae efe he fuuag essue a sous sess eso as he foowg eaos: (70) he seo se o oba he equao fo he ya omoe R, s o mae:, ( ) Eq ( ) Eq (7),,,

33 hs ges:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, (7) Now we mus ae ae ha he hase aeagg oeao ao be smy emue wh he sae a me eaes, oay o he esembe aeagg oeao < > (Eq 4) heefoe we mus ome ba o he esembe aeagg oeao o emue wh he wo eaes: ( ) w ψ ψ ψ ψ ψ ψ ψ (7) ( ),,,,,, ψ ψ ψ ψ ψ ψ ψ (74) We ea ha he oe of he assumo (66), we mae o so bewee he Fae aeage a he hase aeage hs oe, we a wo o he ffee ems of Eq (7) Fs we oe ha: 0,,,,,,,,, (75) beause of he omessby of hase

34 ( ) ( ) ( ) ( ),,,,,,,,,,,,,,,,,,,,,,,,,,,,,, m R R w R & (76) whee he efo (65) has bee use Eq (7) a be ewe: ( ) ( ),,,,,,,,,,,,,,,,,,,,,,,, R R m R R & (77) efg he efaa Reyos sess eso weghe by hase hage by he foowg eao:,,, m R & (78) Eq (77) a be ewe: 4

35 ( R ), R (,,, ),,,,,,,, R R,,, (),,,,, R, (),, () (V) (V) (V) (79) he HS of Eq (79) s smy he aso em of he oeao R, by he mea eoy fe he ffee ems he RHS hae bee umbee: he hysa sgfae of eah of hem s ge heeafe () s ue o hase hage (Reyos sess efaa asfe assoae o mass asfe) () s he egee of he e oeao whh ees o be moee () ae he ouo ems by he mea eoy gaes, hey ee o fuhe moeg (V) ae aoa ems ue o he omessby of he fuuag eoy fe, hese ems o o es omessbe sge-hase fows, hey ae sef o wo-hase fows (V) oa he egee of he essue-eoy oeaos as we as he essue-efomao oeaos A he e, (V) oa he moeua ffuso as we as he sous ssao ems Aohe ofe use baae equaos ubuee moes s he ubue e eegy equao Reag he efo (5) of he ubue e eegy, a be see ha he ae of he Reyos sess eso s we hs ubue e eegy:, ( R ) K R (80) heefoe, he baae equao fo ubue e eegy a be obae by ag haf of he ae of he equao (79): 5

36 ( K ) R,,,,,,,,,, K,, () (V) (V) (V) () K () (8) wh he same eeao of he ffee ems he RHS of Eq (8) he s equaos (79) ae he bass of he wo-hase RSM (Reyos Sess Moe) a he equao (8) s he bass of he wo-hase K-ε moe he same ε equao mus be ee o omee eah of hese moes ue o s omey, hs baae equao w o be eeoe hee (See eg Moe, 995) 4 Hyb aoah fo sese wo-hase fows Whe he fow s sese (eg a bubby o a oe fow), oe of he wo hases (he see o sese hase) s embee o he ohe hase (he ouous hase) ue he fom of see usos hese usos a be fu usos (e bubbes a qu o oes a gas) o so oes (e usy fows fo eame) f he oume fao of he sese hase s ey sma ( << ), he fow s sa o be ue, ohewse s sa o be ese ue o he ssymmey of he wo hases fo hs of fows, s aaageous o esbe he see hase by oos ffee fom hose fo he ouous hase o ow (seos -), he wo hases wee eae symmeay, whou ay assumo o he ommo efae ofguao Hee, we ae a of he esee of hese umeous usos o ee he equaos fo he sese hase fom ouao baaes As hese ouao baaes hae o sese fo he ouous hase, we ae obge o ea he ouous hase as he eeg seo he wo hases ae heefoe esbe ffee maes, hee he ame hyb of he aoah omaso o he symme wo-fu moe eeoe seo, he aaages o oee e ha ae umeous: he equaos fo he sese hase ae moe eaabe, beause hey esembe o he equaos eeoe fo a sge ae 6

37 he esg owege o a sge ae behaou se a fu a be uze o he o he osue of he aeage equaos (eg he foe eee o a ae by he suoug fu, see aagah 7, s efomao ae ) he umbe of feeom egees o esbe he sese hase a be oseaby eue sme ases Fo eame, fo shea g aes, s suffe o oue he hee omoes of he ae asao eoy, ogehe wh he hee omoes of s oao eoy o esbe s moo omeey hs euo of he umbe of feeom egees aows o eue he eso omey whe s ossbe a o ao suous effes yay eouee wh he use of he wo-fu moe Some moa heomea, e e-ae osos o fu aes oaesee a bea-u, ae muh ease o oue he bas oo s he ouo of a sbuo fuo f(ξ;,) o esbe he fu (o so) aes ouao he eo ξ gous he so-ae hase ea ooaes, seaae fom he eea ooaes a by a semoo (;) hs eo oas a he aabes hose o esbe he ouao, e he aes osos, he asaoa eoes, he sze a shae, emeaues We mus sgush bewee a N-ae eso a a ae eso, he seo oe beg a aua ase fom he fs Of ouse, he omey of he eso mus be aae o he omey of he obem ue suy hs s oe of he gea aaages of he meho o aow o sa fom a sme eso a o ogessey ease he omey of hs eso by ag ohe effes, oseg he eousy aque owege We sa by he eso of a ouao of shea aes of ea szes, a sme ase sue eeey by aée (997) a Kaufma (004) 4 eso of a ouao moo-sese sze bu musese eoy 4 efo of aabes a eao of he ma baae equaos hs eso s aaogous o he e heoy of gas moeues We assume a seso of shea aes hag a ommo osa sze, ge by he aus a o he amee a o eah ae s assoae a oso a eoy eazao hase ooaes he aeagg oeao < > mus be uesoo as a esembe aeage ooe o oe ouous hase eazao We assume ha he ae amee s so sma ha fow aou eah ae a be osee as a eeg fow, so ha he Soes ag ge by Eq (84) fo a so ae o he ag (85) fo a ea bubbe a be use hs as smfao s oy fo eagoga uose, a w be eae ae he ae sbuo fuo f(;,) s efe suh ha f(;,) s he obabe umbe of ae ees a me oae a oume eeme aou o a hag a asao eoy bewee a he oao eoy s o osee fo smy hs fs eso e us ose a fe quay ψ (;,) haaezg he sese hase a be a saa, a eo o a eso he assoae aeage fe s efe by he foowg eao: 7

38 whee (, ) ( ;, ) f ( ;, ) ψ ψ (4) ψ s he -weghe aeage of he quay ψ (;,), (,) beg he ae umbe esy whh s efe by: (, ) f ( ;, ) (4) A eame of quay ψ (;,) s he ae eoy sef, whh aeage ges he mea ae eoy: (, ) (, ) (, ) f ( ;, ) (4) he ou of he ae umbe esy a he mea eoy s he ge by he fs oe mome of he eoy sbuo fuo Ohe usefu momes ae he -oe ee momes: (, ) [( )( )] f ( ;, ) (44), he seo oe ee eoy mome s ae he e sess eso a s que aaogous fo he sese hase o he Reyos sess eso fo he ouous hase (see seo ) s ossbe o ee he equao fo f a ey geea mae (eg Aha, 978) Whe he ae eoy s he sge ea hase ooae, hs equao eas:, f f F m ( f ) o f (45) Whee F s he sum of he foes eee o he ae, ug s wegh he ao F/m s F heefoe he ae aeeao, m beg he ae mass, a s he ooa m aeage of he ae aeeao, hag a ae wh eoy he RHS of Eq (45) omes fom e-ae osos Eq (45) s ae a oue-bozma equao Muyg (45) by he quay ψ (;,) a egag oe he eoy sae, he foowg Esog geea equao s eue: ψ C( ψ ) ψ ( ψ ) f o F ψ m C( ψ ) ψ ψ wh : (46) 8

39 whee C( ψ ) esus fom osos wha foows, we ee he equaos fo he zeo-h, fs a seo oe momes of he eoy sbuo fuo hese hee momes oeso o he ae umbe esy baae equao, he ae momeum baae equao a he e sess eso baae equao Pae umbe esy baae equao: ψ Mag ψ Eq (46), he foowg ae umbe esy baae equao s obae: ( ) C() (47) Eug ae bea-u a oaesee, as we as ueao a oase, he RHS of (47) s zeo f hee s o hase hage, he ae mass m s a osa Muyg (47) by m a emag ha m, he ae mass baae equao s eee: ( ) 0 (48) whh s he same ha () whe a assumg o hase hage ( 0) Pae momeum baae equao: ψ Mag ψ Eq (46), he foowg ae momeum baae equao s obae: F ( ) C() m (49) Muyg by he osa ae mass m a ag o aou ha m, Eq (49) a be ewe: ( ) ( ) C(m) F m (40) whee we hae oue he e sess eso (44) whh s a aua fom of Eq Ke sess eso baae equao: ψ ( -, (,)) ( -, (,)) Mag ψ ( -, (,)) ( -, (,)) Eq (46), he foowg e sess eso baae equao s obae: 9

40 ( ) ( ) ( ),,, ) C( m F (4) o, muyg by he osa ae mass m: ( ) ( ) ) C(m m F m F,,, (4) 4 Cosue of he efaa foe fo eeg fows Whe he fow a he ae sae (moso sae) a be osee as a eeg fow, a he assumos of he seo 7 ae gobay sasfe, we a use he esus of he seo 7 o eess he foe F eee by he ouous hase o he ae Assumg a so ae whh s heae ha he suoug fu, he ma foes eee o he ae ae s wegh a he Soes ag ge by Eqs (8)-(84) Mauag hese eessos, he foowg eesso s fou fo he ae aeeao: 8 wh u g m F µ (4) whee s he so-ae ae eaao me he eo u s he ouous hase eoy a he ae oao (he so-ae fu eoy see by he ae) shou be e m ha he esu (4) s a oy whe he ae s a so oe a he fow suoug he ae a be esbe by he eeg fow heoy, hee he ae Reyos umbe efe by (8) mus be muh smae ha oug he esu (4) o he momeum equao (40) ges: ( ) ( ) ) C(m u g (44) whee we hae assume a osa ae eaao me he aeage eoy u s he -weghe aeage of he fu eoy see by he aes s moeg s que eae, a hs quay s ofe eomose by oug he foowg seso (o f) eoy (eg aée, 997): V u (45) whee V s he maoso seso eoy a s he ouous mea eoy ( he usua sese of he wo-fu moe) Hee, Eq (44) a be ewe: 40

41 V ( ) ( ) g C(m) (46) Aog o Jaso (997), he oso em aeag he momeum baae fo he aes a be ewe by usg he foowg ayo eeome: whee ( m s ) : m s C(m) (47) s a s ae aeage osoa sess esos of oe a eseey he equao (47) s a ayo eeome Fo o be a, he suesse ems hs eeome shou beome smae a smae heefoe, assumg ha he seo em Eq (47) s eggby sma omaso o he fs, a subsug he emag em Eq (46) ges: ( ) ( ( s ) g V (48) We a omae Eq (48) o he assa wo-fu moe momeum equao () we fo he sese hase ( ) Assumg ha he mea ae eoy a be assmae o he hase aeage eoy (whh s us aomaey ue beause aes o aou a he ae ea moos as we as he ae oao, whh ae o ae o aou hs usaes he smfaos bough by he e heoy aoah eeoe hee omaso o he wo-fu aoah), we a omae he RHS of Eqs () a (48) he aeage essue a sous sesses fo he ouous hase ae eae by he osoa sesses fo he sese hase he same mae, he Reyos sess eso fo he ouous hase s eae by he e sess eso fo he sese hase A he e, we see ha a aomae fom of he momeum efaa asfe (a ue he assumos eae hs aagah) s ge by: M V (49) he ouous hase efaa momeum asfe em M s he eue fom he efaa momeum baae (0) a a aoae eesso fo he mue momeum soue M m he e sess eso obeys o he baae equao (4) sg Eq (4), he em og he efaa foe (4) a be ewe as: 4

42 u u u u u u m F m F (40) he eao (40) oes he e sess eso as we as he symme a of he fu-ae eoy oeao eso u Subsug (40) o (4) ges: ( ) ( ) (V) ) C(m () u u () (),,, (4) he HS of Eq (4) s smy he aso of he e sess eso omoe by he mea sese hase eoy he fou ems he RHS of Eq (4) hae he foowg hysa sgfae: (): egee of he e eoy oeao he baae equao fo he e eoy oeao ou be ee fom he Esog geea equao (46) hs em s ey sma o he em () he RHS of he Reyos sess eso baae equao (79) eeoe he oe of he wo-fu moe he wo ems () Eq (4) ae ouo ems by he mea eoy gaes hey ee o fuhe moeg hey ae aso ey sma o he ouo ems () he RHS of Eq (79) he em () he RHS of Eq (4) eeses he eao wh he ubue moo of he ouous hase he as em (V) s ue o e-ae osos 4 eso of a ouao mu-sese sze (a eoy) hs seo, a seo ea hase ooae, he ae amee, s oue he fu o so aes ae assume o ea he shea shae, bu he amee aes fom oe ae o aohe oe, a a aso ay aog oe ae s aeoy he ae sbuo fuo f(,;,) s ow efe suh ha f(,;,) s he obabe umbe of ae ees a me oae a oume eeme aou o, hag a amee bewee a a hag a asao eoy bewee a he oue-bozma equao (45) s geeaze o he foowg equao: ( ) o f, f, m F f f f (4) 4

43 he -weghe aeage quay (4) s smy geeaze o he foowg efo: (, ) ψ (, ) ψ (,;, ) f (,;, ) (4) whee Ω s a abbeae oao fo he ea hase sae eeme Whe aes hae ffee szes, he masses ae aso ffee, a oe has aaage o oue he foowg Fae (o mass weghe) aeage fo he sese hase: (, ) (, ) ψ (, ) m( ;, ) ψ (,;, ) f (,;, ) Ω Ω (44) whee a ae eae o he sese hase whou ambguy o he oao, se he sese hase s he oy oe osee hee he ae mass m s ge by: π ( ) (, ) 6 m ;, (45) ue o he shea aes assumo Muyg Eq (4) by mψ a egag he esug equao oe he ea hase sae, he foowg equao s obae: mψ ( ψ ) fω ( ψ ) ( mψ) mψ, fω mψ fω ( ψ), fω C m (46) Whee, s he h omoe of he ae aeeao, s ge by:, F, (47) m Fo geomea momes (quaes o weghe by he ae mass), a equao aaogous o Eq (46) a be obae by muyg Eq (4) by ψ (sea of mψ) he egag hs ges: ψ ( ψ ) fω ( ψ ) ψ, fω ψ ψfω, fω C ( ψ) (48) 4

44 4 Mass a momeum baae equaos oe o ee he mass a momeum baae equaos fo he sese hase, oe mus fs ee eessos fo he agaga eaes of he ae amee / a eoy / aeag Eqs (46) a (48) o f he eas, we w assume ha he osee aes ae gas bubbes a ouous qu he fs eae eeses he amee aao measue aog he bubbe ah hs sze aao s ue o he gas omessby o oe ha, a o he hase hage (aozao o oesao) o he ohe ha he bubbe mass aao m/ s oy ue o hase hage, hee we a we: m π 6 m& π (49) whee m& s efe as he bubbe mass ga e u sufae e u me ue o hase hage m& s he mea aue of (9) oe he bubbe sufae he agaga eae / beg ae a he bubbe eoy, s easy o eue fom (49): m& (40) Fo he sae of geeay, we w aso ee he eao foe moe fo bubbes qu wh ay aue of he bubbe Reyos umbe he suy of he foes eee o a bubbe by he suoug qu s a que ffu as Aog o Me & Peao (00), a geea fom fo he bubbe momeum equao a be oose whh eas ag, essue gae, ae mass a gay foes (see aso Maey & Rey, 98; ago, 98) Aohe moa foe ae he f foe (Auo, 987) s mssg he eso of Me & Peao (00), bu fo ou eagoga uose, s suffe o sa fom he wo of Me & Peao (00) he momeum baae equao of a sge bubbe eas: π 6 π π π u π π u g g C u ( u) CA bubbe wegh geeaze Ahmee's foe ag foe aemass foe (4) Fou foes ae ae o aou he RHS of Eq (4), hese ae he bubbe wegh, he geeaze Ahmee s foe, he ag foe a he ae mass foe he eo fe u s he qu eoy see by he bubbe (he so-ae ueube eoy) he ag foe s a geeaze eesso of he foe aeay see fo eeg fows (seo 7) a oes a ag oeffe C whh ees o be moee as a fuo of he bubbe amee, bubbe Reyos umbe Ema eessos fo C a be fou sh (990) he as foe, he ae mass foe, s oooa o he qu mass sae by he bubbe mmeso ( eog he ouous hase esy, e he qu esy) hs foe s a bae o he bubbe aeeao eaey o he suoug qu he ae mass oeffe C A esseay ees o he bubbe shae Fo shea bubbes, s equa o oe haf (C A ½) g Eq (4) by he bubbe mass a egoug he ems oooa o he bubbe aeeao he HS ges he foowg equao: 44

45 u u ( C A ) g (4) C A whee s a bubbe eaao me aaogous o he oe efe Eq (4) fo Soes ag, a s ge hee as a fuo of he ag oeffe: 4 (4) C u oug he efos of he foowg oeffes: m ( C ) A ˆ CA, b ˆ, ˆ (44) C A C A he foowg fa fom of he bubbe aeeao s obae: u u b g m (45) Now we a ee he aeage mass a momeum baae equaos fo he bubbe swam Mag ψ Eq (46) ges: m ( ) fω ( ) ( m) fω m fω (46) ( ) whee we u C m 0 beause he bubbe mass s a osoa aa oug (40) o (46) a smfyg yes: ( ) π mfω & (47) whh s aaogous o he mass baae equao () obae he oe of he wofu moe Mag ψ Eq (46) ges: m ( ) fω ( ) ( m) m fω m fω C( m) fω (48) Fs subsug (40) o (48) a smfyg ges: 45

46 ( ) ( ) m fω mπ fω C( m & ) (49) he fs em he RHS s he bubbe aeeao em, he bubbe aeeao beg ge by Eq (45) he seo em he RHS s he aeage eo foe a he as em s he oso em whh a be ewe as he egee of a eso (see Eq 47) efg a aeage sese eoy weghe by hase hage: m& π fω (440) a eeog he aeage of he esoa ou of eoes he HS of (49) as he ou of he aeage eoes us he aeage of he ou of fuuag eoes, as has bee oe seos a 4, Eq (49) beomes: ( ) ( ) ( ( s ) wh : oug Eq (45) o he as em of (44), we f: (44) ( ) ( ) ( ( s ) u m b u g (44) he momeum baae (44) s he geeaze fom of (48) s he ase of mu-sze bubbes a ug hase hage a he ae-mass effe s use ees fuhe moeg 4 eomea momes baae equaos efg he h oe mome of he amee sbuo fuo by he foowg eao: S fω (44) he baae equao fo he mome S s obae by mag ψ he geea equao (48) he esu s: S ( ) S mfω C( ) & (444) Seea usefu equaos a be ee fom (444) by ag ffee aues fo s ea fom Eq (44) ha he zeoh oe mome S 0 oesos o he bubbe umbe esy he o fao s oooa o he h oe mome ( πs /6) as a be see by mag ψ Eq (44) he ese oe of he e heoy, he efaa aea 46

47 oeao a (Eq 64) s oooa o he seo oe mome (a πs ) We a aso efe a fe umbe of mea bubbe amees by he foowg eao: q q S (445) S q he mahemaa fom fo he bubbe amee sbuo fuo a he be esume o ose he moeg of he bubbe amees Fo eame, Kam e a (00) hoose o moe he maga sbuo fuo f(;,) by a og-oma aw: [ og( / )] ( ) ( ) 00 f ;,, e (446) πσˆ σˆ whee 00 s he mea amee a σˆ s a wh aamee hese wo aamees ee o he oso a me a, ogehe wh, omeey efe he bubbe amee sbuo fuo a hese oso a me he (ea) osue ees o ow he o fao ogehe wh he wo aua momes S a S hough he eaos: 6S 6 5σˆ / σ ˆ, 00 e (447) πs πs Hee oe ees o auae S a S by usg he baae equaos: S S ( ) S & mfω C( ) 4 ( ) S & mfω C( ) V (448) he equaos (448) a be soe umeay whe a he ems,, a V hae bee moee by aoae osue aws hese ems oeso o () aso of he mome by he mea a fuuag eoes, () gas omessby, () hase hage a (V) oaesee, beau, ueao a oase he moeg of a hese effes s a ome as ha w o be eeoe hs ouo o wo-hase fows Of ouse, he eesso (446) fo he bubbe amee sbuo fuo s oy oe aae Ohe eessos, moe o ess ome, hae bee oose he eaue o moe he seum of he bubbe amees 5 Vaous sea ases 5 ema eoy 47

48 ue o he buoyay (Ahmee) foe, he eae bubbe moo a qu s esseay uwas (bubbe sg eoy) he ema eoy s efe as he aua bubbe sg eoy a equbum bewee he ag foe a he Ahmee foe, e whe a he ohe foes hae esse o a o he bubbe hs s a ey moa oo se aows obag he bubbe ag oeffe by measug he ema eoy whh s aessbe by sua obseao he bewee he ag oeffe a he ema eoy s smy obae by equazg he ag foe a he e buoyay foe (he Ahmee foe ogehe wh he bubbe wegh) Fo a sge (soae) bubbe a qu, hs ges (by smfyg Eq 4): π π (5) 6 4 ( ) g C whee he e eas ha he bubbe s aoe a fe qu meum he ema eoy a be auae ey fom (5) f he ag oeffe s ow: ( ) 4 g (5) C Fo a ey sma bubbe, he eeg fow assumo a be oe (seo 7), a he ag oeffe s ge by Eq (85) Subsuo of (85) o (5) ges he aue of he ema eoy of a ea bubbe eeg fow: ( ) g (5) µ he eaos (5)-(5) hae bee obae fo a sge bubbe ase Now we w seah a equae oo fo a bubby fow wh a age umbe of bubbes sag fom he wo fu moe momeum equaos () We assume ha he wo hases o o aeeae a ha he moeua a Reyos sess esos a be egee We aso assume o hase hage Ou as assumo s o assume ha a he ffee aeage essues aeag Eq () ae equa o a sge aeage essue, a assumo ommoy use wo-fu oes e hese assumos, he equaos () we fo he wo hases egeeae o he foowg oes: 0 0 g M g M (54) We a emae he mea essue gae bewee he wo equaos (54) by muyg he fs oe by, he seo oe by a he subag hs ye: ( ) g M M (55) Negeg he ffeees bewee M a M (whh s ohee wh he assumos eousy mae) a egeg he momeum efaa soue M m (e egeg sufae eso), we a we M - M M hee: 48

49 ( ) g M (56) he HS of Eq (56) oas he buoyay foes he RHS oas he efaa foes (ag, ae mass ) Coseg ha he ag foe s he oy oe o a o eah bubbe, beause we ae seahg fo he aeage ema eoy, he RHS of (56) mus be eae by he aeage ag foe e u oume of he wo-hase mue hs aeage ag foe s ge by he equao (49) ogehe wh he equao (4) Negeg he seso eoy V a eag by he mea ema eoy V, he equao (56) beomes: C ( ) g V (57) 4 whee a he bubbes ae assume o hae he same amee he mea ema eoy a be obae ey fom (57): V 4 ( ) (58) C g he omaso of he wo eessos (5) a (58) of he ema eoy shows he aeaae of a fao / he mea ema eoy, whh was o aea he ema eoy of a sge bubbe Fs shou be emae ha he ase of a soae bubbe a fe qu meum, he qu fao of esee es o hee he ffeee bewee (5) a (58) he sge bubbe ase s eggbe Fo a bubbe swam, shou be emae ha he Ahmee foe eee o a ge bubbe omes fom he bubby mue suoug he osee bubbe a o fom he qu aoe ue o he esee of he ohe bubbes, he bubby mue s ghe ha he qu as s haaeze by he foowg esy: (59) M whh s he aeage fom of he fs equao () Reag he HS of (5) by he mue esy (59), he esu (58) s eee ae of (5), showg he omee omaby bewee he wo fu moe a a baae of foes o a ge bubbe, a eas he sme oos sue hee 6 he eame of he NEPNE_CF oe hs seo, we ge he eame of he NEPNE_CF oe eeoe by EF a CEA fo he umea sues of wo-hase fows sme as we as ome geomees hs oe uses a geeaze wo-fu moe, geeaze meag ha a ose moe ha wo hases Howee, fo he sae of smy, we w ose oy he ase of wo hases hee Nowaays, hs oe has esseay hee yes of aaos: gas-so fows (so aes sese a ouous gas hase), a wo yes of gas-qu fows: safe fows a bog bubby fows oe o f he eas, we w ose oy he 49

50 as aao (bog bubby fows) he bas se of equaos soe o smuae suh a fow s a s-fo oe oas: wo mass baae equaos wo momeum baae equaos wo eegy baae equaos he fom of oa ehay baae equaos hs bas se of equaos s omusoy o auae bog bubby fows o hese s equaos, auay baae equaos a be ae hese aoa baae equaos ae may of wo yes: ubuee equaos use o e he ubuee esg eah hase eomea momes baae equaos use o e he aeage ooogy of he bubbe swam wha foows, we esbe eah of hese equaos he smfe eso use he oe, ogehe wh he ma assumos oe o oba hese smfe equaos fom he ea baae equaos eeoe he eeg seos 6 he mass baae equaos he NEPNE_CF oe he mass baae equaos ae ge by he equao () ogehe wh he um oo () he RHS of hs equao s ue o hase hage (aozao o oesao) he oy aoa eso we mus ge s ha hs RHS s s o wo ffee obuos he fs obuo s he hase hage hough he sufaes of he aeay esg bubbes he seo obuo s he ueao of ew bubbes Nueao a assay be e o wa ueao (heeogeeous ueao) a ueao he qu bu (homogeeous ueao) Hee, oy he heeogeeous ueao s osee a moee heefoe, he equao () s ewe he foowg fom: u (6) u whee s he a of assoae o hase hage ohe ha ueao a s he ueao a he a a eese aozao o oesao bu he ueao u a oy oesos o aozao Obousy, eah a of efes he eao () eeey 6 he momeum baae equaos he NEPNE_CF oe he momeum baae equaos he NEPNE_CF oe ae base o he equao () smfe by some aoa assumos hese assumos ae he foowg oes: (H): No so s mae bewee he ffee aeage essues A he aeage essues aeag Eq () ae assume o be ea o a uque mea essue 50

51 (H): No so s mae bewee he wo aeage eoes a (H): he as em of Eq () s egee e he assumos (H)-(H), Eq () eues o: ( ( ) g M S (6) he as em of Eq (6) has bee ae o ae o aou sea hyss, e fo eame he efuga foe whe he fow oma s o aea Now we mus ge he osue eaos fo he aeage sous sess eso, he Reyos sess eso a he efaa momeum asfe Assumg ha eah hase s a Newoa fu, he aeage sous sess eso a be obae by oey aeagg he osue eao fo he oa saaeous (moso) sous sess eso hs has bee oe by sh (975) a he esu s: µ ( ) µ µ µ ( ) ( ) ( ) (6) whee he foowg assumo has bee mae: (H4): he yam sosy µ s a osa f we ege aso he fuuaos of he esy, he hee s o so bewee he Fae aeage eoy a he hase aeage eoy foowg aoa assumo: (H5): he esy of hase oes o fuuae (e s equa o s aeage aue) he eesso fo he aeage sous sess eso (6) a be smfe: heefoe, ue he µ µ (64) he ems og he gae of he mea eoy ae ae bu efomao eso by sh (975) a he aoa ems og he fuuag eoes ogehe wh he efaes oma a fuo of esee ae ae efaa ea efomao eso by hs auho he NEPNE_CF oe, we mae he foowg aoa assumo: (H6): he efaa ea efomao eso s egee 5