Bubble interactins in liquid/gas flws L. VAN WIJNGAARDEN Technlgical University Twente, Enschede, The Netherlands Abstract. The system f equatins, usually emplyed fr unsteady liquid/gas flws, has cmplex characteristics. This as well as ther facts have led t the search fr a mre accurate descriptin f effects assciated with relative mtin. Fr liquid/bubble systems the fluctuatins resulting frm hydrdynamic interactin between the bubbles may be taken int accunt in the same way as particle interactins in the thery f viscus suspensins. This is illustrated fr the pressure. In a descriptin accurate up till the third pwer f the vid fractin tw-bubble interactins are f primary imprtance. Numerically btained results fr the relative mtin in bubble pairs are presented and interpreted with help f simplified equatins frm which cnclusins can be drawn in an analytic way. 1. Intrductin In recent years the numerical instabilities which arise in the cmputatin f transients in tw-phase flws, have stimulated research n the interactin between phases. Such interactins, inertial r f ther nature, give rise t additinal terms in the equatins f mtin. These terms might, in rendering real characteristics, prevent instabilities. In ur labratry a study is in prgress regarding the interactin effects in a mixture f massless spheres and a perfect liquid. Such a fluid reasnably apprximates a bubbly flw under circumstances in which bubbles are small enugh t be kept spherical by surface tensin and in which surface active agents are absent in the liquid. The latter cnditin means that the flw arund an individual bubble in the mixture can be with gd accuracy apprximated by a ptential flw. The interactin effects prduce in any pint in the fluid fluctuating pressures and velcities. Just as in turbulence ne is interested in mean values. When a distinctin between fluctuatins and mean values is made and sme type f averaging is carried ut (ensemble averaging, vlume averaging r therwise) effects f the fluctuatins n the stress in the fluid remain. These have been calculated by Vinv and Petrv [7] with the use f a cell methd. Such a methd hwever leads t results f unknwn accuracy. Our apprach is similar t that used by Batchelr and his assciates, see e.g. [2] in the thery f suspensins dminated by viscus effects. As an illustratin we deal here with the calculatin f the average r bulk pressure in the inhmgeneus liquid. 331 Applied Scientific Research 38:331-339 (1982) 0003-6994/82/0384-0331 $01.35. 1982 Martinus NifhffPublishers, The Hague. Printed in the Netherlands.
332 2. Bulk pressure in bubbly liquid Cnsider a large vlume V f a suspensin f N bubbles f zer mass in a perfect liquid. A pint in the fluid, whether in a bubble r in liquid, is indicated with its psitin vectr x. We assume that the ensemble average f the velcity u ver all pssible cnfiguratins f the N bubbles is given as U, 1 N! fu (x, CN)P(CN) dcn = U, (1) where P(CN) is the prbability distributin fn bubbles, r (u) (x) = U. (2) The vlume flw U can be divided in a gas flw (assuming that the bubbles are filled with gas f negligible density) aug, ~ being the cncentratin f gas by vlume, and a vlume flw (1 -- cou ~ f liquid, ~U z + (1 --a)(1 -a)u~ = U0. (3) Our aim eventually is t frmulate equatins f mtin fr the averaged quantities. If we carry ut the ensemble äveraging, the average pressure (p) mäkes its appearance. Assuming that it is permitted t replace ensemble averaging by vlume averaging (fr this statistical hrngeneity is required) we have if~ if if pdv+ 2 -~ pdv. (p> = ~. pdv = ~- v t VB In this equatin, V l dentes the vlume ccupied by liquid and V B the vlume f ne f the N identical bubbles. Next we define (P)z as the pressure averaged ver the liquid alne. Then i f (p) -- (1--a)(ph + ~~- pdv? VB 1 f {P -- (P)t } d V. = <p~~ + E -v. v, Upn intrducing the number density n = N/V and the quantity S given by we write (4) as S = ( (P -- (P)l)dV, (5) J VB (P) = (P)t +n(s). (6) The quantity S in (5) is similar t the 'particle stress' discussed in [1]. In particular it was shwn there that (5) may be wfitten as (4)
333 S = ~ P--.(P)t)r'dA, (7) the integratin being ver a surface A B which lies just at the liquid side f the interface between liquid and gas. When ~ ~ 1 we may in a first apprximatin assume each bubble t be alne in the liquid. Far frm the bubble, the pressure is ip) and the velcity U. With bubble velcity Ue, the flw is presented by the ptential q5 = U "r+(u --Uu)'a 3r 2r 3, (8) where a is the radius f a bubble and r gives the psitin f a pint with respect t the centre f the bubble. Upn calculatin f p with Bernulli's Therem (U may depend n time) and upn carrying ut f the integratin in (7), we find fr S, 7ra 3 S ~ S - 3 {IU--Ug[} 2. (9) Accrdingly we find fr the bulk pressure, frm (6), ip) = ip)z-¼ c~p{[u~--u«l} 2 + &), (10) p being the density f the liquid. (Vinv and Petrv [7] find ½ where ccurs in [10].) After a similar calculatin f the 'Reynlds stress', by calculatin f/puu) we btain, see [10], t O (a) z, fr ne-dimensinal flw in x directin p(1 -- a) + t x / - Ox (P)t + 2-Ö ~xx {a(u«- ul) z }, (11) which may be cmpared with the result fund by Van Beek [8] in his cntributin t this Sympsium. In the next rder f apprximatin, we start with the exact expressin fr is), i is) = ~., fs(x0, CN)P(CN/X) dcn, (12) where S is given by (5) fr a bubble with centre in X, and where P(CN/X) is the prbability distributin fr N bubbles, an additinal ne being in X (the s-called cnditinal prbability). We nw replace in each cnfiguratin the N bubbles by just ne bubble situated in xl and with P(xl/X) as the prbability f finding a bubble centred at xl, When there is ne centred in X, we have
334 <S> = S + j'(s--s) (X,Xl)P (xl/x)d3xl. (13) The integratin in (13) is ver all pssible psitins f the secnd bubble. In S, as given by (5) r (7), this time p is the pressure at distance r (see Figure 1) frm the centre f ne bubble, anther ne having its centre in xl. The ptential fr the flw invlving tw spheres can in the present cntext cnveniently be expressed in terms f twin spherical expansins as used in [4] and [9]. When this is dne and the assciated pressure is intrduced in (13) we are cnfrnted with the fact that (S--S) behaves at large values f (xl ±x0) like {Ixl--Xl} -3 and that therefre the integral is nt unifrmly cnvergent. This prblem can be slved by using the nrmalizatin technique, reviewed in [2]. Figure 1. Tw bubbles, in x and x~, immersed in a liquid which acquires a velcity UH(t). Anther prblem is the determinatin f P(xl/x0). In cntrast t inhmgeneus media with a fixed structure, like a prus bed, the prbability distributin is in general affected by the flw itself. The questin whether a statinary distributin is reached can nly be cnsidered after the prblem f determining relative mtin in a pair f bubbles has been slved. Bth fr the calculatin f (S --S) and fr the investigatin fp(xl/x) we turn ur attentin t relative mtin between tw bubbles. 3. Relative mtin in a pair f bubbles T be specific we cnsider pairs f bubbles in a bubbly flw which is at t = 0 instantaneusly accelerated t a velcity U. The equatin f mtin is fr each bubble f pda = 0, (14) A because the bubble mass is neglected. The pressure p can be derived, by using Bernulli's Therem, frm the ptential f the flw. This cnsists, if we cnsider a specific bubble with centre in X, f a part which is regular in X, with gradient ur and a singular part. The latter can be represented by mnples, diples and multiples situated in pints xù within the bubble (fr a spherical bubble x~ cincides with X). The frce n a bubble can be expressed in the strength Mq f these singularities and the derivatives in X f il R.
335 This has been dne recently by Landweber and Milh [6]. Hefe we disregard, fr simplicity, changes f the vlume r f a bubble - inclusin ffers n essentially new prblems - and write using the result in [6] and denting the velcity f a bubble with v, the relatin (14) as with d dt {prv--4rrm1} + F = 0, (15) ~q F = --4np ~Mq--- (Un)x=xs. (16) ~x q The part f the frce which is indicated with F is due t the 'velcity squared' term in Bernulli's Therem. Mq is the singularity f rder q(m1 is a diple, M2 a quadruple etc.) and is multiplied in the expressin fr the frce with the gradient f rder q f un in X. At time t = 0 + nly the terms in the braces in (15) are effective and the resulting velcity is, see [9], the same fr each bubble and f magnitude v = 3U0 + O(~). The M a can be fund frm the ptential mentined abve and equatins fr the velcities f the bubbles in X and in xl can be fund by applying (15) t each f them. These are cmplicated expressins because the Mq cntain the unknwn velcities f the bubbles. Nte that because f the ccurrence f F relative mtin with velcity dr v - (17) dt develpes fr t > 0, R being the distance xl - x between the tw centres. Next by cmbinatin f these equatins an equatin fr V can be cnstructed, which has t be slved numerically. This prgram has been carried ut in [5] by Knibbe. Sme trajectries R = R(t) btained frm this by Biesheuvel are shwn in Figure 2. Since the analysis is quite cmplicated, it is difficult t understand and interprete the results n the basis f the full prblem. A qualitative insight can be btained by taking nly the leading singularities, in terms f the parameter (a/r) int accunt. The leading term in O/Ox(un) in (16) is the gradient f the velcity induced in X by the diple in xl The latter behaves as Ua3/R a and because the initial diple strength is Ua a we can neglect the variatin in the diple strength due t the relative mtin. The leading term therefre in F is F = -- 47rpa3(U V)U R (), (18) where u() indicates the velcity in X and V indicates the gradient with respect t X. The velcity un() is given by
336 ~ -c~ J ù~ ~ H ~ ~ ~ /- ///f--,- ~- /' / / / - ~ OO "# ù' ~ 0~ ~ c~'~~ 0 " c- ~' / e~ Ló L -.g.~ L~ II. 0 m-, ù. L. m +. J--- "N c~: ~~ 00'~ 00"ù OC '~~' OO" OC 0 '~~c.'~ ò.~ 1 ~S I L~ 0
337 U a a. r 1 ur( ) = --V r~ (19) and is in this apprximatin equal t un (1), the velcity induced in Xl. Frm subtractin f the equatins fr V and vl, we find in this apprximatin d Ua a R d-~ {--½pr(v --Vl)} = 87rpa~(U VR)VR Ra (20) With V 1 -- V 0 write this as dr = dr/dt and wrking ut the righlhand side f (20) we can --+VG = 0, dt with G _ 12a 3 U 2 R3 (3 cs 2 0-- 1), (21) 0 being the angle between U and R as indicated in Figure 1 and the dt nr indicating the time derivative. The relatin (21) means that there is a cnstant f the mtin, the energy, which is in spherical plar crdinates ½(/~2 +R2~2)+ G = G,say where accunt has been taken f the initial cnditins n/~ and 0. Frm analytical mechanics it fllws that there is a Lagrangian L, L = ½(R2 +R202)_G, (23) the Euler equatins f which prvide the equatins f mtin (20). These equatins cannt be slved anälytically but sme imprtant cnclusins can nevertheless be drawn. Lack f space prevents t give the analysis here fr which we refer t [3]. Here we summarize these cnclusins, referring t Figure 3. 2 Figure 3. Summary f results fr R(t) accrding t apprximate thery.
338 (i) When at t--0 the separatin vectr R ends in regins I and II, where G>0, the bubbles escape frm each ther. Fr R>~R, /~ ~(2G) 1/2 and 0 ~0. The angle at which G changes sign is 0 = 0e ~ 55. (ii) When initially G < 0, which places R in regins III r IV, the bubbles apprach each ther. The line f centres tends t the vertical in Figure 3. (iii) When initially 0 ~0e, scillatry mtins are pssible. R is apprximately cnstant, which gives, frm (21) and (23) r Ö-~2 2 sin20 = 0, ~22-02 = ~22 (cs 200 --cs 20). 36a 3 U~ Hwever R is nly apprximately cnstant. Eventually the separatin distance becmes either large r small. These qualitative prperties f trajectries agree quite weu with the cmputed trajectries. FinaUy we cnsider the questin f the prbability distributin. The pair prbability distributin P(x, x + R) changes, because f the relative mfin, accrding t bp -- + v. 0t (ve) = 0. It can be shwn, see [3], that V V = O, whence dp --= O. (25) dt This means that if we mve alng a trajectry in R, t space, P remains cnstänt. If therefre P is randm at t = 0, e.g.p = n, the prbability density remains unifrm. This is in cntrast t what happens in suspensins dminated by viscsity where the prbability density is affected by the relative mtin, and in sme cases cannt even be determined wing t the ccurrence f clsed trajectries. R~ References 1. Batchelr GK (1970) The stress system in a suspensin f frce-free particles. J Fluid Mech 41: 545. 2. Batchelr GK (1974) Transprt prperties f tw-phase materials with randm structure. Ann Rev Fluid Mech 6: 227. 3. Biesheuvel A and Van Wijngaarden L (t be published). 4. Jeffrey DJ (1973) Cnductin thrugh a randm suspensin f spheres. Prc R Sc Lndn A 335: 355. 5. Knibbe P (1981) Master's Thesis. Technlgical University Twente, The Netherlands.
339 6. Landweber L and Milh T (1980) Unsteady Lagally therem fr multiples and defrmable bdies. J Fluid Mech 96: 33. 7. Vinv OV and Petrv AG (1977). On the stress tensr in a fluid cntaining disperse particles. PMM 41: 362. 8. Van Beek P (1981) An O(c0-accurate mdel fr liquid-bubble dispersins. Appl Sc Res. 9. Van Wijngaarden L (1976) Hydrdynamic interactin between gas bubbles in liquid. J Fluid Mech 77: 27. 10. Van Wijngaarden L Jn (1980) On the mathematical mdeling f tw-phase flws. Prc IVth Int Meeting n water clumn separatin, Cagliari, 1979.