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2110 D. Gerlach et al. / Chemical Engineering Science 62 (2007) 2109 – 2125

Starting from the one-stage models of Davidson and Schüler(1960a,b) for dynamic formation in viscous and inviscid liquids,improved models have been proposed by means of two-stageand non-spherical models (Tsuge, 1986 ). In these models theinuence of the inow conditions of the gas through the orice,i.e., constant gas or constant pressure regime, were intensively

investigated. Boundary-integral methods were also successfullyemployed to examine the formation process, as by Og̃uz andProsperetti (1993) , Wong et al. (1998) and Higuera (2005) .

In the present approach, full numerical simulations were per-formed using a combination of the volume-of-uid and level-set methods ( Sussman and Puckett, 2000; Son and Hur, 2002 ),where a single set of Navier–Stokes equations in an axisym-metric formulation was solved in the computational domain.The parameters studied were the orice ow rate and radius,the wettability of the orice material and the inuence of sur-face tension and the liquid density and viscosity. The constantorice ow rate was increased stepwise until a transition fromsingle periodic (SP) to double periodic (DP) formation was de-tected, i.e., until the inuence of a bubble on the followingone is such that the second bubble detaches earlier comparedwith the SP case. This phenomenon was also observed experi-mentally by Kyriakides et al. (1997) , Zhang and Shoji (2001)and Tufaile and Sartorelli (2002) . As a consequence, the trail-ing bubble of each pair is smaller than the leading bubble andtwo distinct but constant detachment periods exist which repeatregularly. In this DP regime, the two interacting bubbles maycoalesce some distance above the orice or directly duringformation ( Zhang and Shoji, 2001 ), the latter case occurringat higher ow rates is not considered here. The present work focuses on the periodic bubble formation process, but also

provides information about the conditions, under which theformation regime changes from a SP to a DP regime.

Air and water at 20 ◦ C served as reference working uidsfor the computations. Based on their data, the uid propertieswere varied individually to study their inuence on the forma-tion process. Less information is typically given in the liter-ature about the orice material used, as discussed by Ponterand Surati (1997) , which can make a comparison of differentstudies difcult. Most of the models available are based onthe assumption that the bubble base coincides with the oricerim during formation, which can be an oversimplication de-pending on the orice material employed. Attempts have been

made here to examine the inuence of the orice material bymeans of a static model for the contact line movement. Basedon this background, the present study investigated bubble for-mation at submerged orices under exactly dened conditionsand demonstrated that the present numerical approach can im-prove the understanding of the process of bubble formation. Asan extension to previous studies dealing with the inuence of uid properties on the bubble formation process in the constantow regime (Kumar and Kuloor, 1970; Terasaka and Tsuge,1993; Jamialahmadi et al., 2001 ), the present work considersalso the inuence of the properties in combination with bubbleinteractions (pairing) and the wettability of the orice plate.

The present paper is arranged as follows. In the next sec-tion, the problem studied here is described and non-dimensional

numbers are used to estimate the importance of the forces in-uencing the bubble formation process. A short description of the mathematical model and the numerical method is given.The results of a numerical parameter study are provided inSection 3 considering the inuence of the operating conditionsand the uid properties. The phenomenon of period doubling

of the bubble formation process, i.e., the transition from SP toDP formation, is discussed in Section 3.3. Finally, an existingcorrelation for the bubble volume based on non-dimensionalquantities was used to validate the results with experiments.

2. Analysis

2.1. Problem description

The formation and detachment of gas bubbles at single sub-merged orices is considered. Owing to the gas ow throughthe orice, the bubble volume increases continuously. When

the lift forces exceed the retarding forces (viscous and capil-lary forces) the bubble detaches and rises upwards. For the lowand medium ow rates involved in the present work, the pro-cess of growth and detachment is assumed to be axisymmetric,whereby the origin of the (r, z) coordinate system is placed atthe center of the orice rim as indicated in Fig. 1 . The ow ratethrough the orice of radius R o is constant and thus indepen-dent of the pressure variations in the bubble. This is called theconstant ow rate regime in contrast to the constant pressureregime ( Clift et al., 1978 ). The properties of the liquid phase arethe density l and the viscosity l , the respective properties of the gas phase being g and g . The surface tension betweenthe two uids is assumed to be constant. A static contact anglemodel is applied to characterize the movement of the triplecontact line of the gas–liquid–solid components as described

Fig. 1. Computational domain and boundary conditions.

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The level-set function is introduced in addition to F , becauseconvenient formulae arise for the calculation of the local meancurvature ( ) in the surface tension force term, Eq. (9). istypically dened as a signed distance function from the inter-face (see, e.g., Sussman et al., 1994 ). Liquid regions are regionsin which > 0, whereas in gas regions < 0. The interface is

implicitly given by the = 0 contour. The mean curvature inEq. (9) is

= ·| |

. (11)

Since the uid type remains constant along particle paths, thevoid fraction F is passively advected byj F j t

+ v · F = 0. (12)

Since the motion of the interface is computed as a combinedvolume-of-uid and level-set method, it is useful to solve atransport equation for the level-set function similar to Eq. (12)

j

j t + v · = 0. (13)

The numerical discretization of the above given equations isdone as described in detail in a previous work by the authors(Gerlachet al., 2006 ). In this study, three different volume-of-uidmethods were compared and validated by means of their appli-cation to surface tension dominant two-phase ow problems.The combined volume-of-uid and level-set method used herewas one method under consideration. Hence, only a brief summary and some extensions are described in the follow-

ing. The governing equations are discretized on a Cartesianand equidistant grid, where the ow variables are dened ina staggered manner ( Harlow and Welch, 1965 ). For the bub-ble formation problem studied here, the code ( Gerlach et al.,2006) was extended to allow computations in axisymmetriccoordinates. The components to capture the interface, i.e., thefront-capturing method, the interface reconstruction and themodel to include surface tension forces are provided in Gerlachet al. (2006) . The combined volume-of-uid and level-setmethod takes advantage of a mass conserving volume-of-uiddiscretization of the transport equation to advance the inter-face, and a smooth level-set function across the interface to

determine the interface normal and the curvature (Eq. (11)), assuggested by Sussman and Puckett (2000) and Son and Hur(2002) . For time integration of the continuity and momentumequations (Eqs. (5)–(8)), a second-order Euler time steppingscheme was employed.

A sketch of the computational domain is provided in Fig. 1 .The origin of the axisymmetric ( r , z) coordinate system is placedat the center of the orice rim. The orice radius is denoted byR o . The boundary conditions at the centerline r = 0 and at theright boundary r = R (Fig. 1) are described as symmetry or slipboundary conditions, i.e., u = 0 and j w/ j r = j F/ j r = j / j r = 0.Neumann conditions are dened at the outow (z = Z ), i.e.,the gradient normal to the boundary of all quantities is zero.At the inow , a parabolic prole of the velocity component

in z direction is prescribed as

w(r) = 2wo 1 −r

R0

2

, (14)

where wo = Q̇/ R 2o is the averaged velocity at the orice. The

Reynolds number of the gas ow in the pipe based on the gasproperties is below the transition from the laminar to the tur-bulent regime. Therefore, the velocity prole of a laminar pipeow as given above can be used as inow condition. Further-more, at the inow it is u = 0 and F = 0. At the orice plate,no-slip and impermeability conditions ( wall ) are dened for thevelocity eld ( u = w = 0).

Special treatments are necessary to capture the behavior of the triple contact line at the orice plate. Gibbs (1906) found,that if a three-phase contact line coincides with an edge ona solid surface, the contact angle , measured between thegas–liquid interface and the horizontal plate through the liquid,can take values of s (180◦ − ) + s , where s is thenatural or static contact angle of the three-component systemand is the angle of the solid wedge. This was conrmedtheoretically by Dyson (1988) and was proved experimentallyby Mason and co-workers ( Oliver et al., 1977; Bayramli andMason, 1978 ). For the present case ( = 90◦ ), this means that thecontact angle at the orice rim will not fall below the staticcontact angle s under static conditions. Instead, if < s , thebubble base (contact line) starts to spread outwards, i.e., theorice material essentially dewets. Furthermore, it was assumedin the present study that the moving contact line maintains aconstant contact angle = s , which is independent of thevelocity and the direction of the contact line movement. The

static contact angle model described above was implementedin the computer code by xing the contact line for > s anddening a constant contact angle during contact line movementusing the boundary condition j / j z = − cos s .

The extensions of the computational domain R and Z (Fig. 1)are dened as R = 5D e / 2 and Z = 10D e , where D e is the diam-eter of an equivalent spherical bubble. The width was shown byChen and Fan (2004) to be sufcient to keep the bubble forma-tion process unaffected by the lateral boundaries. The height of the domain was chosen such that the bubble dynamics abovethe orice considered in this work are not inuenced by theoutow boundary condition. The cell width was chosen to be0.25 mm, which was checked by grid renement tests to be suf-ciently small and computationally feasible. The time step wasdened such that the time step restrictions given in Gerlachet al. (2006) are fullled. A typical time step was 10 − 6 s.

2.3. Validation

For the validation of the computer code, it is referred toa previous work ( Gerlach et al., 2006 ), in which the presentfront-capturing method was tested against two other methodsby applying them to a variety of two-phase problems. The com-bined volume-of-uid and level-set method was shown thereto perform well with low computational costs compared tothe other methods considered by Gerlach et al. (2006) . The

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D. Gerlach et al. / Chemical Engineering Science 62 (2007) 2109 – 2125 2113

46 ms

46 ms

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46 ms

0.03

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0-0.01 -0.005 0 0 .005 0 .01

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0.03

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0.005

0-0.01 -0 .005 0 0 .005 0 .01

Fig. 2. Comparison between numerically computed bubble shapes with experiments of Zhang and Shoji (2001) , Q̇ = 100ml/min, D o = 2mm.

bubble formation at an underwater orice under quasi-staticconditions was compared with an analytical solution based onthe Young–Laplace equation ( Gerlach et al., 2006 ). The bubbleshapes during formation were found to agree excellently. Tomaret al. (2005) applied the computer code to the problem of lmboiling including heat transfer and interface mass transfer.

A direct comparison between bubble shapes of the numericalsimulation and experiments of Zhang and Shoji (2001) for Q̇ =100 ml/min and D o = 2 mm has been provided in Fig. 2 .1 Thebubble shapes show very good qualitative agreement, the agree-ment of the total formation time T is excellent.

3. Results

3.1. Inuence of operating conditions

In the following, the effect of the air ow rate through theorice Q̇ , the orice radius R o and the orice material s(wettability) on the bubble formation process is investigated.In these investigations, the uid properties were kept constant

1 Reprinted from Chemical Engineering Science, 56, Zhang, L., Shoji,M., Aperiodic bubble formation from a submerged orice, 5371-5381, Copy-right (2001), with permission from Elsevier.

Table 1Properties of the reference uid: air–water (aw) system at 20 ◦ C

Property Unit Value

l,aw kg/m 3 998.12g,aw kg/m 3 1.188l,aw Pa s 1 .002 × 10− 3

g,aw Pa s 1 .824 × 10− 5

aw N/m 72 .8 × 10− 3

at reference values of an air–water (index aw ) system at 20 ◦ Cas given in Table 1 . Owing to limitations on numerical efforts,orice radii only of R o = 1 and 1.5mm were studied. Q̇ was in-creased stepwise from 1 ml/min by 50 ml/min until a DP forma-tion process was observed. s was varied between 50 ◦ (wettingliquid) and 110 ◦ (non-wetting liquid). Additionally, the caseof a totally wetting liquid ( s = 0◦ ) was examined, for whichthe triple contact line of the bubble is pinned at the orice rimindependent of the other parameters.

3.1.1. Orice materialThe inuence of the wettability ( s ) of the orice material

on the bubble formation process is shown in Fig. 3 , where




t / T = 2 / 3

z [ m ]

-0.005 0

0

0.005

0.01

0.015

r [m]

0.005

z [ m ]

-0.005 0

0

0.005

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r [m]

0.005

z [ m ]

-0.005 0

0

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-0.005 0

0

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0.005

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-0.005 0

0

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-0.005 0

0

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0.005

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-0.005 0

0

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0.005

z [ m ]

-0.005 0

0

0.005

0.01

0.015

r [m]

0.005

z [ m ]

-0.005 0

0

0.005

0.01

0.015

r [m]

0.005

t / T = 1 / 3t / T = 0

Fig. 3. Three instants of bubble formation for the case (a) s = 0◦ (bubble pinned at the orice), (b) s = 70◦ and (c) s = 110 ◦ with Q̇ = 100ml/min,R o = 1 mm. The time period between the pictures is (a) T / 3 = 0.014 s, (b) T / 3 = 0.018s and (c) T / 3 = 0.042s.

sequences of bubble shapes are provided for s = 0◦ (pinnedbubble base or contact line), s = 70◦ and s = 110 ◦ . The owrate and the orice radius are xed at Q̇ = 100 ml/min and R o =1 mm. The time period between the snapshots of a sequence isT / 3 with T being the detachment time, i.e., T / 3 = 0.014, 0.018and 0.042 s for s = 0◦ , 70 ◦ and 110 ◦ , respectively.

As can be seen in the snapshots, the contact angle at thebubble base has large values at the detachment of the bubble.During the formation, decreases and leads to a bubble base

spreading along the orice plate if s (Figs. 3(b) and (c)), inaccordance with the static model of the contact line movement,as described in Section 2.2. In contrast, if s is smaller than thesmallest value of reached during formation, the bubble baseremains at the orice and the bubble volume at detachment istherefore governed by R o , as demonstrated exemplarily for thelimiting case of s = 0◦ in Fig. 3 (a). As the analytical study of Gerlach et al. (2005) showed for the slow formation case, theminimal value of reached during formation decreases with




V B

[ m m

3 ]

0 50 100

0

100

200

300

Q=1ml/minQ=10ml/minQ=50ml/minQ=100ml/minQ=150ml/minQ=200ml/minQ=250ml/minanal.,Fritz(1935)

DP

DP

s [°

]

Fig. 4. Inuence of the wetting conditions of the orice plate ( s ) on thebubble volume depending on Q̇ for R o = 1mm.

decrease in R o . Consequently, orice plates with increasing scause spreading earlier for a given Ro and can lead to forma-tion processes which are dominated by s rather than by R o .Furthermore, the maximal bubble base diameters D B reachedduring formation also increase with increase in s . This re-sults in an increasing capillary force acting downwards, whichto a rst approximation is D B , and thus leads to larger bub-

ble volumes V B . From the bubble shapes and the detachmenttimes given, the strong increase in bubble volume or detach-ment time between s = 0◦ and 110 ◦ (by a factor of 3) becomesapparent. Similar phenomena were observed experimentallyby Lin et al. (1994) .

In Fig. 4 , the simulation results for different s are sum-marized in a V B – s plot. Additionally, a curve representingthe low ow limit data of Fritz (1935) is added. His data arebased on the assumption that the bubble formation occurs witha constant contact angle = s , which is true in his case of the growth of vapor bubbles on a solid surface without an ori-ce. Agreement can be seen for the lowest ow rate in the

present study (1ml/min) in the cases s=

90◦

and 110◦

, wherethe formation is dominated by s . Strong deviations from thes -dominated results of Fritz occur for smaller values of s ,

since the contact line is pinned over a longer time. Finally,for s = 50◦ and 0 ◦ , the bubble volume is independent of s ,because the bubble remains pinned at the orice at any time.The two curves Q̇ = 200 and 250 ml/min need further expla-nation. The branching of these curves indicate the transitionfrom a SP to a DP formation regime. It can be seen that twobubble volumes exist under constant operating conditions. Therst bubble of each pair is typically larger than the second, be-cause the rst one forces the second to detach earlier than inthe SP regime. Only the periodic and the rst DP results areshown here, i.e., DP detachments were found for an orice with

Table 2Comparison of the bubble volume for different static contact angles s atan orice of R o = 1 mm under low ow rates of Q̇ = 1 and 10 ml/min withanalytical data

s (◦ ) V B (mm3)

VOF (ml/min) Analytical data ( Q̇ → 0)

1 10 Gerlach et al. (2005) Fritz (1935)

0 35 39 35 050 35 39 35 1270 39 45 33 3290 69 79 69 69

110 127 139 126 126

R o = 1mm for s < 90◦ with Q̇ = 250 ml/min, for s < 70◦

withQ̇ = 200ml/min and for all values of s considered here,when Q̇ = 300 ml/min (not shown). Although locating the tran-sition points accurately was not the aim of this work, the stepin the volume ux Q̇ brackets the transition. These resultsindicate that for Q̇ > 300 ml/min also an aperiodic formationoccurs. As reported in the literature, further transitions will oc-cur with increasing ow rate, i.e., transitions from DP to tripleperiodic formation and nally to chaotic detachment periods(Zhang and Shoji, 2001 ).

However, for all cases shown the bubble volume increaseswith increasing orice ow rate. Furthermore, V B increasesstrongly with s , when the formation is s dominated. For

s > 100 ◦ (non-wetting liquid) V B is roughly three timeslarger than for s < 50◦ (wetting liquid). This suggests that theinuence of the orice material should be taken into account

to predict accurately the bubble detachment volume under thepresent conditions.

3.1.2. Orice ow rateFirst, the bubble formation at low ow rates near static con-

ditions was investigated. In Table 2 , the bubble volume afterdetachment is given for two ow rates Q̇ = 1 and 10 ml/minwith s being a parameter. The results are compared withthose of the analytical studies of Fritz (1935) and Gerlachet al. (2005) . In these analytical studies, static bubble contoursare computed as a result of the balance of pressure and capil-lary forces (Young–Laplace equation) and boundary conditions.

They are applicable at very low ow rates (quasi-static condi-tions), where viscous forces are negligible. In Fritz (1935) , thesolutions of the Young–Laplace equation were calculated forgasbubbles on solid surfaces maintaining different values of s .It was assumed that the bubble volume increases by means of evaporation. Also, based on the balance of pressure and capil-lary forces, Gerlach et al. (2005) presented a model which ac-counts for the Gibbs condition at the orice rim and the inu-ence of a possible contact line movement away from the oricerim on the bubble formation. The treatment of the orice wasthe same as described in Section 2.2. The bubble volumes of Gerlach et al. (2005) given in Table 2 are the volumes abovethe bubble neck and for the low ow limit, i.e., for Q̇ → 0. Asis known from the literature and will be seen below, the bubble




volume is essentially constant at very low gas ow rates as aresult of the buoyancy and capillary force balance. Changes of Q̇ under such conditions result in a change in the bubble for-mation frequency, but with V B = constant. Since the solution of the Young–Laplace equation provides bubble contours only upto the last static/equilibrium stage, the dynamic necking until

detachment is not included in the analytical approach. How-ever, from Wong et al. (1998) , the time for the neck to pinchoff from a radius equal to the orice radius can be estimated by

t n = 7.6 l,aw Ro

aw= 0.1 ms (15)

for the air–water case at an orice with R o = 1 mm, which underthe present conditions is negligibly small. Hence the additionalgas volume t n Q̇ supplied during pinch off has no signicantinuence on V B for the cases considered in Table 2 .

Excellent agreement of the simulation results with the an-alytical data of Gerlach et al. (2005) can be seen in the caseQ̇ = 1 ml/min, which was shown in Gerlach et al. (2006) tobe valid also for the bubble shapes. Further comparisons withFritz (1935) reveal that two formation modes can be identiedunder the present conditions. For s = 0◦ and 50 ◦ the bubblestays attached to the orice, so that the results agree with staticmodels, which use a pinned bubble base as a boundary condi-tion (Longuet-Higgins et al., 1991; Gerlach et al., 2005 ). Onthe other hand, for s = 90◦ and 110 ◦ the formation processesare governed by the spreading with a constant contact angle andthus agree with Fritz (1935) and Gerlach et al. (2005) . Only theintermediate value of s = 70◦ deviates signicantly. The rea-son is that the analytical approach predicts a s -dominated for-mation, which is opposite to the simulation, where the growth

is dominated by the orice radius.Another interesting issue is to compare the results with the

static force balance equation (1) of capillary and buoyancyforces. From this equation, a bubble volume of 47mm 3 is ob-tained for the uid properties in Table 1 and R o = 1 mm. Eq. (1)is based on the assumption of a bubble forming at the rim of the orice with radius Ro . Compared with the data in Table 2for static conditions or a ow rate of 1 ml/min, the volumediffers by about 33%. From Table 2 , it can also be seen thatalready at Q̇ = 10 ml/min noticeable deviations exist betweensimulation and the analytical results indicating the transitionfrom static to dynamic bubble formation, where the validity of

the analytical approach ends. Similar results to those describedabove were observed for the computations using an orice withR o = 1.5 mm. Those results are summarized in Table 3 , againfor low ow rates.

The inuence of Q̇ on the detachment time T (Fig. 5(a)) andon the detached bubble volume V B (Fig. 5(b)) can be seen to besignicant. As described in the previous section, calculationswere performed with increasing orice ow rates until DP for-mation was detected, i.e., for each curve s = constant in Fig. 5the highest ow rate represents the last SP formation observed.As the distributions in Fig. 5 (a) show for R o = 1 mm, the tran-sition to DP formation is delayed for increasing values of s .It can be assumed that this is a consequence of the fact that thebubble size increases with s , so that the bubble formation time

Table 3Comparison of the bubble volume for different static contact angles s at anorice of R o = 1.5 mm under low ow rates of Q̇ = 1 and 10 ml/min withanalytical data

s (◦ ) V B (mm3)

VOF (ml/min) Analytical data ( Q̇ → 0)

1 10 Gerlach et al. (2005) Fritz (1935)

0 51 57 50 050 51 57 50 1270 52 61 50 3290 67 77 69 69

110 126 137 126 126

is increased and the interaction between successive bubbles isreduced. Also shown in Fig. 5 are estimations for the criticalbubble volume V B,c and time period T c , at which a transition

to DP formation may be expected based on simplied assump-tions (see Section 3.3).

Results of Og̃uz and Prosperetti (1993) using the same radius(R o = 1 mm) and also constant ow conditions are included inFig. 5 (b) for validation. Og̃uz and Prosperetti (1993) investi-gated numerically the bubble formation and detachment froma needle. Although a difference in the bubble volume betweenthe detachment from a needle compared to an orice can beexpected, one can compare their results with those for the case

s = 0◦ , i.e., in the case where the bubble is pinned at theorice. Indeed agreement can be found with the experimentsof Og̃uz and Prosperetti (1993) as shown in Fig. 5 (b). For acomparison of the point of transition (SP → DP), two publica-tions with comparable conguration are available. The transi-tion to a DP formation regime was found by Zhang and Shoji(2001) to occur at about 250ml/min. Kyriakides et al. (1997)reported for a R o = 1 mm nozzle that the transition takes placeat Q̇ = 228 ml/min for water. Both experimental results of thetransition to a DP formation (SP → DP) are comparable to thepresent data in Fig. 5 .

An important feature of the data in Fig. 5 is that for verylow ow rates or static conditions, as has been shown aboveto be valid for 1 ml/min, the bubble volume is almost inde-pendent of Q̇ . As a consequence, T decreases strongly withincreasing Q̇ in this range (Fig. 5(a)). The reason is that atlow ow rates capillary and buoyancy forces govern the bub-ble volume at detachment. An increase in the orice ow rateincreases the bubble formation frequency but leaves the bubblevolume constant. For higher ow rates an opposite behaviorcan be observed: the detachment frequency is almost indepen-dent of the ow rate, but the bubble volume increases sig-nicantly with Q̇ . For liquids of low viscosities and for highow rates, the bubble volume can be predicted by the one-stage model of Davidson and Schüler (1960a) for an inviscidliquid by

V B = 1.378Q̇ 6/ 5

g 3/ 5. (16)

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T [ s ]

0 50 100 150 200 250 300 350 400

10 -1

10 0

10 1

s =0 °

s =50 °

s =70 °

s =90 °

s =110 °Tc

SP

DP

XX

X

V B

[ m 3 ]

100

101

102

1E-07

2E-07

3E-07

4E-07

5E-07

Oguz & Prosperetti (1993)Davidson & Schuler (1960a)V

B,c

X

DPSP

Q [ml/min]Q [ml/min]

s =0°s =50 °

s =70 °

s =90 °

s =110 °

Fig. 5. Inuence of the orice ow rate Q̇ on the bubble formation time T (a) and volume V B (b) for R o = 1 mm. For each curve s = constant, the highestow rate represents the last periodic formation process found using a stepping of 50 ml/min. A critical bubble volume V B,c and time T c can be estimated (seeSection 3.3).

This relation indicates that in the limit of high ow rates,T = V B / Q̇ is only weakly dependent on Q̇ . In the literature,these two regimes are named “constant volume” and “constantfrequency” formation (Clift et al., 1978 ). The data presentedhere are in the constant volume regime and in the transition tothe constant frequency case. Since the present work is restrictedto SP formation, no predictions are given for higher ow rates.

The present results in Fig. 5 (b) indicate that they are converg-ing towards the curve based on Eq. (16) for increasing gas owrates. The inuence of s can be seen to decrease with increasein Q̇ .

For the case Q̇ = 100 ml/min, R o = 1.5mm and s = 0◦ ,Fig. 6 (a) gives an impression of the velocity eld around aforming and rising bubble by means of a vector and streamlineplot. Here the frame of reference is stationary. In Fig. 6 (b) acomparison is provided between the stationary reference frame(right half) and a reference frame (left half), which moves withthe constant rise velocity of the detached bubbles in positive

z-direction. The parameters used in this case are those of ahighly viscous liquid as will be discussed in Section 3.2.2 (

l=

0.15Pas, l = aw , = aw , Q̇ = 100 ml/min, R o = 1 mm ands = 70◦ ). In the moving frame of reference the streamlines

resemble patterns known from creeping ow theory ( Batchelor,1985). For low viscosity liquids, like water (Fig. 6(a)), thebubbles above the orice are wobbling, in accordance with thebubble regime map of Clift et al. (1978) , a representation in amoving reference frame is therefore not pursued here.

3.1.3. Orice diameter Similarly to Table 2 for R o = 1 mm, the bubble volumes

formed at an orice with R o = 1.5 mm under very low ow ratesare summarized in Table 3 . The static results of Fritz (1935)are independent of R o , thus remaining constant compared with

Table 2 . Additionally, data based on the methodused by Gerlachet al. (2005) were calculated for R o = 1.5mm.

The rst observation from a comparison of the results forR o = 1 and 1.5 mm is that the bubble volume is increased witha larger orice radius. This enhancement is almost in agree-ment with the linear dependence of Eq. (1). Second, it is inter-esting that in the case s = 70◦ , R o = 1.5 mm the numerically

computed bubble volume agrees with the cases s = 0◦ and50◦ , in which the bubble forms at the orice rim. Comparingthese results with the data in Table 2 , the conclusion agreeswith Gerlach et al. (2005) that the tendency of the bubble baseto spread outwards and to result in a s -dominated formationas described by Fritz (1935) increases with decrease in oriceradius. During formation, smaller orices reach smaller valuesof , thus favoring spreading.

In Fig. 7 , T vs. Q̇ and V B vs. Q̇ are plotted for R o =1.5 mm, as in Fig. 5 for the smaller orice. SP bubble forma-tions were found for higher ow rates compared with the R o =1 mm case. This is in agreement with the experimental data of Kyriakides et al. (1997) , who observed the transition to occurfor R o = 1.5 mm at around Q̇ = 262 ml/min compared withQ̇ = 228ml/min for Ro = 1 mm. As for the smaller oriceR o = 1 mm, the transition from SP to DP formation happensat lower ow rates for smaller values of s . A closer look atthe s = 90◦ and 110 ◦ curves for both orice radii show thatthey agree for each s value up to Q̇ = 150 ml/min, whichdemonstrate that these cases are s dominated and thus inde-pendent of the orice radius. For Q̇ > 150 ml/min the bubblesproduced for s = 90◦ and 110 ◦ at the orice with R o = 1.5 mmare larger than for R o = 1 mm. In contrast, for small values of

s (0◦ and 50 ◦ ), the bubble volume agrees for a given oriceradius and for most of the ow rates, but increases with theorice radius.




r [m]

-0.01 -0.005 0 0.005 0.01

0

0.005

0.01

0.015

0.02

r [m]

z [ m ]

-0.01 -0.005 0 0.005 0.01

0

0.005

0.01

0.015

0.02

0.025

0.03

fixed ref. framemoving ref. frame

z [ m ]

a

b

Fig. 6. (a) Streamline (left half) and vector (right half) plot for the case Q̇ = 100ml/min, Ro = 1.5mm and s = 0◦ . (b) Streamline plot for a highly viscousliquid case ( l = 0.15Pas, Q̇ = 100ml/min, Ro = 1mm, s = 70◦ , coordinate system moves in z-direction with the constant rise velocity of the detachedbubbles (left half), stationary coordinate system (right half).

T [ s ]

0 50 100 150 200 250 300 350 400

10 -1

10 0

10 1

s =0 °

s =50 °

s =70 °

s =90 °

s =110 °

Tc

SP

DP

V B [ m

3 ]

10 0 10 1 10 2

1E-07

2E-07

3E-07

4E-07

5E-07

s =0 °

s =50 °

s =70 °

s =90 °

s =110 °

VB,c

SPDP

Q [ml/min] Q [ml/min]

a b

Fig. 7. Inuence of the orice ow rate Q̇ on the bubble formation time T (a) and volume V B (b) for R o = 1.5 mm. For each curve s = constant, the highestow rate represents the last periodic formation process found using a stepping of 50 ml/min. A critical bubble volume V B,c and time T c can be estimated (seeSection 3.3).




ρl*

T [ s ]

0.25 0.5 0.75 1 1.25 1.5

400 600 800 1000

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

s =0 °

s =50 °

s =70 °

s =90 °

Tc ,Q=100ml/min

S PD P

ρ l /ρg,aw

T [ s ]

0.25 0.5 0.75 1 1.25 1.5

400 600 800

10 -1

10 0

10 1

anal.,Q=1ml/minQ=10ml/minQ=50ml/minQ=100ml/minQ=200ml/minTc ,Q=200ml/min

DPSP

ρl*

ρ l /ρg,aw

1200 1000 1200

a b

Fig. 8. Inuence of varying liquid density l = l / l,aw (or l / g,aw ) on the bubble formation period for R o = 1 mm depending on the static contact angle

s with Q̇ = 100ml/min (a) and the orice ow rate Q̇ with s = 70◦ (b). A critical time period T c for the transition from single periodic (SP) to doubleperiodic (DP) formation can be estimated (Section 3.3).

3.2. Inuence of uid properties

The inuence of the uid properties on the bubble forma-tion process was studied based on a reference set of param-eters. The reference case is dened by the uid properties inTable 1 , Q̇ = 100 ml/min, R o = 1 mm and s = 70◦ . In the fol-

lowing, the uid properties and operating conditions are takento be the reference values as given above, if not stated other-wise. In Sections 3.2.1–3.2.3 the effects of the liquid density

l , liquid viscosity l and the surface tension are consid-ered. In each section, s and Q̇ are varied as additional pa-rameters. The gas density and viscosity were not varied in thepresent work, since their inuence is assumed to be small (seeTable 1 : g l , g l ).

3.2.1. Inuence of the liquid densityWhen the liquid density is increased and all other properties

remain constant, the buoyancy force increases for a given bub-

ble volume, whereas the surface tension force can be assumedto stay constant. In this case, the bubble volume is expectedto decrease for increasing liquid density in a rst estimation,which is indicated by the simple balance of Eq. (1). This ten-dency is conrmed in Fig. 8 (a), where the inuence of the non-dimensional liquid density l = l / l,aw and the liquid–gasdensity ratio on the detachment time period is depicted. A 1 / lbehavior as suggested by Eq. (1) can be observed. At given Q̇a smaller T implies smaller V B . At small l , the static contactangle s has a strong inuence on T , whereas for increasing lthe effect almost disappears.

In Fig. 8 (b), the inuence of varying liquid density is shownfor different values of Q̇ . The analytical solution, as calculatedby Gerlach et al. (2005) for static formation, is added for a

low ow rate of 1 ml/min, which was validated for l = 1 inTable 2 . As indicated by the branching of the Q̇ = 200 ml/mindistribution, the bubble formation regime was found to changefrom a SP to a DP regime by increasing l from 1 to 1.25 atthis ow rate. A detailed examination of the data in Fig. 8 (b)reveals that the inuence of l on T decreases for increasing Q̇

in accordance with Eq. (1), if this equation is transformed toT = 2 R o /( l gQ̇) . As an example of Fig. 8 (b), T decreases inthe range 0 .25 l 1.5 for Q̇ = 1 ml/min by 93%, whereas forQ̇ = 100 ml/min it is only 77%. Also for the periodic results forthe case Q̇ = 200 ml/min, the same tendency can be observed.

In Fig. 9 , the inuence of l is visualized through snap-shots of the bubble shapes for the cases l = 0.25, 1 and1.5. Again, large differences can be seen clearly by chang-ing l = 1 (b) either to 1.5 (c) or to 0.25 (a). In the rstcase V B differs slightly (12%), whereas in the second case alarge increase in the bubble volume by a factor of about 4is observable.

3.2.2. Inuence of the liquid viscosityIn a review by Kumar and Kuloor (1970) , contradictory re-

sults reported in the literature concerning the inuence of theliquid viscosity on bubble formation process were discussedbased on their theoretical two-stage model. Their conclusionswere that the inuence of the liquid viscosity is large at higherow rates and viscosities and also large when the surface ten-sion effect is small or small orices are used, which can also beconcluded from the capillary number (Eq. (3)). Consequently,at low ow rates the inuence of viscosity is negligible, sincethe only forces are the capillary and the buoyancy force. Basedon these observations and their theoretical models, Kumar andKuloor claried existing contradictions in the literature.




z [ m ]

-0.01 0 0.01

0

0.01

0.02

0.03

z [ m ]

-0.01 0 0.01

0

0.01

0.02

0.03

z [ m ]

-0.01 0

0

0.01

0.02

0.03

r [m] r [m] r [m]

0.01

Fig. 9. Inuence of varying liquid density on the bubble contour at detachment: l = (a) 0.25, (b) 1 and (c) 1.5. Q̇ = 100ml/min, Ro = 1mm and s = 70◦ .

T [ s ]

10 -1 10 0 10 1 10 20.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

s =0 °s =50 °s =70 °s =90 °

Terasaka & Tsuge (1993)Tc ,Q=100ml/min

SP

T [ s ]

10 -1 10 0 10 1 10 210 -2

10 -1

10 0 anal.,Q=1 ml/minQ=10 ml/minQ=50ml/minQ=100 ml/minQ=200 ml/minEq.(34),200 ml/mi nTc ,Q=100 ml/minT

c,Q=200 ml/min

DPDP

μ l* μ l*

a b

Fig. 10. Inuence of varying liquid viscosity l = l / l,aw on the bubble formation period for R o = 1 mm depending on the static contact angle s withQ̇ = 100ml/min (a) and the orice ow rate Q̇ with s = 70◦ (b). A critical time period T c for the transition from single periodic (SP) to double periodic(DP) formation can be estimated (Section 3.3).

The present results are in accordance with the conclusionsdrawn by Kumar and Kuloor (1970) . In Fig. 10 , the inuenceof varying liquid viscosity l = l / l,aw is shown for differentvalues of s (a) and Q̇ (b). The formation time can clearly beseen to increase with increase in viscosity (Fig. 10(a)). How-ever, the increase in T is very small (a factor of 1.7 in thecase s = 70◦ ), although l was increased by a factor of 2000from 0 .1 × 10− 3 to 0.2 Pa s. It is difcult to identify such smalltendencies experimentally for the low and medium ow ratesconsidered here. A large increase in the viscosity of the liquidmay lead to small changes in the surface tension, which in turn

may have an inuence on the detachment frequency, as will beshown in the next section. This may be the reason for contra-dictory results as summarized by Kumar and Kuloor (1970) .From the balance of the viscous and capillary forces ( Ca ), onecan estimate that under the conditions of Fig. 10 (a) the vis-cous force becomes dominant over the inuence of the capillaryforce for l > 44, which is in agreement with the data shown.For validation, one data point extracted from the experimentalresults of Terasaka and Tsuge (1993) is also shown in Fig. 10 (a)for the case l = 109, R o = 1 mm and Q̇ ≈ 100 ml/min using

a stainless-steel orice. Very good agreement for this highly




r [m]

z [ m ]

-0.01 0

0

0.01

0.02

0.03

z [ m ]

-0.01 0 0.01

0

0.01

0.02

0.03

z [ m ]

-0.01 0

0

0.01

0.02

0.03

0.01

r [m] r [m]

0.01

a b c

Fig. 11. Inuence of varying liquid viscosity on the bubble contour at detachment: l = (a) 0.1, (b) 1 and (c) 150. Q̇ = 100ml/min, R o = 1 mm and s = 70◦ .

viscous case and the numerical results of s = 0◦ , 50◦ can beseen.

In Fig. 10 (b) the inuence of varying viscosity is shown fordifferent ow rates. For the low ow rate limit ( Q̇ = 1 ml/min),where viscosity has no effect on the formation, the analyticalsolution is plotted, which is essentially constant for varying

l . In addition, results for 100 and 200 ml/min are provided.Also, the inuence of viscosity on T is small under the presentconditions, as described above, an increase from l = 10 to 50

is sufcient to stabilize the formation process from a DP to aperiodic regime. A comparison of T of periodic formations atQ̇ = 100 and 200 ml/min shows that the increase of T in caseof the higher ow rate is 6% larger than that for the lower,i.e., the importance of viscosity increases with increasing owrate. Also depicted in Fig. 10 is the correlation of Davidsonand Schüler (1960b) for the limit of highly viscous liquids andhigh ow rates:

V B = 6.484 l Q̇

l g

3/ 4

. (17)

The highest ow rate shown in Fig. 10 (b) (200ml/min) can beseen to converge towards the correlation, but is still too smallto agree exactly.

The issues discussed above can further be conrmed by acomparison of bubble shapes during formation for different vis-cosities as provided in Fig. 11 , where contours just before de-tachment are given in the cases l = 0.1 (a), 1 (b) and 150 (c)(Q̇ = 100ml/min, R o = 1 mm and s = 70◦ ). It can already beseen from the bubble shapes that the difference in bubble vol-ume or detachment period is very small, although l differs bya factor of 1500 between the snapshots in Fig. 11 (a) and (c).However, signicant differences in the detached bubbles be-tween Fig. 11 (c) and the other two can be observed concerningtheir shapes. The attached bubble in a very viscous liquid (Fig.

11(c)) can be seen to be signicantly elongated in the verticaldirection compared with the two other cases of lower viscosity((a) and (b)). A reason could be that the pinch off time is di-rectly proportional to the liquid viscosity according to Eq. (15).After the upward forces at the bubble exceed the downwardforces, the bubble remains attached during the slowed neck-ing and becomes elongated. This process resembles the secondstage (detachment stage) of the two-stage models often used intheoretical approaches (Tsuge, 1986 ). Also, the detached bub-

bles show noticeable deviations for different liquid viscosities.These are in agreement with the bubble shape regime map of Clift et al. (1978) . An increase in l means for a freely risingbubble a change in the shape towards a sphere. Almost no dif-ference can be seen between Fig. 11 (a) and (b) demonstratingthe fact that T is almost constant for l < 1 or, in other words,for Ca 1.

3.2.3. Inuence of the surface tensionBased on the reference uid of Table 1 , the surface tension

was varied in the range 0 .41 = / aw 2.75 for differentstatic contact angles s with Q̇ = 100 ml/min (Fig. 12(a)) and

for different ow rates with s=

70◦

(Fig. 12(b)). The sur-face tension can be seen to have a signicant inuence on thebubble detachment period under the conditions of the presentstudy. As indicated by Eq. (1), the bubble volume increaseswith increasing surface tension, e.g. in the case s = 90◦ thedetachment time T increases by a factor of about four in therange of surface tensions considered (Fig. 12(a)). In this gureit can also be seen that the effect is more pronounced for largervalues of s . It is interesting that the detachment mechanismchanges in the three cases s = 0◦ , 50 ◦ and 70 ◦ for < 0.69from a periodic to a DP regime. This is illustrated in Fig. 13 ,where the case of a DP detachment for = 0.41 (a) and aperiodic case at high surface tension = 2.75 (c) are showntogether with the periodic reference case (b).




T [ s ]

0.5 1 2 3

0.05

0.1

0.15

0.2

0.25

0.3

s =0°

s =50 °

s =70 °

s =90 °Tc ,Q=100ml/min

D P

S P

T [ s ]

0.5 1 23

10 -1

10 1

anal.,Q=1 ml/minQ=10 ml/minQ=50 ml/minQ=100 ml/minQ=200 ml/minTc ,Q=100 ml/minTc ,Q=200ml/min

DP DP

σ*

1.5 2.5

10 0

σ*

1.5 2.5

a b

Fig. 12. Inuence of varying surface tension = / aw on the bubble formation period depending on the static contact angle s (a) and the orice owrate Q̇ (b). A critical time period T c for the transition from single periodic (SP) to double periodic (DP) formation can be estimated (Section 3.3).

z [ m ]

-0.01 0

0

z [ m ]

-0.01 0 0.01

0

z [ m ]

-0.01 0 0.01

0

0.01

0.02

0.030.03

0.02

0.01

r [m]

0.01

0.03

0.02

0.01

r [m] r [m]

a b c

Fig. 13. Inuence of varying surface tension on the bubble contour at detachment: (a) = 0.41 with double periodic detachment, (b) = 1 and (c) = 2.75with periodic detachment. Q̇ = 100ml/min, R o = 1mm and s = 70◦ .

The inuence of the gas ow rate is provided in Fig. 12 (b).A closer look at the periodic data in this gure shows thatthe inuence of surface tension decreases signicantly with in-crease in ow rate, since the importance of the viscous effectsincreases compared with the surface tension effects indicatedby capillary numbers of up to 0.45 in Fig. 12 . At very low gasow rates, the bubble volume is entirely given by the balance of surface tension and buoyancy force. With increasing ow rate,the inuence of surface tension decreases and nally disap-pears, as demonstrated, for example, by Davidson and Schüler(1960a) . Their theoretical and experimental work under highgas ow rates revealed no inuence of surface tension. It can

also be seen that an increasing surface tension has a stabilizinginuence on the formation process, i.e., can change the forma-tion from DP to SP (Fig. 12), because an increase in surfacetension results in an increase in bubble volume, which in turnreduces the interaction between successive bubbles.

3.3. Transition from SP to DP formation

Although the present work focuses on periodic bubble for-mation, DP formations were also observed. Two examples of bubble contours for this case of formation are provided inFigs. 13 (a) and 14. In the case of DP formation, the detachment




r [m]

z [ m ]

-0.01 0 0.01

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Fig. 14. Double periodic formation for the case Q̇ = 200ml/min, R o = 1.5 mmand s = 0◦ .

and rise of a bubble have an inuence on the following bubblein such a way that the trailing bubble detaches earlier. As aresult, bubbles rise in pairs and can coalesce at some distance

above the orice or even directly during formation, dependingon the ow rate, whereby the leading bubble has typically alarger volume V B, 1 than the trailing bubble V B, 2 . The exper-imental results of Zhang and Shoji (2001) and Tufaile andSartorelli (2002) showed that period doubling causes a branch-ing of the V B vs. Q̇ dependence and that an increase in Q̇results in an increasing difference V B, 1 − V B, 2. Furthermore,for high enough ow rates, further transitions or branches canoccur to triple periodic formation or nally to a chaotic one.The present results for the transitions described in the previ-ous sections show the same features as found in experiments.Direct comparisons of the point of transition between nu-

merical simulations and experiments ( Kyriakides et al., 1997;Zhang and Shoji, 2001 ) made in Sections 3.1.2 and 3.1.3showed qualitative agreement.

For the results reported here, the transition from SP to DPformation can be estimated based on simplied assumptions. Itcan be assumed that the leading or detached bubble inuencesthe forming bubble, if the time of formation T 2 = V B, 2/ Q̇ isof the order of the time T 1, rise , which the leading bubble needsto rise a certain distance until its wake does not inuence theformationof the trailing bubble. The terminal rise velocity u T of an ellipsoidal bubble of equivalent diameter D e > 1.3 mm in apure liquid system can be approximated from Clift et al. (1978)by u T = (2.14 / l D e + 0.505gD e )0.5. The leading bubble isassumed to rise a certain height, until the wake effect on the

Q [ml/min]

T c

[ s ]

0 100 200 300 400 5000.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

single periodicformation

double periodicformation

Fig. 15. Critical formation period T c depending on the orice ow rate basedon Eq. (18) and the reference uid (Table 1).

trailing bubble disappears. A height of 1 .7D e was found to givegood results for the present numerical data. Equating

T 2 = T 1, rise =1.7D e

u T (18)

and using D e = 2(3V B, 1/( 4 )) 1/ 3, a critical bubble volumeV B,c = V B,

1= V B,

2(or critical formation period T c) can be

calculated, which is equal to both bubble volumes at the point of transition. For the air–water system (Table 1), the distribution isgiven in Fig. 15 in the T c –Q̇ space. If the bubble formation takeslonger than the critical time, i.e., T > T c , then SP formation canbe expected, since the trailing bubble has moved far enough andthus has no inuence on the following bubble. In the other case,for T < T c , period doubling occurs. The critical quantities V B,cand T c according to Eq. (18) were added already in Figs. 5 , 7, 8,10 and 12. Good agreement between the theoretical estimationsand the numerically determined region of period doubling wasfound for the present data.

3.4. Representation of the results in non-dimensionalquantities

In order to validate the present results with literature data ad-ditionally to single data points as in the previous sections, thedata are compared with a general correlation equation in non-dimensional quantities. Jamialahmadi et al. (2001) proposed anon-dimensional correlation based on their comprehensive ex-periments using the functional relationship V B = F ( Fr , Ga , Bo),where the Froude and Galileo numbers are dened in termsof R o by Fr = Q̇ 2/R 5

o g and Ga = 2l R 3

o g/ 2 . With deni-tions of the Bo, Fr and Ga numbers as used in the presentwork, the correlation of Jamialahmadi et al. (2001) takes the




following form:

V BR 3

o=

43

1.119 Bo1.08 + 1.406

Fr 0.36

Ga 0.39 + 0.469Fr 0.51 . (19)

Jamialahmadi et al. (2001) reported an absolute mean average

error of 3.2% between their experiments and the predictions of Eq. (19) for the equivalent bubble diameter of the detached bub-ble, which is claimed to out-perform all other existing correla-tions. The corresponding value for the present data for periodicbubble formation and s = 0◦ is 3.3% (or 9.4% for V B ), where54 data points were generated and compared. This comparisondemonstrates the excellent accuracy of the present approach.The largest errors in the bubble diameter between correlationand numerical simulations of around 6% exist for the low owlimit cases with Q̇ = 1 ml/min, where the analytical results of quasi-static formation already hold (see Section 3.1.2).

4. Conclusions

The results of a comprehensive numerical study of bubbleformation at single submerged orices and under constant in-ow conditions were presented. In the low ow rate limit, thenumerical results were used together with analytical data tostudy the bubble formation under such conditions in detail. Theinuence of various operating conditions on the formation wasexamined with increasing ow rate until a double periodic (DP)formation was detected. For the variations of the uid prop-erties, it was observed that the bubble volume increases withdecreasing liquid density as well as with increasing viscosityand surface tension. With increasing ow rate, the degree of

this inuence decreases for density and surface tension, but in-creases for viscosity. The inuence of the material of the ori-ce plate was incorporated into the numerical method using astatic model for the contact line movement. The numerical re-sults show a strong effect of the contact angle on the bubbleformation process. Generally, an increase in the static contactangle promotes the bubble base to spread outwards, which leadsin turn to larger bubble volumes. This can result in formationprocesses which are governed by the contact angle instead of the orice radius. A systematic experimental study of the in-uence of the wetting properties of the orice material exceptfor the quasi-static case, however, is not yet available in the

literature.In addition to the simulations of periodic bubble formation,where successive bubbles form after regular time intervals, DPformations were also detected, where every second bubble de-taches earlier compared with the single periodic (SP) case dueto the wake effect of the previous bubble. The comparison of the point of transition between both regimes with available ex-perimental data showed reasonable agreement. Based on sim-plied assumptions, it was possible for the present results toestimate a critical bubble volume below which the transitionfrom SP to DP formation occurs. Here, it was assumed thattwo successive bubbles interact when the formation time isshorter than the time needed by the previous bubble to rise outof range.

The results demonstrate that the present numerical approachis a powerful tool for studying the highly dynamic processof bubble formation and detachment. Extensive validations of the computed bubble volumes with experimental results con-rmed this for the case when the bubble base is attached to theorice rim.

Notation

Bo Bond numberCa capillary numberD e diameter of an equivalent spherical bubbleF void fraction functionFr Froude numberf sv surface tension force per unit volumeg gravitational accelerationGa Galileo number

H Heaviside functionN C capacitance numberP pressureQ̇ ow rater radial component of cylindrical polar coordinates

R extension of the computational domain in r -direction

Re Reynolds numberR o orice radiust timeT formation time periodu velocity component in r -directionuT terminal rise velocityv velocity vectorV B bubble volumew velocity component in z-directionwo averaged orice velocity

z vertical component of cylindrical polar coordinates Z extension of the computational domain in z-

direction

Greek letters

angle of solid wedgelocal mean curvaturedynamic viscosity

densitysurface tensionlevel set functioncontact angle

s static contact angle

Subscripts

aw properties of an air–water system at 20 ◦ Cc critical quantity for transition from SP to DPg gas propertyl liquid property

Superscripts

non-dimensional quantity




Abbreviations

DP double periodic formationSP single periodic formation

Acknowledgments

The support of one of the authors (V. Buwa) by a fellow-ship of the Alexander von Humboldt Foundation is gratefullyacknowledged. The authors would like also to thank G. Biswasand G. Tomar for many useful discussions.

References

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