INTERFACE FORCES CALCULATION FOR MULTIPHASE FLOW SIMULATION

Transcription

1 1º Encontro Brasileiro sobre Ebulição, Conensação e Escoamento Multifásico Líquio-Gás Florianópolis, e Abril e 2008 INTERFACE FORCES CALCULATION FOR MULTIPHASE FLOW SIMULATION Clovis R. Maliska, António F. C. a Silva, Ricaro.P. Rezene, Ivan C. Georg Dept. of Mechanical Engineering, Feeral University of Santa Catarina Computational Flui Dynamics Laboratory SINMEC, Postal Office Box 476, Florianopolis, SC, Brazil. maliska@sinmec.ufsc.br ABSTRACT Despite the tremenous growth in computational resources, multiphase flows are far from being simulate with generality an accuracy. Keeping track of all interfaces an applying the proper bounary conitions, for example in an inustrial gas-liqui flow, is not yet foreseeing possible in a near future, if it will, with the numerical backgroun now available. The growth of the use of simulation tools for multiphase flows rely, therefore, in creating interface moels which tries to represent with more fielity the complex physics happening at the interfaces. This is a research topic which has a very large boy of literature available, both on the funamental physics as well as in theoretical analysis of the forces acting at the interfaces. This paper follows this tren, an an interfacial tracking metho (OF) is use to simulate the behavior of a rising bubble immerse in a quiescent liqui, as well as the buoyancy-riven motion of a viscous rop in a constricte channel. A brief review of the important interfacial forces use in CFD simulations of multiphase flows are iscusse. Numerical results are compare with the available experimental ata an the interfacial forces calculate from the numerical ata are compare with numerical an experimental results. Attention given to the transient behavior of the coefficients normally use in CFD simulations. 1. INTRODUCTION Multiphase flows are important in most inustrial applications incluing energy conversion, paper manufacturing, meical an nuclear applications. Many combustion an energy systems involve ilute multiphase flow, ranging from roplet sprays in high-spee gas turbine combustor flow to bubbly pipe flows of chemical reactors. A better unerstaning of the multiphase interactions can lea to increases in performance, reuction in cost an improve safety in several important engineering areas [1]. Up to very recently, the esign of equipments involving multiphase flows relie strongly on correlations obtaine experimentally. This narrows the range of the esign variables. Experimental stuies present a number of ifficulties because the probes can interfere with the flow an because multiphase systems can be optically opaque. Consierable amount of research has been one in eveloping measuring systems base on new optical an laser technologies. Moreover, multiphase flows are well known for the ifficulties in setting up fully controlle physical experiments. Fine measurements, like the flow insie bubbles, or flows near the bubble surface to investigate the existence of local slip flow are not easy to realize. But the only way for improving the interface moels is to set up fine measurements close to the interface. Computer simulations, in the other han, permit to inclue or neglect gravity, account for the effects of the geometrical configuration, process moifications an to perform many others variations in the physical problem. But, the great complexity of scales of the phenomena of practical multiphase flows, with ifferent length an time scales interacting in a broa range is very ifficult to moel mathematically. One example is the coalescence of rops or bubbles where the interfaces are free to eform an break up. The forces in such systems are very complex, of ifferent types an in the spatial-temporal omain, therefore observe over a large range of length scales [2]. Only recently it has become possible to experimentally observe the rapi motions uring the coalescence of two water rops [3]. Generally, the average escription is use to moel multiphase flow in an inustrial level, the so-calle Eulerian moel, where the isperse phase, like the continuous one, is escribe as a continuous flui with appropriate closures (interfacial momentum transfer between the phases, which inclues a number of force contributions like viscous rag, ae mass an lift, among others ). In these moels, only the average local properties are calculate. The properties of each iniviual isperse particle or bubble, can be moele by the Lagrangian formulation, but unfortunately the amount of particles or bubbles that can be tracke is, even toay an in the near future, very limite, [4]. Other metho to escribe multiphase flow is the interface tracking methos. These methos escribe both fluis with one set of equations an solve another equation for the evolution of the interfaces between the fluis. The closures require in the Eulerian moel have to be moele or provie by empirical ata. The problem can be to fin closures that are as simple as possible, while incorporating just enough physics to accomplish this, for a given range or phenomena an a specifie accuracy. It coul be provie by irect numerical experiments. This approach requires transient numerical experiments at the scale of the closure moel. One coul simulate the motion of one bubble, resolving all scales of the flow an then calculate the forces acting on the bubble. The same proceure can be use for two or three bubbles or for an array of bubbles. The forces are then calculate an the relative importance of them coul be etermine an compare with empirical ata. This proceure is easily controlle, in principle, better than the same experiment, proviing better results. However, the ifficulty in realizing fine experiments an also to solve numerically the complex multiphase flows, even for few bubbles, hiners the evelopment of general closure moels to be use with the two-flui approach. In this paper we present a brief review of the important forces entering the interface moels for multiphase flow simulation. We also present some numerical results for the buoyancy bubble flow in a straight channel as well as a

2 upwar rop flow in a wavy uct. These are preliminary results of a research project in which the main goal is to obtain more general closure moels for CFD calculations. This effort will be supporte by fine experimental measurements. 2. BRIEF REWIEW OF THE ACTING FORCES The ability of a multiphase coe to preict the behavior of gas-liqui or gas-soli flow are limite by the closure moels use to represent the transfer of quantities between the phases. Researchers in this fiel have been worke to obtain rational expressions for the forces acting on bubbles (or particles) over the last twenty years [5]. A general transient two-phase flow problem can be formulate by using the wellknown two-flui moel, this moel, as alreay pointe-out, epens on the availability of closure moels for the interfacial forces. The generalize rag force per unit volume, M i,, which is a combination of several interfacial forces is given by, (Hibiki an Ishii [6]) M i D B L W T = φ ( F + F + F + F + F + F ) = M D + M + M B + M L + M W + M whereφ,, F an M are the volume fraction (or voi fraction in a gas-liqui two-phase flow), volume of a typical particle, force, an force per unit of volume respectively. The subscript inicates isperse phase, an the superscripts of D,, B, L, W an T mean stanar rag, virtual mass, Basset, normal lift, wall lift an turbulent ispersion force of a typical single particle, respectively [6]. The first term on the right-han sie is the force ue to viscous an shape effects; the secon term is the force require to accelerate a quantity of mass of the surrouning flui ue to the bubble (or particle) motion. The thir term, known as the history or Basset force, is the effect of the acceleration on the viscous rag an the bounary-layer evelopment (Basset) [7], or the relative acceleration between the two phases [4]. There is not even full agreement of its formulation an when moeling, this force is almost always neglecte. The fourth term is the lift force normal to the relative velocity ue to rotation of the flui (the rotation causes pressure ifference normal to the flow [8]). The fifth term is the wall lift force ue to the velocity istribution change aroun particles near a wall, an the last term is the turbulent ispersion force ue to the concentration graient [9], or the effect of turbulence fluctuations on the effective momentum transfer. Because of the very weak relative ensity of bubbles compare to that of the liqui, almost all the inertia is containe in the liqui, making inertial-inuce hyroynamics forces particularly important in the preiction of bubble motion, [5]. Some ifficulties are encountere with bubbly flows; the shapes of the bubbles can change with the local hyroynamic conitions, aing new egrees of freeom to an alreay complex problem. Following, a brief escription of the acting forces in a bubble/particle are escribe, trying just to point out the ifficulties an uncertainties in moeling them, if they are to be use as closure moels for the interfacial problem in CFD simulations. In fact, there is a single force acting on the interface bubble/liqui, liqui/particle, gas/particle, gas/rop, which is the combination of all effects, the physics of the flow. To split T, (1) this force in several types was one for engineering convenience only. Therefore, in orer to evelop new closure moels for the interfacial force, it seems that more complex flows, even with a single bubble/particle, nee to be solve, generating more general closure moels, as one in [30]. Drag Force Drag force is the result of the viscous effect in the bounary layer an pressure ifferences cause by the shape of the bubble. Except in low-reynols number regime, no theory is generally available for etermining the viscous rag experience by a bluff boy [5]. For free liqui contamination, the bubble-liqui interaction seems to escape this rule because the viscosity of the gas filling the bubble is typically much smaller than the viscosity of the surrouning liqui. In this case, there is the possibility of slip along the surface of the bubble an the bounary conition impose at the bubble surface shoul be of a zero-shear-stress rather than a no-slip one (Batchelor) [10]. It is not fully etermine the influence of this bounary conition on the resulting rag force as a function of the Reynols number. For ilute flows, the interphase momentum transfer ue to shape an viscous rag is moele base on the rag on a single particle in an infinite flui [4], given by F ρ g 3 CD φ g ρl U r U (2) ρl 4 p D = r where, C D is the rag coefficient, p is the average local particle or equivalent bubble iameter, φg is the gas volume fraction, ρg an ρ l are the gas an liqui ensities an U r is the relative local velocity between the flui an the soli or gas phase. The rag coefficient for a single-particle system epens not only on the flow regimes but also on the nature of the particles; i.e., soli particles, rop or bubbles. Therefore, for a multiparticle system these ifferences are also expecte to play central roles in etermining the rag correlation [11]. At sufficiently low particles Reynols numbers [12], rops an bubbles are spherical. High viscosity an/or high surface tension keeps flui particle spherical. In this case, the rag coefficient behaves the same as for soli spherical particles. Even in the case of a single particle, C D is a complex function of the Reynols (Re), Eötvös (Eo), an Morton (Mo) numbers. The shapes of the bubbles vary with size, continuous-flow fiel, an the physical properties of the system. Clift et at. [13], presente correlations for the terminal velocity of single bubbles in various size ranges. When gas fraction increases, the interaction between bubbles becomes more vigorous. Bubbles collie, they coalesce an break up an affect their neighboring bubbles. Drag formulations for single bubbles cannot be expecte to apply in this situation, since the bubbles in this case o not move altogether inepenently of each other. Harper [14] observe that bubbles flowing in-line one after the other, move more rapily than single bubbles of the same size, as a result of the interaction of the particles with the wake flow, inicating that the correlations to be use in CFD simulation for multiple bubbles nees improvements. Ishii an Zuber [15] mae a careful stuy of the bubble rise velocities in isperse flow an evelope rag formulations

3 useful for a whole gas fraction range. The actual local gas fraction is a variable in the rag formulation. They cover flow regimes from the Stokes regime through the istorte particle regime an up to the churn turbulent regime. The approach base on experimental ata is, apparently, the best-justifie way to ajust the generalize rag formulation for high gas concentration. Jakobsen [16] aopte the correlation vali in the churn turbulent regime to moel the flow structure in the turbulent bubble column; it was foun that the physical effect of this relation was to ecrease the relative velocity between the liqui an gas with ecreasing the gas fraction, contrary to experimental measurements reporte in the literature. This iscrepancy clearly inicates that the rag coefficient formulations are not so general even for very similar bubbly flows. The outcome of this brief review shows that it is a challenging task to fin correlations for the rag force, such that it represents the complex interfacial phenomena occurring in multiphase flows when several local flow regimes an local bubble sizes are present. The question, alreay raise in this work, is if one shoul continue trying to moel the rag, an the other forces, in an iniviual form, since what is require is, in fact, is the global force at the interface. irtual Mass Force Drag force takes into account the interaction between liqui an bubbles in a uniform flow fiel uner nonaccelerating conitions. If, however, the bubbles are accelerate relative to the liqui, part of the surrouning liqui has to be accelerate as well. This aitional force contribution is calle the ae mass force or virtual mass force. The concept of virtual mass force can be unerstoo by consiering the change in kinetic energy of the flui surrouning an accelerating bubble/particle. In potential flow the acceleration inuces a resisting force on the sphere equal to one-half the mass of the isplace flui times the acceleration of the sphere [17]. van Wachem an Almste [4] give a general equation for this force, F DUg DU l = φ g ρlcm (3) Dt Dt in which the virtual mass coefficient, C M, is a function of the volume fraction. Maxey an Riley [18] euce theoretically the virtual mass coefficient for spherical rigi particles in potential flow with a value of C M =0.5. Cook an Harlow [19] use a value of 0.25 for eforme bubbles in water. Geary an Rice [20] suggeste 0.69 for spherical bubbles in water. The gasfraction epenency of the virtual mass coefficient has been reporte by [21] an [22], showing a fall in the coefficient with increasing gas fraction from about 0.5 at low gas fractions to the range at gas fractions aroun 0.3 to 0.4. Drew an Lahey [9] erive forces on a sphere in invisci flows. They foun the same relationship for the lift force as given by Eq. (3). Rivero et al. [23] use a numerical proceure to separate the history an virtual mass forces from the total unsteay force. Their investigations of oscillatory an uniformly accelerating flows establishe the invisci result of C M, =0.5 to be vali even at moerate Reynols numbers. As pointe out in [24], in non-uniform flows, the separation of the virtual mass force is complicate by the fact that the non-uniformity not only inuces an inertial force but also moifies the viscous rag as a irect consequence of the changes in the surface vorticity istribution. They performe simulations of straining ambient flows over a soli sphere an a spherical bubble at moerate Reynols number. Base on the compute pressure rag, they evaluate the virtual mass coefficient for the convective acceleration to be 1/2. For instance, in the case of a growing or collapsing, the bubble experiences a virtual mass force even if the relative velocity oes not vary with time [25]. Similarly, when the flui ensity varies temporally ue to rapi compression or expansion, the time-erivative of ρ contributes to the virtual mass force [26]. This compressional force retars the boy uring flui compression an accelerates it uring flui expansion. Geometrically, the virtual mass coefficient is a measure of the bluffness of the boy, an its value increases as the boy (or bubble) becomes oblate because it accelerates or isplaces forwar a larger volume of flui. The virtual mass force plays a central role in bubble hyroynamics because it is weighte against flui ensity an is typically a factor of (ρ/ρ b ) ~10-3 larger than the rate of change of the bubble momentum [5]. Basset Force Basset force or history force is a viscous force ue to the relative acceleration between the two phases. Most often, this force is ignore in continuum moeling, an there is not even full agreement of its formulation even for the single bubble case [4]. Uner steay conitions in a uniform ambient flow, the Basset force becomes zero. Drew an Lahey [27] give an expression for the history force, t B 9 ρlμl a( r, t) F = φl τ π (4) t τ p 0 where a(r,t) is the acceleration between the phases an the (tτ) -1/2 is the generally accepte Basset kernel for short times. Lift Force Lift force represents the transverse force ue to rotational strain, velocity graients, or the presence of walls. A general equation for the lift force cause by rotational strain is given by ( U U ) ω L,r = g l L,r g l F φ ρ C (5) an the equation for the lift force cause by velocity graients is given by ( U g Ul ) ( l L, v = g ρlcl, v U F φ ) (6) where C L,r an C L,v are the lift force coefficients associate with rotational strain an velocity graients, respectively. From laminar, invisci flow, the value of 1/2 can be etermine for the lift force coefficient for rotational strain. In literature, a wie range of values can be foun for the lift force coefficient, as the above equations originate from invisci flow aroun a single sphere. Tomiayama et al. [29] have performe experiments of single bubbles in simple shear flows an have foun positive an negative values for the lift

4 force coefficient, epening upon the specific bubble characteristics. Dany an Dwyer [30] compute numerically the threeimensional flow aroun a sphere in shear flow from continuity an the Navier-Stokes equations. They examine the two contributions to the lateral force, the viscous an the pressure contribution, respectively. The viscous contribution was always foun to be positive. The pressure contribution, however, woul change sign over the surface of the sphere an was foun to be opposite to the classical Magnus force irection, that is, negative, for most Reynols numbers an shear rates examine. The total lift was always foun to be positive, however. Bubbles o not always behave as preicte as moels for rigi spheres. Kariyasaki [31] stuie bubbles, rops, an particles in linear shear flow experimentally an showe that the lift on a eformable particle is opposite to that on a rigi sphere. For Reynols numbers between 0.01 an 8, the rag coefficient coul be estimate by Stokes law, the flui particles woul eform into an airfoil shape. Tomiyama et al. [32] performe both experiments an numerical simulations of the lateral migration of a bubble in a laminar flow an in a quiescent liqui. They stuie the effects of the Eötvös number an liqui volumetric flux on the lateral motion. They foun a lateral force ue to the wall in the near wall region an a lift force ue to circulation aroun the bubble away from the wall. The latter epens strongly on the Eötvös number. In quiescent liqui, they foun that bubbles migrate to the center of the uct an then rose straight. They foun that increasing the liqui flux in either irection enhance the lateral motion an that the lift force was proportional to the liqui flux. For moeling purposes, they use a lift force, Eq. (6). They encountere negative values for lift force coefficients. The explanation for the negative lift force coefficients is the istortion of the bubble shape an the subsequent circulation aroun the bubble. They therefore conclue that the nature of the force is similar to that of the Magnus force. However, it may just a well be classifie as a lift force for inuce rotation ue to eformation. Bubbles exhibit very ifferent motion epening on whether the bubble eforms or not. A spherical bubble experiences a lift to the right, in accorance with theory for rigi spheres, whereas the eformable bubble takes the shape of an airfoil an moves to the left, for example, experiences a negative lift. These finings shows again the ifficulty in separating the effects in a complex flow, inicating that the etermination of more general correlations can be useful for simulating multiphase flows in which the interfaces are not tracke. Turbulent Dispersion Force Turbulence, of course, affects the whole flow an, therefore, affects the resulting force at the interface. The isperse phase can amp the turbulent energy of the flow by means of several effects. The size of the bubbles/roplets, the relative velocity between phases an the turbulence intensity of the continuous phase, are some of the flow characteristics which may affect significantly the resulting force. How to take into account these effects on the closure moels is the motivation for large amount of research available in the literature. In orer to improve the single-phase moels, it can be consiere that the movement of the bubbles relative to the liqui phase creates rag work in the liqui phase in aition to liqui shear, Svensen et al. [33]. An empirical coefficient, C t enotes the fraction of the bubble-inuce rag work creating aitional turbulence prouction in the liqui phase. This parameter generally epens on the bubble size an shape, incluing turbulent kinetic energy on the length (or time) scales of the bubbles. There is no general agreement in the community about these moels. The effect of turbulence on interphase momentum transfer is largely unknown. The Lopez e Bertoano (1991) moel T F = ρ C k φ (7) l T l l is use successfully for air bubbles in water with the turbulent coefficient C T =0.1 to 0.5, however C T is not universal, [12]. Clift et al. [13] notes that turbulence generally increases the rag although there is some isagreement among the various stuies. Soo [34] asserts that the ata preominately shows a ecrease in rag coefficient ue turbulence. Crowe et al. [35] reviewe a number of stuies for turbulence effects on particle rag coefficient an note a wie variety of results (increases an ecreases) over a large range of Reynols number. Issues, which may be responsible for a lack of consistency, inclue experimental uncertainty, the turbulent spectrum variations, particle an wall influences. While there are theoretical results, which explain some of these trens for bubble rise velocity ecrease, [36], no robust empirical correction can be recommene at this point. Universal moels are far from being attaine, of course, since all forces, which are usually moele separately, contain influences from each other. In the case of turbulence, this phenomenon greatly influences the flow, what makes extremely ifficult to measure its influence over the whole flow an to separate these influences over each of the forces, like rag, virtual an so on. In this section, we presente some of the available moels for the interphase momentum transfer in multiphase flows, trying to pointing out that much more work is require in orer to establish reasonable closure moels for CFD simulation of multiphase flows. Certainly, the use of massive numerical simulation for solving complex problems, in which the interfaces can be tracke an its local forces calculate, will greatly help the evelopment of more general closure moels. Fine experiments will support the numerical calculation of the interfacial force, a quantity which can not be experimentally measure. 3. INTERFACE TRACKING METHODS When a sharp interface exists, separating two fluis, ifficulties arise to accurately simulate these flows. It can be attribute to the fact that the interface separating the fluis nees to be tracke accurately without introucing excessive computational smearing, the well known numerical iffusion. In the past ecae a number of techniques, each with their own particular avantages an isavantages, Annalan et al. [37], have been evelope to simulate complex two flui flow problems. One of the most well known methos is the volume of flui (OF) metho [38] an [39]. This metho uses a color function that inicates the fractional amount of flui present (the local volume fraction of one of the phases) at a certain position in a etermine time. This function is unity in computational cells occupie completely by flui of one of the phases, an zero in regions occupie completely by the other phase, an a value between these limits in the cells that contain a free surface. In OF algorithm, the color function is iscontinuous over the interface; however, in the closely

5 relate level-set algorithm, [40] an [41], this function is continuous. The avantage of the level-set algorithm is the simplicity to compute erivatives of the color function, require to calculate the curvature of the surface. However, in flow fiels with appreciable vorticity, or in cases where the interface is significantly eforme, level-set methos suffer from loss of mass [4]. The accuracy of OF methos in calculating the curvature of the interface, by etermining the erivative of the color function, is ifficult from a numerical point of view. Marker particle methos, [42] an [43], in which fictitious particles are place along the interface an use to track the motion of the interface, is a very accurate metho, but computationally expensive because the interface is reconstructe from the location of the particles in each iteration. Coalescence an breakup of interfaces is a problem, because some metho to moel these phenomena nees to be use. Front tracking methos, [44], make use of markers connecte to a set of points to track the interface, whereas a fixe or Eulerian gri is use to solve the Navier-Stokes equations. This metho is extremely accurate but complex to implement because ynamic remeshing of the Lagrangian interface mesh is require an mapping of the Lagrangian ata onto the Eulerian mesh has to be carrie out. A proper sub-gri moel is neee to represent the coalescence an break up. Automatic merging of interfaces o not occur in front tracking metho. The interface tracking methos commente above consier two separate, incompressible flui phases. The two phases are separate by a reconstructe interface, from some color function or another Lagrangian representation of the interface, [4]. Now follows the equation systems use in this work. The governing conservation equations for unsteay, incompressible, Newtonian flow are given by the continuity equation U = 0, (8) the Navier-Stokes equation ρ U + ρ ( UU) = P + Τ + ρg + S (9) t in which ρ is the ensity of the local flui, P the local pressure, S the surface tension, T the viscous stress tensor, an U the velocity fiel. The velocity fiel hols both the liqui an the gas velocity. iscosity an ensity are assume constant in each of the phases, but may vary. Stanar OF an level set methos make use of a transport equation to etermine the evolution of the color function φ + U φ = 0, (10) t whereφ is the color function, representing the liqui or gas phase. The surface tension, S, is inclue in Eq. (9) as a boy force an acts only on the inter-phase bounary surface S= γκ( x) φ, (11) where γ is the surface tension coefficient an κ is the surface curvature given by κ = φ n, n =. (12) φ In this work the color function is mae equal to volume fraction of surrouning flui. In the next section, we present some numerical results using the OF metho escribe above. The iea is to evaluate the moel as a tool for the solution of more complex flows for the sake of improving the closure methos for interfacial forces. 4. NUMERICAL RESULTS To apply the OF metho to calculate the interfacial forces in a two-phase problem two physical situations involving a gas bubble moving in an incompressible quiescent liqui is consiere. The problems are solve in a 2D Cartesian omain, therefore, the shape of the bubbles iffers consierably from their original forms, since one is consiering a planar 2D problem. Full 3D an 2D axisymmetric solutions are being compute for future comparisons. The first test is base on experimental results performe by [45] for single gas bubbles rising in a liqui. The secon test problem, the buoyancy-riven viscous rop in constricte capillaries [46] has numerical an experimental results for comparison. The two-imensional numerical simulations were carrie out using the CFD coe ANSYS CFX release 11 TM, which use an element-base finite volume metho. All simulations are performe using high-resolution scheme for space iscretization an a secon orer backwar ifference scheme for the time. Gas Bubble Moving in a Quiescent Liqui Bhaga an Weber [45] stuie experimentally the rising bubble in a quiescent liqui. They presente experimental results for a wie range of flow regimes. In our numerical experiment, it is use one of the results presente in [45] to compare the shapes, upwar velocity an rag correlation. The lift an virtual mass forces are calculate an compare with the numerical an experimental results. The usual imensionless parameters employe are the Reynols, Eötvös an Morton numbers efine as ρf eut gμf ge ρf Re = ; Mo = ; Eo = ; (13) μ γρ γ f f 4 where U t is the terminal velocity, γ is the surface tension coefficient, e is the equivalent iameter of the bubble, ρf is the liqui ensity an μ f the viscosity of the liqui. It was use the same values of the experimental test for these numbers, (Re=7.16, Mo=41.1 an Eo=116) in the numerical simulations. The physical properties use are shown in Tab. 1. ρf ρf Table 1. Physical properties. 2 μ U f t γ e

6 kg/m 3 kg/m 3 g/cm s cm/s ynes/cm cm A Cartesian computational gri (801x801) was use in a 20x20 cm omain that gives a gri size of cm. The time step use in the simulations was of the orer of 10-4 secon. A circular bubble was inserte in the bottom of the channel fille with liqui at rest an no-slip bounary conition was use for the walls. In Fig. 1 the experimental an numerical shapes of the bubble at terminal velocity is compare. It can be seen that a goo qualitative agreement was obtaine for the general shape of the bubble. (a) (b) Figure 2. Evolution of the bubble rise velocity with its shape for 5 vertical positions. Figure 1. (a) Experimental, from [45] an (b) numerical obtaine in this work. Significant for the purpose of this work is to observe that the interface is well represente, with almost no smearing, what is very common when numerical results using OF methos are calculate. This inicates that gri resolution use is fine enough for representing the interfacial shape of the bubble. It also important to remember that in transient flows, if the interface is not well capture, the errors accumulate along time, being ifficult to preict the right shape of the bubble. The rise velocity of the bubble, b, is efine as its baricentric velocity in the vertical irection, by The magnitue of the local velocity of the gas flow can be observe in Figs 3 an 4, where it is shown, respectively, the contours of the horizontal an vertical velocity component. The symmetry of the flow insie the bubble can be ientifie in these two figures. At the upper region of the bubble, the velocity is greater than in the bottom region. b = U. e g g z φ (14) Figure 3. Contours of gas velocity insie the bubble at the terminal velocity (horizontal component) where U g is the velocity of the gas phase in the vertical irection anφ g is the volume fraction of the gas phase. Time evolution of the bubble rise velocity is shown in Fig. 2 with the shapes of the bubble inicate. The bubble reaches its terminal velocity at about 0.3 secon from the start. Experimental terminal velocity for this case is reporte as m/s an the numerical result obtaine was m/s, with a relative error of about seven per cent. Most probably, this ifference is an effect of the wall bounary conition, since the simplifications one in consiering a 2D planar flow woul increase even more this velocity. More tests with a wier channel an consiering 3D flow nee to be conucte in orer to confirm this. Figure 4. Contours of gas velocity bubble at the terminal velocity (vertical component).

7 irtual mass force, by its turn, appears when the flow is accelerating. A balance between buoyancy an virtual mass forces gives ( ρ ρ ) ρg L g g Cvm = + (17) ρ U L rel ρl t in which the total erivative is efine by U t rel Ug UL = + Ug Ug UL UL. (18) t t Figure 5. Inuce liqui velocity contours at the terminal velocity (horizontal component) In Figs. 5 an 6 contours of the horizontal an vertical components of the inuce liqui velocity is shown. The figure shows a symmetric fiel with opposite signal for the x- component of the liqui velocity, as expecte. Near the bottom of the omain the contours appears eforme, what emonstrates the influence of the bounary of the omain on the results. This is not critical, since velocities in that region are very small. By their turn, lateral walls o not show this behavior. In Eq. (18) the baricentric velocities are consiere. It was observe that, for this case, the spatial erivatives in Eq. (18) plays a minor role in the value of the acceleration. Bhaga an Weber [45] provie experimental results for the rag coefficients. The numerical results of this work for C D an C M are compare in Tab. 2. Table 2. C D an C M compare with experimental (C D ) an the expecte value of C M for the flow over a cyliner C D C M Experimental 5.29[45] Numerical The numerical result for C D is closer to the experimental one consiering the simplifications aopte. C L, by its turn, was foun to be equal to zero, as expecte, since the flow is symmetric. C M, by its turn shows consierable iscrepancy when compare with the value of unity for a 2D cyliner [47]. Some new tests are being conucte in orer to clarify this. Drop buoyancy-riven motion in a constricte capillary Figure 6. Inuce liqui velocity contours at terminal velocity (vertical component) The rag an lift forces, F D an F L, are the components of the flui force acting on the bubble surface in the transversal an in the streamwise irection, an they can be calculate by the sum of the pressure an viscous contributions, by F = F + F = pe ns + n τ e S. (15) i ip iv i i S S The rag an lift coefficients, C D an C L, are efine by Fi Ci =. (16) 2 e ρlu tπ( ) 2 in which F i is the force component acting in the bubble surface in the streamwise an transversal irections alreay integrate in the bubble surface, an U t is the terminal velocity calculate numerically. The forces were integrate in the surface using the output from the simulator. The motion of eformable rops an bubbles through constant an variable cross-section channels are of funamental importance as a prototype problem in many engineering an scientific applications [46]. In constricte channels, flow is inherently unsteay an inertial effects might be important, especially in the case of small capillary numbers or severely constricte channels. In this work qualitative an quantitative comparisons is realize with the results reporte in [46]. These authors use a finite volume front tracking metho to simulate a buoyancy-riven motion of rops in constricte channels. Both, numerical an experimental results are reporte in [46]. The case stuie herein is the flow of a rop with raius R = 2.7 mm in a tube with capillary raius R c =5.00 mm. The non-imensional rop raius k is efine as the ratio of the equivalent spherical rop raius to the average channel raius k=(r /R c ). The value use for k is The amplitue of restricte channel use in this work is α=0.28 an the Bon number, Bo= The physical properties are presente in Tab. 3. ρ rop ρ L Table 3. Physical properties. μ rop μ γ H* L kg/m 3 kg/m 3 mpa.s mpa.s N/m mm *Channel height

8 The simulation was performe using the OF metho escribe above with a computational gri of 121x901 volumes (Δy~0.077 mm an Δx~0.097 mm). Figure 7 illustrate a etail on the throat of the mesh use. As shown in the figure, the refinement was concentrate at the central region of the channel. One can note an almost symmetric behavior of the baricentric velocity profile before an after the throat. Figure 7. Mesh etail on the throat of the channel. A qualitative comparison with the experimental rop shape is shown in Fig. 8. As can be seen in this figure, the compute rop shape compares well with the experimental results, inicating the accuracy of the present simulation an the capability of OF metho to represent the interfacial shape of the rop. It shoul be mentione that the numerical result presente in the left sie of Fig. 8 was obtaine with a channel where the amplitue of the wavy wall is a little larger than the one use in the experimental work. Quantitative comparison with the numerical results, reporte in [46] for the shape of the rop is shown in Fig 9. This figure shows that the OF metho use in this work compares very well with the numerical result (front tracking) reporte for the shape of the rop. Figure 9. Comparison of the rop shape between the numerical results after the throat, reporte by Olgac et al (2006) an the present work. Drag force an virtual mass are calculate for this numerical result using Eqs. 15 to 18. A comparison is mae for rag coefficient an presente in Fig. 11, where the time evolution of C D is shown. At a low particle Reynols numbers, the rag coefficient for flows past spherical particles may be compute by 24 C D =, (19) Re For sphere an also for istorte particles the rag coefficient can be calculate, using an equivalent iameter, by C D 4 g Δρ =. (20) 3 ρ e 2 b L a) b) Figure 8. Qualitative comparison between the numerical an experimental shapes of the rop before entering the throat. Extracte from [46]. The rise velocity of the rop was calculate by Eq. 14 an the profile shown in Fig. 10. This figure shows the rop baricentric velocity an the shapes along the capillary tube. There is a suen acceleration just after the rop is release, emonstrating that the inertia effects are very small. Following, there is a smooth amping of velocity before the throat an after the throat the rop starts to accelerate smoothly again. Insie the throat, there is a moification of the rop shape, with elongation in the streamwise irection. Time [s] ,000 0,001 0,002 0,003 0,004 0,005 0,006 0,007 Drop's Baricentric elocity [m/s] Figure 10. a) Baricentric velocity of the rop; b) Drop s path an its shapes along the wavy channel. In Fig 11 it is epicte the unsteay C D coefficient. Three ifferent ways in calculating the rag coefficient were use. The first one uses Eq. (16) in which the force is calculate

9 through the calculation of the acceleration an the mass involve. The acceleration was calculate using the velocities obtaine as output, an Eq. (18). The mass was compute using the ensity an the volume of the rop consiering it as a sphere, even though the flow was consiere planar. Eq. (20), by its turn, is obtaine from a force balance between buoyancy an rag vali, therefore, for steay state flow. In Fig. 11 Eq. (20) is plotte using as terminal velocity the instantaneously velocity of the rising rop. As can be seen, Eq (16), also consiering the terminal velocity as the instantaneous velocity, is in excellent agreement with Eq. (20), what emonstrates that the transient flow of the rop along the wavy channel behaves as a sequence of several steay-state flows. This is a very important fining, since it allows the use of the correlation for steay state flow in transient regime. rag coefficient with the rag correlation use for rops with non-spherical shape agrees well with numerical results. It was emonstrate that the unsteay rising of the rop in the wavy channel is equivalent of a sequence of several steay-state regimes. This is supporte by the fact that the unsteay rag agrees very well with the steay rag calculation when the steay-state velocity is substitute by the instantaneous velocity. In general, the OF metho use to simulate realistic physical problems gives goo results an can be use as a tool to calculate the interfacial forces in multiphase flows. The accuracy of force calculations will epen on the gri an on temporal refinement, as well as on the orer of the numerical scheme. The numerical results shown were solve with a spatial an temporal refinement that resolves much of the flow scales This Work, Eq.(16) Eq.(19) Eq.(20) ACKNOWLEDGEMENTS C D 10 3 The authors gratefully acknowlege the financial support for this work by CNPq Conselho Nacional e Desenvolvimento Científico e Tecnológico - through a scholarship for Ivan C. Georg. REFERÊNCIAS Time [s] Figure 11. Time evolution of the rag coefficient calculate for this work for sphere (Eq. 19) an for an ellipsoial shape (Eq. 20). 5. CONCLUSIONS This work presente a numerical calculation of forces acting on the interface of two fluis consiering transient effects. The results are preliminary, an are emboie in a more general research goal of eveloping closure moels to be use in CFD simulations. The main objective is to solve complex 3D problems with several effects involve, seeking for more general closure moels, trying to avoi the moeling through the split of the interfacial forces in many types, like rag, lift, virtual mass, among others. With this in min, a brief overview of the important interfacial forces was escribe. The OF metho was use for solving the two test problems, namely the flow of a gas bubble rising in a quiescent liqui an the rising rop in a wavy channel. Shapes of the bubble in its terminal velocity were compare with the experimental results reporte in [45]. Drag, lift an virtual mass coefficients were calculate by integrating the tangential an normal forces on the surface of the bubble. The expecte virtual mass coefficient for the flow over a cyliner oes not match with the present results, probably because the non-slip bounary conition use on the walls an the initial bounary conition affect the flow behavior. The consieration of planar flow may have contribute, an this result nees to be analyze in more etail. A funamental problem, the motion of rops in constricte channels was also solve. ery goo agreement with experimental shapes of the rop was obtaine. The calculate [1] S. Sunaresan, Moeling the hyroynamics of multiphase flow reactor: current status an challenges, AIChE Journal, vol. 46, n. 6, pp , [2] T.J. Hanratty, T. Theofanous, J-M. Delhaye, J. Eaton, J. McLaughlin, A. Prosperetti, S. Sunaresan, G. Tryggvason, Workshop Finings, International Journal fo Multiphase Flow, vol. 9, pp , [3] A. Menchaca-Rocha, A. Martinez-Davalos, R. Nunez, S. Popinet, S. Zaleski, Coalescence of liqui rops by surface tension, Phys. Rev. E, n. 4, , [4] B.G.M. van Wachem, A.E. Almstet, Methos for multiphase computational flui ynamics, Chemical Engineering Journal, vol. 96, pp , [5] J. Magnauet an I. Eames, The motion of high- Reynols-number bubbles in inhomogeneous flows, Annu. Rev. Flui Mech., n. 32, pp , [6] T. Hibiki, M. Ishii, Lift force in bubbly flow systems, Chemical Engineering Journal, vol. 62, pp , [7] A.B. Basset, A Treatise on Hyroynamics,Worshop Finings, vol. 2, Deighton, Bell, Cambriges, (Dover, New York, 1961), [8] H. Enwal, E. Peirano, A.-E. Almstet, Eulerian twophase flow theory applie to fluiization, International Journal fo Multiphase Flow, vol. 22 Suppl., pp , [9] D.A. Drew, R.T. Jr. Lahey, The virtual mass an lift force on a sphere in rotating an straining invisci flow, International Journal fo Multiphase Flow, vol. 13, pp , [10] G.K. Batchelor, An Introuction to Flui Dynamics, Cambrige University Press, New Delhi, [11] M. Ishii an K. Mishima, Two-flui moel an hyroynamic constitutive relations, Nuclear Engineering an Design, vol.82, pp , 1984.

10 [12] A.D. Burns, Computational flui ynamics moeling of multiphase flows. Alpha Beta Numerics, Fulwoo, Preston, PR28At, UK, [13] R. Clift, J.R. Grace an M.E.Weber, Bubbles, rops an particles, Acaemic Press, New York, [14] J.F. Harper, The motion of bubbles an rops through liquis, Av. Appl. Mech., n. 12, pp , [15] M. Ishii, N. Zuber, Drag coefficient an relative velocity in bubbly, roplet or particulate flows, AIChE Journal, vol. 25, pp , [16] H.A. Jakobsen, On the moelling an simulation of bubble column reactors using two-flui moel, Dr. Ing. Thesis 97, NTH, Tronheim, [17] J.B. Joshi, Computational flow moeling an esign of bubble column reactors, Chemical Engineering Journal, vol. 56, pp , [18] M.R. Maxey an J.J.Riley, Equation of motion for a small sphere in a non-uniform flow, Phys. Fluis, vol. 26, pp , [19] T.L. Cook an F.H. Harlow, ortices in bubby two phase flow, International Journal of Multiphase Flow, vol. 12, pp , [20] N.W. Geary, R.G. Rice, Circulation in bubble columns: corrections for istorte bubble shape, American Institute of Chemical Engineering Technology, vol. 37, n. 10, pp , [21] L. van Wijngaaren, Hyroinamic iteraction between gas bubbles in liqui, Journal of Flui Mechanics, vol. 77, pp , [22] Yu.G. Mokeyev, Effect of particle concentration on their rag an inuce mass, Flui. Mech. Sov. Res., vol. 6, pp. 161, [23] M. Rivero, Etue par simulaton nuérique es forces exercées sur une inclusion sphérique par un écoulement accéléré, PhD thesis. Inst. Nat. Polytech. Toulose, Toulose, France, [24] M. Rivero, J. Magnauet, J. Fabre, Quelques resultants nouveaux concernat les forces exercées sur une inclusion sphérique par4 un écoulement accéléré, C. R. Aca. Sci. Paris Série II, n. 312, pp , [25] D. Lhuillier, Forces inertie sur une bulle en expansion se éplaçant ans un fluie, C. R. Aca. Sci. Paris Série II, n. 295, pp , [26] I. Eames, J.C.R. Hunt, Invisci flow aroun boies moving in a weak ensity graient in the absence of buoyancy effects, Journal of Flui Mechanics, vol. 353, pp , [27] D. Drew, R. Lahey, Particulate two-phase flow, Butterworth-Heinemann, Boston, Chapter 16, pp , [28] P. Bagchi an S. Balachanar, Inertial an viscous forces on a rigi sphere in straining flows at moerate Reynols numbers, Journal of Flui Mechanics, vol. 481, pp , [29] A. Tomiyama, H. Tamai, I. Zun, S. Hosokawa, Transverse migration of single bubbles in simple shear flows, Chemical Engineering Science, vol. 57, pp , [30] D.S. Dany, H.A. Dwyer, A sphere in shear flow at finite Reynols number: effect of shear on particle lift, rag, an heat transfer, Journal of Flui Mechanics, vol. 216, pp , [31] A. Kariyasaki, Behavior of a single gas bubble in a liqui flow with a linear velocity profile, Proceeings of the 1987 ASME/JSME Thermal Engineering Conference, n. 384, pp , [32] A. Tomiyama, A. Sou, A. Zun, I. Kanami, T. Sakaguchi, Effects of Eötvös number an imensionless liqui volumetric flux on lateral motion of a bubble in a laminar uct flow. Avances in Multiphase Flow, Elsevier, pp. 3-15, [33] H. Svensen, H.A. Jakobsen, R. Torvik, Local flow structures in internal loop an bubble column reactors, Chemical Engineering Science, vol. 47, pp , [34] S.L. Soo, Multiphase flui ynamics, Alershot- Brookfiel, USA: Gower Technical, [35] C.T. Crowe, M. Sommerfel, Y. Tsuji, Multiphase flows with roplets an particles, CRC Press, New York, [36] R.E.G. Poorte, On the motion of bubbles in active gri generate turbulent flows, PhD thesis, University of Twenty, Netherlans, September, [37] M. van Sint Annalan, N.G. Deen, J.A.M. Kuipers, Numerical simulation of gas bubbles behavior using a tree-imensional volume of flui metho, Chemical Engineering Science, vol. 60, pp , , pp , [38] C.W. Hirt, B.D. Nichols, olume of flui (OF) metho for the ynamics of free bounaries, Journal of Computational Physics, vol. 39, p. 201, [39] S. Popinet, S. Zaleski, A front-tracking algorithm for accurate representation of surface tension, International Journal of Numerical Methos in Fluis, vol. 30, pp , [40] M. Sussman, P. Smereka, S. Osher, A level set approach for computing solutions to incompressible two-phase flow, Journal of Computational Physics, vol. 114, pp , [41] M. Sussman, E. Fatemi, An efficient interfacepreserving level set reistancing algorithm an its application to interfacial incompressible flui flow, SIAM Journal of Scientific Computing, vol. 20, pp , [42] J.E. Welch, F.H. Harlow, J.P. Shannon, B.J. Daly, The MAC metho: a computing technique for solving viscous incompressible transient flui flow problems involving free surfaces. Los Alamos Scientific Laboratory Report LA-3425, [43] W.J. Rier, D.B. Kothe, Reconstructing volume tracking, Journal of Computational Physics, vol. 141, pp , [44] S.O. Unveri, G. Tryggvason, A front tracking metho for viscous, incompressible multi-flui flows, Journal of Computational Physics, vol. 100, pp , [45] D. Bhaga an M.E. Weber, Bubbles in viscous liquis: shapes, wakes an velocities, Journal of Flui Mechanics, vol. 105, pp , [46] U. Olgac, A.D. Kayaalp, M. Muraoglu, Buoyancyriven motion an breakup of viscous rops in constricte capillaries, Journal of Multiphase Flow, vol. [47] W. Dijkhuizen, E.I.. van en Hengel, N.G Deen, M. van Sint Annalan, J.A.M. Kuipers, Numerical investigation of closures for interface forces acting on single air-bubbles in water using olume of Flui an Front Tracking moels, Chemical Engineering Science, n. 60, pp , 2005.