3D volume-of-fluid simulation of a wobbling bubble in a gas-liquid system of low Morton number

Transcription

1 Fourth Internationa Conference on Mutiphase Fow, ICMF-2 New Oreans, Louisiana, U.S.A, May 27 June, 2 3D voume-of-fuid simuation of a wobbing bubbe in a gas-iquid system of ow Morton number W. Sabisch, M. Wörner, G. Grötzbach, D.G. Cacuci Forschungszentrum Karsruhe GmbH, Institut für Reaktorsicherheit, Postfach 364, 762 Karsruhe, Germany Abstract A new computer code has been deveoped for the direct numerica simuation of certain aspects of three-dimensiona bubbe dynamics in a fuid. This code combines the Voume-of-Fuid method for tracking the gas-iquid interface with a new, piecewise inear, interface reconstruction agorithm caed EPIRA. Resuts are presented for singe gas bubbes rising in a pane vertica channe fied with a iquid. Four distinct combinations of Eötvös and Morton numbers, in the range.2 Eö B 243 and M 266, are considered. Three of these combinations yied a spherica, eipsoida, and obate eipsoida cap bubbe, respectivey, each rising steadiy on a rectiinear path. For Eö B = 3.7 and M = 2.5 -, a bubbe with an irreguar, osciating wobbing shape rising on an irreguar spira path, is obtained. Even though the simuations were carried out with a gas-to-iquid density ratio of.5, a of the numerica resuts for bubbe shape, wake, and path characteristics are in cose agreement with the experimenta resuts for gas-iquid systems characterized by the same vaues for Eö B and M, but having density ratios much smaer than.5.. Introduction Athough the fundamenta physica understanding of the rise of gas bubbes in a continuous iquid is of significant practica importance for a variety of engineering appications, neither the interactions between bubbes rising in custers nor the bubbe-induced pseudo-turbuence (i.e., the generation of veocity fuctuations by bubbes and their wakes in an otherwise aminar fow) are fuy understood yet. Modeing the bubbe-induced pseudo-turbuence with the current generation of Computationa Fuid Dynamics (CFD) codes requires detaied information about the fu threedimensiona (3D) veocity fied cose to the bubbe and in its wake. Such information, though, cannot be obtained experimentay yet, since even advanced experimenta techniques, such as partice-image-veocimetry (PIV), can at best yied two-dimensiona (2D) projections of the fow at any given instant in time (Brücker, 999). In the absence of detaied experimenta information, the direct numerica simuation of the fow modeed by the Navier-Stokes equation can provide, in principe, a fu 3D time-resoved veocity data fied. The eading methods currenty used for detaied numerica simuations of the hydrodynamics of singe or mutipe bubbes rising in a continuous iquid, where the gas-iquid interface is deformabe, are the front-tracking method (Tryggvason et a., 998), the eve-set method (Sethian, 999), and the Voume-of-Fuid (VOF) method (Kothe, 998; Scardovei and Zaeski, 999) together with finite-difference discretization of the Navier-Stokes equation. However, there are no reports in the open iterature, to our knowedge, of appying the numerica methods mentioned above to 3D-probems invoving Morton numbers M beow M -, where M is defined as Corresponding author. E-mai address: woerner@irs.fzk.de

2 M ( ρ ρ ) 4 g g µ, () 2 3 ρ σ with ρ, µ, σ, g representing density, dynamic viscosity, surface tension, and gravity, respectivey. The superscript () denotes dimensiona quantities, whie the subscripts and g denote the iquid (continuous) and the gas (dispersed) phases, respectivey. Most gas-iquid systems of practica reevance are characterized by very ow Morton number; for exampe, M - for air bubbes in water, whie the gas-iquid density ratio is about /. To simuate systems characterized by very ow Morton numbers, we have deveoped a new numerica method, which combines the Voume-of-Fuid interface tracking technique with the soution of the non-dimensiona incompressibe Navier-Stokes equations using a finite voume formuation on a staggered grid. For the interface reconstruction, we have deveoped a new piecewise inear agorithm (EPIRA), which reconstructs a inear three-dimensiona interface (i.e. a pane) exacty. The purpose of this paper is to present our new numerica method together with iustrative 3D-resuts for singe bubbes rising in a computationay periodic, pane vertica channe, for a gas-iquid density ratio of.5. In particuar, we present resuts for the case M = 2.5 -, which show a shape-osciating bubbe in the wobbing regime, rising on an irreguar spira. 2. Governing equations Since we use a finite voume method, we derive the governing equations by averaging the oca equations over a voume V. With f denoting the iquid voumetric fraction within V, we define the mixture density and viscosity, respectivey, as ρ f ρ + ( f ) ρ (2) g µ f µ + ( f ) µ g (3) and aso define the center of mass veocity as T u = ( u,v,w ) f ρ u + ( f ) ρ gug. ρ (4) Both the iquid and gas are assumed to be incompressibe Newtonian continua; furthermore, we assume that there is no sip between the phase veocities in interface mesh ces. Thus, the nondimensiona continuity and Navier-Stokes equation can be written as u = (5) ρ u T Eö g aintκ n + ( ρ u u) = p + [ µ ( u + ( u) )] ( f ) +, t Re We g We (6) where g = g 2 = 9. 8ms, and where the various non-dimensiona quantities are defined as foows: x x u tu p g x x y ρ µ ρ = =, u =, t =, ρ =, µ =, p =. 2 L U L ρ µ ρ U z (7) The derivations eading to Eqs. (5) and (6) are detaied in the paper by Wörner et a. (2, this conference). The dimensioness groups in Eq. (6) are the erence Reynods, Eötvös, and Weber numbers, respectivey, defined as

3 Re ρ L U µ, Eö 2 2 ( ρ ρ ) g L σ, We ρ L U σ g. (8) By repacing, in Eq. (8), L by the bubbe equivaent diameter d e and, respectivey, U by the termina rise veocity U T of the bubbe, we obtain the bubbe Reynods (Re B ), bubbe Eötvös (Eö B ), and bubbe Weber number (We B ). The Morton number is reated to the Reynods, Eötvös and Weber numbers by means of the reation M = EöWe Re = EöBWeB ReB. (9) The ast term on the right-side of Eq. (6) is due to surface tension; in that term, a int, κ, and n denote the (non-dimensiona) interfacia area concentration, mean interface curvature and mean unit norma vector to the interface within the averaging voume V. The above set of equations is competed by the transport equation for the iquid voumetric fraction f t + ( f u) = which can be derived from the continuity equation for the iquid phase. 3. Numerica Method 3.. EPIRA agorithm for interface reconstruction We have deveoped a new agorithm, caed EPIRA, which yieds a ineary-accurate interface reconstruction on a 3D structured orthogona non-equidistant fixed grid. The acronym EPIRA stands for Exact Pane Interface Reconstruction Agorithm. The agorithm beongs to the cass of PLIC (Piecewise Linear Interface Cacuation) methods. EPIRA reconstructs a pane 3D-interface exacty, regardess of its orientation; in this sense, EPIRA is second-order accurate. In this subsection, we provide a short description of the EPIRA reconstruction step. Additiona detais can be found in the works of Sabisch et a. (999) and Sabisch (2). In EPIRA, we assume a functiona interface of zero thickness, which can ocay be described by a singe vaued height function ξ = h(ζ,η), where (ζ,η,ξ ) define a oca Cartesian coordinatesystem. Expanding h(ζ,η) in a Tayor-series around the point (ζ,η ) yieds ( ζ, η) ( ζ, η) ( ζ, η) 2 h h h h ( ζ, η) = h + ( ζ ζ ) + ( η η ) + ( ζ ζ )( η η ) () ζ η ζ η We approximate ocay the interface by a pane by retaining ony the first three terms on the rightside of Eq. (), namey: ( ζ, η) h + α( ζ ζ ) + β ( η ) h. (2) η This approximation is reasonabe if (ζ,η) is cose to (ζ,η ) or if the interface curvature is sma. To reconstruct the interface orientation and position, we need to determine a vector n = (n ζ, n η, n ξ ) T norma to the interface, and a point (b ζ, b η, b ξ ) within the interface. For this purpose, we consider a rectanguar ce (i,j,k) with a iquid voumetric fraction satisfying < f i,j,k <. For each of the six faces of the interface-ce (i,j,k), we check to see if the respective face happens to be a gas/iquid (g/)-interface-face. Thus, face E(ast) ocated at x i+/2 is a g/-interface-face if < f i+,j,k < ; the faces W(est) at x i-/2, N(orth) at z k+/2, S(outh) at z k-/2, F(ront) at y j-/2 and B(ack) at y j+/2 are checked simiary. Next, we determine a tangentia vector for each g/-interface-face, as iustrated in Fig. for face E, where, without oss of generaity, we assume that the oca co-ordinate system (ζ,η,ξ ) coincides with the goba co-ordinate system (x,y,z). If the pane h(ζ,η) does not cut the upper or ower faces of the adjacent ces (i,j,k) and (i+,j,k), then N i,j,k, S i,j,k, N i+,j,k and S i+,j,k are not g/-interface-faces. ()

4 Fig. : Ces (i,j,k) and (i+,j,k) with pane h(x,y) representing the interface. The fuid is beow h(x,y). It hence foows that x y i j x + x i i+ j fi, j,k = h( x, y) dydx, fi+, j,k = h( x, y) dydx. (3) xi y j zk x + i y j zk xi Inserting Eq. (2) in Eq. (3), performing the integrations, and subtracting f i,j,k from f i+,j,k yieds k E = i+ 2, j,k = i+, j,k xi + xi+ ( f f ) 2 z α α. (4) i, j,k Since α E represents the sope of the interface in the x-direction, it foows that the unit tangentia vector t E is obtained as t E = 2 + α E. (5) α E Note that this tangentia vector is exact for any panar interface, if the ower and upper integration imits in Eq. (3) are correct. As mentioned above, this requires that N i,j,k, S i,j,k, N i+,j,k and S i+,j,k are not g/-interface-faces. Otherwise, the integration imits must be adjusted to account for the intersections of the g/-interface with the ower upper ce faces, respectivey. Such adjustments woud require the consideration of severa distinct situations and woud be computationay ineffective. Figure 2a depicts a case where, even though the face N i,j,k is a g/-interface-face, α E can sti be determined exacty. For this purpose, we extend the domain for computing α E by considering a pair of doube-ces, (i,j,k)+(i,j,k+) and (i+,j,k)+(i+,j,k+), respectivey (see Fig. 2b). For each doube-ce, Eq. (3) can be used again to obtain α E and t E exacty, by repacing z k with z k + z k+ on the right-side of Eq. (3), and f i,j,k +f i,j,k+ (instead of f i,j,k ) and f i+,j,k +f i+,j,k+ (instead of f i+,j,k ) on the respective eft-sides. A simiar procedure is used when S i,j,k or S i+,j,k are g/-interface-faces. b) y a) Fig. 2: a): Exampe where N i,j,k is a g/-interface-face. b): An extension of the eft and right voumes is performed to obtain the sope α E exacty.

5 When N i,j,k+, N i+,j,k+, S i,j,k- or S i+,j,k- are g/-interface-faces, one upper and/or ower extension is sti insufficient to obtain α E exacty. Since we do not wish to ose the ocaity of the reconstruction, though, we perform at most one extension above and/or one beow the basic pair of ces. If this is not sufficient to determine a vaue for α E, then we guess a vaue for it by setting the sope β in Eq. (2) to zero and thereby simpifying the 3D-probem to a 2D one. By considering four possibe different cases for the resuting 2D-probem (see Figure 3), the sope α E is determined anayticay via a case check diagram according to the 2D FLAIR agorithm of Ashgriz and Poo (99), which was extended by us to non-equidistant grids. Using this approximate (2D-) vaue for α E in Eq. (5) yieds an approximate vaue for t E. Fig. 3: Four cases of 2D FLAIR agorithm. Thus, a unit tangentia vector is computed using the above procedure at each of the (minimum three and maximum six) g/-interface-faces of an interface mesh ce. The unit tangentia vector thus computed is fagged as exact, if the sope is determined via Eq. (4), or as inexact, if it is computed using the extended 2D FLAIR agorithm. For computationa reasons, we set the tangentia vector to the nu vector when a face is not a g/-interface-face. From the tangentia vectors at the different faces of ce (i,j,k), we compute a ce-centered unit norma vector, n i, j, k, for each interface ce. We demonstrate this computation for ce (i,j,k), as iustrated in Fig., where t N and t S are zero, whereas t E, t W, t F and t B are not. For each coordinate direction, we determine a representative tangentia vector as foows: if t E and t W are marked with the same fag (i.e., both exact or both inexact), then t EW is obtained by taking their average, namey tew = ( te + tw ) te + tw. Otherwise, either t E or t W can be seected as representative, athough the exact one is to be perred. We fag t EW as exact if t E, t W, or both are exact. The other representative unit tangentia vectors, t NS these three representative unit tangentia vectors, the (minimum one and maximum three) and t FB, are determined simiary. Using preiminary unit norma vectors are computed as shown beow: tew tns tns tfb tfb tew n =, n2 =, n3 =. (6) t t t t t t EW NS NS FB FB If necessary, each of these preiminary norma vectors is re-orientated to point inside the fuid. By averaging these preiminary norma vectors using the same steps as used for obtaining the representative tangentia unit vectors, we obtain, utimatey, the representative ce-centered unit norma vector. For the situation iustrated in Fig., the vector t NS is zero, which impies that ony the norma vector n 3 is non-zero; no averaging is required. By shifting iterativey the interface-representing pane unti the correct iquid voumetric fraction within the mesh ce is recovered, the interface reconstruction is utimatey competed by determining the point (b ζ, b η, b ξ ). The fuxes of the iquid phase across the faces of the ces are computed in the advection step using the naive unspit method. The iquid fuxes in the three co-ordinate directions are computed simutaneousy, based on ony one reconstruction step for each time step. A cuboid of infuence is defined cose to each face; the cuboid s base is the respective ce-face and its depth is the respective face-centered veocity mutipied by the time step width. EW

6 When the fow is obique to the ce faces, the three cuboids of infuence overap in the naive unspit method, and iquid voume may be advected twice. In such a case, as we as in cases of ow bubbe resoution and/or high interface curvature, it can happen that, after the advection step, there remain interface-ces containing a very sma fraction of fuid, ε, or gas, -ε, and ony two g/interface-faces. Such a configuration is not meaningfu for EPIRA, so we redistribute the fuid or gas to neighboring interface-ces. The vaue of ε is reated to the residuum of the divergence of the veocity fied, which is set to -5 in our simuations Soution strategy The EPIRA agorithm has been impemented in our in-house code TURBIT-VOF, which soves the conservation equation for f together with the non-dimensiona continuity and the Navier-Stokes equations shown in Eqs. (5) and (6). TURBIT-VOF is designed to simuate fows in pane channes. It empoys a staggered grid, uniform in the x- and y-directions (parae to the was), but possiby non-uniform in z-direction (norma to the was). The soution strategy is based on a projection method. The predictor step incudes the convection, friction, and buoyancy terms, namey ~ ( ρ u) T Eö g = ( ρ u u) + [ µ ( u + ( u) )] ( f ) t Re We g (7) = C + F B = L( ρ, µ,u, f ). This predictor step is performed by an expicit third order Runge-Kutta scheme as sketched beow: () () n n n n n ρ u = ρ u + t L ρ n, µ,u, f (2) ρ u (2) 3 n = ρ u 4 n ( ) () + ρ u 4 t L () () () () ( ρ, µ,u, f ) ~ ~ n n 2 (2) (2) 2 (2) (2) (2) (2) ρ u = ρ u + ρ u + t L( ρ, µ,u, f ) For the intermediate time eves, we set f () + 4 (2) n+ () (2) ~ n+ () (2) n+ = f = f, ρ = ρ = ρ = ρ, µ = µ = µ. (9) () Treating the pressure term and surface tension term impicity, the corrector step is n+ n+ ~ ~ ρ u ρ u n + n = p + S +. (2) t Appying the divergence operator to Eq. (2) and introducing the continuity equation u n+ = m yieds the Poisson equation t n+ ρm p = u t + + ρm n+ ~ m n n S + (8). (2) A Conjugate Gradient sover (Schönauer et a., 997) is used to sove the above Poisson equation. Aternativey, the buoyancy term can be incuded in the corrector step rather than in the predictor step; we found that this procedure is numericay advantageous for free surface fows Spatia discretization on a staggered grid A second order centra difference scheme is used to discretize the diffusive and non-inear convective terms in the Navier-Stokes equation. For treating the convective terms, we are currenty deveoping a high resoution WENO (Weighted Essentiay Non Osciatory) discretization scheme

7 with one-dimensiona reconstruction aong each co-ordinate direction. WENO shock capturing schemes were originay deveoped for compressibe singe phase fows. The WENO scheme intended for TURBIT-VOF is envisaged to minimize the numerica smearing when computing the convective terms, for ow gas/iquid density ratios, within the three sub-integration steps of the Runge-Kutta time integration procedure. For the gas/iquid density ratio of.5 considered in the present paper, we found that the resuts obtained with the centra difference scheme agree we with those obtained with the WENO scheme presenty impemented in TURBIT-VOF. Within the staggered grid used in TURBIT-VOF, the density, viscosity and the iquid voumetric fraction f i,j,k are ce-centered, whie the contro voumes for the three components of the Navier- Stokes equation are shifted by haf a mesh-width to obtain the veocity components as u i+/2,j,k, v i,j+/2,k, w i,j,k+/2. For the spatia discretization of the various terms in the Navier-Stokes equation, though, we need veocities at centered positions, and the density and viscosity at shifted positions, respectivey. In the current version of TURBIT-VOF, which does not consider mass transfer, the veocities at centered positions are obtained from a inear interpoation, which is justified since the veocity is continuous across the interface (in the absence of mass transfer). The density and viscosity, however, are discontinuous across the interfaces, so that a inear interpoation may introduce arge errors. For this reason, we compute these quantities by a different method as foows: taking into account the actua interface ocation in ce (i,j,k) and ce (i+,j,k), we compute the iquid voumes within the haf-ces [x i,j,k, x i+/2,j,k ] [y i,j-/2,k, y i,j+/2,k ] [z i,j,k-/2, z i,j,k+/2 ] and [x i+/2,j,k, x i+,j,k ] [y i,j-/2,k, y i,j+/2,k ] [z i,j,k-/2, z i,j,k+/2 ], respectivey. We then add both iquid voumes together and divide the resuting sum by the tota voume of both haf-ces to obtain f i+/2,j,k. Having thus obtained f i+/2,j,k aows us to compute ρ i+/2,j,k. The quantity f i+/2,j+/2,k is computed simiary, by considering four quarter-ces Discretization of the surface tension term In this subsection, we iustrate the dicretization procedure for the first component of the source term due to surface tension in the Navier-Stokes equation; the other components are obtained simiary. In principe, we must compute source terms of the form Si+ 2, j,k = aint;i + 2, j,k i+ 2, j,k ni+ 2,j,k We κ. (22) The EPIRA reconstruction agorithm yieds the interface unit norma vector at ce centers (i,j,k). The staggered interface unit norma vector is obtained from the interpoation ni,j,k + ni+,j,k ni+ 2,j,k =. (23) ni,j,k + ni+,j,k Note that n i,j, k is initiaized as the nu vector in any ce which is not an interface mesh ce. Next, we foow Brackbi et a. (992) to compute the curvature from the gradient of the unit norma vector, κ = n. For the first component of the Navier-Stokes equation, this procedure gives nx;i +,j,k nx;i, j,k n y;i+ 2, j+ 2,k n y;i + 2, j 2,k nz;i + 2, j,k + 2 nz;i + 2, j,k 2 κ i+ 2, j,k = + +. (24) x y zk The staggered vaues of the unit norma vector in Eq. (24) are obtained by interpoation as: ni,j,k + ni,j±,k + ni+,j,k + ni+,j±,k ni,j,k + ni,j,k ± + ni+,j,k + ni+,j,k ± ni+ 2,j± 2,k =, ni+ 2,j,k ± 2 =. (25) n + n + n + n n + n + n + n i,j,k i,j±,k i+,j,k i+,j±,k i,j,k i,j,k ± i+,j,k i+,j,k ± The interfacia area concentration a int;i+/2,j,k is computed by adding the areas of the interfaces in the two haf-ces [x i,j,k, x i+/2,j,k ] [y i,j-/2,k, y i,j+/2,k ] [z i,j,k-/2, z i,j,k+/2 ] and [x i+/2,j,k, x i+,j,k ] [y i,j-/2,k, y i,j+/2,k ] [z i,j,k-/2, z i,j,k+/2 ], and dividing the resut by the tota voume of both haf-ces.

8 3.5. Code verification Severa prototypica interfacia probems have been considered, as detaied by Sabisch (2), to verify TURBIT-VOF. E.g., the computationa accuracy of the surface tension mode has been verified by simuating capiary waves. In this benchmark probem, the surface tension acts as the soe driving force on an initiay sinusoida interface at rest. The tempora deveopment of the osciation damped by viscosity has been computed by TURBIT-VOF, and it agrees we with the respective anaytica soution. Furthermore, the cacuation of the buoyancy force has been verified by simuating both gravity waves and the Rayeigh-Tayor instabiity, starting with a sinusoida interface initiay at rest. Aso in the case of gravity waves, the computed osciation of the interface damped by viscosity agrees we with the respective anaytica soution. Both 2D and 3D configurations have been considered for computing the respective Rayeigh-Tayor instabiities. The computed veocities of the rising bubbe and the faing finger are in exceent agreement with vaues from iterature. As detaied by Sabisch (2), very good agreements between numerica and anaytica soutions have aso been obtained for other interfacia probems used to verify the new numerica procedures impemented in TURBIT-VOF. 4. Simuation parameters 4.. Physica parameters The reevant physica quantities for a bubbe rising with its termina veocity through an infinite iquid are incuded in the foowing equation (Grace, 973) ( g,, µ, σ, ρ,,d,u ) = F ρ g µ g e T. (26) The above equation may be rewritten in terms of five independent dimensioness groups as F ( ReB,EoB,M, Γ ρ, Γµ ) =, (27) where Γ = ρ ρ g ρ and Γ = µ µ g µ are the density and viscosity ratios, respectivey. In the simuations reported in this paper, Eö B, M, Γ ρ and Γ µ have been hed fixed, and the simuations were performed for four different combinations of (Eö B, M) as shown in Tabe. According to the Re B versus Eö B diagram of Cift et a. (978) we expect for cases,, and to obtain a bubbe with a steady shape (eipsoida, obate spherica cap, and spherica, respectivey), rising steadiy aong a rectiinear path. The bubbe for case is expected to be in the wobbing regime. For a bubbe simuations presented in Tabe, the density ratio is Γ ρ =.5 and the viscosity ratio is Γ µ =. Setting h min =min k ( x, y, z k ), the convective, viscous, and capiary time step criterion are 2 3 h h minre Γ min ρ hmin tcfl, tµ min,, tσ We n max 6 u Γµ 4π i, j,k Tabe : Physica and numerica parameters for TURBIT-VOF simuations ( + Γ ) ρ. (28) Case Eö B M Re We L x L y L z N x N y N z d e (a) (b) (c) ½ ½ (d)

9 The formua for t µ in the inequaities above indicates that ow vaues of Γ ρ ead to stiff probems, in the sense that the diffusive time scaes of the gas and iquid phases become significanty distinct from each other. For cases (a-c), for exampe, t CFL, t µ, and t σ are of same order of magnitude ( t max.2) when Γ ρ =.5. For a smaer vaue of Γ ρ, the viscous time step criterion wi become the most stringent among a. Due to the discontinuity at the interface, very ow density ratios aso produce numerica difficuties when computing the convective terms, and woud theore ikey need a much higher mesh-ce resoution of the bubbe than is necessary for Γ ρ = Grid parameters, boundary conditions, and initia conditions Numerica simuations have aso been performed for a bubbe rising between two parae pane vertica was, using a non-dimensiona wa distance L z = L z / L = and no sip boundary conditions at z= and z=. In the vertica (x) and span-wise (y) directions, we have used periodic boundary conditions, with periodicity engths L x and L y, respectivey. The orientation is such that the gravity vector points in the negative x-direction. Thus, the resuts to be presented in section 5 beow wi er to the rise of a reguar array of bubbes, as opposed to the rise of a singe bubbe. The simuations were performed on uniform grids consisting of N x N y N z cubica mesh ces. As Tabe shows, most of the simuations were carried out for a cubica computationa domain discretized by 64x64x64 mesh ces, where L = 4m and U = ms -. This discretization aows a 6-mesh-ces resoution of the non-dimensiona bubbe equivaent diameter d e =.25, and corresponds to a tota gas content of about.8%. The periodic boundary conditions used in the numerica simuations mentioned in Tabe aow ony a quaitative comparison with experiments for singe bubbes rising in an infinite iquid. Aso, since we considered a finite channe, the function F in Eq. (26) woud need to incude additiona dimensiona quantities, since, for bubbes rising in a channe or a pipe, U T is aso infuenced by the ratio of the bubbe equivaent diameter to the channe hydrauic diameter. In our simuations, this ratio is λ =.5 d e L z, and has typicay the vaue /8. According to Cift et a. (987, formua 9-35), the termina-rise veocity in this case is reduced by ess than 3% due to wa effects, as compared to the termina-rise veocity of a freey rising bubbe. Theore, we can negect the wa effects for the cases presented in Tabe. For a of the simuations of case, the fuid and gas were initiay at rest. To save CPU time, the simuations for cases,, and were not initiaized from stagnant conditions but from the fina fow state of run (b). The computations were performed on a singe processor of a vectorparae SNI VPP 3. Due to its transient nature, case is the most expensive computationay; the tota CPU time required was about hours. 5. Resuts In the foowing two subsections, we assess the infuence of the density ratio of.5 used in our simuations by comparing the numerica resuts for cases and with experimenta data for gasiquid systems in which the Morton and Eötvös numbers were the same as used in the numerica simuations, but where the density ratios were of the order of -3. In subsection 5.3, we discuss resuts for case, which simuates a wobbing bubbe. For case, the Morton number corresponds to that of an air-water system. Additiona detais for this case are given by Sabisch (2), who obtained a steady, amost spherica, bubbe that rises rectiineary. 5.. Eipsoida bubbe (Eö B = 3.7, M = ) Based on experiments for fifty-four dispersed-continuous phase systems, Week et a. (966) derived empirica reations for the eccentricity E of non-osciating bubbes and drops, over a wide range of partice Reynods numbers. For Eö B < 4 and M -6, they obtained the reation

10 . 757 ( ) E = Eö B. (29) In our simuation, we have obtained (after a transient from the initia spherica shape) an eipsoida bubbe rising steadiy on a rectiinear path, with an confined recircuation zone at its rear. The bubbe s eccentricity was obtained as E =.7 for case (a); this vaue agrees we with the vaue E =.72 obtained from Eq. (29) for Eö B = 3.7. Aso, we computed the bubbe Reynods number as Re B = 6.5, and this vaue ies we within the range of 5 to 7, as observed in experiments (Cift et a., 978). From these resuts, we concuded that the density and viscosity ratios pay ony minor roes in this case. Cases (b) and (c) depict the infuence of the size of the computationa domain, since we use the same bubbe resoution as in case (a). The ratio L x / d e is a measure for the infuence of the periodic boundary conditions on the vertica distance between consecutive bubbes. The ower the ratio L x / d e, the stronger the infuence of the periodic boundary conditions. Tabe 2 shows that reducing the ratio L x / d e = 8 used in case (a) to L x / d e = 4 in cases (b) and (c) eads to an increase of Re B of about %. This increase of the bubbe rise veocity is to be expected because the bubbe comes coser to the eading bubbe and thus experiences a smaer drag. By using the same computationa domain but having the mesh width of case (b), we study in case (d) the infuence of the grid size and bubbe resoution d e / x. As the resuts presented in Tabe 2 show, the grid inement has ony a very sma effect on Re B and We B. Tabe 2: Re B and We B for the different domain sizes and grids of case case (a) case (b) case (c) case (d) L x / d e d e / x Re B We B Obate eipsoida cap bubbe (Eö B = 243, M = 266) The Eötvös and Morton numbers of case correspond to those found in an experiment by Bhaga and Weber (98), as they investigated an air bubbe rising in aqueous sugar soution with Γ ρ /. Fig. 4 shows the stationary bubbe shape in experiment and simuation, respectivey. The obate eipsoida cap bubbe shows the typica impression at its rear. In the experiment, the bubbe is sighty more obate than in the simuation. The reason is presumaby due to the periodic boundary conditions we have used in the simuation. The important effect of the periodicity ength in the direction in which the bubbe rises was aready discussed in the foregoing. For case, we have not yet investigated the infuence of L x / d e. Theore, we cannot precude that the difference in shape in Fig. 4 may party be due to the difference in density and viscosity ratios between the experiment and our simuation. Fig. 4: Steady bubbe shape for Eö B = 243, M = 266 in experiment (Bhaga and Weber, 98) (eft) and simuation (right).

11 5.3. Wobbing bubbe (Eö B = 3.7, M = ) The simuation for case has been initiaized from the fina state of simuation (b), having the same Eötvös number Eö B = 3.7 but a arger Morton number (M = ). The Morton number has been stepwise decreased by increasing Re stepwise, typicay by one hundred, from to 5. A hundred time steps were computed for each intermediate Re. About 6 additiona time steps were computed after the Morton number reached its end vaue of M= Atogether, this procedure has simuated a vertica trave distance of about 6 times d e or, equivaenty, about four times the height of the computationa domain. The bubbe Reynods number reached in this simuation was about. The trajectory of the bubbe s center of mass is depicted in Fig. 5, which shows that the initiay rectiinear path becomes unstabe as the bubbe rises. Initiay, the bubbe performs a atera motion in a x-z-pane, which aready starts in the initia transient, when M is decreased to Associated with the path instabiity, we observe sma asymmetries in the bubbe shape. Once the bubbe has risen to about the height of the computationa domain, when M is reached, the atera motion is aso observed in the x-y-pane. In the y-z-pane-view presented in Fig. 5 (d), a spira motion is observed in accordance with the irreguar heica path foowed by the bubbe. Overa, this transition occurs very fast, presumaby because of the periodic boundary conditions, which cause the bubbe to interact with its own wake, thereby strongy ampifying sma disturbances. Fig. 6 shows one instantaneous bubbe shape together with veocity vectors and vorticity contour ines in the pane z=.5. The respective visuaization dispays the structure of the wake, which is open and dispays sma vortices associated with the bubbe-induced turbuence. Athough it is we known that bubbes with Re B =O() may undergo irreguar shape osciations and rise on more or ess reguar zigzag, rocking or spira paths (Cift et a., 998; Fan and Tsuchiya, 99), it is very difficut to perform quantitative or even quaitative comparisons of the present resuts with experimenta investigations, such as those of Brücker (999) or Stewart (995), for exampe. This is due not ony to the channe geometry and periodic boundary conditions used in our simuations, but aso to the fundamenta difficuty of describing the compex shape, motion and wake characteristics of wobbing bubbes. Furthermore, the dynamics of such bubbes is strongy infuenced by the presence of surface-active contaminants, an effect that is not taken into account in our simuations. Fig 5. Trajectory of the bubbe s centre of mass: (a) Bird eyes view, (b) - (d) 2D projections. Fig 6. Visuaisation of instantaneous bubbe shape, veocity vectors, and vorticity contour ines in pane z =.5.

12 6. Concusions We have deveoped a computer code for the direct numerica simuation of certain aspects of bubbe dynamics in a fuid by combining the Voume-of-Fuid method for tracking the gas-iquid interface with a new interface reconstruction agorithm. The code simuates we the underying physica phenomena, as we have shown using typica 3D benchmark probems for singe bubbes of different shape, wake, and motion. In particuar, our code can be used to simuate even gas-iquid fow regimes at ow Morton numbers and high bubbe Reynods numbers, where, as evidenced by many experiments, the bubbe shapes and rising paths are irreguar and non-steady, whie the wake is turbuent. Such bubbes are of specia practica reevance for genera technica appications e.g. in power engineering and chemica engineering. Athough the simuations we presented in this paper were performed using a gas-iquid density ratio of.5, they showed that the Eötvös and Morton number are the most important simiarity parameters that determine the bubbe shape and the type of its wake. Since the bubbe-induced turbuence is strongy associated with the fow around the bubbe and the generation of vortices in its wake, it aso foows that the Eötvös and Morton numbers wi be important parameters in modes for bubbe-induced turbuence. Our code is not restricted to density ratios of order. -. Future work is aimed at simuations in which the density ratio is varied but the Eötvös and Morton numbers are kept fixed. A quaitative verification of our code is aso envisaged for bubbe Reynods numbers of about 4, where reguar spiraing bubbes, undergoing no shape-osciations, are expected according to detaied experimenta investigations (Brücker, 999). Longer-term research aims at investigating bubbe-bubbe and bubbe turbuence interactions, by simuating the behavior of custers of bubbes. The overa goa of this research is to contribute towards the deveopment of improved statistica modes to account for phenomena in engineering CFD codes for bubby fows. References Ashgriz, N., Poo, J.Y., 99. FLAIR: Fux ine-segment mode for advection and interface reconstruction. J. Comput. Physics 27, Bhaga, D., Weber, M.E., 98. Bubbes in viscous iquids: shapes, wakes and veocities. J. Fuid Mech. 5, Brackbi, J.U., Kothe, D.B., Zemach, C., 992. A Continuum Method for Modeing Surface Tension. J. Comput. Phys., Brücker, C., 999. Structure and dynamics of the wake of bubbes and its reevance for bubbe interaction. Physics of Fuids, Cift, R., Grace, J.R., Weber, M.E., 978. Bubbes, Drops, and Partices. Academic Press. Fan, L.S., Tsuchiya, K., 99. Bubbe wake dynamics in iquids and iquid-soid suspensions. Butterworths-Heinemann Series in Chemica Engineering. Butterworths, Boston. Grace J.R., 973. Shapes and veocities of bubbes rising in infinite iquids. Trans. Instn. Chem. Eng. 5, 6-2. Kothe, D.B., 998. Perspective on Euerian Finite Voume Methods for Incompressibe Interfacia Fows. In: Free Surface Fows. H.C. Kuhmann, H-J Rath, eds., Springer-Verag, Sabisch, W., 2. Dreidimensionae numerische Simuation der Dynamik von aufsteigenden Einzebasen und Basenschwärmen mit einer Voume-of-Fuid-Methode. Forschungszentrum Karsruhe, Wissenschaftiche Berichte FZKA 6478, Juni 2 ( FZKA_Berichte/FZKA6478.pdf). Sabisch, W., Wörner, M., Grötzbach, G., Cacuci, D.G., 999. An improved voume of fuid method for numerica simuation of custers of bubbes. In: M. Sommerfed, ed., Proc.9 th Workshop on Two-phase Fow Predictions, Merseburg, Germany, Apri 3-6, 999, Scardovei, R., Zaeski, S., 999. Direct numerica simuation of free-surface and interfacia fows. Annu. Rev. Fuid Mech. 3,

13 Schönauer, W., Häfner, H., Weiss, R., 997. LINSOL, a parae iterative inear sover package of generaized CG-type for sparse matrices. In: M. Heath et a., eds., Proc. 8 th SIAM Conference on Parae Processing for Scientific Computing (SIAM, Phiadephia, PA, 997). CD-Rom (ISBN ). Sethian, J.A., 999. Leve set methods and fast marching methods: evoving interfaces in computationa geometry, fuid mechanics, computer vision, and materias science. 2 nd edition. Cambridge University Press, Cambridge, UK. Stewart, C.W., 995. Bubbe interaction in ow-viscosity iquids. Int. J. Mutiphase Fow 2, Tryggvason, G., Bunner, B., Ebrat, O., Tauber, W., 998. Computations of mutiphase fows by a finite difference/front tracking method. I. Muti-fuid fows. Von Karman Institute Lecture Notes, Von Karman Institute, Brusses, Begium. Week, R.M., Agrawa, A.K., Skeand, A.H.P., 966. Shape of iquid drops moving in iquid media. AIChE Journa 2, Wörner, M., Sabisch, W., Grötzbach, G., Cacuci, D.G., 2. Voume-averaged conservation equations for Voume-of-Fuid interface tracking. Proc. 4 th Int. Conference on Mutiphase Fow, ICMF-2, New Oreans, Louisiana, U.S.A., May 27 - June, 2.