Implicit Multigrid Computations of Buoyant Drops Through Sinusoidal Constrictions

Transcription

1 Metin Muradoglu 1 Department of Mecanical Engineering, Koc University, Rumelifeneri Yolu, Sariyer, Istanbul 34450, Turkey mmuradoglu@ku.edu.tr Seckin Gokaltun 2 Computational Science and Engineering Program, Informatics Institute, Istanbul Tecnical University, Maslak, Sariyer, Istanbul 34469, Turkey gokaltunse@itu.edu.tr Implicit Multigrid Computations of Buoyant Drops Troug Sinusoidal Constrictions Two-dimensional computations of dispersed multipase flows involving complex geometries are presented. Te numerical algoritm is based on te front-tracking metod in wic one set of governing equations is written for te wole computational domain and different pases are treated as a single fluid wit variable material properties. Te fronttracking metodology is combined wit a newly developed finite volume solver based on dual time-stepping, diagonalized alternating direction implicit multigrid metod. Te metod is first validated for a freely rising drop in a straigt cannel, and it is ten used to compute a freely rising drop in various constricted cannels. Interaction of two buoyancy-driven drops in a continuously constricted cannel is also presented. DOI: / Introduction Dynamics of dispersed bubbles or drops in capillary flows involving complex geometries as attracted considerable interest due to its applications in enanced oil recovery, azardous waste management, microfluidic devices, and biological systems 1 3. Te presence of deforming pases makes te multipase flow computations a callenging task, and strong interactions between te pases and complex geometries add furter complexity to te problem. Terefore, te progress was rater slow and te computations of multipase flows ave been usually restricted to simple geometries 4 or to moderately complex geometries in te limiting case of creeping flow regime 5,6. Since nearly allmultipase flows of practical importance involve complex geometries, it is of obvious interest to extend te modeling and computational tecniques to treat multipase flows in arbitrarily complex geometries. Te motion of a drop in a constricted capillary tube as been studied experimentally by Olbrict and Leal 1, Olbrict and Kung 7, and Hemmat and Boran 2, and computationally in te creeping flow regime by Tsai and Miksis 5 and Magna 6. Udaykumar et al. 3 performed computations of te motion of droplets in a constricted cannel at finite Reynolds numbers by using a mixed Eulerian-Lagrangian metod. In te present work, a finite-volume/front-tracking FV/FT metod is used to simulate dynamics of two-dimensional drops rising due to buoyancy in various constricted cannels. Te fronttracking FT metod developed by Unverdi and Tryggvason 8 is incorporated into a newly developed finite-volume FV algoritm in order to facilitate efficient and accurate simulations of dispersed multipase flows in arbitrarily complex geometries. Te front-tracking metod is based on writing one set of governing equations for te wole computational domain and treating different pases as a single fluid wit variable material properties. In tis metod, te fronts are explicitly tracked in a Lagrangian frame and te effects of surface tension are accounted for by 1 To wom correspondence sould be addressed. 2 Present address: Department of Mecanical Engineering, Florida International University, EAS 2417, W. Flagler Street, Miami, FL Contributed by te Applied Mecanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in te ASME JOURNAL OF APPLIED ME- CHANICS. Manuscript received by te Applied Mecanics Division, August 25, 2003; final revision; June 17, Associate Editor: T. E. Tezduyar. Discussion on te paper sould be addressed to te Editor, Prof. Robert M. McMeeking, Journal of Applied Mecanics, Department of Mecanical and Environmental Engineering, University of California Santa Barbara, Santa Barbara, CA , and will be accepted until four monts after final publication of te paper itself in te ASME JOURNAL OF APPLIED MECHANICS. treating tem as body forces. Te front tracking metod as been successfully applied to a variety of dispersed multipase flow problems, but all in relatively simple geometries. A detailed description of te front-tracking metod can be found in te review paper by Tryggvason et al. 4. Te FV metod used in te present work is based on te concept of te dual or pseudo timestepping metod and is developed for unsteady computations of incompressible laminar flows. Te dual time-stepping metod uses subiterations in pseudotime and as a number of advantages, suc as direct coupling of te continuity and momentum equations in incompressible flow equations, te elimination of factorization error in factored implicit scemes, te elimination of errors due to approximations made in te implicit operator to improve numerical efficiency, te elimination of errors due to lagged boundary conditions at bot solid and internal fluid boundaries, and te ability to use nonpysical, preconditioned iterative metods for more efficient convergence of te subiterations as discussed by Caugey 9. In order to combine te front-tracking metodology wit te FV metod, an algoritm is developed for tracking te front in curvilinear grids and is found to be very efficient and robust. Te details of te present FV/FT metod can be found in Muradoglu and Kayaalp 10. Te paper begins wit a brief description of te governing equations and te numerical solution algoritm. Te results are ten presented and discussed in Section 4. Te present FV/FT metod is first validated for a freely rising drop in a straigt cannel, and te results are compared wit te results of te finitedifference/front-tracking FD/FT metod implemented in te FTC2D code of Unverdi and Tryggvason 8. It is ten applied to a single drop rising in various constricted cannels. Interactions of two identical drops are also studied in te continuously constricted cannel. Finally, some conclusions are drawn in Section 5. 2 Matematical Formulation Following Unverdi and Tryggvason 8, te Navier-Stokes equations are written for te wole flow field, and different pases are treated wit variable material properties. Te effects of surface tension are modeled as body forces and are included in te momentum equations as functions at te pase boundaries. In te Journal of Applied Mecanics Copyrigt 2004 by ASME NOVEMBER 2004, Vol. 71 Õ 857

2 Cartesian coordinates, two-dimensional time-dependent Navier- Stokes equations for incompressible flow can be written in conservative form as were q f g t x y f v x g v y G xx f nds (1) 0 u v qu f pu g v, 2, uv 2, uv pv 0 0 f v xx xy yy (2) xy, g v and te viscous stresses are given for a Newtonian fluid as xx 2 u x, xy u v y x, yy 2 v y In Eqs. 1 3, u, v, p,, and denote te velocity components in x and y directions, te pressure, te density, and te dynamic viscosity, respectively. Te tird term on te rigt-and side of Eq. 1 represents te body force due to buoyancy wit G being te gravitational acceleration and o, were o is te density of te ambient fluid. Te last term represents te effects of te surface tension and, x f,,, n and ds denote te Dirac delta function, te location of te front, te surface tension coefficient, te curvature, te outward unit normal vector on te interface, and te arc lengt along te interface, respectively. Te fluids in and out of te drop are assumed to be incompressible, and te effects of eat transfer are neglected. Terefore, te viscosity and te density remain constant in eac fluid particle, i.e., D Dt 0, (3) D 0 (4) Dt Te flow regime of bubbly flows is caracterized by four nondimensional parameters as discussed by Clift et al. 11. Tese are te Morton number M o 4 ( o b )G/ o 2 3, te Eötvös number Eo( o b )d e 2 G/, te density ratio b / o, and te viscosity ratio b / o, were d e is te equivalent drop diameter and te subscripts o and b refer to te ambient and te drop fluids, respectively. Te Reynolds number is defined as Re o Vd e / o, were V is te rise velocity. 3 Numerical Procedure As can be seen in Eq. 1, te continuity equation is decoupled from te momentum equations because it does not ave any time derivative term in incompressible flows. To circumvent tis difficulty and to be able to use time-marcing algoritms, pseudotime derivative terms augmented wit a preconditioning matrix are added to Eq. 1 yielding wit 1 w q t f g x y f v x g v y og xx f nds (5) w p u v, u 0 (6) 2 2v 2 0 were is a preconditioning parameter wit dimensions of velocity and is a dimensionless parameter to be determined. In Eq. 5, denotes te pseudotime and te dual time-stepping metod is based on marcing in pseudotime until a convergence is reaced for eac pysical time step. Since te transient solution in pseudotime is not of interest, we are free to use any nonpysical convergence acceleration tecnique, suc as preconditioning, local time-stepping, and multigrid metods. To facilitate treatment of complex geometries, Eq. 5 is transformed into a curvilinear coordinates defined by x,y, x,y (7) Using te relation qi 1 w were te incomplete identity matrix I 1 is defined as I (8) 0 0 and te transformation given by Eq. 7, te transformed equations in te curvilinear coordinates can be written as 1 w I1 w t F G F v G v f b were x y x y is te determinant of te Jacobian of te transformation and F, G, F v, and G v are te transformed convective and viscous fluxes given by Fy fx g, F v y f v x g v, Gy fx g, G v y f v x g v (9) (10) Te vector f b represents te last two terms on te rigt-and side of Eq. 1, namely, te sum of te buoyancy forces and te surface tension. Following Caugey 9, subiterated implicit sceme to solve Eq. 9 can be written as 1 wp1 w p I 1 3wp1 4w n w n1 2t F v G v b p f F p1 G 1 FF v GG v f b n (11) were ( ) p denotes te pt level of te subiteration and ( ) n denotes te nt level of te pysical time step. Te iterations in te pysical and pseudotimes are called te outer and inner iterations, respectively. Te parameter is te implicitness factor wit 1, corresponding to a fully implicit metod in pseudotime. As can be seen in Eq. 11, te viscous and source terms are treated implicitly in te pysical time and explicitly in te pseudotime. Te correction ww p1 w p is computed in eac subiteration according to 858 Õ Vol. 71, NOVEMBER 2004 Transactions of te ASME

3 1 3 I1 1 w p I 3wp 4w n w n1 2t 2t F v G v b p f F p1 G GG v f b n 1 FF v (12) As can be seen in Eq. 12, wen a steady state is reaced in te pseudotime, i.e., w p 0, we ave ( ) p () n1. Terefore te metod is equivalent to te second-order implicit backward Euler metod in te pysical time. To solve Eq. 12, te convective fluxes are linearized in pseudotime according to F p1 F pa wp O 2 (13) G p1 G pb wp O2 were te Jacobian matrices are defined as A F p, B G w w p (14) From Eqs , te linearized equations can be written as I à B w p R (15) were te residual vector is defined as RD À1 I 1 3wp 4w n w n1 2t DÀ1 FF v GG v f b p 1 FF v GG v f b n (16) and ÃD À1 A; B D À1 B; DI 3 2t I1 (17) Following Caugey 9, Eq. 15 is factorized as I à I B w p R (18) wic can be solved efficiently in two steps by using a block tridiagonal solver. However, Eq. 18 can be solved more efficiently using te diagonalization procedure. Te diagonalization is possible because te inviscid part of te preconditioned equations are yperbolic, so tere exist modal matrices Q à and Q B suc tat à Q à 1 ÃQà ; B Q B 1 B Q B (19) and te diagonal matrices aving real eigenvalues. Te diogonalized algoritm is ten given by I à 1 Q à QB I B V p 1 Q à R (20) were V p Q B 1 w p. Equation 20 is solved in two steps using a scalar tridiagonal solver in eac step. Note tat te spatial derivatives are approximated by a cell-centered finite volume metod, wic is equivalent to second-order central differences on a regular Cartesian grid and fourt-order numerical dissipation terms similar to tat of Caugey 9 are added explicitly to te rigt-and side of Eq. 18 to prevent te odd-even decoupling. A front-tracking metod similar to tat of Unverdi and Tryggvason 8 is developed for treatment of te different pases and te surface tension. In tis metod, te interface is divided into small line segments called front elements, and te end points of eac element are tracked explicitly in a Lagrangian frame. Te details of te numerical metod can be found in Muradoglu and Kayaalp 10. Te complete solution procedure can be summarized as follows: In advancing solutions from pysical time level n (t n n t) to level n1, te locations of front points at te new time level n1 are first predicted using an explicit Euler metod, i.e., X n1 f X n n f tv f (21) were X f and V f denote te position of front points and te flow velocity interpolated from te neigboring fixed grid points onto front point X f, respectively. Ten te material properties and surface tension are evaluated using te predicted front position X f, i.e., n1 X n1 f ; n1 X n1 f ; f n1 b f b X n1 f (22) Te velocity and pressure fields at new pysical time level (n 1) are ten computed by solving te flow equations by te FV metod for a single pysical time step, and finally te positions of te front points are corrected as X n1 f X n f t 2 V f n V n1 f (23) After tis step, te material properties and te body forces are also reevaluated using te corrected front position. Cubic B-splines are used for all te interpolations from te fixed curvilinear grid onto front points and from te front points onto fixed curvilinear grid, and for distributing surface tension onto fixed curvilinear grid. Te overall metod is second-order accurate bot in time and space. It is empasized tat te metod is implicit in pysical time and te pysical time step t is solely determined by accuracy considerations. An auxiliary regular Cartesian grid is utilized for tracking te positions of te front points in te curvilinear grid and is found to be very robust and efficient. Details of te tracking algoritm can also be found in Muradoglu and Kayaalp 10. Te auxiliary regular Cartesian grid is also used to determine te material properties using te procedure developed by Unverdi and Tryggvason 8, wic involves solution of a Poisson equation. Bilinear interpolations are used to interpolate te material properties from te regular grid onto te curvilinear grid. Te front marker points are reflected back into te computational domain in te case tat te points cross te solid boundary due to numerical errors. Te surface tension is distributed only onto te neigboring grid points wen te front is close to te solid boundary and care is taken to make sure tat te distributed forces are equivalent to te surface tension. However, no special treatment is done wen two fronts come close to eac oter in te computational domain. We note tat, in addition to te preconditioning metod, a multigrid metod similar to tat of Caugey 9 and a local timestepping metod are used to furter accelerate convergence rate in pseudotime stepping. Te details of te FV metod can be found in Muradoglu and Kayaalp Results and Discussion Te metod is first validated for te test case of a buoyancydriven drop rising in a straigt cannel, and te results obtained wit te present metod and wit te well-tested FD/FT metod of Unverdi and Tryggvason 8 implemented in te FTC2D code are compared. Ten te metod is applied to more callenging test cases of te buoyancy-driven drops in various constricted cannels. Altoug te metod is general and can andle many drop Journal of Applied Mecanics NOVEMBER 2004, Vol. 71 Õ 859

4 Fig. 1 Velocity vectors around a ligt drop rising in a straigt cannel for Eötvös and Morton numbers EoÄ1, M Ä10 À4 top plots, EoÄ4, MÄ4Ã10 À4 middle plots and EoÄ16, MÄ16Ã10 À4 bottom plots at t*ä Present results left plots are compared wit te FTC2D results rigt plot. Grid: 96Ã384, dt*ä interactions, ere only a single drop and two drop cases are considered. Te computational results are expressed in terms of nondimensional quantities. For tis purpose, te lengt, time, and velocity scales are defined as Ld e, Td e /G and V r o / o d e, respectively, and te nondimensional quantities are denoted by*. For example, te x and y coordinates are nondimensionalized as x*x/l and y*y/l, respectively. Altoug a tree orders-of-magnitude reduction in te rms residuals of te subiterations is found to produce essentially te same results; te rms residuals are reduced by four orders of magnitude in eac inner iteration in pseudotime for all te simulations presented in tis paper. 4.1 Freely Rising Drop in a Straigt Cannel. Te metod is first applied to a two-dimensional freely rising drop in a straigt cannel. Te purpose of tis test case is to validate te 860 Õ Vol. 71, NOVEMBER 2004 Transactions of te ASME

5 Fig. 2 Te vertical positions left plot and te rise velocities rigt plot of te drop centroid taken from te simulations of te ligt drop rising in a straigt cannel. Computations are performed for Eötvös numbers 1, 4, and 16. Te solid lines denote te FTC2D results and te symbols are te present calculations. Grid: 96Ã384, dt*ä metod against te FD/FT metod implemented in FTC2D code of Unverdi and Tryggvason 8 tat can only use regular Cartesian grids. Te computational domain is 2d e 8d e, were d e is te initial drop diameter and is resolved by a uniform regular Cartesian grid. No-slip boundary conditions are applied on te side walls i.e., x0 and x2d e ), wile periodic boundary conditions are used in te vertical direction. Te drop is initially an infinitely long circular cylinder centered at (x c *)(1.0,2.0) and starts rising from te rest due to buoyancy forces at time t* 0. Freely rising drops take various sapes depending essentially on te Eötvös number. To sow tis effect, computations are performed for tree different Eötvös and Morton numbers i.e., Eo1, M10 4 ; Eo 4, M410 4 ; and Eo16, M ), wile te viscosity ratio is kept constant at 1. Te corresponding density ratios are 0.975, 0.9, and 0.6, respectively. Te pysical time step is fixed at dt*0.0316, and te residuals are reduced by four orders of magnitude in eac inner iteration in pseudotime. Te drop sapes and te velocity vector field in te vicinity of te drop are plotted in Fig. 1 at time t*9.487 for te cases of Eo1 top plots, Eo4 middle plots, and Eo16 bottom plots. Te results obtained wit te FTC2D code are also sown in te rigt plots of Fig. 1. It can be seen in tis figure tat te present results are in good agreement wit te FTC2D results demonstrating te accuracy of te present metod. It is also observed tat drop deformation increases as te Eötvös number increases as expected. To better quantify te accuracy of te present metod te vertical position of te drop centroid and te drop rise velocity computed wit te present metod as well as wit te FTC2D code are plotted as a function of time in Fig. 2. As can be seen in tis figure, te present results are overall in very good agreement wit te results of te FTC2D code except for te small discrepancies observed between te two results for Eo16, wic is partly attributed to te time-stepping error in te present results. Note tat te time step used in te present metod is about 20 times larger tan tat used in te FTC2D code for te case of Eo 16. Altoug te flow is incompressible, drop volume area canges due to numerical errors and te percentage cange in te drop volume is a good indicator for te accuracy of te metod. Figure 3 sows te percentage cange of te drop area obtained wit te present metod and te FTC2D code for all tree sets of dimensionless numbers. It can be observed in tis figure tat, in contrast wit te FTC2D results, te drop volume reduces in time as te drop rises in te present metod for tis test case, but te overall percentage cange in te drop volume is comparable in magnitude in bot metods. In summary te present results compare well wit te results of te well-tested FD/FT metod of Unverdi and Tryggvason 8 demonstrating te accuracy of te present algoritm. 4.2 Freely Rising Drops in Various Constricted Cannels. After validating te metod for te case of a freely rising drop in a straigt cannel, we now consider buoyancy-driven drops rising in various constricted cannels to sow te ability of te metod for treating dispersed multipase flows in complex geometries were te pases strongly interact wit te solid boundaries. Te first test case concerns a single initially cylindrical drop rising due to buoyancy in a constricted cannel. Te cannel is 2d e wide, extends to 12d e in te y direction, and is constricted sinusoidally at te middle by 75% as sown in Fig. 4b. In Fig. 4a, a portion of a coarse grid containing grid cells is plotted in te vicinity of te constriction to sow te overall structure of te body-fitted curvilinear grid used in te simulations. Te governing nondimensional numbers are set to Eo8, M810 4, 0.8, and 1. No-slip boundary conditions are applied on te solid Fig. 3 Percentage cange in te drop area for Eötvös numbers 1, 4, and 16 in te computations of te freely rising drop in te straigt cannel. Dased curves denote te present results, and te solid curves are te FTC2D results. Grid: 96Ã384, dt* Ä Journal of Applied Mecanics NOVEMBER 2004, Vol. 71 Õ 861

6 Fig. 4 Freely rising drop in a sinusoidally constricted cannel. a A portion of te body-fitted curvilinear coarse grid containing 32Ã192 grid cells. b Snapsots taken at time frames t*ä0, 9.49, 15.81, 22.14, 28.46, 31.62, 37.95, and Time progresses from bottom to top. c Te vertical position top plot and te rise velocity bottom plot of te drop centroid computed wit te pysical time steps dt*ä dotted line, dt*ä dased line, and dt*ä solid line. EoÄ2, MÄ8Ã10 À4, Ä0.8, Ä1. Grid: 128Ã768. walls, and periodic boundary conditions are employed in te vertical direction. Te drop is initially centered at (x c *) (1.0,2.0) and starts rising from te rest due to buoyancy forces. Te snapsots taken at te time frames t*0, 9.49, 15.81, 22.14, 28.46, 31.62, 37.95, and are plotted in Fig. 4b to sow te overall beavior of te drop. Te computations are performed on a grid. In order to better quantify te drop motion, te vertical position and te rise velocity of te drop centroid are plotted in Fig. 4c. Computations are repeated for tree different pysical time steps i.e., dt*0.3162, , and on te same grid to demonstrate te time-stepping error convergence. Te small differences between te results computed Fig. 5 Grid convergence analysis for te freely rising drop in te straigt cannel. Te vertical position left plot and te rise velocity rigt plot of te drop centroid as a function of time computed on te body-fitted curvilinear grids containing 48Ã288 solid line, 96Ã576 dotted line and 128Ã768 dased line grid cells in te time interval t*ä25 and t*ä35. dt*ä0.1581, EoÄ Õ Vol. 71, NOVEMBER 2004 Transactions of te ASME

7 Fig. 6 Effects of grid refinement of te front structure. Before te drop enters left plot, time t*ä12.65 and after it passes rigt plot, time t*ä22.14 te constriction computed on 48Ã288 solid line and 128Ã768 dased line grids. dt*ä0.1581, EoÄ4. Te coarse grid results in wiggles on te front wile te front remains smoot in te case of te fine grid. wit te two smallest pysical time steps indicate tat te timestepping error convergence is acieved and dt* is sufficient for tis test problem. Figures 4b and 4c togeter sow tat te drop motion initially resembles tat of te straigtcannel case before te drop starts feeling te effects of te constriction i.e., before about t*15), but ten it is strongly affected by te presence of te constriction as te drop passes troug te constriction. In Fig. 5, te vertical position and te rise velocity of te drop centroid computed on 48288, 96576, and grids are plotted to sow te grid convergence. Te pysical time Fig. 7 Freely rising drop in a continuously constricted cannel. a A portion of te body-fitted curvilinear coarse grid containing 32Ã192 grid cells. b Snapsots taken at time frames t* Ä0, 10.33, 20.66, 30.98, 41.31, and Time progresses from bottom to top. c Te vertical position top plot and te rise velocity bottom plot of te drop centroid computed wit te pysical time step dt*ä on a 128Ã768 grid. EoÄ18, MÄ8Ã10 À4, Ä0.8, Ä1. Journal of Applied Mecanics NOVEMBER 2004, Vol. 71 Õ 863

8 Fig. 8 Buoyancy-driven two-drop interaction in te continuously constricted cannel. Snapsots taken at t*ä0, 35.78, 53.67, 71.55, 89.44, , , , , and EoÄ2, MÄ8Ã10 À4, Ä0.8 and Ä1. Grid: 96Ã576, dt*ä step is kept constant at dt*0.1581, and te results are sown only in te time interval t*25 and t*35, during wic te drop passes troug te constriction. Te decreasing differences between results obtained on te successively finer grids indicate tat te grid convergence is acieved and te grid is sufficient for tis test case. Figure 6 sows te fronts computed on te and grids just before te drop enters and after it passes te constriction in order to demonstrate te effects of te grid refinement on te front structure. As can be seen in tis figure, te coarse grid results in wiggles on te front wile te front remains very smoot in te case of te fine grid. Note tat results obtained on te grid is not plotted because it is almost indistinguisable from te grid case. Te next test case concerns a buoyancy-driven drop freely rising in a continuously constricted cannel depicted in Fig. 7b. Te cannel is 4d e /3 wide, and extends to 8d e in te y direction, and is constricted along te y-axis by sinusoidal wavy walls. Te constriction ratio is 75% as in te singly constricted cannel case. Te initial and te boundary conditions are te same as te singly constricted cannel case, and te drop is initially centered at (x c *)(0.6667,1.6). Te governing nondimensional numbers are Eo18, M810 4, 0.8, and 1. Computations are performed on a grid wit te constant pysical time step dt* A portion of a coarser grid containing grid cells is plotted in Fig. 7a. Note tat a similar geometry was used by Hemmat and Boran 2 in teir experimental study of te buoyancy-driven drops and bubbles. Te snapsots taken at te time frames t*0.0, 10.33, 20.66, 30.98, 41.31, and are plotted in Fig. 7b to sow te overall evolution of te drop motion. Te strong interactions between te drop and te solid Fig. 9 Buoyancy-driven two-drop interaction in te continuously constricted cannel. Te orizontal left plot and te vertical rigt plot positions of te initially left drop centroid solid line, te initially rigt drop dased line, and te center of te mass of te drop system dotted line. EoÄ2, MÄ8Ã10 À4, Ä0.8, Ä1. Grid: 96Ã576, dt*ä Õ Vol. 71, NOVEMBER 2004 Transactions of te ASME

9 Fig. 10 Buoyancy-driven two-drop interaction in te continuously constricted cannel. Te orizontal left plot and te vertical rigt plot velocities of te initially left drop centroid solid line, te initially rigt drop dased line, and te center of te mass of te drop system dotted line. EoÄ2, MÄ8Ã10 À4, Ä0.8, Ä1. Grid: 96Ã576, dt*ä walls can be clearly seen in tis figure. It is empasized tat, in spite of large deformations, te front remains smoot, wic migt be considered as a good indication for te accuracy of te simulation. Te vertical position and te rise velocity of te drop centroid are plotted in Fig. 7c. It can be seen in tis figure tat te rise velocity becomes periodic after a transient period i.e., after about t*20). Finally, te metod is used to compute two identical drops freely rising in te continuously constricted cannel. In tis case, te drop diameters are relatively small compared to te cannel widt, and te ratio of te initial drop diameter to te maximum cannel widt is d e /d max Te corresponding nondimensional numbers are Eo2, M810 4, 0.8, and 1. Te same grid is employed as used in te single-drop case and te pysical time step is taken as dt* Te drops are initially located at (x c *)(1.0,4.8) and (x c *)(3.0,4.8). Te snapsots taken at te time frames t*0, 35.78, 53.67, 71.55, 89.44, , , , , and are plotted in Fig. 8 to sow te overall beavior of te drops. Te drops strongly interact wit te solid walls as well as wit temselves and deform considerably as tey pass troug te constrictions. It is interesting to observe tat te drops initially rise side by side but te rigt drop passes te first constriction little earlier tan te left drop and ten te left drop catces up and passes te rigt drop. After tat te drops switces teir positions and tis beavior is repeated periodically after eac constriction. Tis beavior can also be seen in Figs. 9 and 10, were te orizontal and vertical locations, te orizontal and vertical rise velocities of te individual drop centroids, and te center of te mass of te drop system are plotted. Note tat neiter breakup nor coalescence is allowed in te present simulations. It can be seen from tese figures tat te motion of a two-drop system becomes periodic after a transient period. Due to relatively poor grid resolution and large pysical time step, drops lose about 20% volume wen tey move about 14 drop diameters wic corresponds to t*215). In spite of very strong interactions between te drops and te solid walls as well as between drops temselves and a relatively coarse grid, te periodic beavior of te two-drop system observed in Figs. 9 and 10 indicates tat te main features of te two-drop interactions are well captured by te present metod. 5 Conclusions Te computations of buoyant drops in constricted cannels ave been reported in tis paper. Te FV/FT metod is first validated for te case of a buoyancy-driven drop in a straigt cannel by comparing te results wit te computations obtained by te well-tested FD/FT metod implemented in FTC2D code 8. Te metod as been successfully applied to buoyant drops in various constricted cannels. Some error convergence studies ave also been performed. It is found tat te present metod is a viable tool to model dispersed multipase flows in arbitrarily complex geometries. References 1 Olbrict, W. L., and Leal, L. G., 1983, Te Creeping Motion of Immiscible Drops Troug a Diverging/Converging Tube, J. Fluid Mec., 134, Hemmat, M., and Boran, A., 1996, Buoyancy-Driven Motion and Deformation of Drops and Bubbles in a Periodically Constricted Capillary, Cem. Eng. Commun., , Udaykumar, H. S., Kan, H.-C., Syy, W., and Tran-Son-Tay, R., 1997, Multipase Dynamics in Arbitrary Geometries on Fixed Cartesian Grids, J. Comput. Pys., 137, Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawai, N., Tauber, W., Han, J., Nas, S., and Jan, Y.-J., 2001, A Front Tracking Metod for te Computations of Multipase Flows, J. Comput. Pys., 169, Tsai, T. M., and Miksis, J. M., 1994, Dynamics of a Drop in a Constricted Capillary Tube, J. Fluid Mec., 274, Manga, M., 1996, Dynamics of Drops in Branced Tubes, J. Fluid Mec., 315, Olbrict, W. L., and Kung, D. M., 1992, Te Deformation and Breakup of Liquid Drops in Low Reynolds Number Flows Troug Capillaries, Pys. Fluids, 4, Unverdi, S. O., and Tryggvason, G., 1992, A Front-Tracking Metod for Viscous, Incompressible Flows, J. Comput. Pys., 100, Caugey, D. A., 2001, Unsteady Flows Past Cylinders of Square Cross- Section, Comput. Fluids, 30, Muradoglu, M., and Kayaalp, A. D., 2004, A Finite Volume/Front-Tracking Metod for Computations of Dispersed Multipase Flows in Complex Geometries J. Comput. Pys. submitted for publication. 11 Clift, R., Grace, J. R., and Weber, M. E., 1978, Bubbles, Drops, and Particles, Academic Press, San Diego, CA. Journal of Applied Mecanics NOVEMBER 2004, Vol. 71 Õ 865