TWO- AND THREE-DIMENSIONAL SIMULATION OF A RISING BUBBLE AND FALLING DROPLET USING LEVEL SET METHOD

Transcription

1 European Conference on Computationa Fuid Dynamics ECCOMAS CFD 2006 P. Wesseing, E. Oñate, J. Périaux (Eds) TU Deft, The Netherands, 2006 TWO- AND THREE-DIMENSIONAL SIMULATION OF A RISING BUBBLE AND FALLING DROPLET USING LEVEL SET METHOD Ching-Biao Liao*, Cheng-Hsin Chen *Department of Water Resources Engineering, Feng Chia University, No. 0 Wenhwa Rd., Seatwen, Taichung, Taiwan 407, R.O.C e-mai: cbiao@fcu.edu.tw Web page: Graduate Institute of Civi and Hydrauic Engineering, Feng Chia University, No. 0 Wenhwa Rd., Seatwen, Taichung, Taiwan 407, R.O.C e-mai: p @fcu.edu.tw Key words: Leve Set Method, Rising Bubbe, Dropet, Runge-Kutta Scheme, WENO Abstract. The numerica simuation of a rising bubbe in iquid and faing dropet is presented. The two- and three-dimensiona incompressibe Navier-Stokes equations are used to sove the gas-iquid two phases fow system with free surfaces. A sharp interface separates fuids of different density and viscosity. The interface between gas and iquid is described as the zero eve set of a smooth function. In order to maintain the eve set function as a smooth signed distance function from the interface, a reinitiaization operation is required. We aso appy a constraint to improve the conservation of mass significanty. A WENO scheme for space derivatives and third-order Runge-Kutta scheme for the time discretization are used to sove the eve set equation. The projection method are used to derive a pressure Poisson equation to ensure the satisfaction of continuity equation. The veocity fied can be obtained by second-order expicit Adams-Bashforth scheme, whie the pressure is soved. The staggered grid system is used in the numerica mode. The mode aows us to simuate a wide range of fow regimes and surface tension effect can be consideration. The effect of deformation and motion of a riging bubbe is investigated by changing the Reynods numbers and Weber numbers. A dropet fas into a tank of a quiescent fuid is aso simuated. Compex numerica simuations show the capabiity of our numerica mode. Our mode can be used to simuate a number of probems and to investigate many physica phenomena invoving mutifuid fows that consider the free surfaces. 1 INTRODUCTION In this work, we present a incompressibe two-phase fow modes. The purposes are to simuate high density and viscosity ratio fows with compex interface changes, such as fows with coinstantaneous occurrence of bubbes and dropets. The motion of bubbes and dropets may be very compex due to high density and viscosity ratios as we as topoogy changes.

2 Computationa fuid dynamic (CFD) is a powerfu too can simuate the detais of bubbe and dropet motion. There are many numerica methods and experiments have been studied. Ryskin and Lea 1 deveoped a numerica method to compute the steady motion of a bubbe rising in the iquid. The numerica resut was compared with the experimenta data by Hnat and Buckmaster 2 show exceent. Bhaga and Weber 3 was determined the shape and termina veocities of bubbes rising in viscous iquids by experiment. Most popuar numerica methods for interface tracking are Voume-of-Fuid (VOF) method and Leve Set method. VOF method is proposed by Hirt and Nichos 4. They were introduced a VOF function as the voume fraction in each ce to represent the interface. The advantage of this method is preserved mass conservation, but the disadvantage is compicated geometric cacuations for the interface reconstruction. Chen and Li 5 were using a modified VOF method to simuate two-phase fows with a varying density. In the Leve Set method, is devised by Osher and Sethian 6, is using a smooth eve set function φ to track the interface. The interface can be found whenφ = 0. This method aows computing for two-phase fows with high density and viscosity ratios. One of the advantages of this method is that we can easiy track and present the interface. But, this method has a disadvantage that mass is not reay conservation. Sussman and Smereka 7 were studied three-dimensiona axisymmetric free boundary probems using eve set method incuding a gas bubbe rising, burst at a gas-iquid interface, and drop impact on a poo of water. The computationa resuts were compared with experimenta data and iteratures show good agreement. Recenty, Sussman and Puckett 8 were presented a couped eve set/voume-of-fuid (CLSVOF) method to combine the advantages and improve the disadvantages in VOF method and Leve Set method. Ohta et a. 9 were used CLSVOF method to simuate three-dimensiona numerica motion of a gas bubbe rising in viscous iquids. Liao and Chen were used the eve set method to simuate the motion of a rising bubbe in iquid in Cartesian coordinate system. In this paper, we combine eve set method and two- and three-dimensiona incompressibe Navier-Stokes equations to simuate the motion of a rising bubbe and a faing dropet. Surface tension is aso considered. The effect of deformation and motion of a riging bubbe is investigated by changing the Reynods numbers and Weber numbers. We were aso simuated a dropet fas into a tank of a quiescent fuid of the same phase. Compex numerica simuations show the capabiity of our method. 2 GOVERNING EQUATIONS 2.1 Leve Set Method The eve set method for moving interfaces was used in Osher and Sethian. An appication of the eve set formuation to incompressibe two-phase fow was used in Sussman and Smereka. The origina ideas of eve set method is to define a smooth function φ ( x, t) that represents the interface at φ xt, = 0. The interface can be captured at any time by ocating ( )

3 the zero eve set, φ < 0 in the gas phase and φ > 0 in the iquid phase. Since the interface moves with the fuid partices the evoution of φ in fow fied is given by r φ + U φ = 0 (1) t where U r is the veocity of fuid. The numerica osciation in the interface may occur due to arge density ratio of gas and iquid phase. In order to avoid this numerica instabiity, the density and viscosity of fuids are repaced by ρ φ = H φ + ρ ρ 1 H φ (2) ( ) ( ) ( g ) ( ( )) ( ) H( ) ( )( 1 H( )) μ φ = φ + μ μ φ (3) g The subscript g and represent in gas and in iquid, respectivey and H ( φ ) is the smooth Heaviside function given by 0 if φ < ε 1 φ 1 πφ H( φ ) = 1+ + sin if φ ε (4) 2 ε π ε 1 if φ > ε The physica concept is that the zero thickness interface expands to 2ε width. In our numerica mode, we take ε = αδ h, where Δh is the minimum grid size and α is a parameter of thickness of the interface, it usuay takes 1.5~3.0. We use α =1.0 in our mode. 2.2 Governing Equations for Fuids Using the Leve Set Method to sove two-phase fow probems, the properties change across the gas-iquid interface. In Euerian Coordinate System, for incompressibe Newtonian fuids with constant of density and viscosity both in gas and iquid, the equations of motion are given by the incompressibe Navier-Stokes equations r U = 0 (5) r r r p ( 2μ( φ) D) σκ ( φ ) φδ ( φ ) r U + U U + = F t (6) ρ( φ) ρ( φ) ρ( φ) where U = ( uvw,, ) is the veocity; p is the pressure; φ is the eve set function; ρ and μ is the density and the viscosity of the fuid, respectivey, which is the function of φ and can be obtained from equation (2) and (3); σ is the surface tension coefficient; 1 r r T φ D = (( U) + ( U ) ) is the rate of deformation tensor; κ( φ) = 2 φ is the curvature of interface; δ ( φ ) is the Dirac deta function that is defined as

4 1 dh ( φ ) ( 1+ cos ( πφ ε) )/ ε if φ < ε 2 ( ) = = r ; = ( 0, 0, g ) δ φ gravity. dφ Initiaization of Leve Set Function otherwise F is the body force, g is the When the initia interface for a fuid probem is much too compicated to hard to define the initia vaue of eve set function. A simpe modification method was proposed in Sussman and Fatemi 11, caed a first-time ony redistance step to obtain the initia eve set function. We appied the method to decide the initia vaue of. This method is executed ony once for the duration of the entire computed process. First, we assume = +1 in the iquid region and = -1 in the gas region. Then we sove equation (7), and compute to t = L, where L is the ength of computationa domain. We can get the initia vaue of eve set function of fow probem. 2.4 Re-initiaization of Leve Set Function Whie φ is initiaized as a distance function, but it doesn t ensure φ as a distance function as time proceeds by equation (1). Thus, we need a re-initiaization process to ensure the condition of φ = 1. We use an iterative procedure proposed by Sussman and Fatemi to reinitiaize the eve set function at each time step to maintain the eve set function as a distance function. The procedure is soved the hyperboic equation to steady state as foows d = sign( φ0)( 1 d ), d ( x,0) = φ0( x) (7) τ where d is the norma distance from x to the interface, τ is the artificia time, sign( φ 0 ) is the smoothed sign function defined as sign( φ ) 2 0 ( H ( φ) 0.5 ) (8) The steady state soution of this probem is the signed distance function from the boundary of φ 0 =0. That is d = 1 for d ε (9) 2.5 Mass Conservation In equation (7), because sign( 0) = 0 theoreticay, the free surface captured by the zero eve set doesn t change the position during the re-initiaization procedure. But this is not ensured in numerica computation. When doing re-initiaization procedure, it may be induced mass error. In order to maintain the mass conservation, the equation (7) is modified as d = sign d f L d f τ ( φ )( 1 ) λ ( φ) ( φ, ) λ ( φ) ()

5 where λ = Ω ( φ) ( φ, 0 ) ( φ) f ( φ) H L d Ω H and f ( ) H ( ) φ φ φ. This correct factor makes the iquid mass conservation Sussman and Fatemi. 2.6 Numerica Procedures The outines of our numerica mode are as foows: 1. Set initia condition and boundary conditions of probem. n 1 2. Use WENO method and Runge-Kutta method to soveφ + from equation (1). 3. Compute re-initiaization and mass conservation by soving equation (). n 4. Sove P + 1 from the pressure Poisson equation. n 1 5. Compute veocity u +, v n+ 1, w n+ 1 by second order Adams-Bashforth method from equation (6). 6. Repeat step 2 to step 5 unti the desired time. 3 COMPUTATIONAL EXAMPLES 3.1 Two Dimensiona Bubbe The computed conditions of bubbe A depict in tabe 1. Other paramets are ρ ρ = , μ μ = , and the dimensioness quantities are Re = 9.8, Fr= , g g and We=7.6. The fuid domain is 3 12 and the grid size is Resuts are shown in Figure 1. We observe that the shapes of the bubbe are simiar to Sussman and Smereka. The dimensioness rise speed shoud be 1. But, in this case, the dimensioness rise speed just have 0.5~0.6, since we are not using the axisymetric coordinate system. For this reason, we extend our mode to truy three-dimensiona formuaitons. 3.2 Three Dimensiona Simuation of Bubbe And Dropet Steady-state Resuts We test the steady-state cases proposed by Hnat and Buckmaster experimentay with rising spherica cap air bubbes. The numerica cacuation aso has been proposed by Sussman and Smereka. In our computation, the domain is 3D 3D 12D (D is the diameter of the bubbe) and the grid size is In the initia setting, a spherica bubbe was imposed at the bottom of the domain and veocity was zero at t = 0. We use cosed computationa domain and non-sip boundary condition on a was. We take bubbe A and B from Tabe 1 of Hnat and Buckmaster as our numerica simuations. The properties of the gas 3 3 and the iquid are ρ g = g cm, ρ = g cm, μ = 0.01 P, and μ g = 1.18 P. The coefficient of surface tension of the iquid is σ = 32.2 dyn cm. The detais of the properties of bubbes are shown in Tabe 1.

6 The voume of bubbe is V. They were observed that V s is steady rising speed. So the effective diameter of the bubbe is D = ( 6 V / π ) 1/3. The dimensiona parameters can aso know from Tabe 1. In numerica computation, the dimensioness rise speed shoud be 1. In our computationa resuts, we find that the bubbe A and B reach a steady state with fina speed of 1.03~1.06. Figure 3 dispays the time evoution of the bubbe A. Figure 4 shows the time history of computed rise speed of bubbe A and B. The numerica resuts are aso shown in good agreement with experimenta resuts of Hnat and Buckmaster Bubbe Shapes In this section, we use the experimenta data of Bhaga and Weber. The properties of the 3 iquid are μ = 0.82~28.0 P, ρ = g cm, σ = 76.9 dyn cm. Computationa domain and grid size are same as ast section. The bubbe voume and rise speed were used bubbe A from Tabe 1. Tabe 2 is the detais of computed cases. Figure 5 shows shape regime map for bubbes in iquids presented by Bhaga and Weber. The dimensioness parameters of Morton (Mo), Eotvos (Eo), and Reynods (Re) number are given by (see Bhaga and Weber) 4 2 gμ Mo = gd ρ 3 Eo = ρσ σ In numerica simuations, we used a fixed density ratio and compared the effect of viscosity on bubbe shape. When we fixed density ratio, the Eo aso must be fixed. So there just compare the effect of Mo. From Tabe 2, at ow Re (high Mo), the bubbe is neary an obate sphericity. Figure 6 shows the fina steady-state bubbe shape. We compare the shapes of the numerica resuts with the diagram of Grace. This comparison enabes us to concude that our numerica method is in a good agreement with experimenta predictions Dropet Impact In this case, a dropet fas into a tank of a quiescent fuid of the same phase. This is a good exampe to show the changes of the free surface. The nondimensiona parameters were given by : Re = 30.0, Fr = 1.0, We = And we set ρg ρ = 0.5, and μg μ = 0.5. The fuid domain is and grid size is The dropet ocated at (0, 0, 5.0) is stationary when t = 0. Figure 6 shows the changes of the free surface at different time. In this figure, we are aso abe to observe the surface tension effects acting over the free surfaces. In the further work, we can simuate the impact of the dropet on different panes, and investigate more information about dropet impact process and spashing. 4 CONCLUSIONS We presented the Leve Set Method and two- and three-dimensiona incompressibe Navier-Stokes equations for simuating the motion of the rising bubbe and the faing dropet. Our computationa resuts compare with experimenta data and other numerica modes show a good agreement. We prove the abiity to simuate the fow incuding the compex free surface probems. In the future, we woud ike to improve our numerica method to make

7 computations easiy. And we wi study two-phase fows with immersed boundary. ACKNOWLEDGEMENT This study was supported by the Nationa Science Counci of the Repubic of China, under grant number NSC E , it is gratefuy appreciated. REFERENCES [1] Ryskin, G. and Lea, L. G., Numerica soution of free boundary probems in fuid Mechanics. Part 2. Buoyancy-driven motion of a gas bubbe through a quiescent iquid, Journa of Fuid Mechanics, 148, 19-35(1984). [2] Hnat, J. G., and Buckmaster, J. D., Spherica cap bubbes and skirt formation, The Physics of Fuids, 19, (1976). [3] Bhaga, D., and Weber, M.E., Bubbes in viscous iquids: shapes, wakes and veocities, Journa of Fuid Mechanics, 5, 61-85(1981). [4] Hirt, C. W. and Nichos, B. D., Voume of Fuid (VOF) Method for the Dynamics of Free Boundaries, Journa of Computationa Physics, 39, (1981). [5] Chen, L. and Li, Y., A Numerica Method for Two-Phase Fows with an Interface, Environ. Mode. Softw., 13, (1998). [6] Osher, S. and Sethian, J. A., Fronts propagating with curvature dependent speed: agorithms based on Hamiton-Jacobi formuations, Journa of Computationa Physics, 79, 12-49(1988). [7] Sussman, M. and Smereka, P., Axisymetric free boundary probems, Journa of Fuid Mechanics, 341, (1997). [8] Sussman, M. and Puckett, E.G., A Couped Leve Set and Voume of Fuid Method for computing 3d and axisymmetric incompressibe two-phase fows, Journa of Computationa Physics, 162, (2000). [9] Ohat, M., S. Haranaka, Y. Yoshida, and M. Sussman, Three-Dimansiona Numerica Simuations of the Motion of a Gas Bubbe Rising in Viscous Liquids, Journa of Chemica Engineering of Japan, 37, (2004). [] Liao, C.B. and Chen, C.H., Three-dimensiona numerica simuation of rising bubbe using eve set method, Computationa Fuid Dynamics JOURNAL, 14(4):53, (2006). [11] Sussman, M. and Fatemi, E., An efficient interface preserving eve set re-distancing agorithm and its appication to interfacia incompressibe fuid fow, SIAM Journa on Scientific Computing, 20, (1999).

8 Bubbe A V(m) ρ(g/m) μ (P) σ(dyn/cm) Vs(cm/s) D(cm) Fr We Re Eo Mo Shape ode Bubbe B V(m) ρ(g/m) μ(p) σ(dyn/cm) Vs(cm/s) D(cm) Fr We Re Eo Mo Shape scc Tabe 1: The properties of bubbes shown in Figure 1 and Figure 2. Bubbe ρ (g/m) μ (P) Fr We Re Eo M Shape oed oec oec oec oe oe Tabe 2: The properties of computed bubbes in section Figure 1: Two dimensiona simuation of a rising bubbe with Re = 9.8, Fr = , We = 7.6.

9 Figure 1: (Continue) Figure 2: (a) Simuated positions and shape variations of a rising bubbe A in gas-iquid fow and time increment is 1.0;(b)-(e) cross sectiona veocity distribution of the bubbe A at four time instants shown in (a).

10 Figure 3: The time history of the rise speed of the top of the bubbe A and B Figure 4: Shape regime map for bubbes in iquids by Bhaga and Weber, the dot is case of this study.

11 Figure 5: Simuated fina shape of six rising bubbes in gas-iquid fow from Tabe 2. Figure 6: Evoution of a faing dropet impact the free surface.

12 Figure 6: (Continue)