OSCILLATION AND ROTATION OF LEVITATED LIQUID DROPLET

Transcription

1 European Conference on Computational Fluid Dynamics ECCOMAS CFD 26 P. Wesseling, E. Oñate, J. Périaux (Eds) TU Delft, The Netherlands, 26 OSCILLATION AND ROTATION OF LEVITATED LIQUID DROPLET Tadashi Watanabe System Computational Science Center, Japan Atomic Energy Agency, Tokai-mura, Naka-gun, Ibaraki-ken, 39-95, Japan Web page: Key words: Droplet, Level set method, Oscillation, Frequency, Amplitude, Rotation Abstract. The oscillation and rotation of a levitated liquid droplet are simulated numerically using the level set method. The frequency and the damping of the oscillation for small amplitude are shown to agree well with those by the linear theory. It is shown that the oscillation frequency decreases as the amplitude increases. The oscillation frequency increases, in contrast to the effect of the amplitude, as the rotation frequency increases. It is found that the oscillation frequency depends on the average pressure difference between the pole and the equator of the droplet. INTRODUCTION A levitated liquid droplet is used to measure material properties of molten metal at high temperature, since the levitated droplet is not in contact with a container, and the effect of the container wall is eliminated for a precise measurement. The measurement technique using levitated droplet is, for instance in the nuclear engineering field, applied to obtain some properties of corium for the analyses of severe accidents. The levitation of liquid droplets, which is also used for containerless processing of materials, is controlled by using electrostatic force 2 or electromagnetic force 3 under the gravitational condition. Viscosity and surface tension are, respectively, obtained from the damping and the frequency of an oscillation of the droplet. Large amplitude oscillations are desirable from the viewpoint of measurement. However, the relation between material properties and oscillation parameters, which is generally used to calculate material properties, is based on the linear theory 4, and small amplitude oscillations are necessary for an estimation of material properties. The effect of the amplitude on the oscillation frequency has been discussed by Azuma and Yoshihara 5. Second order small deviations were taken into account for the linearized solution, and the oscillation frequency was shown to decrease as the amplitude increased. The effect of the rotation has been discussed by Busse 6 and Lee et al. 7 theoretically, and by Wang et al. 8 experimentally. In contrast to the effect of the amplitude, the oscillation frequency was reduced by the rotation. Although qualitative effects were shown by these studies, quantitative estimation is not enough since theoretical approach is based on some assumptions and simplifications. In this study, numerical simulations of an oscillating liquid droplet are

2 performed to study the effects of the amplitude and rotation on the oscillation frequency. Three-dimensional Navier-Stokes equations are solved using the level set method. The level set function, which is the distance function from the droplet surface, is calculated by solving the transport equation to obtain the surface position correctly. Mass conservation of the droplet is especially taken into account in the calculation of the level set function. The staggered mesh system is used and the second-order upwind difference scheme is applied for convective terms. The second-order Adams-Bashforth method is used for time integration. The oscillation of the droplet is studied by changing the amplitude of the initial deformation, which are given by the Legendre polynomial. The period and the damping of the oscillation are obtained for small amplitude, and are compared with the linear theory. Frequency shifts due to the oscillation amplitude and rotation are studied, and the relation between the frequency shifts and the pressure field in the droplet is discussed. 2 NUMERICAL METHOD Governing equations for the droplet motion are the equation of continuity and the incompressible Navier-Stokes equations: and u = () Du ρ = p + ( 2µ D) + F s, Dt where ρ, u, p and µ, respectively, are the density, the velocity, the pressure and the viscosity, D is the viscous stress tensor, and F s is a body force due to the surface tension. The surface tension force is given by F = σκδ φ, (3) s where σ, κ, δ and φ are the surface tension, the curvature of the interface, the Dirac delta function and the level set function, respectively. The level set function is a distance function defined as φ= at the interface, φ< in the liquid region, and φ> in the gas region. The curvature is expressed in terms of φ: φ κ = ( ). φ The density and viscosity are given, respectively, by ρ = ρ + ( ρ ρ H (5) l g l ) (2) (4) and µ = µ + ( µ µ H, (6) l g l ) where H is the Heaviside function defined by

3 H = [ + 2 φ + ε πφ sin( )] π ε ( φ < ε ) ( ε φ ε ) ( ε < φ) where ε is a small positive constant for which φ = for φ ε. The evolution of φ is given by φ + u φ = t All variables are nondimensionalized using liquid properties and characteristic values: x =x/l, u =u/u, t =t/(l/u), p =p/(ρ l U 2 ), ρ =ρ/ρ l, µ =µ/µ l, where the primes denote dimensionless variables, and L and U are representative length and velocity, respectively. The finite difference method is used to solve the governing equations. The staggered mesh is used for spatial discretization of velocities. The convection terms are discretized using the second order upwind scheme and other terms by the central difference scheme. Time integration is performed by the second order Adams-Bashforth method. The SMAC method is used to obtain pressure and velocities. In the nondimensional form of the governing equations, the Reynolds number, ρ l LU/µ l, and the Weber number, ρ l LU 2 /σ, are fixed at 2 and 2, respectively, in the following simulations. In order to maintain the level set function as a distance function, an additional equation is solved: φ = ( φ ) τ 2 φ φ. 2 + α where τ and α are an artificial time and a small constant, respectively. The level set function becomes a distance function in the steady-state solution of the above equation. The following equation is also solved to preserve the mass of the droplet in time 9 : φ = ( A)( κ ) φ τ A o, where A denotes the mass corresponding to the level set function and A denotes the mass for the initial condition. The mass of the droplet is conserved in the steady-state solution of the above equation. 3. RESULTS AND DISCUSSIONS 3. Oscillation of Levitated Droplet The time variation of the droplet radius in the vertical direction is shown in Fig.. The initial droplet radius is. with a deformation given by the Legendre polynomial of order 2. The amplitude of the initial deformation is.2. The size of the simulation region is,, (7) (8) (9) ()

4 3.x3.x3., and xx mesh cells are used. The periodic boundary condition is applied at all side boundaries. It was confirmed that the region size and the cell size did not have an effect upon the simulation results. In Fig., the theoretical decay of amplitude, which is given by Sussman and Smereka based on the Lamb s linear theory 4, is also shown. The agreement between the numerical and the theoretical results is satisfactory as shown in Fig.. The average period for the first three oscillations is. in this case, while the theoretical value is.. The decay of the amplitude depends on the viscosity, and the oscillation frequency is defined by the surface tension. The viscosity and the surface tension are the control parameters for the droplet oscillation. It is, thus, shown in Fig. that our numerical simulations are performed correctly. 3.2 Effect of Amplitude on Oscillation Frequency The effect of the amplitude on droplet oscillation is shown in Fig. 2. The amplitude of initial deformation is varied from.2 to.29, and the time variations of the droplet radius are shown. The curve of the time variation is shown to shift towards the positive direction of the time axis as the amplitude increases. This indicates that the oscillation period becomes large and the oscillation frequency decreases. The variation of oscillation frequency due to the change in amplitude is shown in Fig. 3. The frequency shift indicates the frequency difference normalized by the oscillation frequency for the amplitude of.2. The amplitude is varied from.2 to.76 in Fig. 3. In the linear theory given by Lamb 4, the oscillation frequency is a constant obtained in terms of the surface tension, the density, the radius and the mode of oscillation. The oscillation frequency is, however, shown to be affected largely by the amplitude. This is of importance for experiments, since large oscillations are desirable for measurements. The oscillation frequency decreases as the amplitude increases as shown in Fig. 3. Theoretical curve given by Azuma and Yoshihara 5, which was derived by taking into account a second order deviation from the linear theory, is also shown along with the simulation results. The simulation results agree well with the theoretical curve up to the amplitude of about.3. The theoretical curve overestimates the frequency shift for the amplitude larger than Effect of Rotation on Oscillation Frequency The effect of the rotation on droplet oscillation is shown in Fig. 4. The rotation of the droplet is imposed initially as a rigid rotation with a constant angular velocity around the vertical axis. The amplitude of the initial deformation is.2. The rotation frequency is varied from. to.4, and the time variations of the droplet radius are shown. The curve of the time variation is, in contrast to Fig. 2, shown to shift towards the negative direction of the time axis as the rotation increases. This indicates that the oscillation period becomes small and the oscillation frequency increases. The variation of oscillation frequency due to the change in rotation frequency is shown in Fig. 5. The theoretical curve obtained by Busse 6, which was derived by taking into account a first order deviation from the linear theory due to rotation, is also shown in Fig. 5. Rotation frequency is normalized by the oscillation frequency for the amplitude of.2 without

5 rotation. The rotation frequency is varied from. to. in Fig. 5. It is clearly shown that the oscillation frequency increases as the rotation frequency increases. The simulation results agree well with the theoretical curve up to the rotation frequency of about.5. The theoretical curve overestimates the frequency shift for the rotation larger than Pressure Difference and Frequency Shift The effect of the rotation on the relation between the frequency shift and the amplitude is shown in Fig. 6. For rotating droplets, the steady state shape is not a sphere but an ellipsoid. The initial droplet shape is, thus, assumed to be an ellipsoid in the following simulations. The shape of the ellipsoid at the steady state of rotation is calculated from the variation of droplet radius for the spherical droplet with initial rotation. The average radius in vertical direction is obtained by averaging first three periods of oscillation, and the ellipsoidal shape with this average radius is given as the initial shape for a rotating droplet. The deformation is imposed on the initial shape as the deviation of radius from the average value. The oscillation frequency decreases as the amplitude increases for a constant rotation as shown in Fig. 6, as is the case with the non-rotating droplet shown in Fig. 3. The oscillation frequency is, however, increases as the rotation frequency increases for the same amplitude as shown in Fig. 7. It is found that there exist several combinations between the rotation frequency and the oscillation amplitude which give the frequency shift of zero. The oscillation frequency depends largely on the surface tension. The pressure variation inside the droplet, which is the driving force for the oscillation, is caused by the surface tension and the deformation of the droplet shape. The pressure variation inside the droplet is, thus, of importance for the frequency shift shown in Figs. 6 and 7. The driving force for the oscillation is characterized by the pressure difference between the pole and the equator of the droplet. The average pressure differences between the pole and the equator are shown in Figs. 8 and 9, corresponding to the frequency shifts shown in Figs. 6 and 7. The average pressures under the polar and the equatrial surfaces of the droplet are obtained by averaging over first three oscillation periods, and plotted against the amplitude and the rotation in Figs. 8 and 9, respectively. The pressure difference increases as the amplitude increases as shown in Fig. 8, and decreases as the rotation increases as shown in Fig. 9. Negative and positive pressure differences correspond roughly to the positive and negative frequency shifts. It is found that the frequency shift of zero seems to be given by the pressure difference of zero. 4 CONCLUSIONS Three-dimensional oscillations of a levitated liquid droplet have been simulated numerically using the level set method. The effect of the amplitude on the oscillation frequency was studied by changing the initial amplitude, and it was shown that the oscillation frequency decreased as the initial amplitude increased. The effect of the rotation was also studied, and the oscillation frequency was, in contrast to the effect of the amplitude, shown to increase as the rotation frequency increased. The frequency shift was shown to be overestimated by the theory when the amplitude or the rotation frequency became large. It was found that the oscillation frequency was related to the average pressure difference

6 between the pole and the equator of the droplet, and the frequency shift of zero was corresponding to the pressure difference of zero. REFERENCES [] E. Herviu, N. Coutris and C. Boichon, Oscillations of a drop in aerodynamic levitation, Nucl. Eng. Design, 24, (2). [2] W. K. Rhim and S. K. Chung, Isolation of crystallizing droplets by electrostatic levitation, Methods:A Companion to Methods in Enzymology,, 8-27 (99). [3] V. Shatrov, J. Priede and G. Gerbeth, Three-dimensional linear stability analysis of the flow in a liquid spherical droplet driven by an alternating magnetic field, Phys. Fluid, 5, (23). [4] H. Lamb, Hydrodynamics, Cambridge University Press (932). [5] H. Azuma and S. Yoshihara, Three-dimensional large-amplitude drop oscillations: experiments and theoretical analysis, J. Fluid Mech., 393, (999). [6] F. H. Busse, Oscillation of a rotating liquid drop, J. Fluid Mech., 42, -8 (984). [7] C. P. Lee, M. J. Lyell and T. G. Wang, T. G., Viscous damping of the oscillations of a rotating simple drop, Phys. Fluid, 28, (985). [8] T. G. Wang, A. V. Anilkumar, C. P. Lee and K. C. Lin, Bifurcation of rotating liquid drops: results from USML- experiments in space, J. Fluid Mech., 276, (994). [9] Y. C. Chang, T. Y. Hou, B. Merriman and S. Osher, A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comp Phys, 24, (996). [] M. Sussman and P. Smereka, Axisymmetric free boundary problems, J. Fluid Mech., 34, (997).

7 .3.2 Simulation Theory Radius Time Figure : Time history of droplet radius Radius.3.2. dr=.29 dr=.2 dr=. dr=.2 Frequency Shift Simulation Theory Time Amplitude Figure 2: Effect of amplitude on droplet oscillation Figure 3: Frequency shift due to amplitude.5.6 Radius.95.9 Ω=.4.85 Ω=.3 Ω=.2 Ω= Time Frequency Shift Simulation Theory Rotation Figure 4: Effect of rotation on droplet oscillation Figure 5: Frequency shift due to rotation

8 .2.35 Frequency Shift.5..5 Ω=.5 Ω=.3 Ω=. Frequency Shift dr=.38 dr=.2 dr= Amplitude Figure 6: Effect of rotation on frequency shift Rotation Figure 7: Effect of amplitude on frequency shift..2 Pressure Difference Ω=.5 Ω=.3 Ω= Amplitude Pressure Difference dr=.38 dr=.2 dr= Rotation Figure 8: Effect of rotation on pressure difference Figure 9: Effect of amplitude on pressure difference