OSCILLATION AND ROTATION OF LEVITATED LIQUID DROPLET
Tadashi Watanabe
System Computational Science Center, Japan Atomic Energy Agency,
Tokai-mura, Naka-gun, Ibaraki-ken, 319-1195,Japan
e-mail: watanabe.tadashi66@jaea.go.jp
Web page: http://www.jaea.go.jp/index.shtml/
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.
1 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 accidents1. The levitation of liquid droplets, which is also used for containerless processing of materials, is controlled by using electrostatic force2 or electromagnetic force3 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 theory4, and small amplitude oscillations are necessary for an estimation of material properties.
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:
0 = ⋅ ∇ u (1) and s F D p Dt Du =−∇ +∇⋅ + ) 2 ( µ ρ , (2)
where ρ, u, p and µ, respectively, are the density, the velocity, the pressure and the viscosity, D is the viscous stress tensor, and Fs is a body force due to the surface tension. The surface tension force is given by
φ σκδ∇ =
s
F , (3)
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=0 at the interface, φ<0 in the liquid region, and φ>0 in the gas region. The curvature is expressed in terms of φ:
) | | ( φ φ κ ∇ ∇ ⋅ ∇ = . (4)
The density and viscosity are given, respectively, by
H l g l (ρ ρ ) ρ ρ = + − (5) and H l g l (µ µ ) µ µ = + − , (6)
= H 1 )] sin( 1 1 [ 2 1 0 ε πφ π ε φ + + ) ( ) ( ) ( φ ε ε φ ε ε φ < ≤ ≤ − < , (7)
where ε is a small positive constant for which ∇φ =1 for |φ|≤ε. The evolution of φ is given by 0 = ∇ ⋅ + ∂ ∂φ φ u t . (8)
All variables are nondimensionalized using liquid properties and characteristic values: x’=x/L, u’=u/U, t’=t/(L/U), p’=p/(ρlU2), ρ’=ρ/ρl, µ’=µ/µl, 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, ρlLU/µl, and the Weber number, ρlLU2/σ, are fixed at 200 and 20, respectively, in the following simulations.
In order to maintain the level set function as a distance function, an additional equation is solved: 2 2 |) | 1 ( α φ φ φ τ φ + ∇ − = ∂ ∂ , (9)
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 time9:
| | ) 1 )( ( κ φ τ φ ∇ − − = ∂ ∂ A Ao , (10)
where A denotes the mass corresponding to the level set function and A0 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 3.1 Oscillation of Levitated Droplet
3.0x3.0x3.0, and 100x100x100 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. 1, the theoretical decay of amplitude, which is given by Sussman and Smereka10 based on the Lamb’s linear theory4, is also shown. The agreement between the numerical and the theoretical results is satisfactory as shown in Fig. 1. The average period for the first three oscillations is 10.1 in this case, while the theoretical value is 10.0. 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. 1 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 0.02 to 0.29, 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 0.02. The amplitude is varied from 0.02 to 0.76 in Fig. 3. In the linear theory given by Lamb4, 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 Yoshihara5, 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 0.3. The theoretical curve overestimates the frequency shift for the amplitude larger than 0.3.
3.3 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 0.02. The rotation frequency is varied from 0.1 to 0.4, 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.
rotation. The rotation frequency is varied from 0.1 to 1.0 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 0.5. The theoretical curve overestimates the frequency shift for the rotation larger than 0.5.
3.4 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
between the pole and the equator of the droplet, and the frequency shift of zero was corresponding to the pressure difference of zero.
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0.98 0.99 1 1.01 1.02 1.03 0 5 10 15 20 25 30 35 40 Simulation Theory R adiu s Time
Figure 1: Time history of droplet radius
0.8 0.9 1 1.1 1.2 1.3 0 5 10 15 20 25 30 35 40 dr=0.29 dr=0.20 dr=0.11 dr=0.02 R a di us Time -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Simulation Theory Fr eq u e nc y S h ift Amplitude
Figure 2: Effect of amplitude on droplet oscillation Figure 3: Frequency shift due to amplitude
0.8 0.85 0.9 0.95 1 1.05 0 5 10 15 20 25 30 35 40 Ω=0.40 Ω=0.30 Ω=0.20 Ω=0.10 Rad ius Time -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 Simulation Theory F requ e n cy S h if t Rotation
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 Ω=0.50 Ω=0.30 Ω=0.10 Frequ e n cy S h ift Amplitude -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.2 0.4 0.6 0.8 dr=0.38 dr=0.20 dr=0.02 Frequency S h ift Rotation
Figure 6: Effect of rotation on frequency shift Figure 7: Effect of amplitude on frequency shift
-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0 0.2 0.4 0.6 0.8 Ω=0.50 Ω=0.30 Ω=0.10 P ressure Diffe renc e Amplitude -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0 0.2 0.4 0.6 0.8 dr=0.38 dr=0.20 dr=0.02 Pre s su re Diffe re n c e Rotation