Improvement of a Simple Coupled VOF with LS (S-CLSVOF) Method

Figures & data

Figure 1. Variation of ${\varphi }_x$ as a function of α .

Figure 2. Location of ${\varphi }_x,$ ${\varphi }_y$ and $\varphi .$

Figure 3. Schematic of Test A.

Figure 4. Two-dimensional test for $\varphi$ (Run A). (a) Result of run A. (b) Zoom of Figure 4 (a).

Figure 5. Two-dimensional test for $\varphi$ (Run B). (a) Schematic of test B. (b) Result of run B.

Figure 6. Variable arrangement in a cell.

Figure 7. Calculation procedure.

Table 1. Numerical schemes in the present study.

Term of calculation	Scheme
Advection of $\mathit{V}$	3rd order CIP scheme (Constrained Interpolation Profile Scheme)
Viscosity	Central difference
Time difference	2nd order Adams-Bashforth
Advection of $\alpha$	THAINC
Poisson equation of $P$	BiCGStab (Biconjugate Gradient Stabilized method)

Table 2. Method for calculation of surface tension.

Method	Equations
A (CSF with VOF function)	Eq. (25)
B (Albadawi et al. 2013)	Eq. (26) with (11)
C (Proposed)	Eq. (26) with (17)

Table 3. Physical properties.

Unit	Bubble	Drop
${\rho }_1$[kg/m^3]	$1$	$1000$
${\rho }_2$[kg/m^3]	$1000$	$1$
${\mu }_1$[Pa･s]	$1\times {10}^{-5}$	$1\times {10}^{-3}$
${\mu }_2$[Pa･s]	$1\times {10}^{-3}$	$1\times {10}^{-5}$
$\mathrm{\sigma }$[N/m]	$1.0\times {10}^{-2}$	$1.0\times {10}^{-2}$

Table 4. Grid size resolution.

Level	Number of cells	$\mathrm{\Delta }x,\mathrm{\Delta }y$ [m]
L1	$50\times 50$	$0.001$
L2	$100\times 100$	$0.0005$
L3	$150\times 150$	$0.001/3$
L4	$200\times 200$	$0.00025$

Figure 8. Velocity profiles at t = 0.1 s (Bubble case, resolution:L2). (a) Numerical results obtained by the original Method A. (b) Numerical results obtained by Method B. (c) Numerical results obtained by the Method C.

Table 5. Comparison of the value of two errors.

(a) Bubble case
Method	Resolution	$\left|V_{\max }\right|$	$\left|V_{\mathrm{ave}}\right|$
A	L1	$5.41\times {10}^{-2}$	$2.93\times {10}^{-3}$
	L2	$6.97\times {10}^{-2}$	$2.62\times {10}^{-3}$
	L3	$9.78\times {10}^{-2}$	$2.45\times {10}^{-3}$
	L4	$1.32\times {10}^{-1}$	$3.29\times {10}^{-3}$
B	L1	$1.26\times {10}^{-2}$	$4.75\times {10}^{-4}$
	L2	$2.75\times {10}^{-2}$	$5.34\times {10}^{-4}$
	L3	$3.51\times {10}^{-2}$	$5.45\times {10}^{-4}$
	L4	$3.61\times {10}^{-2}$	$3.57\times {10}^{-4}$
C	L1	$8.82\times {10}^{-3}$	$3.83\times {10}^{-4}$
	L2	$6.62\times {10}^{-3}$	$1.37\times {10}^{-4}$
	L3	$9.30\times {10}^{-3}$	$1.42\times {10}^{-4}$
	L4	$9.19\times {10}^{-3}$	$1.05\times {10}^{-4}$
(b) Drop case
Method		$\left|V_{\max }\right|$	$\left|V_{\mathrm{ave}}\right|$
A	L2	$7.35\times {10}^{-2}$	$3.37\times {10}^{-3}$
	L3	$9.63\times {10}^{-2}$	$4.27\times {10}^{-3}$
B	L2	$2.83\times {10}^{-2}$	$6.23\times {10}^{-4}$
	L3	$7.05\times {10}^{-2}$	$9.66\times {10}^{-4}$
C	L2	$4.54\times {10}^{-3}$	$1.50\times {10}^{-4}$
	L3	$4.67\times {10}^{-3}$	$1.61\times {10}^{-4}$

Figure 9. Boundary conditions.

Table 6. Condition of calculation.

Nozzle diameter d	$1.7\mathrm{ }\mathrm{mm}$
Averaged velocity of gas U	$0.0881\mathrm{ }\mathrm{m}/\mathrm{s}$
Liquid viscosity	$0.126\mathrm{Pa}\cdot \mathrm{s}$
Liquid density	$1223.8\mathrm{kg}/{\mathrm{m}}^3$
Gas viscosity	$1.822\times {10}^{-5}\mathrm{ }\mathrm{Pa}\cdot \mathrm{s}$
Gas density	$1.205\mathrm{kg}/{\mathrm{m}}^3$
Surface tension coefficient	$6.6\times {10}^{-2}\mathrm{ }\mathrm{N}/\mathrm{m}$
Cell size	$0.03125\mathrm{ }\mathrm{m}$

Figure 10. A sequence of calculated bubble formations (left: CLSVOF by Ohta et al. (2011), right: Proposed method, $T$ is dimensionless time).

Figure 11. A sequence of pinch-off.

Albadawi A, Donoghue DB, Robinson AJ, Murray DB, Delauré YMC. 2013. Influence of surface tension implementation in volume of fluid and coupled volume of fluid with level set methods for bubble growth and detachment. Int J Multiphase Flow. 53:11–28. doi: 10.1016/j.ijmultiphaseflow.2013.01.005

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Ohta M, Kikuchi D, Yoshida Y, Sussman M. 2011. Robust numerical analysis of the dynamic bubble formation process in a viscous liquid. Int J Multiphase Flow. 37:1059–1071. doi: 10.1016/j.ijmultiphaseflow.2011.05.012

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