For the present study, we developed a three-dimensional numerical method based on the level set method that is applicable to two-phase systems with high-density ratio. The present solver for the Navier-Stokes equations was based on the projection method with a non-staggered grid. We improved the treatment of the convection terms and the interpolation method that was used to obtain the intermediate volume flux defined on the cell faces. We also improved the solver for the pressure Poisson equations and the reinitialization procedure of the level set function. It was shown that the present solver worked very well even for a density ratio of the two fluids of 1:1000. We simulated the coalescence of two rising bubbles under gravity, and a gas bubble bursting at a free surface to evaluate mass conservation for the present method. It was also shown that the volume conservation (i.e., mass conservation) of bubbles was very good even after bubble coalescence.

Issue Section:
Technical Papers
Keywords:
flow simulation, two-phase flow, Navier-Stokes equations, convection, interpolation, Poisson equation, bubbles, gravity waves
Topics:
Bubbles, Density, Two-phase flow, Convection, Pressure, Interpolation, Poisson equation, Navier-Stokes equations, Numerical analysis, Gravity (Force)
1.
Hirt
,
C. W.
, and
Nichols
,
B. D.
,
1981
, “
Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries
,”
J. Comput. Phys.
,
39
, pp.
201
–
225
.
2.
Unverdi
,
S. O.
, and
Tryggvason
,
G.
,
1992
, “
A Front-Tracking Method for Viscous, Incompressible, Multi-fluid Flows
,”
J. Comput. Phys.
,
100
, pp.
25
–
37
.
3.
Juric
,
D.
, and
Tryggvason
,
G.
,
1998
, “
Computations of Boiling Flows
,”
Int. J. Multiphase Flow
,
24
, pp.
387
–
410
.
4.
Osher
,
S.
, and
Sethian
,
J. A.
,
1988
, “
Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations
,”
J. Comput. Phys.
,
79
, pp.
12
–
49
.
5.
Sussman
,
M.
,
Smereka
,
P.
, and
Osher
,
S.
,
1994
, “
A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow
,”
J. Comput. Phys.
,
114
, pp.
146
–
159
.
6.
Sussman
,
M.
, and
Smereka
,
P.
,
1997
, “
Axisymmetric Free Boundary Problems
,”
J. Fluid Mech.
,
341
, pp.
269
–
294
.
7.
Son
,
G.
, and
Dhir
,
V. K.
,
1998
, “
Numerical Simulation of Film Boiling Near Critical Pressures with a Level Set Method
,”
ASME J. Heat Transfer
,
120
, pp.
183
–
192
.
8.
Sussman
,
M.
et al.,
1998
, “
An Improved Level Set Method for Incompressible Two-Phase Flows
,”
Comput. Fluids
,
27
, pp.
663
–
680
.
9.
Sussman
,
M.
et al.,
1999
, “
An Adaptive Level Set Approach for Incompressible Two-Phase Flows
,”
J. Comput. Phys.
,
148
, pp.
81
–
124
.
10.
Himeno
,
T.
, and
Watanabe
,
N.
,
1999
, “
Numerical Analysis of Two-Phase Flow under Microgravity Condition
,”
Trans. Jpn. Soc. Mech. Eng., Ser. B
,
65
, pp.
2333
–
2340
(in Japanese).
11.
Sussman
,
M.
, and
Puckett
,
E. G.
,
2000
, “
A Coupled Level Set and Volume-of-Fluid Method for Computing 3D and Axisymmetric Incompressible Two-Phase Flows
,”
J. Comput. Phys.
,
162
, pp.
301
–
337
.
12.
Fedkiw
,
R. P.
et al.,
1999
, “
A Non-Oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (The Ghost Fluid Method)
,”
J. Comput. Phys.
,
152
, pp.
457
–
492
.
13.
Zang
,
Y.
et al.,
1994
, “
A Non-staggered Grid, Fractional Step Method for Time-Dependent Incompressible Navier-Stokes Equations in Curvilinear Coordinates
,”
J. Comput. Phys.
,
114
, pp.
18
–
33
.
14.
Zang
,
Y.
, and
Street
,
R. L.
,
1995
, “
A Composite Multigrid Method for Calculating Unsteady Incompressible Flows in Geometrically Complex Domains
,”
Int. J. Numer. Methods Fluids
,
20
, pp.
341
–
361
.
15.
Patankar, S. V., 1980, “Numerical Heat Transfer and Fluid Flow,” McGraw-Hill/Hemisphere.
16.
Van der Vorst
,
H. A.
,
1992
, “
Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems
,”
SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput.
,
13
, pp.
631
–
644
.
17.
Fujino
,
S.
, and
Zhang
,
S.-L.
,
1996
, “
Mathematics of Iterative Methods
,” Asakura Publishing Company, Tokyo (in Japanese).
18.
Takahira, H., and Banerjee, S., 2000, “Numerical Simulation of Three Dimensional Bubble Growth and Detachment in a Microgravity Shear Flow,” In Microgravity Fluid Physics and Heat Transfer, V. Dhir, Ed., pp. 80–87, Begell House, New York, NY.
19.
Hnat
,
J. G.
, and
Buckmaster
,
J. D.
,
1976
, “
Spherical cap bubbles and skirt formation
,”
Phys. Fluids
,
19
, pp.
182
–
194
.
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