Abstract

In this paper, the growth of a rising vapor bubble in superheated water was numerically studied using an advanced interface tracking method, called the intersection marker (ISM) method. The ISM method is a hybrid Lagrangian–Eulerian front-tracking algorithm that can model an arbitrary three-dimensional (3D) surface within an array of cubic control volumes (CCV). The ISM method has cell-by-cell remeshing capability that is volume conservative, maintains surface continuity, and is suited for tracking interface deformation in multiphase flow simulations. This method was previously used in adiabatic bubble rise simulation with no heat and mass transfers to or from the bubble were considered. This work will extend the ISM method's application to simulate vapor bubble growth in superheated water with the inclusion of additional physics, such as the convective heat transfer mechanism and the phase-change. Coupled with an in-house variable-density and variable-viscosity single-fluid flow solver, the method was used to simulate vapor bubble growth due to the convective action. The forces such as the surface tension and the buoyancy were included in the momentum equation. The source terms for the mass transfer were also modeled in the computational fluid dynamics governing equations to simulate the growth. Bubble properties such as size, shape, velocity, drag coefficient, and convective heat transfer coefficient were predicted. Effects of surface tension and temperature on the bubble characteristic were also discussed. Obtained numerical results were compared against the analytical and past works and found to be in good agreement.

References

1.
Prosperetti
,
A.
,
2017
, “
Vapor Bubbles
,”
Annu. Rev. Fluid Mech.
,
49
(
1
), pp.
221
248
.10.1146/annurev-fluid-010816-060221
2.
Samkhaniani
,
N.
, and
Ansari
,
M. R.
,
2017
, “
Numerical Simulation of Superheated Vapor Bubble Rising in Stagnant Liquid
,”
Heat Mass Transfer
,
53
(
9
), pp.
2885
2899
.10.1007/s00231-017-2031-6
3.
Faghri
,
A.
, and
Zhang
,
Y.
,
2020
,
Fundamentals of Multiphase Heat Transfer and Flow
,
Springer Nature
, Cham,
Switzerland
.
4.
Mikic
,
B. B.
,
Rohsenow
,
W. M.
, and
Griffith
,
P.
,
1970
, “
On Bubble Growth Rates
,”
Int. J. Heat Mass Transfer
,
13
(
4
), pp.
657
665
.10.1016/0017-9310(70)90040-2
5.
Robinson
,
A. J.
, and
Judd
,
R. L.
,
2004
, “
The Dynamics of Spherical Bubble Growth
,”
Int. J. Heat Mass Transfer
,
47
(
23
), pp.
5101
5113
.10.1016/j.ijheatmasstransfer.2004.05.023
6.
Holman
,
J. P.
,
1992
,
Heat Transfer
,
McGraw-Hill International (UK) Limited
, London, UK.
7.
van Sint Annaland
,
M.
,
Deen
,
N. G.
, and
Kuipers
,
J. A. M.
,
2005
, “
Numerical Simulation of Gas Bubbles Behaviour Using a Three-Dimensional Volume of Fluid Method
,”
Chem. Eng. Sci.
,
60
(
11
), pp.
2999
3011
.10.1016/j.ces.2005.01.031
8.
Hua
,
J.
,
Stene
,
J. F.
, and
Lin
,
P.
,
2008
, “
Numerical Simulation of 3D Bubbles Rising in Viscous Liquids Using a Front Tracking Method
,”
J. Comput. Phys.
,
227
(
6
), pp.
3358
3382
.10.1016/j.jcp.2007.12.002
9.
Balcazar
,
N.
,
Lehmkuhl
,
O.
,
Jofre
,
L.
, and
Oliva
,
A.
,
2015
, “
Level-Set Simulations of Buoyancy-Driven Motion of Single and Multiple Bubbles
,”
Int. J. Heat Fluid Flow
,
56
, pp.
91
107
.10.1016/j.ijheatfluidflow.2015.07.004
10.
Ho
,
M.
,
Yeoh
,
G. H.
,
Reizes
,
J. A.
, and
Timchenko
,
V.
,
2018
, “
Bubble Flow Simulations Using the Intersection Marker (ISM) Interface Tracking Method
,”
Int. J. Numer. Methods Heat Fluid Flow
,
28
(
1
), pp.
118
137
.10.1108/HFF-09-2017-0385
11.
Jeon
,
S. S.
,
Kim
,
S. J.
, and
Park
,
G. C.
,
2011
, “
Numerical Study of Condensing Bubble in Subcooled Boiling Flow Using Volume of Fluid Model
,”
Chem. Eng. Sci.
,
66
(
23
), pp.
5899
5909
.10.1016/j.ces.2011.08.011
12.
Bothe
,
D.
,
Koebe
,
M.
,
Wielage
,
K.
,
Pruss
,
J.
, and
Warnecke
,
H.-J.
,
2004
, “
Direct Numerical Simulation of Mass Transfer Between Rising Gas Bubbles and Water
,”
Bubbly Flows: Analysis, Modelling and Calculation (Heat and Mass Transfer)
,
M.
Sommerfeld
, ed.,
Springer
,
New York
, pp.
159
174
.10.1007/978-3-642-18540-3_13
13.
Harlow
,
F. H.
, and
Welch
,
J. E.
,
1965
, “
Numerical Calculation of Time-Dependent Viscous Incompressible of Fluid With Free Surface
,”
Phys. Fluids
,
8
(
12
), pp.
2182
2189
.10.1063/1.1761178
14.
Hirt
,
C. W.
, and
Nichols
,
B. D.
,
1981
, “
Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries
,”
J. Comput. Phys.
,
39
(
1
), pp.
201
226
.10.1016/0021-9991(81)90145-5
15.
Youngs
,
D. L.
,
1982
, “
Time Dependent Multimaterial Flow With Large Fluid Distortion
,”
Numerical Methods for Fluid Dynamics
,
K. M.
Morton
and
M. J.
Baines
, eds.,
Academic Press
,
New York
.
16.
Rudman
,
M.
,
1997
, “
Volume-Tracking Methods for Interfacial Flow Calculations
,”
Int. J. Numer. Methods Fluids
,
24
(
7
), pp.
671
691
.10.1002/(SICI)1097-0363(19970415)24:7<671::AID-FLD508>3.0.CO;2-9
17.
Noh
,
W. F.
, and
Woodward
,
P.
,
1976
, “
SLIC (Simple Line Interface Calculation)
,”
Proceedings of Fifth International Conference on Fluid Dynamics
(Lecture Notes in Physics, Vol. 59),
A. I.
van de Vooren
and
P. J.
Zandbergen
, eds.,
Springer
,
Berlin
, pp.
330
340
.10.1007/3-540-08004-X_336
18.
Osher
,
S.
, and
Sethian
,
J.
,
1988
, “
Fronts Propagating With Curvature-Dependant Speed: Algorithms Based on Hamilton-Jacobi Formulations
,”
J. Comput. Phys.
,
79
(
1
), pp.
12
49
.10.1016/0021-9991(88)90002-2
19.
Sussman
,
M.
,
Smereka
,
P.
, and
Osher
,
S.
,
1994
, “
A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow
,”
J. Comput. Phys.
,
114
(
1
), pp.
146
159
.10.1006/jcph.1994.1155
20.
Tryggvason
,
G.
,
Sussman
,
M.
, and
Hussaini
,
M. Y.
,
2007
, “
Immersed Boundary Methods for Fluid Interfaces
,”
Computational Methods for Multiphase Flow
,
A.
Prosperetti
and
G.
Tryggvason
, eds.,
Cambridge University Press
,
Cambridge, UK
.
21.
Unverdi
,
S. O.
, and
Tryggvason
,
G.
,
1992
, “
A Front-Tracking Method for Viscous, Incompressible Multi-Fluid Flows
,”
J. Comput. Phys.
,
100
(
1
), pp.
25
37
.10.1016/0021-9991(92)90307-K
22.
Esmaeeli
,
A.
, and
Tryggvason
,
G.
,
1998
, “
Direct Numerical Simulation of Bubble Flows. Part I. Low Reynold Number Arrays
,”
J. Fluid Mech.
,
377
, pp.
313
345
.10.1017/S0022112098003176
23.
Esmaeeli
,
A.
, and
Tryggvason
,
G.
,
1998
, “
Direct Numerical Simulation of Bubble Flows. Part II. Moderate Reynolds Number Arrays
,”
J. Fluid Mech.
,
385
, pp.
325
358
.10.1017/S0022112099004310
24.
Tryggvason
,
G.
,
Bunner
,
B.
,
Esmaeeli
,
A.
,
Juric
,
D.
,
Al-Rawahi
,
N.
,
Tauber
,
W.
,
Han
,
J.
,
Nas
,
S.
, and
Jan
,
Y.-J.
,
2001
, “
A Front Tracking Method for the Computations of Multiphase Flow
,”
J. Comput. Phys.
,
169
(
2
), pp.
708
759
.10.1006/jcph.2001.6726
25.
Ladd
,
A. J. C.
,
1994
, “
Numerical Simulations of Particulate Suspensions Via a Discretised Boltzmann Equation. Part 1. Theoretical Foundation
,”
J. Fluid Mech.
,
271
, pp.
285
309
.10.1017/S0022112094001771
26.
Ladd
,
A. J. C.
,
1994
, “
Numerical Simulations of Particulate Suspensions Via a Discretised Boltzmann Equation. Part 2. Numerical Results
,”
J. Fluid Mech.
,
271
, pp.
311
339
.10.1017/S0022112094001783
27.
van Sint Annaland
,
M.
,
Dijkhuizen
,
W.
,
Deen
,
N. G.
, and
Kuipers
,
J. A. M.
,
2006
, “
Numerical Simulation of Behavior of Gas Bubbles Using a 3-D Front-Tracking Method
,”
AIChE J.
,
52
(
1
), pp.
99
110
.10.1002/aic.10607
28.
Ho
,
M.
,
Yeoh
,
G.
,
Reizes
,
J.
, and
Timchenko
,
V.
,
2016
, “
The Intersection Marker Method for 3D Interface Tracking of Deformable Surfaces in Finite Volumes
,”
Int. J. Numer. Methods Fluids
,
81
(
4
), pp.
220
244
.10.1002/fld.4182
29.
Tryggvason
,
G.
,
Scardovelli
,
R.
, and
Zaleski
,
S.
,
2011
,
Direct Numerical Simulations of Gas-Liquid Multiphase Flows
,
Cambridge University Press
,
Cambridge, UK
.
30.
Aulisa
,
E.
,
Manservisi
,
S.
, and
Scardovelli
,
R.
,
2004
, “
A Surface Marker Algorithm Coupled to an Area-Preserving Marker Redistribution Method for Three-Dimensional Interface Tracking
,”
J. Comput. Phys.
,
197
(
2
), pp.
555
584
.10.1016/j.jcp.2003.12.009
31.
Yeoh
,
G. H.
, and
Tu
,
J.
,
2010
,
Computational Techniques for Multi-Phase Flows
,
Butterworth-Heinemann
,
Oxford, UK
.
32.
Sharif
,
S. A.
,
Ho
,
M. K. M.
,
Timchenko
,
V.
, and
Yeoh
,
G. H.
,
2019
, “
Gravity-Driven Bubble Rise Simulation
,”
Handbook of Multiphase Flow Science and Technology
,
G. H.
Yeoh
and
J.
Joshi
, eds.,
Springer
,
Singapore
.
33.
Ranz
,
W. E.
, and
Marshall
,
W. R.
,
1952
, “
Evaporation From Drops
,”
Chem. Eng. Prog.
,
48
, pp.
141
146
.
34.
Whitaker
,
S.
,
1972
, “
Forced Convection Heat Transfer Correlations for Flow in Pipes, Past Flat Plates, Single Cylinders, Single Spheres, and for Flow in Packed Beds and Tube Bundles
,”
AIChE J.
,
18
(
2
), pp.
361
370
.10.1002/aic.690180219
35.
Hughmark
,
G. A.
,
1967
, “
Mass and Heat Transfer From Rigid Spheres
,”
AIChE J.
,
13
(
6
), pp.
1219
1221
.10.1002/aic.690130638
36.
Akiyama
,
M.
,
1973
, “
Bubble Collapse in Subcooled Boiling
,”
Bull. JSME
,
16
(
93
), pp.
570
575
.10.1299/jsme1958.16.570
37.
McAdams
,
W. H.
,
1954
,
Heat Transmission
,
McGraw-Hill Book Company
,
New York
.
38.
Lorensen
,
W. E.
, and
Cline
,
H. E.
,
1987
, “
Marching Cubes: A High Resolution 3D Surface Construction Algorithm
,”
ACM SIGGRAPH Comput. Graphics
,
21
(
4
), pp.
163
169
.10.1145/37402.37422
39.
Peskin
,
C. S.
,
1977
, “
Numerical Analysis of Blood Flow in the Heart
,”
J. Comput. Phys.
,
25
(
3
), pp.
220
252
.10.1016/0021-9991(77)90100-0
40.
Patankar
,
S. V.
, and
Spalding
,
D. B.
,
1972
, “
A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Parabolic Flows
,”
Int. J. Heat Mass Transfer
,
15
(
10
), pp.
1787
1806
.10.1016/0017-9310(72)90054-3
41.
Peskin
,
C. S.
,
2002
, “
The Immersed Boundary Method
,”
Acta Numer.
,
11
, pp.
479
517
.10.1017/S0962492902000077
42.
Tornberg
,
A. K.
, and
Engquist
,
B.
,
2004
, “
Numerical Approximations of Singular Source Terms in Differential Equations
,”
J. Comput. Phys.
,
200
(
2
), pp.
462
488
.10.1016/j.jcp.2004.04.011
43.
Deen
,
N. G.
,
van Sint Annaland
,
M.
, and
Kuipers
,
J. A. M.
,
2004
, “
Multi-Scale Modelling of Dispersed Gas–Liquid Two-Phase Flow
,”
Chem. Eng. Sci.
,
59
(
8–9
), pp.
1853
1861
.10.1016/j.ces.2004.01.038
44.
Griffith
,
B. E.
, and
Peskin
,
C. S.
,
2005
, “
On the Order of Accuracy of the Immersed Boundary Method: Higher Order Convergence Rates for Sufficiently Smooth Problems
,”
J. Comput. Phys.
,
208
(
1
), pp.
75
105
.10.1016/j.jcp.2005.02.011
45.
Zhang
,
N.
,
Zheng
,
Z. C.
, and
Eckels
,
S.
,
2008
, “
Study of Heat-Transfer on the Surface of a Circular Cylinder in Flow Using an Immersed-Boundary Method
,”
Int. J. Heat Fluid Flow
,
29
(
6
), pp.
1558
1566
.10.1016/j.ijheatfluidflow.2008.08.009
46.
Incropera
,
F. P.
,
Dewitt
,
D. P.
,
Bergman
,
T. L.
, and
Lavine
,
A. S.
,
2007
,
Principles of Heat and Mass Transfer
,
Wiley
, Hoboken,
NJ
.
47.
Clift
,
R.
,
Grace
,
J. R.
, and
Weber
,
M. E.
,
1978
,
Bubbles, Drops, and Particles
,
Academic Press
,
New York
.
48.
Zhang
,
Y.
,
McLaughlin
,
J. B.
, and
Finch
,
J. A.
,
2001
, “
Bubble Velocity Profile and Model of Surfactant Mass Transfer to Bubble Surface
,”
Chem. Eng. Sci.
,
56
(
23
), pp.
6605
6616
.10.1016/S0009-2509(01)00304-9
49.
Hua
,
J.
, and
Lou
,
J.
,
2007
, “
Numerical Simulation of Bubble Rising in Viscous Liquid
,”
J. Comput. Phys.
,
222
(
2
), pp.
769
795
.10.1016/j.jcp.2006.08.008
50.
Kulkarni
,
A. A.
, and
Joshi
,
J. B.
,
2005
, “
Bubble Formation and Bubble Rise Velocity in Gas-Liquid Systems: A Review
,”
Ind. Eng. Chem. Res.
,
44
(
16
), pp.
5873
5931
.10.1021/ie049131p
51.
Liu
,
L.
,
Yan
,
H.
,
Zhao
,
G.
, and
Zhuang
,
J.
,
2016
, “
Experimental Studies on the Terminal Velocity of Air Bubbles in Water and Glycerol Aqueous Solution
,”
Exp. Therm. Fluid Sci.
,
78
, pp.
254
265
.10.1016/j.expthermflusci.2016.06.011
52.
Mendelson
,
H. D.
,
1967
, “
The Prediction of Bubble Terminal Velocities From Wave Theory
,”
AIChE J.
,
13
(
2
), pp.
250
252
.10.1002/aic.690130213
53.
Wanchoo
,
R. K.
,
1993
, “
Terminal Velocity Equation for a Two Phase Bubble in an Immiscible Liquid Medium
,”
Chem. Eng. Commun.
,
120
(
1
), pp.
111
117
.10.1080/00986449308936128
54.
Sideman
,
S.
, and
Taitel
,
Y.
,
1964
, “
Direct-Contact Heat Transfer With Change of Phase: Evaporation of Drops in an Immiscible Liquid Medium
,”
Int. J. Heat Mass Transfer
,
7
(
11
), pp.
1273
1289
.10.1016/0017-9310(64)90068-7
55.
Beard
,
K. V.
, and
Pruppacher
,
H. R.
,
1969
, “
A Determination of the Terminal Velocity and Drag of Small Water Drops by Means of a Wind Tunnel
,”
J. Atmos. Sci.
,
26
(
5
), pp.
1066
1072
.10.1175/1520-0469(1969)026<1066:ADOTTV>2.0.CO;2
56.
Clift
,
R.
, and
Gauvin
,
W. H.
,
1970
, “The Motion of Particles in Turbulent Gas-Streams,”
Proc. Chemeca
,
70
(
1
), pp.
14
28
.
57.
Bird
,
R. B.
,
Stewart
,
W. E.
, and
Lightfoot
,
E. N.
,
2005
,
Transport Phenomena
,
Wiley
,
Singapore
.
58.
Chen
,
J. C.
,
1966
, “
Correlation for Boiling Heat Transfer to Saturated Fluids in Convective Flow
,”
Ind. Eng. Chem. Process Des. Dev.
,
5
(
3
), pp.
322
329
.10.1021/i260019a023
59.
Soh
,
G. Y.
,
Yeoh
,
G. H.
, and
Timchenko
,
V.
,
2016
, “
An Algorithm to Calculate the Interfacial Area for Multiphase Mass Transfer Through the Volume-of-Fluid Method
,”
Int. J. Heat Mass Transfer
,
100
, pp.
573
581
.10.1016/j.ijheatmasstransfer.2016.05.006
60.
Vries
,
A. W.
,
2001
, “
Path and Wake of a Rising Bubble
,” Ph.D. dissertation, Mechanical Engineering Department,
Twente University
,
Enschede, The Netherlands
.
61.
Grace
,
J. R.
,
1973
, “
Shapes and Velocities of Bubbles Rising in Infinite Liquids
,”
Trans. Inst. Chem. Eng.
,
51
, pp.
116
120
.
62.
Bhaga
,
D.
, and
Weber
,
M. E.
,
1981
, “
Bubbles in Viscous Liquids: Shapes, Wakes and Velocities
,”
J. Fluid Mech.
,
105
(
1
), pp.
61
85
.10.1017/S002211208100311X
63.
National Bureau of Standards
,
1976
, “
Release of Surface Tension of Water Substance
,”
The International Association for the Properties of Steam
,
Washington, DC
, Report No. IAPS-76.
64.
Maldonado
,
M.
,
Quinn
,
J. J.
,
Gomez
,
C. O.
, and
Finch
,
J. A.
,
2013
, “
An Experimental Study Examining the Relationship Between Bubble Shape and Rise Velocity
,”
Chem. Eng. Sci.
,
98
, pp.
7
11
.10.1016/j.ces.2013.04.050
65.
Quinn
,
J. J.
,
Maldonado
,
M.
,
Gomez
,
C. O.
, and
Finch
,
J. A.
,
2014
, “
Experimental Study on the Shape–Velocity Relationship of an Ellipsoidal Bubble in Inorganic Salt Solutions
,”
Miner. Eng.
,
55
, pp.
5
10
.10.1016/j.mineng.2013.09.003
66.
Forster
,
H. K.
, and
Zuber
,
N.
,
1955
, “
Dynamics of Vapor Bubbles and Boiling Heat Transfer
,”
AIChE J.
,
1
(
4
), pp.
531
535
.10.1002/aic.690010425
67.
Dittus
,
F. W.
, and
Boelter
,
L. M. K.
,
1930
,
Heat Transfer in Automobile Radiators of the Tubular Type
, Vol.
2
,
University of California Publications in Engineering
,
Berkeley, CA
, pp.
443
461
.
68.
Collier
,
J. G.
,
1981
, “
Forced Convection Boiling
,”
Two-Phase Flow and Heat Transfer in the Power and Process Industries
,
A. E.
Bergles
,
J. G.
Collier
,
J. M.
Delhaye
,
G. F.
Hewitt
, and
F.
Mayinger
, eds.,
Hemisphere Publishing Corp
.,
Washington, DC
.
69.
Yu
,
W.
,
France
,
D. M.
,
Wambsganss
,
M. W.
, and
Hull
,
J. R.
,
2002
, “
Two Phase Pressure Drop, Boiling Heat Transfer, and Critical Heat Flux to Water in a Small-Diameter Horizontal Tube
,”
Int. J. Multiphase Flow
,
28
(
6
), pp.
927
941
.10.1016/S0301-9322(02)00019-8
70.
Kolev
,
N. I.
,
2007
,
Multiphase Flow Dynamic 2: Thermal and Mechanical Interactions
,
Springer Verlag
,
Berlin
.
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