The behavior of air bubble clusters rising in water and the induced flow field are numerically studied using a three-dimensional two-way coupling algorithm based on a vortex-in-cell (VIC) method. In this method, vortex elements are convected in the Lagrangian frame and the liquid velocity field is solved from the Poisson equation of potential on the Eulerian grid. Two-way coupling is implemented by introducing a vorticity source term induced by the gradient of void fraction. Present simulation results are favorably compared with the measured results of bubble plume, which verifies the validity of the proposed VIC method. The rising of a single bubble cluster as well as two tandem bubble clusters are simulated. The mechanism of the aggregation effect in the rising process of bubble cluster is revealed and the transient processes of the generation, rising, strengthening, and separation of a vortex ring structure with bubble clusters are illustrated and analyzed in detail. Due to the aggregation, the average rising velocity increases with void fraction and is larger than the terminal rising velocity of single bubble. For the two tandem bubble cluster cases, the aggregation effect is stronger for smaller initial cluster distance, and both the strength of the induced vortex structure and the average bubble rising velocity are larger. For the 20 mm cluster distance case, the peak velocity of the lower cluster is about 2.7 times that of the terminal velocity of the single bubble and the peak average velocity of two clusters is about 2 times larger. While for the 30 mm cluster distance case, both the peak velocity of the lower cluster and two clusters are about 1.7 times that of the terminal velocity of the single bubble.

References

References
1.
Mudde
,
R. F.
,
2005
, “
Gravity-Driven Bubbly Flows
,”
Annu. Rev. Fluid Mech.
,
37
, pp.
393
423
.10.1146/annurev.fluid.37.061903.175803
2.
Cheng
,
M.
,
Hua
,
J. S.
, and
Lou
,
J.
,
2010
, “
Simulation of Bubble-Bubble Interaction Using a Lattice Boltzmann Method
,”
Comput. Fluids
,
39
(
2
), pp.
260
270
.10.1016/j.compfluid.2009.09.003
3.
Chiang
,
T. P.
,
Sheu
,
W. H.
, and
Hwang
,
R. R.
,
1997
, “
Three-Dimensional Vortex Dynamics in a Shear-Driven Rectangular Cavity
,”
Int. J. Comput. Fluid D.
,
8
(
3
), pp.
201
214
.10.1080/10618569708940805
4.
Cottet
,
G. H.
, and
Poncet
,
P.
,
2003
, “
Advances in Direct Numerical Simulations of Three-Dimensional Wall-Bounded Flows by Vortex-in-Cell Methods
,”
J. Comput. Phys.
,
193
(
1
), pp.
136
158
.10.1016/j.jcp.2003.08.025
5.
Sohn
,
S. I.
, and
Hwang
,
W.
,
2005
, “
Numerical Simulations of Vortex Sheet Evolution in Stratified Shear Flow
,”
J. Phys. Soc. Jpn.
,
74
(
5
), pp.
1472
1478
.10.1143/JPSJ.74.1472
6.
Borthwick
,
A. G. L.
, and
Barber
,
R. W.
,
1992
, “
Numerical-Simulation of Jet-Forced Flow in a Circular Reservoir Using Discrete and Random Vortex Methods
,”
Int. J. Numer. Meth. Fl.
,
14
(
12
), pp.
1453
1472
.10.1002/fld.1650141207
7.
Ogami
,
Y.
, and
Fukumoto
,
K.
,
2010
, “
Simulation of Combustion by Vortex Method
,”
Comput. Fluids
,
39
(
4
), pp.
592
603
.10.1016/j.compfluid.2009.10.008
8.
Chen
,
B.
,
Wang
,
C.
,
Wang
,
Z. W.
, and
Guo
,
L. J.
,
2009
, “
Investigation of Gas-Solid Two-Phase Flow Across Circular Cylinders With Discrete Vortex Method
,”
Appl. Therm. Eng.
,
29
(
8–9
), pp.
1457
1466
.10.1016/j.applthermaleng.2008.06.025
9.
Akbari
,
M. H.
, and
Price
,
S. J.
,
2010
, “
Wake Synchronization for a Pair of Staggered Cylinders in Cross-Flow With the Upstream Cylinder in Transverse Oscillation
,”
Eng. Appl. Comp. Fluid
,
4
(
1
), pp.
58
70
.
10.
Mustto
,
A. A.
, and
Bodstein
,
G. C. R.
,
2011
, “
Subgrid-Scale Modeling of Turbulent Flow Around Circular Cylinder by Mesh-Free Vortex Method
,”
Eng. Appl. Comp. Fluid
,
5
(
2
), pp.
259
275
.
11.
Chen
,
B.
, and
Wang
,
Z. W.
,
2009
, “
Simulation of Interaction Between Two Vortex Rings
,”
Prog. Comput. Fluid Dyn.
,
9
(
3–5
), pp.
292
299
.10.1504/PCFD.2009.024831
12.
Wang
,
Z. W.
, and
Chen
,
B.
,
2012
, “
Numerical Investigation of the Evolution of Elliptic Vortex Ring
,”
Prog. Comput. Fluid Dyn.
,
12
(
1
), pp.
19
26
.10.1504/PCFD.2012.044851
13.
Uchiyama
,
T.
, and
Matsumura
,
S.
,
2010
, “
Three-Dimensional Vortex Method for the Simulation of Bubbly Flow
,”
ASME J. Fluids Eng.
,
132
(
10
), p.
101402
.10.1115/1.4002574
14.
Wang
,
Z. W.
,
Uchiyama
,
T.
, and
Chen
,
B.
,
2013
, “
Numerical Simulation of the Interaction Between Vortex Ring and Bubble Plume
,”
Appl. Math. Model.
,
37
(
24
), pp.
10007
10026
.10.1016/j.apm.2013.05.061
15.
Milane
,
R. E.
, and
Abdolhosseini
,
R.
,
2004
, “
Development of a Three-Dimensional Vortex-in-Cell Method for a Spatially Growing Uniformly Sheared Flow
,”
Int. J. Comput. Fluid D.
,
18
(
1
), pp.
47
69
.10.1080/1061856021000010766
16.
Leonard
,
A.
,
1980
, “
Vortex Methods for Flow Simulation
,”
J. Comput. Phys.
,
37
(
3
), pp.
289
335
.10.1016/0021-9991(80)90040-6
17.
Greengard
,
L.
, and
Rokhlin
,
V.
,
1987
, “
A Fast Algorithm for Particle Simulations
,”
J. Comput. Phys.
,
135
(
2
), pp.
280
292
.10.1006/jcph.1997.5706
18.
Qian
,
L.
, and
Vezza
,
M.
,
2001
, “
A Vorticity-Based Method for Incompressible Unsteady Viscous Flows
,”
J. Comput. Phys.
,
172
(
2
), pp.
515
542
.10.1006/jcph.2001.6835
19.
Christiansen
,
J. P.
,
1973
, “
Numerical Simulation of Hydrodynamics by the Method of Point Vortices
,”
J. Comput. Phys.
,
13
(
3
), pp.
363
379
.10.1016/0021-9991(73)90042-9
20.
Cheng
,
M.
,
Chew
,
Y. T.
, and
Luo
,
S. C.
,
1997
, “
A Hybrid Vortex Method for Flows Over a Bluff Body
,”
Int. J. Numer. Meth. Fl.
,
24
(
3
), pp.
501
523
.10.1002/(SICI)1097-0363(19970215)24:3<253::AID-FLD490>3.0.CO;2-C
21.
Abdolhosseini
,
R.
, and
Milane
,
R. E.
,
2000
, “
On the Effect of Vortex Grid Density in the Vortex-in-Cell Simulation of Mixing Layers
,”
Int. J. Comput. Fluid D.
,
13
(
2
), pp.
161
183
.10.1080/10618560008940896
22.
Liu
,
C. H.
,
2001
, “
A Three-Dimensional Vortex Particle-in-Cell Method for Vortex Motions in the Vicinity of a Wall
,”
Int. J. Numer. Meth. Fl.
,
37
(
5
), pp.
501
523
.10.1002/fld.180
23.
Uchiyama
,
T.
, and
Naruse
,
M.
,
2004
, “
Numerical Simulation for Gas-Particle Two-Phase Free Turbulent Flow Based on Vortex in Cell Method
,”
Powder Technol.
,
142
(
2–3
), pp.
193
208
.10.1016/j.powtec.2004.04.038
24.
Chorin
,
A. J.
,
1973
, “
Numerical Study of Slightly Viscous Flow
,”
J. Fluid Mech.
,
57
(
4
), pp.
785
796
.10.1017/S0022112073002016
25.
Cottet
,
G. H.
, and
Koumoutsakos
,
P.
,
2000
,
Vortex Methods: Theory and Practice
,
Cambridge University Press
,
Cambridge, UK
.
26.
Creusé
,
E.
,
Giovannini
,
A.
, and
Mortazavi
,
I.
,
2009
, “
Vortex Simulation of Active Control Strategies for Transitional Backward-Facing Step Flows
,”
Comput. Fluids
,
38
(
7
), pp.
1348
1360
.10.1016/j.compfluid.2008.01.036
27.
Beaudoin
,
A.
,
Huberson
,
S.
, and
Rivoalen
,
E.
,
2003
, “
Simulation of Anisotropic Diffusion by Means of a Diffusion Velocity Method
,”
J. Comput. Phys.
,
186
(
1
), pp.
122
135
.10.1016/S0021-9991(03)00024-X
28.
Milane
,
R. E.
,
2004
, “
Large Eddy Simulation (2D) Using Diffusion–Velocity Method and Vortex-in-Cell
,”
Int. J. Numer. Meth. Fluids
,
44
(
8
), pp.
837
860
.10.1002/fld.673
29.
Cottet
,
G. H.
,
Koumoutsakos
,
P.
, and
Salihi
,
M. L. O.
,
2003
, “
Vortex Methods With Spatially Varying Cores
,”
J. Comput. Phys.
,
162
(
1
), pp.
164
185
.10.1006/jcph.2000.6531
30.
Degond
,
P.
, and
Mas-Gallic
,
S.
,
1989
, “
The Weighted Particle Method for Convection-Diffusion Equations
,”
Math. Comput.
,
53
(
188
), pp.
485
507
.10.2307/2008716
31.
Winckelmans
,
G. S.
, and
Leonard
,
A.
,
1993
, “
Contributions to Vortex Particle Methods for the Computation of Three-Dimensional Incompressible Unsteady Flows
,”
J. Comput. Phys.
,
109
(
2
), pp.
247
273
.10.1006/jcph.1993.1216
32.
Koumoutsakos
,
P.
,
Leonard
,
A.
, and
Pépin
,
F.
,
1994
, “
Boundary Conditions for Viscous Vortex Methods
,”
J. Comput. Phys.
,
113
(
1
), pp.
52
61
.10.1006/jcph.1994.1117
33.
Ploumhans
,
P.
, and
Winckelmans
,
G. S.
,
2000
, “
Vortex Methods for High-Resolution Simulations of Viscous Flow Past Bluff Bodies of General Geometry
,”
J. Comput. Phys.
,
165
(
2
), pp.
354
406
.10.1006/jcph.2000.6614
34.
Eldredge
,
J. D.
,
Leonard
,
A.
, and
Colonius
,
T.
,
2002
, “
A General Deterministic Treatment of Derivatives in Particle Methods
,”
J. Comput. Phys.
,
180
(
2
), pp.
686
709
.10.1006/jcph.2002.7112
35.
Ploumhans
,
P.
,
Winckelmans
,
G. S.
,
Salmon
,
J. K.
,
Leonard
,
A.
, and
Warren
,
M. S.
,
2002
, “
Vortex Methods for Direct Numerical Simulation of Three-Dimensional Bluff Body Flows: Application to the Sphere at Re = 300, 500, and 1000
,”
J. Comput. Phys.
,
178
(
2
), pp.
427
463
.10.1006/jcph.2002.7035
36.
Schrader
,
B.
,
Reboux
,
S.
, and
Sbalzarini
,
I. F.
,
2010
, “
Discretization Correction of General Integral PSE Operators for Particle Methods
,”
J. Comput. Phys.
,
229
(
11
), pp.
4159
4182
.10.1016/j.jcp.2010.02.004
37.
Crowe
,
C. T.
,
Gore
,
R. A.
, and
Troutt
,
T. R.
,
1985
, “
Particle Dispersion by Coherent Structures in Free Shear Flows
,”
Part. Sci. Tech.
,
3
(
3–4
), pp.
149
158
.10.1080/02726358508906434
38.
Brecht
,
S. H.
, and
Ferrante
,
J. R.
,
1989
, “
Vortex-in-Cell Simulations of Buoyant Bubbles in Three Dimensions
,”
Phys. Fluids A
,
1
(
7
), pp.
1166
1191
.10.1063/1.857341
39.
Chen
,
H.
, and
Marshall
,
J. S.
,
1999
, “
A Lagrangian Vorticity Method for Two-Phase Particulate Flows With Two-Way Phase Coupling
,”
J. Comput. Phys.
,
148
(
1
), pp.
169
198
.10.1006/jcph.1998.6116
40.
Walther
,
J. H.
, and
Koumoutsakos
,
P.
,
2001
, “
Three-Dimensional Vortex Methods for Particle-Laden Flows With Two-Way Coupling
,”
J. Comput. Phys.
,
167
(
1
), pp.
39
71
.10.1006/jcph.2000.6656
41.
Yang
,
X. G.
,
Thomas
,
N. H.
,
Guo
,
L. J.
, and
Hou
,
Y.
,
2002
,
Two-Way Coupled Bubble Laden Mixing Layer
,”
Chem. Eng. Sci.
,
57
(
4
), pp.
555
564
.10.1016/S0009-2509(01)00416-X
42.
Yang
,
X. G.
,
Huang
,
X. B.
,
Rielly
,
C.
,
Zouaoui
,
Z.
, and
Guo
,
L. J.
,
2005
, “
The Modifications of a Plane Mixing Layer by Condensed Bubbles
,”
Chem. Eng. Sci.
,
60
(
24
), pp.
6876
6886
.10.1016/j.ces.2005.06.007
43.
Uchiyama
,
T.
, and
Degawa
,
T.
,
2006
, “
Numerical Simulation for Gas-Liquid Two-Phase Free Turbulent Flow Based on Vortex in Cell Method
,”
JSME Int. J. B-Fluid. T.
,
49
(
4
), pp.
1008
1015
.10.1299/jsmeb.49.1008
44.
Uchiyama
,
T.
,
2013
, “
Numerical Simulation of Particle-Laden Gas Flow by Vortex in Cell Method
,”
Powder Tech.
,
235
, pp.
376
385
.10.1016/j.powtec.2012.10.051
45.
Uchiyama
,
T.
, and
Yagami
,
H.
,
2008
, “
Numerical Simulation for the Collision Between a Vortex Ring and Solid Particles
,”
Powder Technol.
,
188
(
1–2
), pp.
73
80
.10.1016/j.powtec.2008.03.015
46.
Monaghan
,
J. J.
,
1985
, “
Extrapolating B Splines for Interpolation
,”
J. Comput. Phys.
,
60
(2), pp.
253
262
.10.1016/0021-9991(85)90006-3
47.
Bergdorf
,
M.
, and
Koumoutsakos
,
P.
,
2006
, “
A Lagrangian Particle-Wavelet Method
,”
Multiscale Model. Sim.
,
5
(
3
), pp.
980
995
.10.1137/060652877
48.
Koumoutsakos
,
P.
,
1997
, “
Inviscid Axisymmetrization of an Elliptical Vortex
,”
J. Comput. Phys.
,
138
(
2
), pp.
821
857
.10.1006/jcph.1997.5749
49.
Clift
,
R.
,
Grace
,
J. R.
, and
Weber
,
M. E.
,
1978
,
Bubbles, Drops and Particles
,
Academic Press
,
New York
.
50.
Zheng
,
L.
, and
Yapa
,
P. D.
,
2000
, “
Buoyant Velocity of Spherical and Nonspherical Bubbles/Droplets
,”
ASCE J. Hydraul. Eng.
,
126
(
11
), pp.
852
854
.10.1061/(ASCE)0733-9429(2000)126:11(852)
51.
Alam
,
M.
, and
Arakeri
,
V. H.
,
1993
, “
Observations on Transition in Plane Bubble Plumes
,”
J. Fluid Mech.
,
254
, pp.
363
375
.10.1017/S0022112093002174
52.
Hunt
,
J. C. R.
,
Wray
,
A. A.
, and
Moin
,
P.
,
1988
, “
Eddies, Stream, and Convergence Zones in Turbulent Flows
,” Center for Turbulence Research, Stanford University, Report No. CTR-S88, pp.
193
208
.
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