In this paper, we review and compare the level set, phase-field, and immersed boundary methods for incompressible two-phase flows. The models are based on modified Navier–Stokes and interface evolution equations. We present the basic concepts behind these approaches and discuss the advantages and disadvantages of each method. We also present numerical solutions of the three methods and perform characteristic numerical experiments for two-phase fluid flows.
References
1.
Caboussat
, A.
, 2005
, “Numerical Simulation of Two-Phase Free Surface Flows
,” Arch. Comput. Method Eng.
, 12
, pp. 165
–224
.10.1007/BF030445182.
Prosperetti
, A.
, and Tryggvason
, G.
, 2007
, Computational Methods for Multiphase Flow
, Cambridge University Press
, New York
.3.
Clayton
, T. C.
, Schwarzkopf
, J. D.
, Sommerfeld
, M.
, and Tsuji
, Y.
, 2011
, Multiphase Flows With Droplets and Particles
, CRC Press, Boca Raton, FL
.4.
Worner
, M.
, 2012
, “Numerical Modeling of Multiphase Flows in Microfluidics and Micro Process Engineering: A Review of Methods and Applications
,” Microfluid. Nanofluid.
, 12
, pp. 841
–886
.10.1007/s10404-012-0940-85.
Brackbill
, J. U.
, Kothe
, D. B.
, and Zemach
, C.
, 1992
, “A Continuum Method for Modeling Surface Tension
,” J. Comput. Phys.
, 100
, pp. 335
–354
.10.1016/0021-9991(92)90240-Y6.
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
.10.1016/0021-9991(88)90002-27.
Shu
, C.-W.
, and Osher
, S.
, 1988
, “Efficient Implementation of Essentially Non-Oscillatory Shock Capturing Schemes
,” J. Comput. Phys.
, 77
, pp. 439
–471
.10.1016/0021-9991(88)90177-58.
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
.10.1006/jcph.1994.11559.
Chang
, Y. C.
, Hou
, T. Y.
, Merriman
, B.
, and Osher
, S.
, 1996
, “A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows
,” J. Comput. Phys.
, 124
, pp. 449
–464
.10.1006/jcph.1996.007210.
Sussman
, M.
, Fatemi
, E.
, Smerekc
, P.
, and Osher
, S.
, 1998
, “An Improved Level Set Method for Incompressible Two-Phase Flows
,” Comput. Fluids
, 27
pp. 663
–680
.10.1016/S0045-7930(97)00053-411.
Sethian
, J. A.
, 1999
, Level-Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science
, Cambridge University Press
, New York
.12.
Sussman
, M.
, Almgren
, A.
, Bell
, J.
, Colella
, P.
, Howell
, L. H.
, and Welcome
, M.
, 1999
, “An Adaptive Level Set Approach for Incompressible Two-Phase Flows
,” J. Comput. Phys.
, 148
, pp. 81
–124
.10.1006/jcph.1998.610613.
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
.10.1006/jcph.2000.653714.
Kang
, M.
, Fedkiw
, R. P.
, and Liu
, X. D.
, 2000
, “A Boundary Condition Capturing Method for Multiphase Incompressible Flow
,” J. Sci. Comput.
, 15
, pp. 323
–360
.10.1023/A:101117841762015.
Smith
, K. A.
, Solis
, F. J.
, Tao
, L.
, Thornton
, K.
, and De La Cruz
, M. O.
, 2000
, “Domain Growth in Ternary Fluids: A Level Set Approach
,” Phys. Rev. Lett.
, 84
, pp. 91v94
.16.
Osher
, S.
, and Fedkiw
, R. P.
, 2001
, “Level Set Methods: An Overview and Some Recent Results
,” J. Comput. Phys.
, 169
, pp. 463
–502
.10.1006/jcph.2000.663617.
Enright
, D.
, Fedkiw
, R.
, Ferziger
, J.
, and Mitchell
, I.
, 2002
, “A Hybrid Particle Level Set Method for Improved Interface Capturing
,” J. Comput. Phys.
, 183
, pp. 83
–116
.10.1006/jcph.2002.716618.
Smith
, K. A.
, Solis
, F. J.
, and Chopp
, D. L.
, 2002
, “A Projection Method for Motion of Triple Junctions by Level Sets
,” Interfaces Free Bound.
, 4
, pp. 263
–276
.10.4171/IFB/6119.
Osher
, S.
, and Fedkiw
, R. P.
, 2003
, Level Set Methods and Dynamic Implicit Surfaces
, Springer-Verlag
, New York
.20.
Sethian
, J. A.
, and Smereka
, P.
, 2003
, “Level Set Methods for Fluid Interfaces
,” Annu. Rev. Fluid Mech.
, 35
, pp. 341
–372
.10.1146/annurev.fluid.35.101101.16110521.
Xu
, J.-J.
, and Zhao
, H.-K.
, 2003
, “An Eulerian Formulation for Solving Partial Differential Equations Along a Moving Interface
,” J. Sci. Comput.
, 19
, pp. 573
–594
.10.1023/A:102533691617622.
Sussman
, M.
, 2003
, “A Second Order Coupled Level Set and Volume-of-Fluid Method for Computing Growth and Collapse of Vapor Bubbles
,” J. Comput. Phys.
, 187
, pp. 110
–136
.10.1016/S0021-9991(03)00087-123.
Smith
, K. A.
, Ottino
, J. M.
, and de la Cruz
, M. O.
, 2004
, “Encapsulated Drop Breakup in Shear Flow
,” Phys. Rev. Lett.
, 93
, pp. 204501-1
–204501-4
.24.
Olsson
, E.
, and Kreiss
, G.
, 2005
, “A Conservative Level Set Method for Two Phase Flow
,’’ J. Comput. Phys.
, 210
, pp. 225
–246
.10.1016/j.jcp.2005.04.00725.
Majumder
, S.
, and Chakraborty
, S.
, 2005
, “New Physically Based Approach of Mass Conservation Correction in Level Set Formulation for Incompressible Two-Phase Flows
,” ASME J. Fluids Eng.
, 127
, pp. 554
–563
.10.1115/1.189917226.
Tanguy
, S.
, and Berlemont
, A.
, 2005
, “Application of a Level Set Method for Simulation of Droplet Collisions
,” Int. J. Multiphase Flow
, 31
, pp. 1015
–1035
.10.1016/j.ijmultiphaseflow.2005.05.01027.
Enright
, D.
, Losasso
, F.
, and Fedkiw
, R.
, 2005
, “A Fast and Accurate Semi-Lagrangian Particle Level Set Method
,” Comput. Struct.
, 83
, pp. 479
–490
.10.1016/j.compstruc.2004.04.02428.
Shepel
, S. V.
, and Smith
, B. L.
, 2006
, “New Finite-Element/Finite-Volume Level Set Formulation for Modelling Two-Phase Incompressible Flows
,” J. Comput. Phys.
, 218
, pp. 479
–494
.10.1016/j.jcp.2006.02.00829.
Grooss
, J.
, and Hesthaven
, J. S.
, 2006
, “A Level Set Discontinuous Galerkin Method for Free Surface Flows
,” Comput. Meth. Appl. Mech. Eng.
, 195
, pp. 3406
–3429
.10.1016/j.cma.2005.06.02030.
Hong
, J. M.
, Shinar
, T.
, Kang
, M.
, and Fedkiw
, R.
, 2007
, “On Boundary Condition Capturing for Multiphase Interfaces
,” J. Sci. Comput.
, 31
, pp. 99
–125
.10.1007/s10915-006-9120-x31.
Olsson
, E.
, Kreiss
, G.
, and Zahedi
, S.
, 2007
, “A Conservative Level Set Method for Two Phase Flow II
,” J. Comput. Phys.
, 225
, pp. 785
–807
.10.1016/j.jcp.2006.12.02732.
Sussman
, M.
, Smith
, K. M.
, Hussaini
, M. Y.
, Ohta
, M.
, and Zhi-Wei
, R.
, 2007
, “A Sharp Interface Method for Incompressible Two-Phase Flows
,” J. Comput. Phys.
, 221
, pp. 469
–505
.10.1016/j.jcp.2006.06.02033.
Croce
, R.
, Griebel
, M.
, and Schweitzer
, M. A.
, 2010
, “Numerical Simulation of Bubble and Droplet Deformation by a Level Set Approach With Surface Tension in Three Dimensions
,” Int. J. Numer. Meth. Fluids
, 62
, pp. 963
–993
.34.
Ausas
, R. F.
, Dari
, E. A.
, and Buscaglia
, G. C.
, 2011
, “A Geometric Mass-Preserving Redistancing Scheme for the Level Set Function
,” Int. J. Numer. Meth. Fluids
, 65
, pp. 989
–1010
.10.1002/fld.222735.
Pathak
, M.
, 2012
, “Numerical Analysis of Droplet Dynamics Under Different Temperature and Cross-Flow Velocity Conditions
,” ASME J. Fluids Eng.
, 134
, pp. 044501-1
–044501-6
.10.1115/1.400642736.
Starovoitov
, V. N.
, 1994
, “Model of the Motion of a Two-Component Liquid With Allowance of Capillary Forces
,” J. Appl. Mech. Tech. Phys.
, 35
, pp. 891
–897
.10.1007/BF0236958237.
Chella
, R.
, and Viñals
, J.
, 1996
, “Mixing of a Two-Phase Fluid by Cavity Flow
,” Phys. Rev. E
, 53
, pp. 3832
–3840
.10.1103/PhysRevE.53.383238.
Gurtin
, M. E.
, Polignone
, D.
, and Vinals
, J.
, 1996
, “Two-Phase Binary Fluids and Immiscible Fluids Described by an Order Parameter
,” Math. Models Methods Appl. Sci.
, 6
, pp. 815
–831
.10.1142/S021820259600034139.
Anderson
, D. M.
, McFadden
, G. B.
, and Wheeler
, A. A.
, 1998
, “Diffuse-Interface Methods in Fluid Mechanics
,” Annu. Rev. Fluid Mech.
, 30
, pp. 139
–165
.10.1146/annurev.fluid.30.1.13940.
Lowengrub
, J.
, and Truskinovsky
, L.
, 1998
, “Quasi-Incompressible Cahn–Hilliard Fluids and Topological Transitions
,” Proc. Roy. Soc. London Ser. A
, 454
, pp. 2617
–2654
.10.1098/rspa.1998.027341.
Jacqmin
, D.
, 1999
, “Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling
,” J. Comput. Phys.
, 155
, pp. 96
–127
.10.1006/jcph.1999.633242.
Jacqmin
, D.
, 2000
, “Contact-Line Dynamics of a Diffuse Fluid Interface
,” J. Fluid Mech.
, 402
, pp. 57
–88
.10.1017/S002211209900687443.
Verschueren
, M.
, Van de Vosse
, F. N.
, and Meijer
, H. E. H.
, 2001
, “Diffuse-Interface Modelling of Thermocapillary Flow Instabilities in a Hele-Shaw Cell
,” J. Fluid Mech.
, 434
, pp. 153
–166
.10.1017/S002211200100356144.
Lee
, H.-G.
, Lowengrub
, J. S.
, and Goodman
, J.
, 2002
, “Modeling Pinchoff and Reconnection in a Hele-Shaw Cell. I. The Models and Their Calibration
,” Phys. Fluids
, 14
, pp. 492
–513
.10.1063/1.142584345.
Lee
, H.-G.
, Lowengrub
, J. S.
, and Goodman
, J.
, 2002
, “Modeling Pinchoff and Reconnection in a Hele-Shaw Cell. II. Analysis and Simulation in the Nonlinear Regime
,” Phys. Fluids
, 14
, pp. 514
–545
.10.1063/1.142584446.
Boyer
, F.
, 2002
, “A Theoretical and Numerical Model for the Study of Incompressible Mixture Flows
,” Comput. Fluids
, 31
, pp. 41
–68
.10.1016/S0045-7930(00)00031-147.
Badalassi
, V. E.
, Ceniceros
, H. D.
, and Banerjee
, S.
, 2003
, “Computation of Multiphase Systems With Phase Field Models
,” J. Comput. Phys.
, 190
, pp. 371
–397
.10.1016/S0021-9991(03)00280-848.
Liu
, C.
, and Shen
, J.
, 2003
, “A Phase Field Model for the Mixture of Two Incompressible Fluids and Its Approximation by a Fourier-Spectral Method
,” Phys. D
, 179
, pp. 211
–228
.10.1016/S0167-2789(03)00030-749.
Kim
, J.
, Kang
, K.
, and Lowengrub
, J.
, 2004
, “Conservative Multigrid Methods for Cahn-Hilliard Fluids
,” J. Comput. Phys.
, 193
, pp. 511
–543
.10.1016/j.jcp.2003.07.03550.
Sun
, Y.
, and Beckermann
, C.
, 2004
, “Diffuse Interface Modeling of Two-Phase Flows Based on Averaging: Mass and Momentum Equations
,” Phys. D
, 198
, pp. 281
–308
.10.1016/j.physd.2004.09.00351.
Khatavkar
, V. V.
, Anderson
, P. D.
, Duineveld
, P. C.
, and Meijer
, H. H. E.
, 2005
, “Diffuse Interface Modeling of Droplet Impact on a Pre-Patterned Solid Surface
,” Macromol. Rapid Commun.
, 26
, pp. 298
–303
.10.1002/marc.20040047852.
Kim
, J.
, 2005
, “A Diffuse-Interface Model for Axisymmetric Immiscible Two-Phase Flow
,” Appl. Math. Comput.
, 160
, pp. 589
–606
.10.1016/j.amc.2003.11.02053.
Yue
, P.
, Feng
, J. J.
, Liu
, C.
, and Shen
, J.
, 2005
, “Diffuse-Interface Simulations of Drop Coalescence and Retraction in Viscoelastic Fluids
,” J. Non-Newton. Fluid Mech.
, 129
, pp. 163
–176
.10.1016/j.jnnfm.2005.07.00254.
Kim
, J.
, 2005
, “A Continuous Surface Tension Force Formulation for Diffuse-Interface Models
,” J. Comput. Phys.
, 204
, pp. 784
–804
.10.1016/j.jcp.2004.10.03255.
Badalassi
, V. E.
, and Banerjee
, S.
, 2005
, “Nano-Structure Computation With Coupled Momentum Phase Ordering Kinetics Models
,” Nucl. Eng. Des.
, 235
, pp. 1107
–1115
.10.1016/j.nucengdes.2005.02.00856.
Yang
, X.
, Feng
, J. J.
, Liu
, C.
, and Shen
, J.
, 2006
, “Numerical Simulations of Jet Pinching-Off and Drop Formation Using an Energetic Variational Phase-Field Method
,” J. Comput. Phys.
, 218
, pp. 417
–428
.10.1016/j.jcp.2006.02.02157.
Yue
, P.
, Zhou
, C.
, and Feng
, J. J.
, 2007
, “Spontaneous Shrinkage of Drops and Mass Conservation in Phase-Field Simulations
,” J. Comput. Phys.
, 223
, pp. 1
–9
.10.1016/j.jcp.2006.11.02058.
Ding
, H.
, and Spelt
, P. D. M.
, 2007
, “Wetting Condition in Diffuse Interface Simulations of Contact Line Motion
,” Phys. Rev. E
, 75
, p. 046708
.59.
Khatavkar
, V. V.
, Anderson
, P. D.
, Duineveld
, P. C.
, and Meijer
, H. H. E.
, 2007
, “Diffuse-Interface Modelling of Droplet Impact
,” J. Fluid Mech.
, 581
, pp. 97
–127
.10.1017/S002211200700554X60.
He
, Q.
, and Kasagi
, N.
, 2008
, “Phase-Field Simulation of Small Capillary-Number Two-Phase Flow in a Microtube
,” Fluid Dyn. Res.
, 40
, pp. 497
–509
.10.1016/j.fluiddyn.2008.01.00261.
Kim
, J.
, 2009
, “A Generalized Continuous Surface Tension Force Formulation for Phase-Field Models for Multi-Component Immiscible Fluid Flows
,” Comput. Methods Appl. Mech. Eng.
, 198
, pp. 3105
–3112
.10.1016/j.cma.2009.05.00862.
Shen
, J.
, and Yang
, X.
, 2009
, “An Efficient Moving Mesh Spectral Method for the Phase-Field Model of Two-Phase Flows
,” J. Comput. Phys.
, 228
, pp. 2978
–2992
.10.1016/j.jcp.2009.01.00963.
Acar
, R.
, 2009
, “Simulation of Interface Dynamics: A Diffuse-Interface Model
,” Visual Comput.
, 25
, pp. 101
–115
.10.1007/s00371-008-0208-164.
Ceniceros
, H. D.
, Nós
, R. L.
, and Roma
, A. M.
, 2010
, “Three-Dimensional, Fully Adaptive Simulations of Phase-Field Fluid Models
,” J. Comput. Phys.
, 229
, pp. 6135
–6155
.10.1016/j.jcp.2010.04.04565.
Chiu
, P. H.
, and Lin
, Y. T.
, 2011
, “A Conservative Phase Field Method for Solving Incompressible Two-Phase Flows
,” J. Comput. Phys.
, 230
, pp. 185
–204
.10.1016/j.jcp.2010.09.02166.
Kim
, J.
, 2012
, “Phase-Field Models for Multi-Component Fluid Flows
,” Commun. Comput. Phys.
, 12
, pp. 613
–661
.67.
Lee
, H. G.
, Choi
, J. W.
, and Kim
, J.
, 2012
, “A Practically Unconditionally Gradient Stable Scheme for the N-Component Cahn–Hilliard System
,” Physica A
, 391
, pp. 1009
–1019
.10.1016/j.physa.2011.11.03268.
Peskin
, C. S.
, 1977
, “Numerical Analysis of Blood Flow in the Heart
,” J. Comput. Phys.
, 25
, pp. 220
–252
.10.1016/0021-9991(77)90100-069.
Peskin
, C. S.
, and McQueen
, D. M.
, 1980
, “Modeling Prosthetic Heart Valves for Numerical Analysis of Blood Flow in the Heart
,” J. Comput. Phys.
, 37
, pp. 113
–132
.10.1016/0021-9991(80)90007-870.
Unverdi
, S. O.
, and Tryggvason
, G.
1992
, “A Front-Tracking Method for Viscous, Incompressible, Multi-Fluid Flows
,” J. Comput. Phys.
, 100
, pp. 25
–37
.10.1016/0021-9991(92)90307-K71.
Peskin
, C. S.
, and Printz
, B. F.
, 1993
, “Improved Volume Conservation in the Computation of Flows With Immersed Elastic Boundaries
,” J. Comput. Phys.
, 105
, pp. 33
–46
.10.1006/jcph.1993.105172.
Sheth
, K. S.
, and Pozrikidis
, C.
, 1995
, “Effects of Inertia on the Deformation of Liquid Drops in Simple Shear Flow
,” Comput. Fluids
, 24
, pp. 101
–119
.10.1016/0045-7930(94)00025-T73.
Stockie
, J. M.
, 1997
, “Analysis and Computation of Immersed Boundaries, With Application to Pulp Fibres
,” Ph.D. thesis, University of British Columbia, British Columbia, Canada.74.
Udaykumar
, H. S.
, Kan
, H. C.
, Shyy
, W.
, and Tran-Son-Tay
, R.
, 1997
, “Multiphase Dynamics in Arbitrary Geometries on Fixed Cartesian Grids
,” J. Comput. Phys.
, 137
, pp. 366
–405
.10.1006/jcph.1997.580575.
Kan
, H. C.
, Udaykumar
, H. S.
, Shyy
, W.
, and Tran-Son-Tay
, R.
, 1998
, “Hydrodynamics of a Compound Drop With Application to Leukocyte Modeling
,” Phys. Fluids.
, 10
, pp. 760
–774
.10.1063/1.86960176.
Roma
, A. M.
, Peskin
, C. S.
, and Berger
, M. J.
, 1999
, “An Adaptive Version of the Immersed Boundary Method
,” J. Comput. Phys.
, 153
, pp. 509
–534
.10.1006/jcph.1999.629377.
Kan
, H. C.
, Shyy
, W.
, Udaykumar
, H. S.
, Vigneron
, P.
, and Tran-Son-Tay
, R.
, 1999
, “Effects of Nucleus on Leukocyte Recovery
,” Ann. Biomed. Eng.
, 27
, pp. 648
–655
.10.1114/1.21478.
Peskin
, C. S.
, 2002
, “The Immersed Boundary Method
,” Acta Numer.
, 11
, pp. 479
–517
.10.1017/S096249290200007779.
Francois
, M.
, and Shyy
, W.
, 2003
, “Computations of Drop Dynamics With the Immersed Boundary Method, Part 1: Numerical Algorithm and Buoyancy-Induced Effect
,” Numer. Heat Tran. B
, 44
, pp. 101
–118
.10.1080/71383634780.
Francois
, M.
, Uzgoren
, E.
, Jackson
, J.
, and Shyy
, W.
, 2004
, “Multigrid Computations With the Immersed Boundary Technique for Multiphase Flows
,” Int. J. Numer. Meth. Heat Fluid Flow
, 14
, pp. 98
–115
.10.1108/0961553041051165881.
Mittal
, R.
, and Iaccrino
, G.
, 2005
, “Immersed Boundary Methods
,” Annu. Rev. Fluid Mech.
, 37
, pp. 239
–261
.10.1146/annurev.fluid.37.061903.17574382.
Uhlmann
, M.
, 2005
, “An Immersed Boundary Method With Direct Forcing for the Simulation of Particulate Flows
,” J. Comput. Phys.
, 209
, pp. 448
–476
.10.1016/j.jcp.2005.03.01783.
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
, pp. 75
–105
.10.1016/j.jcp.2005.02.01184.
Griffith
, B. E.
, Hornung
, R. D.
, McQueen
, D. M.
, and Peskin
, C. S.
, 2005
, “An Adaptive, Formally Second Order Accurate Version of the Immersed Boundary Method
,” J. Comput. Phys.
, 223
, pp. 10
–49
.10.1016/j.jcp.2006.08.01985.
Newren
, E.
, Fogelson
, A. L.
, Guy
, R. D.
, and Kirby
, R. M.
, 2007
, “Unconditionally Stable Discretizations of the Immersed Boundary Equations
,” J. Comput. Phys.
, 222
, pp. 702
–719
.10.1016/j.jcp.2006.08.00486.
Shin
, S. J.
, Huang
, W.-X.
, and Sung
, H. J.
, 2008
, “Assessment of Regularized Delta Functions and Feedback Forcing Schemes for an Immersed Boundary Method
,” Int. J. Numer. Meth. Fluids
, 58
, pp. 263
–286
.10.1002/fld.170687.
Kim
, Y.
, and Peskin
, C. S.
, 2008
, “Numerical Study of Incompressible Fluid Dynamics With Nonuniform Density by the Immersed Boundary Method
,” Phys. Fluids
, 20
, p. 062101
.88.
Yang
, X.
, Zhang
, X.
, Li
, Z.
, and He
, G.-W.
, 2009
, “A Smoothing Technique for Discrete Delta Functions With Application to Immersed Boundary Method in Moving Boundary Simulations
,” J. Comput. Phys.
, 228
, pp. 7821
–7836
.10.1016/j.jcp.2009.07.02389.
Chen
, K. Y.
, Feng
, K. A.
, Kim
, Y.
, and Lai
, M. C.
, 2011
, “A Note on Pressure Accuracy in Immersed Boundary Method for Stokes Flow
,” J. Comput. Phys.
, 230
, pp. 4377
–4383
.10.1016/j.jcp.2011.03.01990.
Li
, Y.
, Jung
, E.
, Lee
, W.
, Lee
, H. G.
, and Kim
, J.
, 2012
, “Volume Preserving Immersed Boundary Methods for Two-Phase Fluid Flows
,” Int. J. Numer. Methods Fluids
, 69
, pp. 842
–858
.10.1002/fld.261691.
Li
, Y.
, Yun
, A.
, Lee
, D.
, Shin
, J.
, Jeong
, D.
, and Kim
, J.
, 2013
, “Three-Dimensional Volume-Conserving Immersed Boundary Model for Two-Phase Fluid Flows
,” Comput. Meth. Appl. Mech. Eng.
, 257
, pp. 36
–46
.10.1016/j.cma.2013.01.00992.
Cahn
, J. W.
, and Hilliard
, J. E.
, 1958
, “Free Energy of a Nonuniform System. I. Interfacial Free Energy
,” J. Chem. Phys.
, 28
, 258
–267
.10.1063/1.174410293.
LeVeque
, R. J.
, and Li
, Z.
, 1994
, “The Immersed Interface Method for Elliptic Equations With Discontinuous Coefficients and Singular Sources
,” SIAM J. Numer. Anal.
, 31
, pp. 1019
–1044
.10.1137/073105494.
LeVeque
, R. J.
, and Li
, Z.
, 1997
, “Immersed Interface Methods for Stokes Flow With Elastic Boundaries or Surface Tension
,” SIAM J. Sci. Comput.
, 18
, pp. 709
–735
.10.1137/S106482759528253295.
Berthelsen
, P. A.
, 2004
, “A Decomposed Immersed Interface Method for Variable Coefficient Elliptic Equations With Non-Smooth and Discontinuous Solutions
,” J. Comput. Phys.
, 197
, pp. 364
–386
.10.1016/j.jcp.2003.12.00396.
Li
, Z.
, Ito
, K.
, and Lai
, M. C.
, 2007
, “An Augmented Approach for Stokes Equations With a Discontinuous Viscosity and Singular Forces
,” Comput. Fluids
, 36
, pp. 622
–635
.10.1016/j.compfluid.2006.03.00397.
Ye
, T.
, Mittal
, R.
, Udaykumar
, H. S.
, and Shyy
, W.
, 1999
, “An Accurate Cartesian Grid Method for Viscous Incompressible Flows With Complex Immersed Boundaries
,” J. Comput. Phys.
, 156
, pp. 209
–240
.10.1006/jcph.1999.635698.
Fadlun
, E. A.
, Verzicco
, R.
, Orlandi
, P.
, and Mohd-Yusof
, J.
, 2000
, “Combined Immersed-Boundary Finite-Difference Methods for Three-Dimensional Complex Flow Simulations
,” J. Comput. Phys.
, 161
, pp. 35
–60
.10.1006/jcph.2000.648499.
Deng
, J.
, Shao
, X. M.
, and Ren
, A. L.
, 2006
, “A New Modification of the Immersed-Boundary Method for Simulating Flows With Complex Moving Boundaries
,” Int. J. Numer. Meth. Fluids
, 52
, pp. 1195
–1213
.10.1002/fld.1237100.
Sheu
, T. W. H.
, Ting
, H. F.
, and Lin
, R. K.
, 2008
, “An Immersed Boundary Method for the Incompressible Navier–Stokes Equations in Complex Geometry
,” Int. J. Numer. Meth. Fluids
, 56
, pp. 877
–898
.10.1002/fld.1558101.
Shyam Kumar
, M. B.
, and Vengadesan
, S.
, 2012
, “Influence of Rounded Corners on Flow Interference Due to Square Cylinders Using Immersed Boundary Method
,” ASME J. Fluids Eng.
, 134
, pp. 091203-1
–091203-23
.10.1115/1.4007015102.
Harlow
, F. H.
, and Welch
, J. E.
, 1965
, “Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid With Free Surface
, Phys. Fluids.
, 8
, pp. 2182
–2189
.10.1063/1.1761178103.
Chorin
, A. J.
, 1968
, “Numerical Solution of the Navier–Stokes Equations
,” Math. Comput.
, 22
, pp. 745
–762
.10.1090/S0025-5718-1968-0242392-2104.
Trottenberg
, U.
, Oosterlee
, C. W.
, and Schüller
, A.
, 2001
, Multigrid
, Academic Press
, New York.105.
Eyre
, D. J.
, 1998
, “Unconditionally Gradient Stable Time Marching the Cahn–Hilliard Equation
,” Mater. Res. Soc. Symp. Proc.
, 529
, pp. 39
–46
.10.1557/PROC-529-39106.
Kim
, J. S.
, and Bae
, H. O.
, 2008
, “An Unconditionally Gradient Stable Adaptive Mesh Refinement for the Cahn–Hilliard Equation
,” J. Korean Phys. Soc.
, 53
, pp. 672
–679
.10.3938/jkps.53.672Copyright © 2014 by ASME
You do not currently have access to this content.
