Interface dynamics of two-phase flow, with relevance for leakage of oil retained by mechanical oil barriers, is studied by means of a two-dimensional (2D) lattice-Boltzmann method (LBM) combined with a phase-field model for interface capturing. A multirelaxation-time (MRT) model of the collision process is used to obtain a numerically stable model at high Reynolds number flow. In the phase-field model, the interface is given a finite but small thickness, where the fluid properties vary continuously across a thin interface layer. Surface tension is modeled as a volume force in the transition layer. The numerical model is implemented for simulations with the graphic processing unit (GPU) of a desktop personal computer. Verification tests of the model are presented. The model is then applied to simulate gravity currents (GCs) obtained from a lock-exchange configuration, using fluid parameters relevant for those of oil and water. Interface instability phenomena are observed, and obtained numerical results are in good agreement with theory. This work demonstrates that the numerical model presented can be used as a numerical tool for studies of stratified shear flows with relevance to oil-boom failure.

References

References
1.
Fingas
,
M.
,
2001
,
The Basics of Oil Spill Cleanup
,
2 ed.
,
CRC Press LCC
,
Boca Raton
, Chap. 6.
2.
Delvigne
,
G. A. L.
,
1989
, “
Barrier Failure by Critical Accumulation of Viscous Oil
,”
Int. Oil Spill Conf. Proc.
,
1989
(
1
), pp.
143
148
10.7901/2169-3358-1989-1-143.
3.
Leibovich
,
S.
,
1976
, “
Oil Slick Instability and Entrainment Failure of Oil Containment Booms
,”
ASME J. Fluids Eng.
,
98
(
1
), pp.
98
103
.10.1115/1.3448229
4.
Thorpe
,
S. A.
,
1969
, “
Experiments on the Instability of Stratified Shear Flows: Immiscible Fluids
,”
J. Fluid Mech.
,
39
(1), pp.
25
48
.10.1017/S0022112069002023
5.
Holmboe
,
J.
,
1962
, “
On the Behavior of Symmetric Waves in Stratified Shear Layers
,”
Geophys. Publ.
,
24
, pp.
67
113
.
6.
Browand
,
F. K.
, and
Winant
,
C. D.
,
1973
, “
Laboratory Observations of Shear-Layer Instability in a Stratified Fluid
,”
Boundary-Layer Meteorol.
,
5
(
1–2
), pp.
67
77
.10.1007/BF02188312
7.
Lawrence
,
G. A.
,
Browand
,
F. K.
, and
Redekopp
,
L. G.
,
1991
, “
The Stability of a Sheared Density Interface
,”
Phys. Fluids
,
A3
(
10
), pp.
2360
2370
10.1063/1.858175.
8.
Pouliquen
,
O.
,
Chomaz
,
J.
, and
Huerre
,
P.
,
1994
, “
Propagating Holmboe Waves at the Interface Between Two Immiscible Fluids
,”
J. Fluid Mech.
,
266
, pp.
277
302
.10.1017/S002211209400100X
9.
Smyth
,
W. D.
, and
Moum
,
J. N.
,
2000
, “
Length Scales of Turbulence in Stably Stratified Mixing Layers
,”
Phys. Fluids
,
12
(
6
), pp.
1327
1342
.10.1063/1.870385
10.
Grilli
,
S. T.
,
Hu
,
Z.
,
Spalding
,
M. L.
, and
Liang
,
D.
,
1997
, “
Numerical Modeling of Oil Containment by a Boom/Barrier System: Phase II
,” Department of Ocean Engineering, University of Rhode Island, Kingston, RI.
11.
He
,
X.
, and
Luo
,
L.-S.
,
1997
, “
Lattice Boltzmann Model for the Incompressible Navier–Stokes Equation
,”
J. Stat. Phys.
,
88
(
3–4
), pp.
927
944
.10.1023/B:JOSS.0000015179.12689.e4
12.
Bhatnagar
,
P. L.
,
Gross
,
E. P.
, and
Krook
,
M.
,
1954
, “
A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems
,”
Phys. Rev.
,
94
(
3
), pp.
511
525
.10.1103/PhysRev.94.511
13.
Lallemand
,
P.
, and
Luo
,
L.-S.
,
2000
, “
Theory of the Lattice Boltzmann Method: Dispersion, Dissipation, Isotropy, Galilean Invariance, and Stability
,” NASA Langley Research Center, Hampton, VA, Technical Report No. 2000-17.
14.
Brackbill
,
J. U.
,
Kothe
,
D. B.
, and
Zemack
,
C.
,
1992
, “
A Continuum Method for Modeling Surface Tension
,”
J. Comput. Phys.
,
100
(
2
), pp.
335
354
.10.1016/0021-9991(92)90240-Y
15.
Jacqmin
,
D.
,
1999
, “
Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling
,”
J. Comput. Phys.
,
155
(
1
), pp.
96
127
.10.1006/jcph.1999.6332
16.
Cahn
,
J. W.
, and
Hilliard
,
J. E.
,
1958
, “
Free Energy of a Nonuniform System. I. Interfacial Free Energy
,”
J. Chem. Phys.
,
28
(
2
), pp.
258
267
.10.1063/1.1744102
17.
Huang
,
J. J.
,
Shu
,
C.
, and
Chew
,
Y. T.
,
2009
, “
Mobility-Dependent Bifurcations in Capillarity-Driven Two-Phase Fluid Systems by Using a Lattice Boltzmann Phase-Field Model
,”
Int. J. Numer. Methods Fluids
,
60
(
2
), pp.
203
225
.10.1002/fld.1885
18.
Fakhari
,
A.
, and
Rahimian
,
M. H.
,
2010
, “
Phase-Field Modeling by the Method of Lattice Boltzmann Equations
,”
Phys. Rev. E
,
81
(
3 Pt. 2
), p.
036707
.10.1103/PhysRevE.81.036707
19.
Shao
,
J. Y.
,
Shu
,
C.
, and
Chew
,
Y. T.
,
2012
, “
A Hybrid Phase-Field Based Lattice-Boltzmann Method for Contact Line Dynamics
,”
Int. J. Mod. Phys.: Conf. Ser.
,
19
, pp.
50
61
10.1142/S2010194512008586.
20.
He
,
X.
,
Shan
,
X.
, and
Doolen
,
G.
,
1998
, “
Discrete Boltzmann Equation Model for Nonideal Gases
,”
Phys. Rev.
,
57
(
1
), pp.
13
16
10.1103/PhysRevE.57.R13.
21.
He
,
X.
,
Chen
,
S.
, and
Zhang
,
R.
,
1999
, “
A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh-Taylor Instability
,”
J. Comput. Phys.
,
152
(
2
), pp.
642
663
.10.1006/jcph.1999.6257
22.
He
,
X.
, and
Luo
,
L.-S.
,
1997
, “
Theory of the Lattice Boltzmann Method: From the Boltzmann Equation to the Lattice Boltzmann Equation
,”
Phys. Rev.
,
56
(
6
), pp.
6811
6817
10.1103/PhysRevE.56.6811.
23.
Pooley
,
C. M.
, and
Furtado
,
K.
,
2008
, “
Eliminating Spurious Velocities in the Free-Energy Lattice Boltzmann Method
,”
Phys. Rev. E
,
77
(
4
), p.
046702
.10.1103/PhysRevE.77.046702
24.
Schlichting
,
H.
, and
Gersten
,
K.
,
2000
,
Boundary Layer Theory
,
Springer
,
Berlin
, pp.
126
128
10.1007/978-3-642-85829-1.
25.
Lamb
,
H.
,
1932
,
Hydrodynamics
,
6th ed.
,
Dover Publications
,
New York
, pp.
461
462
.
26.
Lowe
,
R. J.
,
Rottman
,
J. W.
, and
Linden
,
P. F.
,
2005
, “
The Non-Boussinesq Lock-Exchange Problem. Part 1. Theory and Experiments
,”
J. Fluid Mech.
,
537
, pp.
101
124
.10.1017/S0022112005005069
27.
Härtel
,
C.
,
Eckart
,
M.
, and
Necker
,
F.
,
2000
, “
Analysis and Direct Numerical Simulation of the Flow at a Gravity-Current Head. Part 1. Flow Topology and Front Speed for Slip and No-Slip Boundaries
,”
J. Fluid Mech.
,
418
, pp.
189
212
.10.1017/S0022112000001221
28.
Benjamin
,
T. B.
,
1968
, “
Gravity Currents and Related Phenomena
,”
J. Fluid Mech.
,
31
(
Pt. 2
), pp.
209
248
.10.1017/S0022112068000133
29.
Marino
,
B. M.
,
Thomas
,
L. P.
, and
Linden
,
P. F.
,
2005
, “
The Front Condition for Gravity Currents
,”
J. Fluid Mech.
,
536
, pp.
49
78
.10.1017/S0022112005004933
You do not currently have access to this content.