numerical study of interaction between internal solitary
TRANSCRIPT
Numerical Study of Interaction between Internal Solitary Waves and Uniform Slope by MPS Method
Xiao Wen1, Decheng Wan1*, Changhong Hu2
1 Computational Marine Hydrodynamics Lab (CMHL), State Key Laboratory of Ocean Engineering,
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China 2 Research Institute for Applied Mechanics, Kyushu University, Fukuoka, Japan
*Corresponding author
ABSTRACT
In present study, the interaction between internal solitary waves (ISWs)
and a uniform slope is numerically studied based on the moving
particles semi-implicit (MPS) method. Aiming at the stratified flows, a
multiphase MPS method is firstly developed on the basis of the original
single-phase MPS method, by introducing a series of special interface
treatments. These treatments include the density smoothing scheme,
inter-particle viscosity model and surface tension model. Then, the
multiphase MPS method is validated through the numerical simulation
of the ISWs generation and propagation. The results obtained by
present method achieve good agreement with the experimental data and
other numerical methods on the interfacial displacements and wave
profiles. Moreover, the calculated pressure and velocity fields are
smooth and reflect the actual physical features. Finally, the multiphase
MPS method is applied to investigate the ISWs interaction with the
submarine slopes with different inclinations, and a qualitatively
analysis of the different breaking mechanisms is performed.
KEY WORDS: MPS; Multiphase flows; Interfacial solitary
wave; Slope; Interaction; Wave breaking
INTRODUCTION
Due to the rapid changes of temperature and salinity, the density
stratification phenomena have been frequently observed in the ocean
environment and tend to cause some complex internal flow
characteristics. When the stable stratification of the ocean is altered by
any disturbance, the internal waves with large amplitudes may be
induced. Especially, the internal solitary waves (ISWs) are likely to
form in the stratified zone when the tidal currents flow past the bottom
topography. As an evolution of the internal gravity waves (IGWs), the
ISWs have an important role in the ocean and coastal engineering.
When the ISWs propagate along the continental shelf/slope, violent
interactions are possible to happen between the ISWs and the
shelf/slope, which bring complicate influence to the stable profiles of
the ISWs and even result in the breaking of the ISWs in some particular
conditions. Moreover, the interaction can further induce the suspension
and redistribution of the bed materials, which has great affects on the
seabed stability.
Over the past several decades, many researches focusing on the
interaction between ISWs and the continental shelf/slope have been
implemented by both experimental and numerical methods. Helfrich
(1992) conducted a study on the shoaling of ISWs of depression in a two-
layer system on a uniform slope and described the kinematics of the
breaking of the ISWs. In the research of Cheng et al. (2009, 2011), both
numerical approach and laboratory experiments are employed to study
the evolution of an ISW of depression type propagating over double
triangular obstacles and a trapezoidal cross-section representing a shelf-
slope condition, respectively. Xu et al. (2018) performed a direct
numerical simulation to study the scalar transport induced by the ISWs of
depression propagating over slope-shelf topography, and it is found that
the ISW-induced scalar transport consists of four stages, including the
slip transport, the wash transport, the vortex transport, and the secondary
transport. La Forgia et al. (2018) investigated the ISWs with large
amplitude by laboratory experiments, in which the empirical relations
between the initial setting parameters and the generated ISWs kinematic
and geometric main features are developed, and the different breaking
mechanisms of ISWs are developed.
In general, a certain amount of achievements have been made in the
field of interaction between ISWs and the slope. However, when the
interaction become violent, the flow field may greatly change and lead
to severe deformation of the two-phase interface of the layered system,
which bring great challenges to the traditional grid-based methods. For
example, additional functions, such as Volume of Fluid (VOF) and
Level-Set methods, are necessary to be carried out in every time step to
capture the deformation of phase interface. Fortunately, in recent years,
the development of mesh-free particle methods (Gingold and
Monaghan, 1977; Lucy, 1977; Koshizuka and Oka, 1996; Koshizuka et
al., 1998) provides an alternative approach to overcome the limitations
of the grid-based approaches. In mesh-free particle methods, the
calculation domain is discretized by a set of unordered particles which
1986
Proceedings of the Thirtieth (2020) International Ocean and Polar Engineering ConferenceShanghai, China, October 11-16, 2020Copyright © 2020 by the International Society of Offshore and Polar Engineers (ISOPE)ISBN 978-1-880653-84-5; ISSN 1098-6189
www.isope.org
can freely move according to the governing equations. Therefore, the
phase interface can be directly obtained from the particles distribution,
without the need of special capturing algorithm. Therefore, the large
interface deformation and fragmentations in the process of ISWs-slope
interactions can be better handled. Besides, the movement of particles
follows the Navier-Stokes equation in Lagrangian description, thus the
numerical diffusion in the discretization of advection term can be
eliminated by use of the substantial derivative.
The aim of this paper is to employ one of the most popular mesh-free
particle methods, moving particle semi-implicit (MPS) method, to
numerically investigate the interaction between ISWs and a uniform
slope. The MPS method is originally proposed by Koshizuka and Oka
(1996) for fully incompressible flows. Different from another widely
used mesh-less method, the smoothed particle hydrodynamics (SPH)
method, MPS method adopts a semi-implicit algorithm and the pressure
field is solved by the Poisson Pressure Equation (PPE). Therefore, the
numerical stability of MPS method can be better guaranteed, and seems
to be more suitable for the multiphase simulations where instability
may be caused by the discontinuity across phase interface. However,
due to the lack of multiphase model, the MPS was mainly used to solve
single-phase flow problems with large deformations of free surface in
its early stage of development (Khayyer and Gotoh, 2008; Lee et al.,
2011a; Lee et al., 2011b; Zhang and Wan, 2017). In recent years, various
multiphase MPS methods are gradually been developed and the
advantages of MPS on multiphase simulation begin to be exploited
(Shakibaeinia and Jin, 2012; Khayyer and Gotoh, 2013; Duan et al.,
2017). However, until now, the applications of MPS method on ISWs
and its interactions with different submarine topographies are still
rarely found.
Therefore, in present paper, a multiphase MPS method is firstly
developed on the basis of the original single-phase MPS method by
introducing a series of special interface treatments, including the
density smoothing scheme, inter-particle viscosity model and surface
tension model. Then, the generation and propagation of internal solitary
waves are simulated and compared with experimental date and other
numerical results in open literature, to examine the accuracy and
stability of present multiphase MPS method in stratified flows
problems. With the multiphase MPS method validated, it is finally
applied to investigate the ISWs interaction with the submarine slopes
with different inclinations, and a qualitatively analysis of the different
breaking mechanisms is performed.
Multiphase MPS Method
Governing Equations
In present multiphase MPS method, the density stratified system is
treated as a single-fluid system with multi-density and multi-viscosity,
and the governing equations for different fluids have a uniform
expression, including the continuity and momentum equations as
follow:
( )D
Dtu
(1)
+ V B SDP
Dt
uF F F (2)
where , u, and P are the density, velocity and pressure, respectively.
FV, FB and FS denote the viscosity, external body, and surface tension
forces, respectively.
Particle Interaction Models
In MPS method, the differential operators in governing equations are
discretized by particle interaction models, including gradient,
divergence and Laplacian models, defined as
0 2( ) (| |, )
| |
j i
i j i j i e
j i j i
DW r
nr r r r
r r
(3)
2
(| |, )| |
j i
i j i j i e0j i j i
DW r
n
Φ ΦΦ r r r r
r r
(4)
2
0
2( ) (| |, )i j i j i e
j i
DW r
nr r
(5)
where ϕ is an arbitrary scalar function, Ф is an arbitrary vector, D is the
number of space dimensions, n0 is the particle number density at initial
arrangement, λ is a parameter defined as
2(| |, ) | |
(| |, )
j i e j i
j i
j i e
j i
W r
W r
r r r r
r r
(6)
which is applied to keep the variance increase equal to that of the
analytical solution. The W(|rj―ri|, re) is the kernel function applied to
determine the strength of particle interaction with respect to particle
distance. In present paper, the improved kernel function presented by
Zhang and Wan (2017) is applied:
- 1 (0 )
0.85 0.15,
0 ( )
ee
ee
e
rr r
r rW r r
r r
(7)
where r and re represent the particle distance and the support radius of
the particle interaction, respectively.
Semi-Implicit Algorithm
In MPS method, the incompressibility of fluid is guaranteed by keeping
particle number density equal to its initial value:
0(| |, )i j i e
j i
n W r nr r
(8)
In order to ensure that the above equation holds true during the
simulation, a semi-implicit algorithm (Koshizuka and Oka, 1996) is
adopted, in which each time step is subdivided into two steps. The first
step, also called prediction step, is an explicit calculation where a
temporal velocity field is obtained based on the viscosity, gravity, and
surface tension forces. Then, a correction step is performed, where the
pressure force is calculated by solving the Poisson Pressure Equation
(PPE) and used to update the particle velocity and location implicitly.
In this paper, the improved PPE with a mixed source term, which is
proposed by Tanaka and Masunaga (Tanaka and Masunaga, 2010) and
rewritten by Lee et al. (2011b), is employed:
* 0
2 1 *
2 0(1 )n i
i i
n nP
t t nu
(9)
1987
where γ is a blending parameter with a value less than 1. The range of
0.01≤ γ ≤0.05 is recommended according to the numerical tests
implemented by Lee et al. (2011b).
Multi-Density and Multi-Viscosity Models
For multiphase flows, the mathematical discontinuity of density and
viscosity at phase interface causes a discontinuous acceleration field
and accordingly numerical instabilities. To deal with this discontinuity,
a multi-density model based on density smoothing technique is firstly
employed for particles near interface, based on a simple spatial
averaging as follow:
(| |, )
(| |, )
j j i e
j I
i
j i e
j I
W r
W r
r r
r r
(10)
where I includes the target particle i and all its neighboring particles.
As for the multi-viscosity model, a mutual inter-particle viscosity is
defined to tackle the viscosity discontinuity across the interface.
According to the numerical tests of Shakibaeinia and Jin (2012), the
inter-particle viscosity with a harmonic mean form presents the most
compatible results with the analytical solutions, defined as
2 i j
ij
i j
(11)
Using the above inter-particle viscosity, the viscous force term in Eq. 2
can be discretized as
2
0
2= ( ) (| |, )V
ij j i j i e
j i
DW r
nF u u u r r
(12)
Surface Tension Model
The surface tension effects play an important role on multiphase flows,
especially for the accurate capturing of phase interface deformations.
The surface tension force has a great influence on the simulation of
some multiphase flows, such as the bubble dynamics problems. And for
the multiphase problems where the surface tension force is less
dominant, such as the present study, the surface tension force is still
beneficial to maintain a natural and clear phase interface and prevent
the unphysical inter-penetration of particles belonging to different
phases. In present method, the continuum surface force (CSF) method
proposed by Brackbill et al. (1992) is used, in which surface tension
force is converted into a body force based on the following equation:
-S CF (13)
Where σ represents the surface tension coefficient, κ is the interface
curvature, and C is the gradient of a color function, which is defined
as
0 if particle belongs to the specified phase
1 if particle belongs to the other phasei
iC
i
(14)
The most important parameter of interface tension model is the
interface curvature κ. In present study, the contoured continuum surface
force (CCSF) model developed by Duan et al. (2015) is used. The main
ideal of CCSF model is to approximate the phase interface by the use of
the contours of smoothed color function. In the first step, the smoothed
color function f at an arbitrary location (x, y) is obtained by performing
a spatial weighted averaging through the implementation of a Gaussian
kernel function G:
s
s
( , ),
( , )
j jj
jj
C G rf x y
G r
r r
r r
(15)
2
s 2 2
99( , ) exp
ij
ij
s s
rG r r
r r
(16)
where rs represents the effect radius of the Gaussian kernel function.
Once the smoothed color function is known, the local contour passing
through particle i can be obtained through a Taylor series expansion.
According to Duan et al. (2015), the equation of the local contours can
be derived as
2
, , ,
2
, ,
1( ) ( ) ( ) +
2
1 ( )( )+ ( ) 0
2
x i i y i i xx i i
xy i i i yy i i
f x x f y y f x x
f x x y y f y y
(17)
where the subscripts x and y represent the partial derivatives with
respect to x and y at particle i, respectively.
Finally, the interface curvature at particle i can be analytically
calculated as
2 2'', , , , , , ,
3/2 3/2' 2 2
, ,
2
1
x i y i xy i x i yy i y i xx i
i
i x i y i
f f f f f f fy
y f f
(18)
Boundary Conditions
In MPS method, the kinematic boundary condition is automatically
satisfied in Lagrangian method, while the dynamic free surface
boundary condition is implemented by setting zero pressure on the free
surface particles. To impose the dynamic free surface boundary
condition, free surface particles should be detected firstly. The
improved detection technique proposed by Zhang and Wan (2017) is
employed in this paper. In this method, a function based on the
asymmetric arrangement of neighboring particles of the center particle
is defined as:
(| |, )ij
i j i e
j i ij
DW r
n r
rr r
F (19)
Particles satisfying
0
0.9i F F (20)
are judged as free surface particles, where 0
F is the value at initial
arrangement.
As for wall boundary condition, no-slip condition is imposed by
introducing mirror particles. At each time step, locations and velocities
of the mirror particles are rearranged according to corresponding fluid
particles near the wall. The locations of mirror particles are
1988
symmetrical to corresponding fluid particles about the wall, and their
velocities are decided as
(2 )mirror w iu t u u t , (2 )mirror w iu n u u n (21)
where n and t are the normal and tangential vectors to the wall
boundary, respectively. The subscripts mirror, w and i represent the
mirror, corresponding fluid and wall particles, respectively.
NUMERICAL SIMULATION
Validation of the Multiphase MPS Method
To validate its effectiveness, the multiphase MPS method presented in
this paper is firstly employed to simulate the generation and
propagation of internal solitary waves. For comparison purposes, the
numerical model adopted in this section keeps completely consistent
with the experiment set-ups of Kodaira et al. (2006) and the one chosen
by Zheng and Chen (2019) in their SPH study. As shown in Fig. 1, two
layers of fluids with different densities, water and silicone oil, are
contained in a rectangular wave tank. The silicone oil layer is located at
the upper part with the density ρ2 = 856 kg/m3 to a depth h1 = 0.05 m,
and the water layer is located at the lower part with the density ρ1 = 996
kg/m3 to a depth h2 = 5h1 = 0.3 m. A vertically removable sluice gate
was mounted at x = 0 m, which divided the wave tank into two parts. To
generate the internal solitary waves, an interfacial displacement di = 3h1
= 0.015 m exists between the two sides of the sluice gate, thus the free
surface on the right side of the gate has an small elevation of ds to keep
the balance of the initial static pressures of the two sides.
In present numerical simulation, the initial particle distance is 0.0015 m
and a total of 1242526 particles are used, in which 1183328 fluid
particles are included. to record the interfacial wave elevations, five
wave probes are arranged at x = -1.5, -2.5, -3.5, -4.5, -5.5 m,
respectively.
(a) t = 1 s
(b) t = 3 s
(c) t = 6 s
(d) t = 12 s
(e) t = 18 s
Fig. 2 Consecutive snapshots of generation and propagation of internal solitary wave simulated by the multiphase MPS method.
In Fig. 2, the consecutive snapshots of generation and propagation of internal solitary waves simulated by the multiphase MPS method are
Fig. 1 Numerical model for generation and propagation of the internal
solitary waves
1989
presented. When the sluice gate is removed, the gravity collapse
induces an internal dam-break flow and the locked lighter fluid on the
right hand of the gate begins to move forward into the ambient fluid. As
the lighter fluid continues to move left, a clear and stable internal
solitary wave of depression type is gradually formed and observed. In
general, the internal solitary wave is well simulated by present
multiphase MPS method, and the phase interface of the stratified fluids
is clearly captured.
In order to further verify the accuracy of MPS in generating internal
isolated waves, the interface displacements measured at five wave
probes are compared with the experimental data and SPH results, as
shown in Fig. 3. It can be seen that the calculated MPS results are
consistent with the experiment and SPH method, especially with the
latter. The amplitude of the leading solitary wave is observed to
decrease as it propagates downstream, in particular to the second and
the last wave probes. For the second wave probes, the initial decrease
of the amplitude implies that the solitary wave needs some time before
it reaches a steady state. Then the wave height remains approximately
unchanged. However, at the last wave probe, a smaller unexpected
wave height appears for the MPS and SPH results, which may be
caused by the numerical diffusion of method and it would be the aim of
the future work to solve it.
Fig. 3 Comparison of interfacial displacements at five wave probes.
Moreover, the wave profiles observed when the internal solitary wave
travels to x = -3.5 m is compared in Fig. 4. It can be seen that in the
front/left part of the wave, the MPS method, MCC-KL method and
MCC-FS method are in good agreement with the experiment. However,
in the back/right part, the profile obtained by experiment is
unsymmetrical and is relatively broader compared with the numerical
methods, which is caused by the slight unsteadiness of the experiment,
as pointed by Kodaira et al. (2006).
Fig. 4 Comparison of the wave profiles in experiment and different
numerical methods.
Fig. 5 Pressure field obtained by multiphase MPS method at t = 12 s.
Fig. 6 x-velocity field obtained by multiphase MPS method at t = 12 s.
In Figs. 5~6, the pressure and x-velocity fields obtained by the
multiphase MPS method at t = 12 s are presented, respectively. It can
be observed from the pressure field that, when the internal solitary
wave passes through, the lighter fluid distributes deeper in the vertical
direction and cause a slight upward bulge of the pressure contour. As to
the x-velocity field, it can be observed that the velocities of the upper
lighter fluid and lower heavier fluid are completely opposite, which
demonstrates that the internal solitary waves occur a characteristic of
shear flows. On the whole, both the pressure and x-velocity fields are
extremely smooth and the pressure oscillations commonly existing in
mesh-less particle methods are not found, validating the stability of
present method.
ISWs interaction with a uniform slope
With the multiphase MPS method validated, the interaction between
internal solitary waves and a uniform slope is numerically investigated
in this section. The numerical model employed here is similar with the
one used in previous section. The difference is that a uniform slope is
located on the other side of the wave tank, as shown in Fig. 7. In
addition, for the reason of saving computation cost, the length of the
wave tank is only half of the model used in previous section. In the
simulations, five slopes with different inclinations are studied,
respectively, including s = 1, 0.8, 0.6, 0.4, 0.2. The inclination of the
slopes s is defined as
s = tan (θ) (22)
where θ represents the angle between the slope and horizontal direction,
as shown in Fig. 7.
1990
Fig. 7 Numerical model for interaction between internal solitary waves
and a uniform slope.
Fig. 8 shows the consecutive snapshots of interaction between internal
solitary waves and the uniform slope with a relatively larger inclination,
s = 1. In this case, the shoaling process of the wave is similar with the
interaction between the wave and a vertical wall. When the wave
reaches the slope, almost all the wave energy is reflected back because
of no presence of waves breaking. Moreover, due to the influence of the
slope, it is observed that several secondary waves with small
amplitudes formed following the reflected wave. The presence of the
secondary waves could be explained by the x-velocity field and velocity
vector. In the later stages of the reflection process (t = 10.5 s), a gravity
current of the heavier fluid flows up the slope and forms a vortex flow
with the lighter fluid reflected, and the vortex flow further induces the
secondary waves. As presented in Fig. 8, the second waves are also
observed in the study of La Forgia et al. (2018).
The numerical results for slopes with s = 0.8, 0.6, 0.4 are presented in
Figs. 9~10, respectively. With the inclination of the slope decreases, the
interaction between internal solitary waves and the uniform slope
becomes violent, and the phenomena of wave breaking can be observed.
The breaking mechanisms in the three cases are similar with the type of
collapsing breakers defined by La Forgia et al. (2018). During the
shoaling processes of internal solitary waves, the heavier fluid trapped
between the wave and the slope quickly leaves its original position and
move downward. And compared with the dominant downward motion of
the confined fluid, the steeping of the trailing edge slowly occurs but
without inducing any observable instability. After passing the wave
trough, the downward velocity of the heavier fluid quickly decreases
under the effects of gravitational force, and start to move upward along
the slope boundary fast, giving rise to an anticlockwise motion that
causes a large amount of mixing in the breaking location. From Fig. 10, it
can be seen that with the inclination of the slope decreases, the upward
velocity of the heavier fluid become larger, inducing a more violent
wave breaking and fluid mixing.
When the inclination continues to decrease to s = 0.2, the shoaling
process of the internal solitary wave shows some difference from the
above three cases. As shown in Fig. 11, the shoaling district is larger
due to the relatively gentle slope, which makes the downward motion
of the confined heavier fluid last longer. Therefore, it can been from the
velocity field that, when the downward velocity of some heavier fluid
decreases to zero and start to move upward, there is still some heavier
fluid moving downward. These two groups of fluid make a collision in
the middle part of the slope, which leads to the presence of a complex
turbulent structure, consisted of violent vortex motions and wave
breakings. Almost all the wave energy is dissipated in this process and no
reflected wave is generated.
(a) t = 8.5 s
(b) t = 10.0 s
(c) t = 10.5 s
(d) t = 12.5 s
Fig. 8 Interaction between ISWs and a uniform slope with s = 1 (Left: experimental date; Middle: multiphase MPS method; Right: x-velocity and
velocity vector).
1991
(a) t = 8.5 s
(b) t = 10.0 s
(c) t = 10.5 s
(d) t = 11.0 s
(e) t = 11.5 s
(f) t = 12.0 s
Fig. 9 Interaction between ISWs and a uniform slope with s = 0.8, 0.6, 0.4, from left to right.
(a) t = 10.0 s
(b) t = 11.0 s
(c) t = 11.5 s
Fig. 10 x-velocity and velocity vector calculated with s = 0.8, 0.6, 0.4, from left to right.
CONCLUSIONS
The interaction between ISWs and a uniform slope is studied in present
study based on the MPS method. A multiphase MPS method is
developed from the original single-phase MPS method and validated
through simulations of the ISWs generation and propagation. The
interfacial displacements and wave profiles obtained by present method
achieve good agreement with the experimental data and other
numerical methods, and the calculated pressure and velocity fields are
smooth and reflect the actual physical features. The multiphase MPS
method is applied to investigate the ISWs interaction with submarine
slopes with different inclinations, and a qualitatively analysis of the
breaking mechanisms is performed. It is found that with the inclination
of the slope decreases, the interaction between internal solitary waves
and the uniform slope becomes violent, and the phenomena of wave
breaking can be observed. Moreover, it is found that the breaking
mechanisms with the parameters in this paper are similar with the type
of collapsing breakers defined by La Forgia et al. (2018).
ACKNOWLEDGEMENT
This work is supported by the National Natural Science Foundation of
China (51879159), The National Key Research and Development
Program of China (2019YFB1704200, 2019YFC0312400), Chang
Jiang Scholars Program (T2014099), Shanghai Excellent Academic
Leaders Program (17XD1402300), and Innovative Special Project of
Numerical Tank of Ministry of Industry and Information Technology of
China (2016-23/09), to which the authors are most grateful.
1992
REFERENCES
Brackbill, JU, Kothe, DB, and Zemach, C (1992). “A Continuum Method
for Modeling Surface Tension,” Journal of Computational Physics,
100, 335-354.
Cheng, MH, Hsu, RC, Chen CY, and Chen CW (2009). “Modelling the
Propagation of an Internal Solitary Wave across Double Ridges and a
Shelf-Slope,” Environmental Fluid Mechanics, 9(3), 321-340.
Cheng, MH, Hsu, RC, and Chen CY (2011). “Laboratory Experiments on
Waveform Inversion of an Internal Solitary Wave over a Slope-Shelf,”
Environmental Fluid Mechanics, 11(4), 353-384.
Duan, G, Koshizuka, S, and Chen, B (2015). “A Contoured Continuum
Surface Force Model for Particle Methods,” Journal of Computational
Physics, 298, 280–304.
Duan, G, Chen, B, Koshizuka, S, and Xiang, H (2017). “Stable
Multiphase Moving Particle Semi-implicit Method for Incompressible
Interfacial Flow,” Comput Methods Appl Mech Engrg, 318, 636–666.
Gingold, RA, and Monaghan, JJ (1977). “Smoothed Particle
Hydrodynamics-Theory and Application to Non-Spherical Stars,”
Mon. Not. R. Astron. Soc., 181, 375–389.
Helfrich, KR (1992). “Internal Solitary Wave Breaking and Run-Up on a
Uniform Slope,” Journal of Fluid Mechanics, 243, 133-154.
Khayyer, A, and Gotoh, H (2008). “Development of CMPS Method for
Accurate Water-Surface Tracking in Breaking Waves,” Coastal Eng,
50(2), 179–207.
Khayyer, A, and Gotoh, H (2013). “Enhancement of Performance and
Stability of MPS Mesh-free Particle Method for Multiphase Flows
Characterized by High Density Ratios,” J Comput Phys, 242, 211–233.
Kodaira, T, Waseda, T, Miyata, M, and Choi, W (2016). “Internal
Solitary Waves in a Two-Fluid System with a Free Surface,” Journal
of Fluid Mechanics, 804:201-223.
Koshizuka, S, and Oka, Y (1996). “Moving-Particle Semi-Implicit
Method for Fragmentation of Incompressible Fluid,” Nucl Sci Eng, 123,
421-434.
Koshizuka, S, Nobe, A, and Oka, Y (1998). “Numerical Analysis of
Breaking Waves Using the Moving Particle Semi-implicit Method.”
Int J for Numer Methods Fluids, 26(7), 751-769.
La Forgia, G, Adduce, C, and Falcini, F (2018). “Laboratory
Investigation on Internal Solitary Waves Interacting with a Uniform
Slope,” Advances in Water Resources, 120, 4-18.
Lee, BH, Park, JC, Kim, MH, and Hwang, SC (2011a). “Moving Particle
Simulation for Mitigation of Sloshing Impact Loads Using Surface
Floaters,” Comput Model Eng Sci, 75(2), 89-112.
Lee, BH, Park, JC, Kim, MH, and Hwang, SC (2011b). “Step-By-Step
Improvement of MPS Method in Simulating Violent Free-Surface
Motions and Impact-Loads,” Computer Methods in Applied Mechanics
and Engineering, 200, 1113-1125.
Lucy, LB (1977). “A Numerical Approach to the Testing of the Fission
Hypothesis,” Astron. J., 82, 1013–1024.
Shakibaeinia, A, and Jin, Y (2012). “MPS Mesh-free Particle Method for
Multiphase Flow.” Comput Methods Appl Mech Eng, 229-232, 13-26.
Tanaka, M, and Masunaga, T (2010). “Stabilization and Smoothing of
Pressure in MPS Method by Quasi-Compressibility,” Journal of
Computational Physics, 229, 4279-4290.
Xu, J, Wang, LL, Tang, HW, Zhu, H, and Williams, JJR (2018). “Scalar
Transport by Propagation of an Internal Solitary Wave over a Slope-
Shelf,” Journal of Hydrodynamics, 31(10), 317-325.
Zhang, YL, and Wan, DC (2017). “Numerical Study of Interactions
between Waves and Free Rolling Body by IMPS Method,” Comput
Fluids, 155, 124–133.
Zhang, YL, and Wan, DC (2017). “Numerical Study of Interactions
between Waves and Free Rolling Body by IMPS Method,” Comput
Fluids, 155, 124–133.
Zheng, BX, and Chen, Z (2019). “A Multiphase Smoothed Particle
Hydrodynamics Model with Lower Numerical Diffusion,” Journal of
Computational Physics, 382, 177-201.
(a) t = 6.0 s
(b) t = 7.5 s
(c) t = 9.0 s
(d) t = 10.5 s
(e) t = 12.0 s
Fig. 11 Interaction between ISWs and a uniform slope with s = 0.2 (Left: phase distribution; Right: x-velocity and velocity vector).
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