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

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(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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