ORI GIN AL PA PER

A corrected 3-D SPH method for breakingtsunami wave modelling

Jinsong Xie • Ioan Nistor • Tad Murty

Received: 10 August 2010 / Accepted: 2 May 2011� Springer Science+Business Media B.V. 2011

Abstract A 3-D large eddy simulation model that was first transformed to smoothed

particle hydrodynamics (LES-SPH)-based model was employed to study breaking tsunami

waves in this paper. LES-SPH is a gridless (or mesh-free), purely Lagrangian particle

approach which is capable of tracking the free surface of violent deformation with frag-

mentation in an easy and accurate way. The Smagorinsky closure model is used to simulate

the turbulence due to its simplicity and effectiveness. The Sub-Particle Scale scheme, plus

the link-list algorithm, is applied to reduce the demand of computational power. The

computational results show that the 3-D LES-SPH model can capture well the breaking

wave characteristics. Spatial evolution features of breaking wave are presented and visu-

alized. The detailed mechanisms of turbulence transport and vorticity dynamics are

demonstrated as well. This application also presents an example to validate the SPH model.

Keywords Smoothing particle hydrodynamics � Large eddy simulation �Breaking tsunami waves � Turbulence modelling � Fragmentation �Lagrangian method � Free surface tracking

1 Introduction

Nearshore behaviour of tsunami waves represents one of the most complex physical

phenomena and one of the least understood aspects of this natural disaster. Breaker height,

depth and water particle velocities are of considerable interest for hydrodynamic

description and sediment transport. While it took decades to understand tsunami wave

dynamics, several unknowns still remain, especially with respect to the tsunami wave

breaking mechanism in the breaker zone and subsequent behaviour in the surf zone. Due to

its long wave length/period, it is technically difficult and also very expensive to employ a

physical model to examine such phenomena. It is also challenging to numerically simulate

J. Xie (&) � I. Nistor � T. MurtyUniversity of Ottawa, 161 Louis Pasteur, Ottawa, ON K1N 6N5, Canadae-mail: jxie036@uottawa.ca

123

Nat HazardsDOI 10.1007/s11069-011-9954-x

tsunami wave breaking due to the fact that the numerical model has to include fully 3-D

turbulence modelling and free surface wave tracing techniques.

Tsunami waves, which are long gravity waves, have small amplitudes in deep ocean

conditions, and thus, shallow water equation (SWE) model is the common tool to inves-

tigate the properties of long waves due to the characteristics of their relative water depth

(d/L � 1) and/or their relative wave steepness (H/L � 1, where L is wavelength, H is

wave height, and d is water depth). Such a model is computationally efficient and effective

for large domain simulation during wave propagation. More specifically, when both of the

above conditions are satisfied, small amplitude wave theories can be used to investigate

tsunami wave properties. Xie et al. (2007) performed a systematic study of the tsunami

wave propagation using a shallow water equations numerical model. As a tsunami wave

approaches the shore, its wave height increases quickly, while the wave profile becomes

steeper. As documented in the literature, the initiation of breaking starts when the ratio of

breaker height to depth is in the range of 0.6–1.3. Moreover, the breaking criterion from

solitary wave theory, first derived by McCowan (1894), later updated by Williams (1981),

is given by:

Hb ¼ 0:78db ð1Þ

in which Hb is defined as the maximum wave height, and db, breaker depth, is defined as

the still water depth over the beach profile at the location of maximum wave height. In the

process of wave breaking, generally speaking, there are three typical types of wave

breakers in shallow water, namely spilling breaker, plunging breaker and surging breaker,

which depend on the incident wave steepness and beach slope. The classification of various

breakers can be made based on the surf similarity parameter nb, which is defined as

(Battjes 1974):

nb ¼tan bffiffiffiffiffiffiffiffiffiffiffiffiffi

Hb=L0

p ð2Þ

where b is the beach slope angle, Hb is the breaker height, and L0 is the deep water

wavelength of the incident periodic wave. Since the wave length and wave period of a

solitary wave are theoretically infinite, Grilli et al. (1997) introduced the following

dimensionless slope parameter for solitary waves:

S0 ¼sL0

h0

¼ 1:521sffiffiffiffiffiffi

H00

q ð3Þ

in which s is the beach slope, h0 is the offshore water depth, and H00 ¼ H0

h0is the dimen-

sionless incident solitary wave height H0 is the deep water wave height. When S0 \ 0.025,

waves break in the form of the spilling breakers which form a series of aerated wave crests

on the beach, a small jet locally generates small eddies in the vicinity of a steep wave crest

(see Fig. 1). When 0.025 B S0 \ 0.3, plunging breakers occur. The wave crest curls

downwards and impinges on the wave trough in the front, air bubbles are subsequently

entrained, the horizontal vortex roller dissipates the energy dramatically, and the flow

regime changes from irrotational to rotational, accompanied by the generation of strong

turbulence. Small waves are regenerated and splashed up, and they behave as large and

small rollers or eddies (see Fig. 2). To some extent, the small waves restabilize and

transform into a turbulent bore, with the height of the broken wave then remaining at an

approximate fixed proportion to the mean water depth and moving onshore in a bore-like

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Fig. 1 Tsunami wave breaking simulation (spilling breaker)

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Fig. 2 Plunging breaker simulation-breaking, development and formation of plunging jet (Li 2000, thegrey-coloured pictures)

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shape (see Fig. 3). The area from the initiation of wave breaking to this particular point is

defined as breaker zone. The present paper focuses on the breaker zone, somewhat dif-

ferent from the traditional definition of a portion of the surf zone. When 0.3 \ S0 \ 0.37,

waves break in the form of collapsing or surging breakers, during which most of the wave

energy is reflected back from the beach, with a small amount of wave energy lost in the

Fig. 3 A plunging breaker post-breaking and splash-up—qualitative comparison with the laboratoryphotographs (Li 2000, the grey-coloured pictures)

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breaking process. When the slope is very steep, the surging wave experiences near com-

plete reflection, with little breaking after it runs up to a maximum elevation.

Significant advances have been made in both theoretical and experimental studies on

wave breaking, especially for short waves. Therefore, the literature in short wave breaking

is extensive and detailed. However, the difficulty of measuring turbulent velocity due to the

presence of many air bubbles entrained by plunging jets has obstructed experimental

studies on wave breaking. On the other hand, progress on numerical modelling of breaking

waves is still far from complete, and simulation of breaking tsunami waves remains

surprisingly sparse, especially for 3-D modelling. Turbulence modelling and free surface

tracking are two key issues in the study of breaking tsunami waves. Traditionally, Eulerian

method–based wave breaking model requires the incorporation of two mechanisms to

simulate wave breaking. One is the trigger mechanism for the initiation of wave breaking,

while the other one is the energy dissipation mechanism. A prescribed wave breaking

criterion (when and where the wave breaks) must be established first so that the dissipation

term can be triggered and added to the momentum equations. Three types of energy

dissipation methods are often employed: the surface roller model (Madsen et al. 1997, the

vorticity model (Svendsen et al. 2000) and the eddy viscosity model (Kennedy et al. 2000).

The difficulties in calculating the free surface profiles during and after wave breaking are

enormous when one uses the volume of fluid (VOF) method (Hirt and Nichols 1981) and

probability density function (PDF) method (Watanabe 1996). In the aerated area, the

conventional assumptions that the free surface is continuous and in the form of a material

surface are no longer valid. Compared with Eulerian methods, Lagrangian methods do not

require the initiation mechanism for wave breaking but focus on the energy dissipation

modelling. Additionally, there is no special need to treat free surface profile, while frag-

mentation can be easily simulated. However, because numerical accuracy of high-order

numerical methods and high spatial and temporal resolution are indispensable to restrict

computational errors and keep computation stable, there have been few papers on 3-D

wave breaking simulation due to the high computational effort required.

Although wave breaking is essentially a 3-D phenomenon, some researchers have

assumed that if the incident wave perpendicularly enters the computational domain,

breaking wave motion can be simplified as a 2-D problem. In the literature, there are three

kinds of models to simulate complicated turbulence flow, namely the Reynolds-averaged

Navier–Stokes (RANS) model, the large eddy simulation (LES) model and the direct

numerical simulation (DNS) model, enumerated in their increasing order of accuracy and

computational capability demand. Lin and Liu (1998) performed a quantitative comparison

between the spilling and plunging breakers by using an advanced 2-D RANS model; Shao

and Ji (2006) employed an incompressible 2-D SPH-LES model with SPS scheme to study

plunging wave breaking. Their computations were compared with the experimental data of

Ting and Kirby (1995), and good agreement was found between the numerical and

experimental results. However, 2-D SPH-LES model results were shown to be more

effective in describing large free surface deformation and better than those obtained by

using RANS model. Direct numerical simulation (DNS) for 2-D water wave breaking was

carried out by Watanabe and Saeki (1996). Overturning jets, plunging jets and large-scale

eddies were rather accurately simulated. These models are 2-D approximations of the 3-D

flow and hence are less computational demanding than fully 3-D models. However, using

these models, it is difficult to study the details of 3-D flow features such as those observed

during the wave breaking process. Watanabe and Saeki (1999) also carried out a 3-D large

eddy simulation (LES) of breaking waves and found that there was a 3-D coherent eddy

structure comprised of horizontal, helical and vertical eddies following breaking. Helical

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and vertical eddies are generated simultaneously when the overturning jets plunge at first,

while the positive and negative vorticities are arranged by turns in the lateral direction and

are stretched in the wave direction as the wave front progresses. The fully stretched eddies

are then gradually broken down into small fractions. Although many air bubbles entrained

by breaking waves potentially affect turbulent evolutions in actual wave breaking process,

the effects of bubbles are ignored for the sake of simplicity in the aforementioned papers.

2 A corrected 3-D LES-SPH-SPS model

Meshless methods have been developed to solve the complex surface geometry that cannot

be readily handled by mesh-based methods. Smoothed particle hydrodynamics (SPH) is

one of the earliest gridless methods invented in mid-1970s for the study of astrophysics.

Due to its simplicity and robustness, this numerical method has been extended to the study

of solid mechanics for dynamic phenomena and more recently to complex fluid mechanics

problems such as dam breaking and wave propagation. Substantial development and

improvement of meshless methods started in mid-1990s, after which there was a significant

development of meshless methods in computational physics and mechanics. At present,

SPH is the main meshless method that has been successfully applied in fluid flow mod-

elling. In SPH, a computational domain is represented by a set of interpolation points

called particles where the fluid medium is discretized by interaction between particles

rather than grid cells. The basic concept of SPH is that continuous media are represented

by discrete particles with volume, density and mass. The particles have a kernel function to

define their range of interaction, and the hydrodynamic variables are approximated by

integral interpolations. Mesh-based numerical methods limit their applications in many

complex problems such as grid generation, which is not always a straightforward process

and can constitute an expensive task, both in terms of computational time and mathe-

matical complexity. Mesh-free methods are easier to treat large deformations, disconti-

nuities and singularities compared with grid-based methods. In the SPH computations, the

free surface can be easily and accurately tracked by following water particles without

numerical diffusion.

The standard SPH code (SPHysics version 2.2) is an open-source code that has been

released in September 2010, which was jointly developed by researchers at the Johns

Hopkins University (USA), the University of Vigo (Spain), the University of Manchester

(UK) and the University of Rome La Sapienza (Italy). A corrected, improved and opti-

mized 3-D SPH model has been developed specifically for free surface hydrodynamics in

the present research.

In SPH, the fundamental principle is to approximate any function in discrete notation at

a particle a by

Aa ¼X

b

mbAb

qb

Wab ð4Þ

where Wab ¼ W r!a � r!b; h� �

is the weighting function (or kernel), h is called the

smoothing length, and the mass and density are denoted by mb and qb, respectively. The

weighting function should satisfy several conditions such as positivity, compact support

and normalization. Also, it must be monotonically decreasing with increasing distance

from particle a and should behave in the form of a Dirac-delta function, as the smoothing

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length h tends to zero (Liu and Liu 2003). Monaghan (1989) obtained a form of the

gradient for a scalar field through numerical experiments:

ðrAÞa ¼ qa

X

b

mbAa

q2a

þ Ab

q2b

� �

rWab ð5Þ

Divergence of a vector field can be written in SPH formalism:

ðr � AÞa ¼ qa

X

b

mbAa

q2a

þ Ab

q2b

� �

r �Wab ð6Þ

The above equations are antisymmetric with respect to the a and b subscripts. Conse-

quently, the contribution of particle b to ðrAÞa (or ðr � AÞa) is identical to the contribution

of particle a to ðrAÞb (or ðr � AÞb). From a computational point of view, these consider-

ations are important as the calculation time can be reduced by employing them. There is no

need to calculate the contribution of particle a to particle b if the contribution of particle b to

a is already computed. In SPHysics, the user can choose from one of the following four

different kernels: (1) the modified Gaussian (Colagrossi and Landrini 2003), (2) quadratic

(Dalrymple and Rogers 2006), (3) cubic spline (Monaghan 1992) and (4) quintic (Wendland

1995). The commonly used weighting functions in the research project are cubic spline

W r; hð Þ ¼ ad

1� 32

q2 þ 34

q3 0� q� 114ð2� qÞ3 1� q� 2

0 q� 2

8

<

:

ð7Þ

where ad ¼ 10=ð7ph2Þ in 2-D and ad ¼ 1=ðph3Þ in 3-D, q = r/h. The second derivative of

this kernel is continuous, and the leading truncation error term is O(h2). The finiteness of

the kernel support means that only a limited number of neighbouring particles b play a role

in all the sums of conservation equations. This is used to reduce the computational time by

building a link list between particles at each time step.

There are various forms in which the N–S equations are represented in SPH (Monaghan

1992). The most commonly used form is:

Dqa

Dt¼X

b

mb ua � ubð Þrwab ð8Þ

(compressible form)

D u!a

Dt¼ � 1

qr!Pþ H

!þ g!¼ �X

b

mbpa

q2a

þ pb

q2b

þ Rab

� �

rwab þ g! ð9Þ

where H!

refers to the diffusion terms.

As noted in the LES model, one deals with two different scale levels—the larger eddies,

which represent the mean flow features and the boundary conditions, and the smaller

eddies, which are universal everywhere. The former are much larger than the later ones.

The computational domain must be large enough to represent the energy-containing

motions, while the grid spacing must be small enough to resolve the dissipative scales. In

addition, the time step used to advance the solution is limited by considerations of

numerical accuracy. This requires that the time step in the model is reasonably small;

otherwise, this will make the numerical model unstable and cause large computational

perturbation errors. As a result, huge computational capacity is required, and in some case,

this is extremely computationally intensive. In order to avoid solving the complex

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Poisson’s equation, which demands significant computing power, the fluid pressure in the

SPH formalism is treated as weakly compressible. The compressibility is adjusted by using

Courant Fredrich Levy (CFL) condition to slow the speed of sound (pressure wave) such

that the time step is reasonable. Monaghan (1989) also pointed out that the sound speed

should be about ten times faster than the maximum fluid velocity, thereby keeping density

variations within less than 1%.

P ¼ Bqq0

� �c

�1

� �

ð10Þ

where c = 7, and B ¼ c20q0=c being q0 ¼ 1,000 kg m�3 for the reference density and

c0 ¼ c q0ð Þ ¼ffiffiffiffiffiffiffiffiffiffi

oPoq

r �

�

�

�

q0

the speed of sound at the reference density. For turbulence mod-

elling, several complicated formulations have been proposed to model the sub-grid scale

(SGS) turbulence. The Sub-Particle Scale (SPS) approach is very similar to the concept of

large eddy simulation (LES), which was first described by Gotoh et al. (2001) to represent

the effects of turbulence in their moving particle simulation (MPS) model. The concept

comes from the fact that large eddies vary from flow to flow, while small eddies are

universal everywhere. The idea is that we only need to compute the resolvable large-scale

turbulence and the mean flow, while the unresolvable small-scale turbulence can be

modelled based on some closure models and can be thus removed from the intensive

calculations. In the recently upgraded SPHysics version 2.2, three different options for

diffusion can be used: (1) artificial viscosity (Monaghan 1992), (2) laminar viscosity (Lo

and Shao 2002) and (3) full viscosity (laminar viscosity?Sub-Particle Scale turbulence).

The artificial viscosity proposed by Monaghan (1992) has been used very often due to its

simplicity. However, it is a scalar viscosity which cannot take into account the flow

directionality and does not conserve angular momentum. In addition, it has been proved

that it leads to strong dissipation in some case of flow simulation (Dalrymple and Rogers

2006) such as in the case of complex shearing flows where a too large vorticity decay and

unphysical momentum transfer are present. Following Dalrymple and Rogers (2006),

momentum equation with full viscosity can be written as:

d u!a

dt¼ �

X

b

mbPb

q2b

þ Pa

q2a

� �

r!aWab þ g!þX

b

mb4v0 r!abr

!aWab

ðqa þ qbÞ r!ab

�

�

�

�

2

!

u!ab

þX

b

mbsb

q2b

þ sa

q2a

� �

u!aWab ð11Þ

3 Add source code for solitary wave generation

In order to simulate breaking waves, a source term has to be added into the computational

domain. It should be noted that until now, most of the wave theories were based on velocity

potential formula. A solitary wave (which consists of a single volume of fluid propagating

entirely above the undisturbed free surface and is regarded as a weakly non-linear and

dispersive wave) is often used to approximate the leading wave front of a tsunami. While

wave non-linearity tends to make wave front steeper, wave dispersion will counterbalance it.

For this reason, a solitary wave can travel a long distance without significant shape distortion

and energy loss. For a tsunami generated from far field, phase dispersion and amplitude

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dispersion lead to wave frequencies separation. As a result, short waves are damping while

the wave travels in deep ocean and only long waves approach the shore. Normally, only the

first few waves are of concern. For a tsunami generated in the near field, due to the limited

water depth, group velocity and phase velocity are about equal and, generally, no wave

separation happens. Therefore, it is acceptable to use a single wave in both numerical and

physical models for tsunami wave research in the nearshore region. Also, it is noticeable that

today’s computing power still does not allow large area simulation for breaking waves

though our numerical technologies have advanced considerably. Scaling methods can be

used in numerical models to scale down the prototype as it would be in physical models, and

this again brings scaling effects and boundary effects into numerical computation. However,

one can utilize some numerical skills to minimize those impacts on results.

The usual procedure for long wave and more specifically solitary wave generation

consists in matching the paddle velocity at each position in time with the vertically

averaged horizontal velocity of the wave. For long waves, it is reasonable to assume that

the horizontal velocities are nearly constant over the depth. Equating the wave board speed

to the depth-averaged particle velocity on the face of the board gives the expression

dX0ðtÞdt¼ uðX0;tÞ ð12Þ

in which X0(t) is the wave board location. It can be derived from continuity considerations

that the depth-averaged velocity for shallow waters of permanent form can be written as

(Svendsen 1984):

u x; tð Þ ¼ Cgðx; tÞd þ gðx; tÞ ð13Þ

where C is wave celerity, and g(x, t) is sea surface elevation. Substituting into Eq. 12

gives:

dX0ðtÞdt¼ CgðX0; tÞ

d þ gðX0; tÞð14Þ

Following Eq. 14, Goring and Raichlen (1980) assumed that the sea surface elevation

could be represented as a wave height multiplied by some function of phase angle; Syn-

olakis (1990) recommended integrating the Eq. 14 directly using the Runge–Kutta

numerical integration method. He stated this technique converges faster, particularly for

cnoidal waves. It is well known for a solitary wave at the position of a piston-type wave

board over the range -?\ t \? that the wave profile can be written in the form:

gs x; tð Þ ¼ H sec h2 hð Þ ¼ H sec h2½k Ct � X0ð Þ� ð15Þ

where k ¼ffiffiffiffiffi

3H4d3

q

and C ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gðd þ HÞp

. Substituting Eq. 15 into Eq. 14, one can have an

implicit equation for the wave board time history

X0 tð Þ ¼ H

kdtan h½k Ct � X0ð Þ�: ð16Þ

The motion of the wavemaker was thus programmed to generate a solitary wave by

forcing a (horizontal) velocity field. The authors adopted a procedure similar to that

introduced by Goring and Raichlen (1980); however, the authors followed the suggestion of

Dalrymple (private correspondence, 2011) that the total stroke duration time be estimated as

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the time it takes for the wave paddle to reach 0.954 times its final position. The compu-

tational domain, as shown in Fig. 4, has the following dimensions: length 6 m, width 0.2 m

and height 0.55 m with the beach slope 1:13. The length of the flat bottom before the mildly

sloping beach is 1.0 m, and the still water depth is 0.3048 m. Table 1 summarizes the

incident wave conditions (one corresponds to a spilling breaker, the other to a plunging

breaker). Since the plunging breaker has stronger turbulence than spilling breaker, more

works are focused on plunging breaker. Based on this, a model of tsunami wave with

relative wave height H/d = 0.3 is produced. The initial condition was set by specifying the

particles motion near the computational entrance (see Fig. 5).

4 Numerical implementation

Four numerical schemes are implemented in resolving SPH: (1) Predictor–Corrector

algorithm (Monaghan 1989), (2) Verlet algorithm (Verlet 1967), (3) Symplectic algorithm

(Leimkuhler et al. 1996) and (4) Beeman algorithm (Beeman 1976). The user can choose

one of them according to the needs of application. The Symplectic time integration

algorithm (Leimkuhler et al. 1996) is time reversible in the absence of friction or viscous

effects and hence represents a very attractive option for meshless particle schemes. First,

the values of density and acceleration are calculated at the middle of the time step as:

qnþ1=2a ¼ qn

a þDt

2

dqna

dt; rnþ1=2

a ¼ rna þ

Dt

2

drna

dt; Pnþ1=2

a ¼ Bq

nþ12

a

q0

!c

�1

" #

ð17Þ

where the superscript n denotes time step and t = nDt. In the second stage, momentum

change provides the velocity and hence the positions of particles at the end of each time step:

0 1 2 3 4 5 6 7 8 90 0.10.2

0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Y(m)X(m)

Initial Particle ConfigurationZ

(m)

Fig. 4 Initial particle configuration and computational domain

Table 1 Wave parametersWater depth (d), m 0.3048 0.1

Wave height (H), m 0.09144 0.015

Wave number (k) 1.4916 3.354

Total stroke duration time (t), s 1.0 1.109

Beach slope (h) 1:13 1:15

Breaker type Plunging Spilling

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ðVaqauaÞnþ1 ¼ ðVaqauaÞnþ1=2 þ Dt

2

dðVaqauaÞnþ1=2

dt; rnþ1

a ¼ rnþ1=2a þ Dt

2unþ1

a ð18Þ

where V is the volume of particle a. At the end of the time step,dqnþ1

a

dtis calculated using the

updated values of unþ1a and rnþ1

a (Monaghan 2005). No matter which scheme is chosen,

time step control is dependent on the CFL condition, the forcing terms and the viscous

diffusion term. According to Monaghan and Kos (1999), the time step is constrained by

dt ¼ min 0:4h

c0

; 0:25

ffiffiffiffi

h

F

r

; 0:125h2

t

!

ð19Þ

where c0 ¼ 10 Umax and F are the characteristic velocity and force experienced by a

particle, respectively, Umax being the maximum flow speed. The particle position at each

time step is expressed as:

dri

dt

� �

a

¼ ðuiÞa: ð20Þ

For free surface flows, Monaghan (1994) suggests the use of XSPH equation to update

the particle position. This XSPH form is given by:

dri

dt

� �

a

¼ ðuiÞa þ �X

b

mb

qab

½ uið Þb� uið Þa�Wab ð21Þ

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

u (m/s)

T (

s)

wave paddle velocity

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

X (m/s)

T (

s)

wave paddle trajectory

Fig. 5 Wavemaker moving velocity and trajectory during simulation

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where � ¼ 0:5 by default. This method moves the particle with a velocity that is close to

the average velocity occurring in its neighbourhood. This form helps maintain an orderly

movement of the particles and prevents penetration of particles associated with different

fluids as in a multi-phase flow.

In reality, water waves propagate in a viscous fluid over an irregular bottom of varying

permeability. A remarkable fact, however, is that, in most cases, the main body of the fluid

motion is nearly irrotational. This is because the viscous effects are usually concentrated in

thin ‘boundary’ layer near the bottom. The flow region under a water wave train can be

decomposed into two parts, namely the bottom boundary layer, within which the fluid

viscous effect is significant, and the flow outside the boundary layer where the viscous

effect is negligible. The thickness of the bottom boundary layer induced by a wave train

can be estimated by the following simple formula derived from dimensional analysis:

d�Oðffiffiffiffiffiffi

mTpÞ ð22Þ

where m is the kinematic viscosity of the fluid, and T is the wave period. Given the typical

value of m = 1.0 9 10-6 m2/s for water and T = 1–30 min (from wind waves to tsunami

waves), the corresponding boundary layer thickness ranges from 1.4 mm to 4.5 cm. In the

surf zone where the water depth is from tens of metres to hundreds of metres, the bottom

boundary layer thickness is much smaller than entire water depth and one can treat the

whole flow region as one entity. Three boundary conditions can be applied in SPH:

(1) dynamic boundary conditions (Crespo et al. 2007, Dalrymple and Knio 2000), (2)

repulsive boundary conditions (Monaghan and Kos 1999, Rogers and Dalrymple 2008) and

(3) periodic open boundary conditions. Numerical parameters used in the paper are

summarized in the Table 2 later.

5 Evaluation of the 3-D velocity field

As mentioned previously, the wave flow is essentially irrotational before the wave breaks.

This can be seen clearly from Fig. 6. If the initial wave is irrotational, the wave will

maintain its original form as it propagates. Only when there are factors such as wave

breaking and wave-structure interaction, the initial wave form changes from irrotational to

Table 2 Numerical modelparameters

Model parameters Value

Order of the kernel 3rd order

Smoothing length ratio 0.03

Fluid particles 45,600

Moving wall particles 290

Wall particles 3,177

Total particle number 49,067

Initial particle spacing dt (m) 0.02

Lateral boundary condition Periodic

Other boundary condition Repulsive

XSPH parameter 0.5

Wall viscosity 8.0e-4

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rotational. Due to the strong directionality of the solitary wave, the horizontal scale is

much larger than the vertical scale. The velocity magnitude in Y-direction, as well as in

Z-direction, is small and, therefore, negligible compared to the horizontal (X-direction)

velocity magnitude. It is evident that fluid motion can be simplified into two dimensions

and the flow is irrotational, thus non-breaking wave can be modelled effectively and

efficiently by using 2-D models. When the wave breaking initiates, the vertical velocity

obviously becomes significant (Fig. 7) and the transverse velocity suddenly increases just

when the breaking wave front impinges on the water body in front of the wave. Moreover,

Fig. 6 shows that when wave breaks, the generation of the transverse velocity component

(red colour) has transformed the flow from 2-D to 3-D, while the vertical velocity

Fig. 6 Velocity variation in Y-direction (normal to the page) before and after wave breaking

Fig. 7 Vorticity variation in Y-direction (xy)

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magnitude has been increased significantly. Figure 8 shows that when the breaking wave

jet impinges the bottom of front wave body, the transverse velocity increases in both ways

(in Y and -Y-directions), something that transfers more momentum and energy into the

transverse direction. Figure 9 demonstrates that the momentum and energy are transferred

back and forth with a net transfer in the direction of wave propagation. The breaking

process shows, therefore, a strong transport and turbulent mix of energy in the entire fluid,

and all flow variables in 3-D need to be considered at this stage. Finally, the variables of

interest in each direction are at the same order and hence become equally important.

Therefore, 3-D modelling is a must in order to capture the breaking wave process.

Fig. 8 Velocity variation in Y-direction (normal to the page) before and after breaking jet impinges thebottom of front wave body

Fig. 9 Velocity direction and magnitude at the breaking front

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6 Modelling results of breaking tsunami waves

Figure 2 shows snapshots of wave breaking from the wave initiation to slump down and

then impact the wave front at the beach side. The maximum horizontal velocity is located

at the free surface (top front). When the wave phase velocity surpasses the group velocity,

the wave tends to separate from the wave group to which it initially belonged. Once

separated, the wave moves forward in the way. Once breaking occurs, the overturning jet

hits the forward water surface and rebounds back with a net forward momentum as shown

in Figs. 3, 7, and 8. During this process, air bubbles are entrained and impinged into the

water, causing an energy flux transfer between the fast-running wave front and the slowly

moving water body. The rebounding jet repeatedly plunges and transits to a bore-like front

which moves further inshore. When the wave breaks and reforms several times, the energy

is finally distributed evenly among the whole water body during this period. However, the

horizontal velocity gradually attenuates as the front progresses, while the lateral velocity

and vertical velocity intensify during the course of the wave breaking. It is obvious that the

wave energy has partially transferred from horizontal direction to the lateral and vertical

directions except for the energy dissipation due to viscosity. The boundary friction and

surface resistance have been ignored in the present model but will be considered in the

future development of the model. Figure 10 displays three particles—44,869 at the surface,

40,970 at the bottom and 60,660 in the middle depth—which are chosen to reflect their

density and pressure variation with time. The figure represents the random motion of the

water particle during the passage of wave. Figure 11 provides particle pressure and density

variation during simulation time (from t = 0 to t = 5 s ended at time step N = 198) for

the same particle shown in Fig. 10. From these figures, it is clear that the density variation

is quite small and within the assumption range (less than 1%), while the pressure is a

function of density, its perturbation reflects the wave propagating process in the compu-

tational domain, with the maximum pressure occurrence at the moment of maximum wave

height. Once the wave passes by, the pressure fluctuates around the hydrostatic pressure.

During the wave breaking and splash-up process, a significant fluctuation can be observed

from those figures. Compared with 2-D SPH simulation results (Lo and Shao 2002;

Khayyer et al. 2008), Figs. 2 and 3 demonstrate that the 3-D SPH modelling has very well

reproduced the plunging jet development and the resultant splash-up development, as

observed in the laboratory photographs of Li (2000). Figure 9 shows the velocity direction

and magnitude of the breaking front, and it clearly shows that due to the curl down of the

wave front, the water in the front of the wave moves back to form a portion of the curved

wave body, and then, it swirls up and sweeps down again into the wave body, while portion

of the water particles spins out of the swirl and splashes away from the body. The long

arrow represents the larger speed of water particles. This indicates that the splashed par-

ticles carry lots of energy away from the wave body. That energy is attenuated by the

interaction between the water particles and the air.

7 Examination of the 3-D vortices of the breaking waves

The splash-up is the result of the plunging jet penetration and the associated resulting

momentum and energy exchange between the fast moving overturning jet and the wedge-

shaped water located ahead of the wave. The splash-up is responsible for the generation of

large-scale vortices and plays a significant role in the momentum transfer and energy

dissipation. Temporal and spatial changes in each vorticity component are investigated to

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visualize the generation and evolution of 3-D large-scale eddies. The vortices are com-

puted as an angular velocity cross-multiplied by particle mass in three directions

ðxx;xy;xzÞ. The vorticity of each particle is defined as (Monaghan 1992):

xi ¼X

mjðui � ujÞ riWij ð23Þ

Figure 7 shows several vortex pictures: it can be seen that the positive vorticity

(clockwise turning) and negative vorticity (counterclockwise turning) dominate the

rebounding jets. The vortex rollers associated with the transverse eddy are stretched

during propagation of the breaking wave front. The fully stretching eddies are finally cut-

off and decomposed into small fractions due to the instability of the vortex structures,

and splashed particles are successfully reproduced and modelled. Hence, the capacities of

the present SPH model in the simulation of large wave deformations and fragmentation

have been demonstrated through a qualitative comparison with the laboratory photo-

graphs (Li 2000).

Fig. 10 Variation of the three particles (44,869 at the surface, 40,970 at the bottom and 60,660 in themiddle depth) velocity

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Fig. 11 Particle pressure and density variation during simulation time (from t = 0 to t = 5 s, ended at timestep N = 198) for the particles numbered in Fig. 10

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8 Conclusion

The computational results show that the 3-D LES-SPH model can better capture the

breaking wave characteristics compared to other traditional (2-D) methods. The evolution

of the breaking surface profile and large-scale eddies during wave breaking has been

successfully tracked with ease and reliability and was shown to be similar to those

observed during the experiments of Li (2000). The spatial evolution of the breaking wave

is also presented. The detailed mechanisms of turbulence transport and vorticity dynamics

are shown to be well reproduced. It has been confirmed that the 3-D features of breaking

wave can be accurately modelled with a 3-D model. In addition to this work, further

investigations will focus on the accurate free surface tracking of discontinuous waves and

of the air-entrained multi-phased flow. The dissipative ability of Smagorinsky turbulence

closure in combination with rather dissipative SPH model will be further examined, as

several few researchers have shown that such models are highly dissipative.

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