International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 2

Number 2

June 2008

137

Numerical Simulation of Nonlinear Water

Wave Problems

D.C. Lo

Institute of Navigation Science and Technology, National Kaohsiung Marine

University, Kaohsiung, Taiwan

Jia-Shen Hu

Department of Shipping Technology, National Kaohsiung Marine University,

Kaohsiung, Taiwan

I-Fu Lin

Institute of Navigation Science and Technology, National Kaohsiung Marine

University, Kaohsiung, Taiwan

ABSTRACT: The main purpose of present paper aims at the establishment of a numerical model for solving

the nonlinear water wave problems. The model is based on the Navier-Stokes equations with the consideration

of a free-surface through the streamfunction-vorticity formulation. The main advantage of the streamfunction-

vorticity formulation is that pressure field can be eliminated from the Navier-Stokes equations. To

demonstrate the model feasibility, the present studies are first concentrated on problems including the

collision of two solitary waves with different amplitudes, and the overtaking collision of two solitary waves.

Then, the model is also applied to a solitary wave passes over the submerged obstacle in a viscous fluid.

Finally, the application of present study is also to simulate the generation of solitary waves by underwater

moving object. All examples give very promising results, those applications reveal that present formulation is

a very powerful approach to simulate the fully nonlinear water wave problems.

1 INTRODUCTION

The applications of free-surface flows find wide

range of navigation science in the design of ship-

waves interaction problems. The last one decade

have seen growing importance placed on research in

free-surface flows simulating. Many analytical

methods have been developed to solve traveling and

ship-wave problems associated with inviscid fluids

as well as for the solution of linear and low order

nonlinear problems. Cao and Beck [1] has employed

a numerical method to calculate solitary waves

generated in an inviscid fluid by a submerged cylinder

using the Laplace equation and fully nonlinear free

surface boundary conditions. A Taylor-Galerkin

method was proposed by Ambrosi and Quartapelle

[2] to compute the evolution of water waves with

moderate curvature of the free-surface shallow water

equations. However, their proposed scheme was

extended to study 2D solitary wave passing over a

circular cylinder in an inviscid fluid. Lo and Young

[3] developed a numerical mode based on velocity-

vorticity formulation to solve several free-surface

flow problems. The aim of present study is to

develop a more general approach to handle inviscid-

viscous free-surface flows. When the inviscid model

is used, the Poisson-type streamfunction equation

and fully-nonlinear free surface boundary conditions

are governed forthe solution of the free-surface

flow. Moreover, it is necessary to deal with viscous

free-surface model in order to comprehend the

complexities of the phenomenon including the

viscous effects. Thus the vorticity transport equation,

Poisson-type streamfunction equation and fully-

nonlinear free surface boundary conditions

should to be solved. The accuracy of the solution

for the inviscid-viscous model depends on the

138

representation of actual free surface unknown

values.

The numerical treatments of free-surface

problems involve the tracking of the moving free-

surface boundary during the flow transients. The

arbitrary Lagrangian-Eulerian (ALE) coordinate

system [4] is used to track the free-surface position

on the moving free-surface and the interior

computational domain at each time step. It is

important to keep the computation of a free-surface

position in perspective. In this study, the nodal points

can be arbitrarily controlled in order to get finer

mesh distribution during computations.

The main difficulty for the solution of the

viscous, incompressible free-surface flows indicate

that the free-surface boundary conditions are not

known priori. In this study, we delivered an accurate

and effective scheme based on streamfunction-

vorticity formulation and its application to the

simulation of wave-wave and wave-structure

interaction. In the computational fluid dynamics, the

fractional step method is an accurate and efficient

scheme for the solution of Naver-Stokes eqations. In

the present study, the fractional step is used for

the discretization of the governing equations and

free-surface boundary conditions (FSKBC, FSDBC).

The above mentioned of governing flow equations,

those equations are coupled and nonlinear, hence an

iterative numerical scheme is adopted in order that

we could obtain a significant saving in computer

memory.

We present an inviscid-viscous free-surface

model using the finite element discretization for the

interior of the domain and finite difference

discretization for a free-surface. The detailed

contents were addressed in the following.

2 MATHEMATICAL FORMULATION

2.1 Incompressible viscous flow

The partial differential equations of the viscous

incompressible fluid are governed by the Navier-

Stokes equations. The corresponding non-

dimensional streamfunction-vorticity form of

Navier-Stokes equations in ALE form can be

expressed as follows

Streamfunction Poisson equation

ψω

∇=−

(1)

Vorticity transport equation

ˆ ˆ

( )( )

ty x x y

ωψωψω

∂∂∂∂∂

+−+−−=∇

∂∂ ∂ ∂ ∂

(2)

where

is the streamfunction,

is the vorticity,

is the Reynolds number,

ˆ ˆ

(,)uv

is the mesh

velocity in the

,xy

direction, respectively.

Equations (1) and (2) are known as the Navier-

Stokes equations in streamfunction-vorticity form.

We seek a solution in the domain

Ω

, which satisfies

the initial conditions, with no-slip or slip velocity

boundary conditions depending on the need on the

solid boundary

Ω

2.2 Boundary conditions

Consider a two-dimensional, viscous and

incompressible fluid. In the present study, the free-

surface flow is considered as a two-phase flow. It is

assumed that the adjacent fluids are impermeable at

the interface with constant density and viscosity. The

main difficulty of the free-surface flow is that the

position of the free-surface boundary is not known a

priori. The boundary-fitted coordinates system is

used to solve the equations for the free-surface

boundary conditions in streamfunction-vorticity

form. The relationships among velocity,

streamfunction and vorticity can be defined as

(,) ( , )

ψψ

= −

. The arbitrary

physical space

(, )xy

can be general mapped to a

normalized computational domain

(,)

ξη

with the

help of the boundary-fitted coordinates system.

According to the statement of a material surface,

a particle on the surface must remain on the surface

itself. By satisfying the above kinematic condition in

an arbitrary frame of reference to be

()

h v u uh=−−

(3)

In the above equation, the frame of reference

moves with a free surface in the vertical direction

and

is either zero or equal to

The free surface dynamic boundary condition

(FSDBC) represents the continuity of the stresses on

the free-surface and is obtained by force balance

equations. Neglecting the surface tension effect on

the free-surface, the FSDBC in an arbitrary frame of

reference can be obtained by combining momentum

equations to yield

ˆ ˆ

( ( ) ( ) ( ))

( ( ) ( ) ( )) 0

t x y xx yy

u u uu v vu u u x

v u uv v vv v v y

+− +− − + +

+− +− + − + =

(4)

As far as initial condition is concerned, an initial

solitary wave is imposed as present in Gtimshaw’s

third order formulas [5] in the free-surface profile

and streamfunction.

139

3 NUMERICAL FORMULATION

For the solution of 2D free-surface flow using

streamfunction-vorticity formulation, the

quadrilateral finite element is used to discretize the

Poisson-type streamfunction equation and the

vorticity transport equation. The free-surface

boundary conditions are solved by the FDM.

Table 1. Comparison of present and Su and Mirie results [6] of

the maximum amplitude run-up during the collision

,AB

0.2,0.2

0.3,0.2

0.4,0.2

0.5,0.2

0.6,0.2

Present

0.4298

0.5436

0.6604

0.7866

0.9420

Su and

Mirie

0.4240

0.5375

0.6520

0.7675

0.8840

Error

(%)

1.37

1.13

1.29

2.49

6.56

4 RESULTS AND DISCUSSIONS

4.1 Validation of the numerical model for two

solitary waves head-on collision

It is interesting to note that the solitary waves which

retain their identity upon collision are called

solitons. The topics of solitons had a large impact on

various branches of modern physics, and this topic

attracts many investigators to study. In this section,

the results for collisions of two opposite solitary

waves using the present numerical scheme were

verified. Su & Mirie [6] study the head-on collisions

of two solitary waves generated using a perturbation

method. The flow is an inviscid and incompressible

fluid. In a similar work, Mirie & Su [7] depicted the

collisions of two solitary waves to confirm the

existence of a secondary wave as predicted in their

previous work.

0.00 10.00 20.00 30.00 40.00

(a)

0.00 10.00 20.00 30.00 40.00

(b)

Fig. 1. Evolution of free surface position of different

amplitudes. (a)

0.3, 0.2AB= =

(b)

0.6, 0.2AB= =

We consider the collision of two solitary waves

with different amplitudes. Calculations were carried

out for initial heights of a solitary wave at the left-

hand side given by the

expression,

=0.2,0.3,0.4,0.5,0.6. The initial

amplitude of a solitary wave at the right-hand side

was assumed to be equal to 0.2. The grid size was

0.1, 0.02xt∆= ∆=

and the range consisted of 8000

quadrilateral elements. The Reynolds number is

Re 60,000=

for this case. In this moving grid, the

interval of

x∆

is fixed but

y∆

is changed by the

FSKBC. The maximum amplitudes of the head-on

collision of two solitary waves were shown in Table

1. Good agreement with the solution obtained by Su

& Mirie [6] was depicted in above table. Table 1

provides the list of 5 values that were computed for

verifying the model in this study. The present

numerical model could predict the maximum run-up

with less than 2% error in the cases of

0.2,0.3,0.4A =

and

0.2B =

. However, the error

turned out to be a 6.56% at

0.6A =

0.2B =

. A

partial explanation for this may lie in the fact that the

computation of present case the fully nonlinear

boundary conditions are considered. However, the

result obtained by Su & Mirie [6] was to assume the

weakly nonlinear effect on the free surface for this

case. Figure 1 shows the evolution of free surface

profiles of the two solitary waves during the

collisions. The high nonlinear interaction has been

observed during the collisions between the two

solitary waves. We also found that after the

collision, the solitary waves recuperate almost all of

their original amplitudes in the time range of the

transient. Also, the solitary waves have experienced

phase change and were trailed by a dispersive wave

train when the two solitary waves head-on collide in

our computation.

140

0.00 40.00 80.00 120.00 160.00 200.00

120

Fig. 2. Evolution of free surface position of different

amplitudes. (

0.6, 0.2AB= =

)

4.2 The overtaking collision of two solitary waves

As mentioned before of head-on collision of two

solitary waves, now we consider the simulation of

the 2D overtaking of two solitary waves of

amplitudes 0.6, 0.2. The initial amplitudes of the two

solitary waves were assumed to be equal to 0.6 on

the left-hand side and 0.2 on the right-hand side. The

positions of the two solitary waves were initially

fixed at

10x =

and

20x =

in the same direction.

The present simulation of free-surface flow was

tested for inviscid model. In the present

computations, the non-dimensional parameter with

finite depth, D equal to 1 and length in the

direction equal to 200 were assumed, respectively. A

total number of 16,000 elements and 17,017 nodes

(1001, 17 divisions in the

x −

y −

directions,

respectively) with a time step

t∆

=0.02 were used for

computation in the present case. Figure 2 depict the

time evolution of the interaction of two solitary

waves. The highly nonlinear behavior during the

overtaking process between two opposite solitary

waves is presented in Figure 2. However, both

solitary waves emerge completely unscathed.

0.00 10.00 20.00 30.00 40.00

Fig. 3. Evolution of free surface elevation at different time for

Re 60,000=

4.3 Solitary wave passing over a submerged

structure

The computation for a fluid-structure interaction

with a free-surface was carried out for a solitary

wave passing through the submerged structure. We

assume the dimensionless length of the mean water

level as 1 and length as 42 as well as initial

amplitude of the solitary wave as 0.4. The initial

condition of a solitary wave was fixed in

10x =

the water channel. The flow was considered to be

viscous. For the present case of viscous model, the

fluid is assumed to have small viscosity (Re=60,000,

Fr=1), our goal is the observation of the separated

flow pattern for the viscous model. Figure 3 lists the

time evolution of a solitary wave passing through the

submerged structure in present simulation.

In order to plot the vortex motion generated

during wave-structure interactive process. The

streamline were plotted and were shown in Figure 4.

The local velocity distribution near the structure was

given in Figure 5. The gradual development of the

recirculating flow regimes at the downstream of the

structure and was clearly predicted by the present

numerical study. The recirculating flow generated at

the trailing edge of the structure was clearly

observed in Figure 5. The results reflected in Figure

5 indicate that the viscosity plays an important

role to affect the interactive processes between

the solitary wave and the solid structure. In general,

the vortical flow is always grown up on the rear

surface of the structure while the solitary wave

passes through the submerged structure in viscous

fluid.

16 18 20 22 24 26

t=8

16 18 20 22 24 26

t=12

16 18 20 22 24 26

t=14

Fig. 4. Evolution of streamline distribution at different time

141

t=10

t=12

t=16

Fig. 5. Local velocity distribution at different time

4.4 The generation of solitary wave by underwater

moving object

The final test case illustrates the capability of 2D

free surface algorithm for a succession of solitary

waves generated by underwater moving object. The

submerged object is moving with a constant velocity

in the negative x-coordinate direction in shallow

water. The layout of geometry and conditions is well

illustrated in Figure 6. We assume the constant

velocity of underwater object is moving in a channel

[ ] [ ]

0,70 0,1×

with the object as shown in Figure 6,

initially the region of object is

max

cos ( 40) , ( 40)

(, )

Bx x

bxy

otherwise



− −<− <

 









(5)

For the case of present used study, we assume

max

0.15B =

and

1L =

. The computation for a

submerged moving object in a channel is assumed to

be 70 in the

direction and with the upstream

dimensionless depth of 1 in the

direction.

The computation of the present case is used with

a total number of 9,600 elements and 10,217 nodes

with a time step

0.02t∆=

and

Re 30,000=

Figure 7 depicts the local velocity and vorticity

distributions at t = 100 for the flow past the

submerged object. The flow circulation behind

the object is seen in Figure 7, in which the region

of the vortex structure near the object is delineated.

Further, the streamline contours at different time are

also revealed in figure 8. Finally, the evolution

of free surface elevation at different time for

the propagation of solitary waves is shown in

figure 9. It can be observed that the generation and

propagation of the upstream advancing solitary wave

become more dominated as time increases.

u=1

Fig. 6. Layout of an underwater moving object

40 40.5 41 41.5 42

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

t=100

142

40 40.5 41 41.5 42

0.25

0.5

0.75

1.25

1.5

1.75

24.4444

8.88889

-6.66667

-22.2222

-37.7778

-53.3333

-68.8889

-84.4444

-100

t=100

Fig. 7. Local velocity and vorticity distribution at t=100

5 CONCLUSIONS

A numerical model by combing FEM and FDM for

free-surface flow problems was proposed using the

Navier-Stokes equations in streamfunction-vorticity

form. The present study has extended to an inviscid-

viscous model for free surface flow problems.

The accuracy of results obtained we have is

excellent. Also, both inviscid and viscous models

play an important role to free surface flows.

However, the key issue in the viscous model is to

find the accurate solution for separated flow pattern

of a solitary wave passing over the submerged

structure. It shows that the streamfunction-vorticity

formulation gives accurate and correct results for

flow problems with a free surface. We can conclude

with certainty that both inviscid and viscous models

are able to simulate non-linear water waves in

present work.

ACKNOWLEDGMENTS

The work reported in this chapter was supported

by the National Science Council, Taiwan under

the grant no. 95-2221-E-022-020. It is greatly

appreciated.

t=30

t=60

Fig. 8. Streamline distributions at different time

0.00 20.00 40.00 60.00

120

Fig. 9. The evolution of free surface elevation at different time

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