第 33 屆海洋工程研討會論文集 國立高雄海洋科技大學 2011 年 12 月

Proceedings of the 33rd

Ocean Engineering Conference in Taiwan

National Kaohsiung Marine University, December 2011

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Numerical study on the dynamic response of the

density-stratified fluid in a submarine rectangular

trench

Han-Lun Wu1 Shih-Chun Hsiao

2 Hwung-Hweng Hwung

3

1 Ph.D. Student, Department of Hydraulic and Ocean Engineering, National Cheng Kung University 2 Associate Professor, Department of Hydraulic and Ocean Engineering, National Cheng Kung University

3 Professor, Department of Hydraulic and Ocean Engineering, National Cheng Kung University

ABSTRACT

The dynamic response of generated internal waves in a submarine rectangular trench by incident

surface waves is investigated through numerical results, using the three-dimensional numerical model

named Truchas. The volume of fluid (VOF) method is used to trace the free surface and

incompressible flows with multi-fluid interfaces motion. The numerical results are compared with

experimental data. Two typical distinct phenomena of internal waves both were observed in the

numerical results, which are standing internal waves and the traveling internal waves. In this paper,

the numerical model is further employed to investigate the dynamic response of density-stratified

fluid in a submarine rectangular trench under various wave conditions.

Keywords: density-stratified fluid; trench; VOF

波浪通過矩形凹槽內密度分層流體之波動機制數值研究

吳漢倫 蕭士俊 黃煌煇

摘要

本研究應用數值模式模擬並探討波浪通過矩形凹槽內具有密度分層流體時所生成的內波。

應用流體體積法追蹤自由液面和內波界面之運動現象，並和過去的實驗資料做比較藉此探討本

文數值模式之適用性。經由分析結果，在數值和實驗部分皆可以觀察到兩種不同型態的內波現

象 (standing internal waves 及 traveling internal waves)。本文中使用六種不同的波浪條件，進而

探討在不同表面波頻率對於內波運動所造成之反應。

關鍵詞：密度分層、矩形凹槽、流體體積法

1. Introduction

The internal waves can be generated by surface

waves or moveing ship propagate the interface

between two superposed fluids of different densities in

the ocean. The phenomenon of the internal waves is

one of the important engineering problems. However,

a situation may exist in navigational channels and

harbors where bottom is composed of very fine

sediments. It is observed that when the frequency of

the incoming surface waves corresponds to the natural

frequency of the internal waves in the trench, the

amplitude of the internal waves becomes large

compared to the amplitude of the surface waves, so

that large amplitude of internal wave results in large

velocity near boundary, and then enhances the process

of erosion and affects ship maneuverability. The

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density-stratified fluid in these channels can have an

influence on the kinematics around the perimeter of

the trench as a result of internal waves generated in the

trench. Therefore, when waves propagate across

density-stratified fliud in a submarine trench, the

generation of internal wave is of importance topic in

many engineering applications.

The motion of the internal waves in the trench is

a major interest in this study. Thorpe (1968) has made

an extensive study, both theoretical and experimental,

of standing internal waves at the interface of two

fluids and in a continuously stratified fluid. For the

two-layer problem, his method of analysis was similar

to the perturbation scheme in Stokes waves. The

propagation of periodic water waves past a rectangular

trench with a homogeneous fluid in the trench has

been studied by Lee & Ayer (1981) and Kirby &

Dalrymple (1983). Primary interests in those studies

are related to the phenomenon of wave scattering, in

which strong reflection of the incident waves can

occur for suitable dimensions of the trench relative to

the wavelength of the incident waves. It was found

that for a particular symmetric trench where the water

depths before and after the trench were equal, there

existed an infinite number of discrete wave

frequencies at which the incident wave energies were

fully transmitted, the maximum and minimum values

of the transmission and reflection coefficients

appeared periodically, but the effect of the trench on

wave energy transmission decreased monotonically as

the wavelength decreased. Based on the formulation

by Lee & Ayer (1981), Ting & Raichlen (1986)

showed that the wave energies trapped within the

trench were very small compared to the energies in the

incident waves. The excitation of internal waves in a

rectangular trench by normally incident surface waves

has also been studied experimentally and theoretically

by Ting & Raichlen (1988). Their analysis dealt with

small-amplitude waves and an inviscid two-layer fluid

in the trench, whereas fresh water and salt water were

used to create the density stratification in the

experiments. It was observed that at resonance the

amplitude of the internal waves was large compared to

the amplitude of the surface waves. However, the

effects of the internal waves on the surface waves

were not measurable. The theoretical solutions

predicted the wave motions quite well even for

relatively large amplitude waves in the trench.

In the above studies, the stratified fluid in the

trench was created using fresh water and salt water. A

homogeneous fluid was used by Hwung et al., (2009).

Two typical distinct phenomena of internal waves

were captured in the experiments, which are standing

internal waves usually, occurred as the incident waves

range from 0.6~0.8 times the length of trench, and the

traveling internal waves usually occurred outside this

range.

In this paper, the numerical model (Truchas) is

used to simulate the dynamic response of

density-stratified fluid in a submarine rectangular

trench. The results will be discussed and compared

with experimental data including the free surface

elevation and the internal waves.

2. Numerical Setup

The present numerical model, Truchas which is

developed by Los Alamos National Lab (LANL) in the

United States, is the three-dimensional numerical

model. Current version is further refined and added

more modules to extend the application, including

turbulence model, wave generation, and so on. This

model solves Navier-Stokes equations by two-step

projection method with finite volume method (FVM).

In this section, the surface and interface treatment,

boundary conditions and Computational domain are

summarized as follows.

The computational domain of numerical model in

the present study is shown in Fig. 1. Both the free

surface and the interface between water and sugar

fluid are described by volume of the fluid (VOF)

originally developed by Hirt and Nichols (1981) in the

problem. An equation for the filled fraction of each

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numerical cell f is solved in the VOF method, so that f

= 1 in filled cells and f = 0 in empty cells, and 0 < f <1

indicates the interface of each fluid. The piecewise

linear interface calculation (PLIC) was used by Rider

and Kothe (1998) to improve the spatial accuracy of

the VOF method. In the present model, we will adopt

the improved VOF method to trace the free surface

and incompressible flows with multi-fluid (Newtonian

fluid) interfaces motion.

The boundary condition at the numerical tank

bottom and all the rigid body is treated as no-slip

condition that it physically means that fluid is stuck on

the solid wall boundary. Due to the two-dimensional

nature of the problems to be solved, the boundary

condition at the numerical tank sidewalls is treated as

free-slip condition that it physically means the wall is

impermeable and there is no tangential shear stress

generated on the wall. The sponge layer condition is

applied to the upstream and downstream of the

numerical wave tank. The initial conditions of the

velocities, hydrodynamic pressure, and surface

displacements are set to zero at time t=0.

In the Truchas model, the monochromatic waves

are generated through the internal wave-maker by

using the source function inside the numerical domain.

A mass source can be added through a source term in

the mass conservation equation. When the mass is

changed due to the source function, the free surface

will response, and waves will be generated by gravity.

In the study, the internal source is used to generate the

second-order Stokes wave. The corresponding source

function was derived by Lin and Liu (1999). In order

to dissipate wave energy and to eliminate possible

reflected waves, a passive wave absorbing system that

absorbs waves is adopted by using the numerical

sponge layer in the upstream and downstream of the

numerical domain. More detailed about numerical

schemes can be referred to Wu (2004).

Following the experimental setup, the numerical

wave flume is taken to be as shown in Fig. 1. The

structure meshes were used and divided into five

different sub-regions using different mesh sizes in

present computational domain for efficiency. The

finest cells (Δx=5mm and Δy =0.5mm) can be

deployed in region IV for capturing the interface of the

internal waves. In order to save the computational time,

we choose put only one cell in the span-wise direction.

The computational domain is 25.4 m longer than the

laboratory wave flume. This is due to the fact that a

sponge layer has been placed in region I and V in

order to absorb the wave energy.

Fig. 1 Schematic diagram of the numerical wave

flume.(a) x-z plane; (b) x-y plane

Table 1. wave conditions

Case no. T (s) L (m) HS (m) HI (m) kW

Case 1 0.91 1.02 0.0490 0.0054 6.653

Case 2 1.15 1.32 0.0491 0.0099 5.141

Case 3 1.35 1.60 0.0594 0.0243 4.241

Case 4 1.45 1.71 0.0620 0.0385 3.955

Case 5 1.50 1.80 0.0607 0.0255 3.770

Case 6 1.60 1.93 0.0674 0.0241 3.507

3. Results

In the section, the model is applied to simulate

wave over density-stratified fluid in a submarine

rectangular trench and the numerical results are

compared with the experimental data. Various wave

conditions including incident wave period, wave

height and wavelengths of the free surface after passed

the slope are performed numerically (see Table. 1).

Among these cases, the wave conditions of Case1,

Case2, and Case3 were implemented in the experiment.

Particularly, the time history of free surface elevation

and the snaphots of the internal wave were compared

between simulation and experimental data for Case1

x = 25.4 m

Internal

Wavemaker

x = 21.56 m x = 25.4 m

0.3 m

x = 0 m x = 13.8 m

x

x = 16.32 m x = 17.4 mx = 10.3 mx = 2.8 m

1:23

III

z

Sponge

Layer

Sponge

Layer

I

II V

Sugar

Fluid 0.08 m

0.15 m

0.3 m

x = 0 m x = 13.8 m

x

x = 16.32 m x = 17.4 mx = 10.3 mx = 2.8 m

y

Sponge

Layer

Sponge

Layer

Sugar

Fluid1:23 slope

IV

(a)

(b)

1 2 543

1.5 m

0.36 m

Equal spacing

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and Case2. It is because that Case1 and Case2

represent two typical distinct phenomena of internal

waves that are the traveling internal waves and

standing internal waves in the Case1 and Case2,

respectively.

Fig. 2. Comparisons the time histories of the free surface displacement at five locations between experimental

data (○) and computational results (─), respectively. (Right: T=0.91 s, Life: T=1.15 s)

Figure 2 compare the time histories of free

surface elevation at several measuring points shown in

Fig. 1 beween the numerical results and the

experimental data. When the waves pass over the

submerged rectangular trench, the wave shape

gradually changes. The variation of the wave shape in

Case2 is more apparent than that in Case1. More

specifically, the secondary wave crest can be observed

clearly at wave gauge 4 in Case2. This phenomenon

results from the fact that low frequency wave can

experience the topographic change than high

frequency wave as waves propagate over the trench.

Clearly, the computed free surface elevation shows

good agreement with experimental data.

The snapshots of internal waves at four time

instants with equal time interval (t / T =1/4) are

compared between the experimental data and

numerical results as shown in Fig. 3. Because the

images of internal waves in the experiment were only

captured on the right half side of the trench, snapshot

of internal waves in the right half side of trench are

presented. The phase and amplitude of internal waves

exhibit a little discrepancy between numerical results

and experimental data in Fig. 3. We shall emphasize

here that the experimental data were measured after

three minutes of wave generations, whereas numerical

simulation time is stopped when quasi-steady state is

reached. This means that the discrepancy between

them may be in part due to the reflection of wave

flume and re-reflection of wave-maker in the

experiments. It is noted that two typical distinct

phenomena of internal waves both were observed in

the numerical and experimental results, which are the

traveling internal waves and the standing internal

waves in the Case1 and Case2, respectively. The

interface of traveling internal waves in the trench are

10 15 20 25

t (s)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

(

m)

Wave gauge 1

10 15 20 25

t (s)

-0.06

-0.04

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0

0.02

0.04

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(

m)

Wave gauge 2

10 15 20 25

t (s)

-0.06

-0.04

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0

0.02

0.04

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(

m)

Wave gauge 3

10 15 20 25

t (s)

-0.06

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0

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(

m)

Wave gauge 4

10 15 20 25

t (s)

-0.06

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-0.02

0

0.02

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(

m)

Wave gauge 5

10 15 20 25

t (s)

-0.06

-0.04

-0.02

0

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0.06

(

m)

Wave gauge 1

10 15 20 25

t (s)

-0.06

-0.04

-0.02

0

0.02

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0.06

(

m)

Wave gauge 2

10 15 20 25

t (s)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

(

m)

Wave gauge 3

10 15 20 25

t (s)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

(

m)

Wave gauge 4

10 15 20 25

t (s)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

(

m)

Wave gauge 5

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due to the pressure force exerted by the surface waves

on the density interface that displaces the heavier fluid

and brings buoyancy effects into action. Its motion

looked like a ‗ hump‗ of fluid travelling back and forth

inside the trench. However, the standing internal

waves are an exciting oscillation within the submarine

trench. It is found that there are several wave nodes

along the trench. In particular, the standing surface

wave had a antinode at the end wall. The complex

internal wave phenomenon is well captured by present

numerical model, which proves that the numerical

model is capable of simulating such phenomenon and

can be used for further study with confidence.

Fig. 3. Comparisons snapshots of the internal waves at four times between experimental data (○) and

computational results (─), respectively. (Case1: a(1)-a(4) , Case2: b(1)-b(4))

An important aspect of the interaction of surface

waves with a density-stratified fluid in a submerged

trench is the oscillations of internal waves in the

trench. To investigate how the incoming surface

wavelength affects the internal waves, various wave

conditions were performed. In figure 4, the internal

wave height in the trench, HI, normalized by the wave

height of the surface wave after passed the slope, HS,

is plotted as a function of kW., where k is the wave

number of the surface wave in the constant-depth

above false bottom obtained using the linear inviscid

theory; W is the width of the trench. However, not all

wave conditions induce a standing internal wave. The

Zero up-corss method is used to calculate the internal

wave height inside the middle of the trench for the

traveling internal wave, and the maximum quasi-node

under the standing internal wave conditions,

respectively. The numerical results of amplitude ratio

show good agreement with available experimental data

for Case1, Case2, and Case3. It is important to note

that the response curve obtained from numerical

results is extremely peaked with the amplitude ratio

about 0.62 for kW being 3.955, to either side of the

case the response decrease significantly. The similar

16.86 16.92 16.98 17.04 17.1 17.16 17.22 17.28 17.34 17.4

x (m)

-0.02

0.00

0.02

0.04

0.06

0.08

(

m)

16.86 17.04 17.22 17.4

-0.05

0.00

0.05

S(

m)t / T = 1/4

16.86 16.92 16.98 17.04 17.1 17.16 17.22 17.28 17.34 17.4

x (m)

-0.02

0.00

0.02

0.04

0.06

0.08

(

m)

16.86 17.04 17.22 17.4

-0.05

0.00

0.05

S(

m)t / T = 1/4

16.86 16.92 16.98 17.04 17.1 17.16 17.22 17.28 17.34 17.4

x (m)

-0.02

0.00

0.02

0.04

0.06

0.08

(

m)

16.86 17.04 17.22 17.4

-0.05

0.00

0.05

S(

m)t / T = 2/4

16.86 16.92 16.98 17.04 17.1 17.16 17.22 17.28 17.34 17.4

x (m)

-0.02

0.00

0.02

0.04

0.06

0.08

(

m)

16.86 17.04 17.22 17.4

-0.05

0.00

0.05

S(

m)t / T = 2/4

16.86 16.92 16.98 17.04 17.1 17.16 17.22 17.28 17.34 17.4

x (m)

-0.02

0.00

0.02

0.04

0.06

0.08

(

m)

16.86 17.04 17.22 17.4

-0.05

0.00

0.05

S(

m)t / T = 3/4

16.86 16.92 16.98 17.04 17.1 17.16 17.22 17.28 17.34 17.4

x (m)

-0.02

0.00

0.02

0.04

0.06

0.08

(

m)

16.86 17.04 17.22 17.4

-0.05

0.00

0.05

S(

m)t / T = 3/4

16.86 16.92 16.98 17.04 17.1 17.16 17.22 17.28 17.34 17.4

x (m)

-0.02

0.00

0.02

0.04

0.06

0.08

(

m)

16.86 17.04 17.22 17.4

-0.05

0.00

0.05

S(

m)t / T = 4/4

16.86 16.92 16.98 17.04 17.1 17.16 17.22 17.28 17.34 17.4

x (m)

-0.02

0.00

0.02

0.04

0.06

0.08

(

m)

16.86 17.04 17.22 17.4

-0.05

0.00

0.05

S(

m)t / T = 4/4

a (1)

a (2)

a (3)

a (4)

b (1)

b (2)

b (3)

b (4)

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results also were found in previous studies including

experimental and theorical results (Ting & Raichlen,

1986; Hwung et al, 2009 ).

Fig. 4. Variation of normalized internal wave height

with relative wave number for all wave conditions

5. Conclusions

The dynamic reponse of the density-stratified

fluid in a submarine rectangular trench was performed

using the three-dimensional numerical model. The

motion of internal waves as surface wave propagating

over a submarine trench is traced using the VOF

method. Two typical distinct phenomena of internal

waves both were observed in the numerical results,

which are standing internal waves and the traveling

internal waves, and the numerical results have good

agreement with experimental data. Besides, the

interaction between the surface wave and interfacial

motion for various wave conditions was further

performed numerically. It is found that hat the

response curve obtained from numerical results

reaches maximum and the amplitude ratio is about

0.62 for kW being 3.955.

Acknowledgements

This research was supported by National Science

Council in Taiwan and Tainan Hydraulics Laboratory

(THL) for assistance with the experiments (Grant No.

NSC-96-2628-E-006-249-MY3). The authors wish to

express the gratitude to Prof. T. R. Wu for the

assistance on the numerical model.

References

1. Hwung, H. H., Hsu, W. Y., Liu, C. M and Yang, R.

Y., (2009) ―Experimental investigation on the

dynamic response of density-stratified fluid in a

submarine trench‖, Proc. 28th International

Conference on Ocean, Offshore and Arctic

Engineering, Hawaii, Vol. 28, OMAE2009-80059,

pp 591-596.

2. Lee, J. J. and Ayer, R. M., (1981) ―Wave

propagation over a rectangular trench‖, J. Fluid

Mech, Vol.. 133, pp 475-63

3. Lin, P. and Philip Liu, L. F., (1999) ―Internal

wave-maker for Navier-Stokes equations models‖,

Journal of Waterway, Port, Coastal, and Ocean

Engineering, Vol.. 125, No. 4, pp 489-528.

4. Kirby, J T. and Dalrymple, A., (1983)

―Propagation of obliquely incident water waves

over a trench‖, J. Fluid Mech, Vol. 32, pp

489-528.

5. Thorpe, SA., (1968) ―On standing internal gravity

waves of finite amplitude‖, J. Fluid Mech, Vol..

32, pp 489-528.

6. Ting, C. K. F. and Raichlen F., (1986) ―Wave

interaction with a rectangular trench‖, Journal of

Waterway, Port, Coastal, and Ocean Engineering,

Div., ASCE 112, pp 454-460

7. Ting, C. K. F. and Raichlen. F., (1988) ―Wave

interaction with rectangular trench in density

stratified Fluid‖, Journal of Waterway, Port,

Coastal, and Ocean Engineering, Div., ASCE 114,

pp 615-636.

8. Ting, C. K. F., (1992) ―On forced internal waves

in a rectangular trench‖, J. Fluid Mech, Vol.. 114,

pp 615-636.

9. Wu, T. R., (2004) ―A numerical study of

three-dimensional breaking waves and turbulence

effects‖, Ph. D dissertation, Cornell University,

Ithaca.

3 4 5 6 7kW

0

0.2

0.4

0.6

0.8

HI/

HS

Numerical Model

Experiments