Establish random wave surface by a suitable spectrum in the Vietnam’s sea
THE INTERNATIONAL CONFERENCE ON MARINE SCIENCE AND TECHNOLOGY 2016
Establish random wave surface by a suitable spectrum
in the Vietnam’s sea
Nguyen Thi Thu Le,
Le Hong Bang, Do Quang Khai
Vietnam Maritime University,
Abstract
In fact, the sea wave is considered as a stochastic process. The random
characteristics of the sea wave are described by a suitable spectrum. Offshore structures
analyzing on random concept may be determined by creating a random wave surface from a
defined wave spectrum for the sea area of consideration. This paper presents the method to
establish random wave surface in Vietnam’s sea areas based on any suitable wave spectrum.
Hereby, hydro -dynamical characteristics of waves can be determined for further calculation
of wave loads on offshore structures.
Keywords: Gravity wave, spectrum, wave period, wave amplitudes, wave frequency,
wave phases, wave scatter diagram, random wave surface.
1. Introduction
The previous researches indicate that wave load can cause very large dynamic forces
on the offshore structures. So far, Morison equation has been using to analyze wave load
acting on offshore structures [1]. To solve Morison equation, dynamic characteristics of wave
must be determined [2]. The wave are known as a random process because of the difference
of wave amplitudes, wave propagating directions, height of sea surfaces, the roughness of sea
bottoms, topographical characteristics, etc. Random characteristics of wave are described by
a suitable spectrum. This problem was mentioned in foreign scientific researches [3]. In
Vietnam, researches widely concentrate in applying spectral theory for calculation of offshore
structure hydrodynamics. However, in order to solve general problems, this paper propose a
method for further calculation of wave loads on offshore structure in real waves, which is
simulated based on Pierson - Moskowitz spectrum for Vietnam’s sea.
2. Wave characteristics in the Vietnam’s sea
The monsoon, tropical depressions and storms are main reasons affecting waves in
Vietnam’s sea. We can summarize as follows [4].
2.1. The wave in the winter
The wave in the winter often appears from November of the previous year to March
of the following year. The main direction of waves is North - East, the following directions
are the Northward and the Eastward. In the North of Vietnam’s sea, frequency of wave
towards North - East is about 70 - 85% and in the South of Vietnam’s sea, it is about 60 -
75%. In general, the directions of wind and wave towards the North - East are stable and
strong.
2.2. The wave in the summer
The wave in the summer often appears from June to August every year. The wave
direction follows West - South monsoon. The strength of wave towards West - South is
weaker than that toward North - East. In the South of Vietnam’s sea the frequency of wave
towards West - South is about 60 - 70% and in the North about 50 - 60%.
In general, the strength of wave and wind in summer is weaker than that in the winter,
except in case of storms.
HỘI NGHỊ QUỐC TẾ KHOA HỌC CÔNG NGHỆ HÀNG HẢI 2016
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In winter, the average height of wave is 2 - 3 m, period is 7 s - 10 s. In the summer,
the average wave height is about 1,2 m or more, period is 5 s - 9,3 s.
3. Establish random wave surface in Vietnam’s sea
3.1. Wave spectrum
The energy of non-sinusoidal waves is defined by the expression [1]:
π
∞
⁄
2
(
)
E = ρg
Sηη ω, α dωdα
(1)
(2)
∫ ∫
π
−
⁄
0
2
Where Sηη: Wave spectrum
(
)
( ) ( )
Sηη can be expressed as: Sηη ω, α = Sηη ω M α
α: Main wave direction;
g: Acceleration of gravity;
ρ: Water density;
(
)
Sηη ω, α : Directional wave spectrum;
( )
M α ∶ Spreading function;
2
2
( )
Follow Lloyd Germany: M α = cos α
(3)
π
3.2. A suitable wave spectrum is applied for Vietnam’s sea
Different forms of the spectrum at its various generation stages have obtained. Two
such empirical spectra are the Pierson - Moskowitz and JONSWAP spectra. In addition, there
are Gaussian and User defined wave spectra. Wave spectra are introduced in scientific
researches, such as:
( )
-The Pierson - Moskowitz spectrum SPM ω is given by [5, 7]:
−4
5
5
ω
2
4
SPM ω = HSωP. ω−5exp {− 4 (ω ) }
(4)
( )
16
P
Where ω: Wave frequency;
2π
ωP: Angular spectral peak frequency, ωP =
;
⁄
TP
Hs: Significant wave height;
TP: Peak wave period
The Pierson - Moskowitz spectrum is a special case for a fully developed long crested
sea. The former represents fully developed sea states. The Pierson - Moskowitz spectrum is
formulated in terms of two parameters of the significant wave height and the average wave
period.
( )
-The JONSWAP spectrum formula SJS ω is given by [5,7]:
4
2
a
5ω
αg γ
p
( )
SJS ω =
exp (
)
(5)
5
4
ω
4ω
Where ω: Wave frequency;
ωp: Spectral peak frequency;
γ: Peak enhancement factor;
α: Relates to the wind speed and the peak frequency of wave spectrum;
g: Acceleration of gravity.
The JONSWAP spectrum can take into account the imbalance of energy flow in the
wave system (for instance, when seas are not fully developed). Spectral energy imbalance is
nearly always the case when there is a high wind speed.
In analysing offshore structures, assuming that waves propagate on the main
direction, in Vietnam’s sea, North - East is main wave propagating direction. Vietnam’s sea
area is a part of the Pacific Ocean located at the easts of Viet Nam. It is limited by mainland,
HỘI NGHỊ QUỐC TẾ KHOA HỌC CÔNG NGHỆ HÀNG HẢI 2016
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THE INTERNATIONAL CONFERENCE ON MARINE SCIENCE AND TECHNOLOGY 2016
islands, and marine archipelago, but it connects with Indian Ocean and Pacific Ocean. Thus,
Vietnam’s sea represents opened sea, and the waves represent fully - developed waves, wave
characteristics are rather varied. So a Pierson - Moskowitz spectrum can be used to describe
the spectral characteristics in Vietnam’s sea.
3.3. Analyze wave parameters
Facts indicate that wave is caused mainly by wind. The wind blows on the water
surface. Friction between the air and water particle, gravitation of water have been causing
waves. These waves are called gravity waves.
We consider drift of water elements at (x, z) coordinates, the horizontal and vertical
components water particle velocity (u and w) as well as those of acceleration (ax and ay) are
defined [4,5]:
푎
푔푘
(
)
(
)
푢 = 휔 푐표푠ℎ 푘푑 cosh 푘 푧 + 푑 . cos 푘푥 − 휔푡
(6)
(7)
(8)
(9)
푎
푔푘
(
)
(
)
푤 = 휔 푐표푠ℎ 푘푑 sinh 푘 푧 + 푑 . sin 푘푥 − 휔푡
푔푘
(
)
(
)
푎푥 = 푎. cosh 푘푑) . cosh 푘 푧 + 푑 . sin 푘푥 − 휔푡
(
푔푘
(
)
(
)
푎푧 = −푎. cosh 푘푑) . sinh 푘 푧 + 푑 . cos 푘푥 − 휔푡
(
Where a: Wave amplitude;
ω: Wave frequency;
k: Wave number;
d: Depth of the water;
x: Horizontal coordinates
z: Vertical coordinates
3.4. The random wave surface by using calculation program
Wind generating ocean wave is random in nature. Normally it is described
mathematically as the summation of a large number of sinusoids.
( )
Consider a random wave surface η( ), wave spectrum 푆 ω of η( ) can be expressed
x,t
x,t
as equation (4)
The random water surface can be expressed in complex value form as [5, 6]:
푁
(
)
∑
(
)
η 푥, 푡 = ꢀ=1(푎ꢀ. cos 푘ꢀ. 푥 − 휔ꢀ. 푡 + 훼ꢀ )
(10)
η( ): Free surface waves;
x,t
N: Number of waves;
ai: Wave amplitude;
ωi: Radian wave frequency;
ki: wave number;
αi: Phase angle of wave components.
Values of wave spectrum are given in distance from ωs to ωf with n period ∆ω.
휔푛 = 푛. 훥휔
Base on wave data from wave scatter diagrams, if significant wave height 퐻s and
average wave period 푇 are known
o
By default, starting frequency and finishing frequency are defined [6]:
Starting frequency (in rad/s): 휔푠 = 0,58. 2푇휋
표
Finishing frequency (in rad/s): 휔푓 = 5,1101. 2푇휋
표
( )
From equation (4), 푆PM ω will has been defined
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When ∆ω is tiny, approximates ωi follow [6]:
2
( )
0,5푎ꢀ = 푆 휔ꢀ 훥휔
(11)
(12)
( )
Thus: 푎ꢀ = √2. 푆 휔ꢀ . 훥휔
2휋
휔
With ωi of wave component, the vibration period is defined as [6] 푇 =
;
2
푔.푇
The wave length is defined as [6] 퐿 =
tanh(푘푑);
2휋
2휋
The wave number is defined as [6] 푘 =
;
퐿
The random phase angle αi of a wave component is distributed in range of [0;2π], is
determined by function in MATHCAD.
Therefore random wave surfaces are established.
Example:
Base on wave scatter diagrams in the South of Vietnam’s sea (area 40 according to
Global Wave Statistics [4]).
Significant wave height (for 50 years record) 퐻s = 8,5m;
Average period 푇 = 7,5 푠 [1]
표
These figures show the results by using calculation program, which we established.
4
3
S_PM()
2
1
0
1
2
3
4
Figure 1. Pierson-Moskowitz spectrum
4
2
0
(x0)
2
4
3
0
200
400
600
800
110
x
Figure 2. Extension of random wave surface record
3.5. Establish dynamical parameters of random wave by using calculation program
Horizontal water particles velocity:
푎
푔.푘
푖
푖
푁
(
)
∑
(
)
(
)
푢 푥, 푧, 푡 = ꢀ=1 [휔 . cosh 푘 .푑) . cosh 푘ꢀ. 푧 + 푑 . cos 푘ꢀ. 푥 − 휔ꢀ. 푡 + 훼ꢀ ]
(13)
(
푖
푖
Vertical water particle velocity:
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THE INTERNATIONAL CONFERENCE ON MARINE SCIENCE AND TECHNOLOGY 2016
푎
푔.푘
푖
푖
푁
(
)
∑
(
)
(
)
푤 푥, 푧, 푡 = ꢀ=1 [휔 . cosh 푘 .푑) . sinh 푘ꢀ. 푧 + 푑 . sin 푘ꢀ. 푥 − 휔ꢀ. 푡 + 훼ꢀ ]
(
푖
푖
(14)
Horizontal water particle acceleration:
푔.푘
푖
푁
(
)
∑
(
)
(
)
푥 푥, 푧, 푡 = ꢀ=1 [푎ꢀ. cosh 푘 .푑) . cosh 푘ꢀ. 푧 + 푑 . sin 푘ꢀ. 푥 − 휔ꢀ. 푡 + 훼ꢀ ]
(15)
(16)
(
푖
Vertical water particle acceleration:
푔.푘
푖
푁
(
)
∑
(
)
(
)
푎푧 푥, 푧, 푡 = − ꢀ=1 [푎ꢀ. cosh 푘 .푑) . sinh 푘ꢀ. 푧 + 푑 . cos 푘ꢀ. 푥 − 휔ꢀ. 푡 + 훼ꢀ ]
(
푖
Figure 3. Water particle velocity and acceleration components
4. Conclusion
Base on wave data from wave scatter diagrams in Vietnam’s sea, this paper shows
that Pierson - Moskowitz spectrum is a suitable spectrum to describe random wave surfaces
in Vietnam’s sea. Using MATHCAD calculation program, the random wave surface records
had establishing by Pierson - Moskowitz spectrum. The random wave surface based on the
record of South Vietnam’s sea had presenting, too. Therefore the dynamic characteristics of
random wave: water particle velocity and acceleration are determined. The result of this study
will combine with Morison equation in establishing the program which in order to calculate
the wave loads on offshore structures.
References
[1]. Nguyễn Xuân Hùng, Động lực học công trình biển, NXB Khoa học và kỹ thuật, Hà
Nội, 1999.
[2]. Joseph W. Tedesco, William G. McDougal, C. Allen Ross, Structure dynamics,
Addison-Wesley, 1998.
[3]. Aqwa User’s Manual Realease 16.0, 2003.
[4]. Vũ Uyển Dĩnh, Môi trường biển tác động lên công trình, NXB Xây dựng, Hà Nội,
2002.
[5]. J.M.J Journee and W.W.Massie, Offshore hydromechanics, Delft university of
Technology, 2001.
[6]. Deo M C, Wave and structures, Indian institute of technology Bombay Powai
Mumbai, 2013.
[7]. Frigaard, Peter Bak, Hogedal, Michael, Christensen, Morten, Wave Generation
Theory, Aalborg University Denmark, 1993.
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