2017 5th International Conference on Mechanics and Mechatronics (ICMM 2017)
ISBN: 978-1-60595-541-4
aCorresponding author: [email protected]
Design of Single-Point Mooring System with Moored Buoy
Jun WANGa, Yi TANG, Zhu-Yuan YANG and Yu-Mei SHE
School of Mathematics & Computer Science, Yunnan Minzu University, Kunming, Yunnan, China
Abstract. The single-point mooring system is currently under a wide range of applications in marine engineering,
marine observation, marine farming and other engineering fields. So this paper intends to investigate into the single-
point mooring system with moored buoy that is made up of a buoy, four steel pipes, a steel drum, a heavy ball, an
anchor chain and an anchor. The aim here is to design a single-point mooring system mentioned above, which can be
used off shore. Namely, the following parameter values have to be determined: the type and length of chain, the mass
of compact ball, such that do not only both the buoy draught and the moving range, but also the vertical angle of steel
drum reach as small as possible. For the above purpose, firstly, analyse each component of this system by means of
the static force method. Secondly, establish a mathematical model to calculate under the different wind speeds all the
important parameters, including the vertical tilt angle of the steel drum, the horizontal tilt angle of each steel pipe, the
buoy draught, the buoy moving range and the chain shape. Thirdly, on the basis of the above work, a multi-objective
programming model is established to obtain the design schemes in the diverse environmental conditions involving
wind speed, seawater velocity and sea depth. Finally, by simulation it has turned out that these design schemes
obtained above are reasonable and applicable. Therefore, they have definite theoretical significance and practical
values.
1 Introduction
The single-point mooring system with moored buoy is
one of the systems studied earliest by people. And it is
used widely in marine engineering, marine observation,
marine breeding and other fields because of its simple
structure, good direction and low cost. At present, the
study on single-point mooring system with moored buoy
focuses mainly on two aspects: the computational
analysis of and the stability of system. The former mainly
considers tension response of mooring cables and
coupling motion of mooring system. When computational
analysis is performed on the cables, they are generally
recognized as a flexible structure, no withstanding shear
stress, thereby no transmitting torques. The latter studies
mainly the bifurcation and chaotic motion of system and
other dynamic problems. The computational analysis can
be subdivided into the static method and the dynamic
method. The static method [1] neglects the inertial force
of mooring system and involves catenary method, neutral
buoyancy cable method, polygon approximation method,
etc. Smith et al. [2] used the catenary method to do the
static analysis to the mooring system that included two
cables. Therefore, they converted the computational
problem into solving a multi-degree-of-freedom
polynomial equation, which could be solved quickly by
computer. Pan et al. [3] proposed the two-dimensional
static model for a single-point mooring system with
moored buoy, in which the deformation of mooring line
and the change of seawater flow speed were considered.
However, this model was only effective on catenary,
semi-tensioned and tensioned single-point buoy systems.
Wang [4] established the two-dimensional static model
for underwater submerged buoy system, on which the
effect of seawater flow was considered but on which the
effect of the elastic deformation of cable was neglected.
By the static equilibrium, the author calculated some
parameters such as seawater depth, tension, and
inclination tilt angle, horizontal offset and so on, and
developed the calculation software for system design and
deployment. Tang et al. [5] used centralized mass method
for modelling and proposed the calculation method of
mooring tension for deep ocean mooring system. This
method considered gravity, buoyancy, tension, water
current force, submarine support force etc. The results
showed that the greater change of the seabed topography
had a certain influence on the tension of the mooring line.
Lan et al. [6] described in detail the static calculation
steps and methods of submerged buoy system, and
posture analysis. They considered the elongation of cable,
established a three-dimensional static model, and pointed
out that the depth of the submerged buoy system had to
iteratively calculate. Zhang et al. [7] calculated the
tension-span curve for each anchor line based on one
dimensional optimization thought and the catenary
equation method. And the restoring force of mooring
system was calculated by Lagrange interpolation method.
However, this model was only applicable on catenary and
semi-tensioned single-point buoy systems. Wang [8]
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applied the polygon approximation method based on the
concentrated mass on the static analysis of cable single-
point mooring system and established the static model.
They took into account the elastic stiffness of the cable,
analysed the stress characteristics of typical system
components, including buoy, and established the static
equilibrium equation. For the initial conditions of
uncertainty, the iteration method was used to design the
calculation flow, and the static calculation program was
programmed. The feasibility of this method was verified
by the static calculation of a loose type single point buoy
system. Subsequently, the mooring system was carried
out dynamics analysis and modelling. Wang et al. [9]
studied the deep-ocean single-point mooring system with
moored buoy, performed the static analysis, and
established the static model in which the change of
seawater flow speed with the depth of the water was
taken into account. Recently, some researchers [10-13]
built up the dynamical models for all kinds of mooring
systems to analyse their nonlinear dynamical properties
because of the actual requirements of engineering.
In this paper, an offshore single-point mooring system
with moored buoy is investigated. It is made up of a
buoy, four steel pipes, a steel drum, a heavy ball, a chain
and an anchor. The aim here is to determine the type and
length of chain, and the mass of heavy ball to ensure the
buoy draught, the buoy moving range and the vertical tilt
angle of steel drum are all as small as possible. For this
purpose, the statics analysis is done to the above system
by static method, meantime considering the wind speed,
the seawater speed and the moment of force generated by
four steel pipes and steel drum. Then, to obtain the
optimal design schemes, construct a few mathematical
models to analyse and optimize all kinds of parameters,
including the buoy draught, the buoy moving range and
the vertical tilt angle of steel drum. In the end, it is
verified by simulation that our obtained design schemes
of single-point mooring system with moored buoy are
both reasonable and applicable; thereby they have a
certain practical significance and reference value.
2 The Components of System
2.1 Description of the System Investigated
The seawater depth of the offshore observation network
is between 16m to 20m. Its transmission nodes are
composed of buoy, mooring and underwater acoustic
communication systems. The buoy system is simplified to
a cylinder with mass 1000kg, bottom diameter 2m and
height 2m. The single-point mooring system with moored
buoy consists of four steel pipes (for short “pipe” below),
a steel drum (for short “drum” below), a heavy ball (for
short “ball” below), a welded chain (for short “chain”
below) and an anti-drag anchor (for short “anchor”
below). Each pipe is 1m long, 50mm in diameter, and
10kg in mass. The anchor is 600kg in mass. Usually, a
chain uses common chain link to connect. Its types used
often and the parameters are shown in Table 1. If the
angle between the seabed and the tangential direction of
chain at its end does not exceed 16°, the entire mooring
system will stay there without moving. Otherwise, the
anchor will be dragged such that the nodes are shifted and
hence lost. The underwater acoustic communication
system is installed in a sealed cylindrical drum whose
length and diameter are 1m and 30cm respectively, and
the mass of the whole device is 100kg. The top of drum is
connected with the 4th pipe and its end is connected with
the chain. If the drum is in a vertical state, the underwater
acoustic communication system works best. If it is tilted,
the effect of communication system gets worse. As the
angle between it and the vertical line (referred as vertical
tilt angle) exceeds 5°, the communication system works
very poorly. In order to control the size of vertical tilt
angle, we can hang a heavy ball at the junction of drum
and chain (See Figure 1).
Table 1. Chain types and parameters.
Type (mm)l (kg/m) Type (mm)l (kg/m)
I 78 3.2 IV 150 19.5
II 105 7 V 180 28.12
III 120 12.5
Note 1: The length refers to the one of each chain link.
Steel
Steel Drum
Vertical Line
1
2
3
4
5
Buoy
Heavy Ball Anchor
16 20m m
Sea Level
Seabed
Anchor Chain
3L
1L
2L
4L
5L
Figure 1. The system studied and some symbol descriptions.
2.2 Symbols Description
For the convenience of simplicity, some symbols need to
be introduced. The meaning and its unit of every symbol
(if it existed) are shown in Table 2.
2.3 Fundamental Assumptions
For the sake of simplicity, we gave the following five
fundamental assumptions.
Hypothesis 1: As the mooring system reaches a static
equilibrium, the buoy, 4 pipes, drum, ball, chain and
anchor are all in the same plane.
Hypothesis 2: The volumes of chain can be negligible,
hence both its buoyancy and water current force are zero,
i.e., 6F 6 0A .
Hypothesis 3: 20.625 (N)wF v S approximately.
Here, 2(m )S is the projected area of object on the plane
198
perpendicular to the wind direction, and (m/s)v is the
wind speed.
Table 2. Symbol descriptions.
Symbol Units Meaning
g 2m/s The gravitational acceleration, 9.81g .
H m The water depth.
3kg/m The sea water density, 30251. × 10 .
kg/m The linear density of chain.
Ac
N The current force.
A N The water current force acting on ball.
0 7~A A N The water current force acting on the buoy, the 1st~4th pipe, drum, chain and anchor in sequence.
0 7~m m kg The mass of the buoy, the 1st~4th pipe, drum, chain and anchor in sequence, with 1 2 3 4m m m m .
0 7~G G N The gravity of the buoy, the 1st~4th pipe, drum, chain and anchor in sequence.
1 7~F F N The buoyancy of the 1st~4th pipe, drum, chain and anchor in sequence.
1 5~L L m The lengths of the 1st~4th pipe and drum in sequence, with = , 1 5iL L i .
1 5~D D m The diameters of the 1st~4th pipe and drum in sequence, with =1 2 3 4= =D D D D .
1 5~ The horizontal tilt angles of the 1st~4th pipe and drum in sequence.
0 , ,P Q P The locations of the top 0 0( , )x y , the end 0( , )X -H and arbitrary point ( , )x y of chain respectively.
, , The angles of the sea level and the tangential line at 0, ,P P Q respectively.
0,s s m The lengths of chain from Q to P and to 0P respectively, 0 0s l X .
,h R m The buoy draught and the buoy moving range respectively.
0 0,H D m The height and diameter of buoy respectively.
6,T T N The pulling force of drum acting on chain and on ball respectively.
,G F N The gravity and buoyancy of ball respectively.
7T N The pulling force of chain acting on anchor.
,v u m/s The wind speed and seawater current speed respectively.
( , )x yT N The tension of chain at P .
1T N The pulling force of buoy acting on the first pipe.
iT N The pulling force of the (i-1)-th pipe acting on the i-th pipe, 1 4i .
5T N The pulling force of the 4-th pipe acting on drum.
i
The acute angle formed by the line on which Ti lies and the sea level, 1 4i .
199
The vertical tilt angle of drum, 52
.
0( )F h N The buoyancy of buoy, it is a function with respect to h .
Fw
N The wind force acting on the buoy.
l m The length of chain.
0X m The length of chain “lying down” on the seabed.
m kg The mass of ball.
S 2m The projected area of the object in the normal plane.
( )y f x The shape of chain.
i sin , 1 5T i
i i i
Hypothesis 4: 2374 (N)cA u S approximately. Here,
2(m )S is the projected area of object on the plane
perpendicular to the water current direction, and
(m/s) is the seawater flow speed.
Hypothesis 5: Ocean current is a planar flow field,
and u does not change with the water depth; hence, it is
a constant [9].
Hypothesis 6: The gravity of ball is much larger than
its buoyancy and water current force. So, its buoyancy
and water current force can be negligible, i.e.,
0F A .
Hypothesis 7: The water current force of each
component acts on its centroid.
2.4 The Establishment of Coordinate System
As is shown in Figure 2, a planar coordinate system is
established. According to Hypothesis 1, there exists the
plane where the system reaches the static equilibrium
state. We take it as the coordinate plane.
Take the intersecting line of the coordinate plane and
the sea level as the X-axis whose positive direction is just
the wind direction. And the Y-axis, whose positive
direction is just sticking straight up, passed through the
joint of anchor and chain and is perpendicular to the X-
axis. The intersection of the X-axis and Y-axis is the
origin denoted by O .
H
y
x
R
X
Y
,p x y
0
Anchor
Buoy
0X
Anchor Chain
Sea Level
Seabed
h
Figure 2. The coordinate system.
2.5 Design Process
Step 1: Transmission node uses Type 2 of chain whose
length 22.5ml and the mass of ball m 1200kg .
The anchor is placed on the flat seabed where the water
depth 18mH and its density
3 31.025 10 kg/m . Assuming that 0u , construct a
model to calculate all the parameter values , , , ih R
and determine the shapes ( )f x of chain as 12m/sv
and 24m/sv respectively.
Step 2: According to Hypotheses 1-7, calculate all the
parameter values , , , ih R and determine the shapes
( )f x of chain as 36m/sv . If the anchor is dragged
or the communication system works poorly, then adjust
the mass of ball m, satisfying the conditions that
0 5 & 0 16 .
Step 3: The device is placed in the sea where the
water depth is between 16m and 20m and the maximal
water current speed 1.5m/su and the maximal wind
speed 36m/sv . The design scheme of mooring
system is given according to different wind speed, water
current speed and seawater depth.
Step 4: Simulation. For the designed mooring system,
simulate the static equilibrium system in the following
three cases of 12,24,36m/sv respectively and u
1.5m/s . Namely, determine the reasonability of the
parameter values , , , ih R and the shapes of chain in
the three cases above, as the mooring system is equalized.
Subsequently, verify whether the parameters meet the
demands of objective reality. If not, the design scheme
will be redesigned.
200
3 The Implementation of Step 1
The force analysis of each component is done and then
their force and torque equilibrium equations are achieved
by the above fundamental hypotheses. And so they
constitute a system of equations, through which the
expressions of , , , (1 4)ih R i , and ( )f x can
be deduced. Model 1 is constructed using these
expressions. Then Model 1 is solved and all the
parameter values of , , , ih R and ( )f x is obtained
under different conditions. Now let’s begin with the force
analysis on each component.
3.1 Force Analysis of Each Component
3.1.1 The Force Analysis of Buoy, Pipe and Drum (Including Ball)
By Hypotheses 2 and 6, and Figures 3-5, the force
equilibrium equations of the buoy, 4 pipes (1 4)i
and drum are respectively obtained as follow.
1 1 0 0sin ( )T F h G (1)
1 1cos wT F (2)
1 1sin sini i i i i iT T G F (3)
1 1cos cosi i i iT T (4)
6 5 5 5 5sin sinT T F G G (5)
6 5 5cos cosT T (6)
Further, select respectively as the fulcrum the centroid
C of each pipe and of the drum. Then by Hypothesis 2
and Figures 4-5, their torque equilibrium equations are
also respectively obtained as follow.
1 1sin( ) sin( )i i i i i iT T (7)
5 5 5 6 5 5sin( ) sin( ) cos 0T T G (8)
Wind Direction
Sea Level1
h1T
0F h
0G
wF
Figure 3. Force analysis of buoy.
1
2
1
2
2T
LC
1T1
F
1G
2
Figure 4. Force analysis of the first pipe.
7T
G
3F
6G
5T
5
5
6T
C
7T
Figure 5. Force analysis of drum and ball.
3.1.2 The Global Force Analysis of Chain
The force of chain can be analysed according to the
following two cases:
0 0, 0X ; (2) 0 0, 0X .
By Figure 6, it follows that 0 0s l X . So by
Hypothesis 2, the chain has the following global force
equilibrium equations.
7 6 0sin sin ( )T T g l X
(9)
1 1cos wT F (10)
3.1.3 The Local Force Analysis of Chain
By Figure 7, the local force equilibrium equations are got
as follow.
0
2
01 , 0x
Xs y dx s l X (11)
7 ( , )sin sinx ygs T T
(12)
7 ( , )cos cosx yT T (13)
0 0
, tanx X x Xy H y (14)
Note 2: If 0 0X , then 0 .
201
6T
7T
0gs
0s
0
1 0, 0X
0gs
0s
0
2 0, 0X
0X
0 0 0,P x y
0 0 0,P x y
7T
6T
Figure 6. The global force analysis of chain.
,x yT
7T
gs
0
1 0, 0X
s
0
2 0, 0X
0X
,x yP
gs
,x yT
,x yP
s
7T
Figure 7. The local force analysis of chain.
3.2 Some Basic Relations
By Hypothesis 3 and some basic physical knowledge, the
following basic relations hold.
2
0 00.625 ( )wF D H h v (15)
2
0 0( ) 4F h gD h (16)
24 , 1 5i iF gD L i (17)
6 6
, 1 5;
,
i iG m g i
G m g gl G mg
(18)
3.3 The Establishment of Model 1 and the Results on Solution 3.3.1 The Expression of Relevant Parameters
By Equations 0 0s l X and (1) (18) , the
expressions of , , , (1 4)ih R i and the shape
( )f x of chain are obtained immediately. Let’s discuss
them as follow.
(a) Case 1: 0v . At this time, 0wF . It can be seen
easily that 2 (1 4)i i and 0 . So, it
has the following results :
0 1 5
2 2 2 2
0 1 5 0
00 2
0
00
00
4
4 5 4
4
5
m m m m l
D L l D L D L D HX
D
h H L l X
R X
And the shape ( )f x of chain is composed of the
following two line segments 1C and 2C .
0000
1 2
00
0: :
x Xx t t XC C
y t H t X ly H
(b) Case 2: 0v . Then, 0wF . So it has the
following system of equations.
1 5 0 0 0
0 1 5 6
0 2 2
0 0
4 ( )
4
/ 4 0.625tan
F F F H gX
G G G G Gh H
gD D v
(19)
1 warctan 2 , 1 4i i i F i (20)
5 5 5 5 warctan 2 2F G F (21)
52 (22)
5
0
1
sin i
i
y h L
(23)
wa g F (24)
2
0 01 ( sec ) 1 tans a ay aH (25)
0 0X l s (26)
0 0
2
0 0(tan ) 1 tan1ln
sec tan
x X
s a s a
a
(27)
5
0
1
cos i
i
R L x
(28)
0 00, 0; 0, 0X X (29)
0
0
0 0 0
, 0 ;
seccosh ( )
tansinh ( )
sec,
H x X
a x Xa
ya x X X x x
a
Ha
(30)
Note 3: sini i iT (1 5)i can be obtained by
Equations (1) and (3).
202
3.3.2 The Establishment of Model 1
Now, only consider the case 0v . Combining
Equations (1), (3), (15), (16) and (19) ~ (30), Model 1 is
obtained.
3.3.3 The Solution of Model 1
For Model 1, using the step-search method, solve all the
parameter values , , , (1 4)ih R i as the mooring
system is equalized under different wind speed. Further,
one can ascertain ( )f x . So, one has the following
conclusion.
The mooring system does not shift as 12m/sv and
24m/sv respectively. At this time, all the parameter
values are shown in Table 3, and the shapes ( )f x of
chain shown in Figure 8.
4 The Implementation of Step 2
4.1 Determine the State of Mooring System
4.1.1 The Condition for Mooring System in Static Equilibrium State
The necessary and sufficient condition that the mooring
system reaches a static equilibrium state is 0 0s X l .
4.1.2 Determine the State of Mooring System
Assuming that the mooring system is in a static
equilibrium state as 36m/sv , employ the way used in
Model 1 and finally get 0 0X &0 0.6480s l ,
which does not meet the condition of mooring system in
static equilibrium state.
Table 3. The results of Model 1 ( 12m/sv and 24m/sv ).
(m/s)v 12 24 (m/s)v 12 24
( )h m 0.7348 0.7489 ( )1 89.0236 86.2677
( )R m 14.2983 17.4199 ( )2 89.0178 86.2465
( ) 1.0073 3.8460 ( )3 89.0120 86.2250
( )4 89.0061 86.2033
Figure 8. The shapes ( )f x of chain ( 12 m /s 24 m /sv = ,v = ).
4.2 Determine the Best Mass of Ball
4.2.1 The Establishment of Model 2
According to the above force analysis, getting a model
determining the smallest mass of ball, which is shown
below, called Model 2.
0 0 1 5 6
1 5 0
min 4
tan 4w
G F h G G G G
F F F gX
(31)
0 0
0 5
. . 0 16
1200 ,
(1), (3), (15), (16), (20) (28)
X
s t
m M G mg
Take 2000M as the stop condition for the step-
search algorithm solving Model 2.
4.2.2 The Solution of Model 2
As 36m/sv , the least mass of ball should be
1802.7kg , and at this time, 4.9178 5 and
14.15 16 . The other parameter values are
shown in Table 4.
Table 4. The results of Model 2 ( 36m/sv ).
(m/s)v 36 (m/s)v 36
(m/s)v 36
(m)h 0.9504 ( )1 85.1815 3
( ) 85.1440
(m)R 18.485 ( )2 85.1628 ( )4 85.1250
5 The Implementation of Step 3 — the Design of Mooring System
The design of the mooring system is to determine the
type and length of chain, and the mass of ball, such that
not only do the buoy draught and moving range reach the
minimum, but also the vertical angle of drum do so if the
203
mooring system is in an equilibrium state. Therefore, it is
a multi-objective programming problem.
Based on practical applications, the water current
force has to be taken into account in the system design.
So, it is necessary to make the following modification on
the force equilibrium equations of the buoyant, the drum,
and all the pipes.
5.1. The Modified Equation
5.1.1 The Force Equilibrium Equations of the Buoyant, 4 Pipes and the Drum
There being the water current force, i.e., 0u , so,
according to Hypotheses 2-7, only the horizontal force
equilibrium equations of the buoyant, the 4 pipes and the
drum, i.e., Equations (2), (4) and (6), have to be modified
as follow.
1 1 0 wcosT A F
(32)
1 1cos cos , 1 4i i i i iT T A i (33)
6 5 5 5cos cosT T A (34)
Here,
2
0 0374A D u h (35)
2374 sin , 1 5i i iA D u L i (36)
5.1.2 Modification of the Corresponding Equations
Because of Equations (32), (33) and (34), the other
corresponding equations have to be modified as follow.
1
w 0
arctan ,2
1 4
i ii
i iF A A A
i
(37)
5 5 5
5
w 0 5 5
2arctan
2
F G
F A A A
(38)
w 0 1 5
ga
F A A A
(39)
0 0 1 1 5 5
5
6 0
1
4 4
tan w i
i
G F H G G F F G
F A G gX
(40)
0 1 5 6 1 5
52
0 0 0
1
2 2
0 0
2
0
4 4
tan 0.625
4 0.625 tan
374 tan
i
i
G G G G G F F
gX H D v A
hgD D v
D u
(41)
5.2 The Establishment and Solution of Model 3
5.2.1. The Establishment of Model 3
A multi-objective programming model is built up as
follows.
min ,min &minh R
0 0
0 5. .
0 16
(1), (3), (15), (16), (22) (28), (35) (41)
X
s t
According to Equation (41), for any given ,h is
increasing with respect to G . Similarly, for any given
G , h is increasing with respect to as 36m/sv &
u 1.5m/s . So, convert the multi-objective
programming to the following single-objective
programming.
1 2
min16 5
G RW
M M
0 0
0 5. .
0 16
(1), (3), (15), (16), (22) (28), (35) (41)
X
s t
Here, take 1 6000M & 2 40M .
5.2.2 The Solution of Model 3 and the Design Schemes of Mooring System
Due to 0 5 , it has sin 1i and hence,
2374 , 1 5i iA D u L i . It means that the influence
of i on iA can be negligible (1 5i ).
Set 36m/sv and /s1.5mu in Model 3. If
16,17,18,19,20mH respectively, and 3.2,7,
12.5, 19.5,28.12kg/m , then still use the step-search
method to solve Model 3. Finally, get 25 sub-optimal
solutions of Model 3 under the above 25 kinds of cases.
Select 5 optimal solutions from them as the design
schemes of mooring system, which are shown in Table 5.
204
Table 5. The design schemes of mooring system.
(m)H
16 17 18 19 20
(kg/m)
28.12 28.12 28.12 28.12 28.12
Type V V V V V
(m)l
18.532 19.760 20.964 22.148 23.315
(kg)m
3729.5 3694.9 3661.1 3627.8 3595.0
(m)h
1.6490 1.6490 1.6490 1.6490 1.6490
( )
4.9707 4.9707 4.9707 4.9707 4.9707
(m)R
15.605 16.317 16.988 17.621 18.221
6 The Simulation
In order to verify the theoretical design schemes in Table
5, the simulation is done to observe all the parameter
values as 1.5m/su and 12,24,36m/sv
respectively. Here, it should be pointed out that in the
simulation, the changes of water current force acting on
the 4 pipes and the drum are not taken into account, and
that they are regarded as the maximum values of their
own water current forces instead. Here, the shapes of
chain at different speeds are shown in Figure 9
respectively for the seawater depth 16m in the design
scheme.
Figure 9. The shapes ( )f x of chain.
( 12m/s, 24m/s& 36m/sv v v )
Conclusions
Under the same seawater depths, all the parameters meet
the practical requirements, judging from all the obtained
parameter values, including the shapes of chain, at
different speeds based on the above simulation. So, from
the view of static analysis, the simulation results have
shown that the theoretical design schemes of mooring
system in Table 5 are reasonable and applicable, even if
the hurricane reaches to 36m/s and the seawater current
does 1.5m/s. So they have a certain practical significance
and reference value.
Acknowledgement
This work is financially supported by National Natural
Science Foundation Projects (61462096&11361076),
Yunnan Provincial Department of Education Science
Research Fund Project (2016YJS078), and Yunnan
Minzu University Overseas Master’s Program (3019901).
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