Hydrodynamics of Unconventional SWATH Vessels
in Waves
by
MASSACHUSETTS INSTrTuTE
OF TECHNOLOLGY
Abiodun Timothy Olaoye
JUL 3 0 2015
B.S., Petroleum & Gas Engineering
University of Lagos (2010)
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Submitted to the Department of Mechanical Engineering
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Master of Science in Mechanical Engineering
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@ Massachusetts Institute of Technology 2015. All rights reserved.
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-- --....................---- -- -- -- -- -Abiodun Timothy Olaoye
Department of Mechanical Engineering
May 7, 2015
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Certified by.................................................
Assista
Stefano Brizzolara
Research Scientist and Lecturer
irector for Research at MIT Sea Grant
4 T0hesis Supervisor
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A ccepted by ..............
.......
. ........................
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Chairman, Committee on Graduate Students
Department of Mechanical Engineering
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Hydrodynamics of Unconventional SWATH Vessels in Waves
by
Abiodun Timothy Olaoye
Submitted to the Department of Mechanical Engineering
on May 7, 2015, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
The motion responses of unconventional Small Water-plane Area Twin Hull (SWATH)
vessels are unique in the sense that viscosity has significant non-linear effects on their
hydrodynamic parameters. The parametric optimization of the hull shape of these
vessels to reduce the total resistance in waves yields an interesting hull form where
viscous effects become significant and this kind of problem is generally more difficult
to solve. This study aims to investigate the motion response of these special kind of
ships in waves using both numerical and experimental approach with some theoretical
simplifications to better understand the hydrodynamics of the ship. The two modes of
motion of interest in this study are heave and pitch motions which were chosen in order
to focus on the degrees of freedom which significantly contributes to the resistance of
the ship in head waves. The vessel under investigation is an unmanned surface vessel
(USV) proposed to be used to monitor a team of autonomous underwater vehicles. A
scaled version of this model is built and some experiments were conducted at the MIT
towing tank at zero speed. Additionally, the numerical methods are implemented
using 2D and 3D potential flow solvers. As this is an ongoing project, the results
obtained so far including the study of the effects of the inertial characteristics of the
ship on the response amplitude operator (RAO) are presented.
Thesis Supervisor: Stefano Brizzolara
Title: Research Scientist and Lecturer
Assistant Director for Research at MIT Sea Grant
2
Acknowledgement
To almighty God be all the glory for the good health and sound mind that sustained
me through this Masters program.
With a feeling of accomplishment, I would like to express my sincere gratitude and
appreciation to the Federal government of Nigeria and particularly to the President
and Commander-in-Chief of the Nigerian Armed forces, President Goodluck Ebele
Jonathan (GCFR) for initiating the Presidential Scholarship Scheme for Innovation
and Development (PRESSID) under which I was fully funded to pursue a Master
of Science degree in Mechanical Engineering at MIT. I would also like to thank the
PRESSID implementation committee at the National Universities Commission, Abuja
for every support that I have received through out this Masters program.
Additionally, this thesis was actualized with the academic support provided by my
Advisor, Professor Stefano -Brizzolara and other members of the MIT i-Ship group
including Luca Bonfigilo and Guliano Vernengo. The experimental part of the thesis
carried out at the MIT towing tank was aided by the following research students:
Jacob Israelevitz, Jeff Dusek, Amy Gao, and Stephanie Steele. Also, I would like
to thank Giovani Diniz, and Heather Beem for their friendliness which really helped
me through out the program. Finally, I would like to thank my family especially my
parents, Mr. Joel Adebisi Olaoye and Mrs. Oluyemisi Beatrice Olaoye for their moral
support in the course of this program.
3
Contents
Title Page
1
Abstract
2
Acknowledgement
3
Table of Contents
4
List of Figures
6
List of Tables
7
List of Symbols
8
1 Introduction
13
1.1
Literature Review ........
.............................
14
1.2
Thesis Objectives ........
.............................
17
2 Theoretical study of Motion of a Floating Body in Waves
2.1
Theory of Ship Motions in Regular Waves
. . . . . . . . . . . . . . .
19
2.1.1
Ship-Motion Strip Theory . . . . . . . . . . . . . . . . . . . .
21
2.1.2
Equations of Motion of a SWATH . . . . . . . . . . . . . . . .
25
2.1.3
Radiation and Diffraction Problems of SWATH Vessels in Regular Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4
2.2
19
26
Wave Excitation Forces and Moments on SWATH Ships: Strip
Theory M ethod . . . . . . . . . . . . . . . . . . . . . . . . . .
28
Second-order Non-linear Forces Acting on Floating Bodies in Waves .
30
4
2.2.1
Added Resistance of Ships in Waves . . . . . . . . . . . . . . .
31
2.2.2
Salvesen's Method for Predicting Added resistance in Waves .
32
2.2.3
Other Methods of Predicting Added Resistance of a Ship in Waves 35
2.2.4
Comparison between the Integrated Pressure Method and Salvesen's
M ethod
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5
Results of Added Resistance Computation for a Series 60 hull
2.2.6
Viscosity effects on the hydrodynamics of an unconventional
SWATH ........
2.2.7
...
.. .. ..
..
.. ..
..
..
3
Analytical Investigation of the effects of inertial characteristics
Sum m ary
40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Computational Study of the Motion of a Floating Body in Waves
45
3.1
2D Strip Theory Method (PDSTRIP and CAT 5D) . . . . . . . . . .
46
3.2
3D Potential Flow Solver (WAMIT) . . . . . . . . . . . . . . . . . . .
47
3.3
Numerical Investigation of the Effects of Inertial Characteristics on
3.4
4
37
.. ... ...39
on SWATH RAOs . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
36
motion RAOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.3.1
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
Experimental Study of the Responses of SWATH USV in Regular
Head Waves
55
4.1
The SWATH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.1.1
. . . . . . . . . . . . . . . . . . . . . .
55
4.2
Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
4.3
Instrumentation and Procedure
. . . . . . . . . . . . . . . . . . . . .
60
4.3.1
Laser Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.3.2
Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
74
4.4.1
76
4.4
Assembling the Model
D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5 Conclusion
77
5.1
Summary of Results ........
5.2
Recommendations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...........................
78
78
Biblography
79
Appendices
81
List of Figures
2-1
Transverse Section of a Theoretical SWATH Vessel
2-2
Added Drag Versus Encounter Frequency for Series60 with Cb=0.6 at
. . . . . . . . . .
F,=0.283 in Head Waves . . . . . . . . . . . . . . . . . . . . . . . . .
2-3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Comparison of WAMIT and PDSTRIP Heave RAOs for Series 60 ship
with Cb=0.7 at Zero Speed
3-2
38
Added Drag Versus Wavelength for Series60 with Cb=0.7 at Zero Speed
in Head Waves
3-1
26
. . . . . . . . . . . . . . . . . . . . . . .
50
Comparison of WAMIT and PDSTRIP Pitch RAOs for Series 60 ship
with Cb=0.7 at Zero Speed
. . . . . . . . . . . . . . . . . . . . . . .
50
3-3
Effect of Varying the pitch Gyradius on Heave RAO at Constant VCG
51
3-4
Effect of Varying the pitch Gyradius on Pitch RAO at Constant VCG
51
3-5
Effect of Varying the VCG on Heave RAO at Constant Pitch Gyradius
52
3-6
Effect of Varying the VCG on Pitch RAO at Constant Pitch Gyradius
52
4-1
Profile View of Model during Experiment . . . . . . . . . . . . . . . .
56
4-2
Frontal view of model during experiment . . . . . . . . . . . . . . . .
57
4-3
Plan view of Model during experiment
. . . . . . . . . . . . . . . . .
57
4-4
Heave and Pitch Motion Time History for Wave Test No. 1 . . . . . .
62
4-5
Heave and Pitch Motion Time History for Wave Test No. 2 . . . . . .
63
6
4-6
Heave and Pitch Motion Time History for Wave Test No. 3 . . . . . .
64
4-7
Heave and Pitch Motion Time History for Wave Test No. 4 . . . . . .
65
4-8
Heave and Pitch Motion Time History for Wave Test No. 5 . . . . . .
66
4-9
Heave and Pitch Motion Time History for Wave Test No. 6 . . . . . .
67
4-10 Heave and Pitch Motion Time History for Wave Test No. 7 . . . . . .
68
4-11 Heave and Pitch Motion Time History for Wave Test No. 8 . . . . . .
69
4-12 Heave and Pitch Motion Time History for Wave Test No. 9 . . . . . .
70
4-13 Heave and Pitch Motion Time History for Wave Test No. 10 . . . . .
71
4-14 Heave and Pitch Motion Time History for Wave Test No. 11 . . . . .
72
4-15 Cut Heave and Pitch Responses of SWATH USV
73
. . . . . . . . . . .
4-16 Comparison of Experimental and Computational Predictions of Heave
Motions of SWATH USV . . . . . . . . . . . . . . . . . . . . . . . . .
74
4-17 Comparison of Experimental and Computational Predictions of Pitch
Motions of SWATH USV . . . . . . . . . . . . . . . . . . . . . . . . .
75
List of Tables
4.1
Main Model Parameters
. . . . . . . . . . . . . . . . . . . . . . . . .
58
4.2
Tank Dimensions and Model Test Factors . . . . . . . . . . . . . . . .
58
4.3
Wave Parameters for Different Cases of Model Test . . . . . . . . . .
59
7
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8
List of Symbols
aij
- Sectional added mass coefficient
A
- Wave amplitude
Aij
- Added mass coefficient
AW
- Water plane area
bij
- Sectional damping coefficient
Bij (w)
- Damping coefficient
Cb
- Block coefficient
cij
- Sectional restoring coefficient
Cij
- Restoring coefficient
da
- Stern displacement
df
- Bow displacement
F(t)
- Excitation force due to waves
Fi
- Complex amplitude of excitation force
F
- Froude-Krylov force on port side due to waves
FP
- Froude-Krylov force on starboard side due to waves
TFf"
- Hydromechanic force acting in ith direction
(Fji)*
- Complex conjugate of the Froude-Krylov part of the exciting force
Fn
- Froude number
Fs
- Second-order mean steady force
g
- Acceleration of gravity
GML
- Metacentric height
hj (t)
- Harmonic motion of ship in the jth mode of motion
Hij
- Complex analytical function of added mass and damping coefficients
Iij
- Ship's mass moment of inertial for rotational modes of motion
9
k
- Wave number
(Kv)
- Pitch gyradius
11
- Distance between bow sensor and LCG
12
- Distance between stern sensor and LCG
13
- Longitudinal distance between stern sensor and aftmost point of the ship
L
- Length overall of ship
LS
- Distance between sensors
M
- Mass of ship
Paw
- Energy dissipated by the ship during one encountering period
Pi
- Linear hydrodynamic pressure obtained from Bernoulli's equation
R
- Added resistance
RAOmax
- RAO at resonance
R,
- Incident wave contribution to the added drag.
Rf
- Diffraction contribution to the added drag.
R7
- A function of the sectional damping coefficients in heave and sway.
t
- time co-ordinate
UP3
- Wave particle orbital velocity
g3
- Wave orbital acceleration
V
- Velocity vector in two or three dimensions
Vz
- Relative vertical velocity
VCG
- Distance from waterline to center of gravity. Positive above the waterline
w
- Circular wave frequency
we
- Wave encountering frequency
Wn
- Natural Frequency
x
- Longitudinal axis of ship
Xb
- X-coordinate of a given section
y
- Transverse axis of ship
Yb
- Y-coordinate of a given section
10
YO
- Half the distance between center planes
z
- Vertical axis of ship
Zb
- Z-coordinate of a given section
Zb
- Vertical distance between top of bow and CG in deflected position
Zh
- Heave displacement
OP
- Pitch angle
ZS
- Vertical distance between top of stern and CG in deflected position
0
- Angle between direction of wave propagation and ship heading
'Ei
- Phase difference
r1
- Regular wave elevation
7
- Damping ratio
A
- Wavelength
P
- Dynamic viscosity
V
- Spatial derivative
#j
- Radiated wave potential
#0
- Incident wave potential
#7
- Diffraction potential
IT
- 2D time domain linear potential
p
- Density of water
0
- Angular displacement of body
0
- Total fluid potential
OB
- Potential due to body disturbance
0*
- conjugate of the incident wave potential
(j
- Response of ship(in the jth mode of motion) to sinusoidal waves
(j (w)
- Complex amplitude of ship motion in the jth degree of freedom
- Ship acceleration
- Ship velocity
11
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12
Chapter 1
Introduction
The hydrodynamics of floating structures in waves is unique because of the frequency
dependence of hydrodynamic parameters such as added mass, damping and the wave
exciting force. For a SWATH, the effect of viscosity on the added mass and damping
is non-negligible and must be taken into consideration when predicting the motion
response of this kind vessels in waves.
This paper presents the theoretical, numerical and experimental study of the motion responses of floating bodies in waves with focus on an unconventional SWATH
USV proposed to be used to monitor a team of autonomous underwater vehicles
(Brizzolara, 2011).
The use of this watercraft is expected to reduce maintenance
costs associated with manned ships and fixed platforms. Additionally, using remotely
operated surface vessels such as this would guarantee safety and make very remote
areas more readily accessible. The design operations of this vessel involves recharging
batteries of AUVs and retrieving the AUVs back to onshore locations. A small version
of this vessel can retrieve at least one autonomous vehicle per trip. A literature review
of the advancement made in the optimization of this class of water crafts is presented
as a basis on which future research is being carried out to study the hydrodynamics
of this vessel.
The objectives of this thesis are to compute the motion responses of the SWATH
USV using 2D and 3D potential flow solvers, report experimental results obtained
so far and attempt to validate the numerical results. The importance of this work
13
extends to other classes of multi-hull ships as there exists only few available data in
this range of crafts relative to well established mono-hull ships. Additionally, the effect
of the inertial characteristics of the model, vertical position of the center of gravity
(VCG) and gyradius in pitch (K.) on the RAO is investigated both analytically and
numerically.
1.1
Literature Review
Strip theory originally developed by Korvin-Krokowsky and Jacobs (1957) formed
the basis of most numerical methods for computing the motions of various kinds of
ships in waves. This was further extended to account for wave-induced vertical shear
forces and bending moments according to Jacobs (1958). The original strip theory
gave very good agreement with experiments especially for regular cruiser stern ships
operating at moderate speeds in head waves (Salvesen et. al., 1970).
Gerritsma and Beukelman (1967) further extended the method by proposing a
modified strip theory which predicts the motions of high speed destroyer hull in head
seas very well with respect to experiments.
The strip theory method for heave and pitch motions in head waves in which
the forward speed terms satisfy the symmetry relationship proved by Timman and
Newman (1962) was properly derived independently by Soding (1969), Netsvetayev
(1969) and Tasai and Takaki (1969).
Following the significant improvements made in the computation of the sectional
added mass and damping coefficients using closed-fit methods, Smith and Salvesen
(1970) showed that this method can be used to predict the motions of high speed
hulls with large bulbous bows in head seas with satisfactory accuracy.
Major advancements in the computation of ship hydrodynamics using the strip
theory method were proposed in the famous work of Salvesen, Tuck and Faltinsen
(1970). Their method can predict satisfactorily the heave, pitch, sway, roll and yaw
motions as well as wave-induced vertical and horizontal shear forces and bending
moments and torsional moments for a ship advancing at constant forward speed with
14
III
I
I
Pi"op"1111IT IIIP1,11II,
I"'I"11 II
I-IIIIIII III II
arbitrary heading in regular sinusoidal waves.
The lateral symmetry of the ship
guarantees that the equations of motion in the six degrees of freedom can be written
into two sets of coupled equations of motion as follows: one set of three coupled
equations for surge, heave and pitch as well as another set of three coupled equations
for sway, roll and yaw. It was proved that in the first set of coupled equations, the
surge motion may be neglected if the ship is a slender body in addition to having
lateral symmetry.
Hence, the equations of motion in the first set reduces to two
coupled equations for heave and pitch. Of course, the strip method defined assumes
that viscous effects are negligible and the assumption is credible because viscous
damping is insignificant for vertical ship motions.
Soding (1987) further improved the strip theory but the predictions were still
limited to linear equations of motion, rigid-body responses to periodic exciting forces
and moments due to free surface waves with small slopes. This method also assumes
that the ship maintains mean forward speed in infinite water and a straight mean
course.
The strip theory method offers an immediate insight into the hydrodynamics of
a ship saving computational costs at the preliminary design stage. The Numerical
implementation of this method has been carried out using two-dimensional and threedimensional approaches. Both approaches give reasonably accurate predictions for the
response amplitude operator of the ship motions; however, the latter is more efficient
for predicting hydrodynamic pressures and forces.
SWATHs are special types of watercrafts which are designed basically to reduce
their motions in waves by reducing the water plane area of the sections interacting
with surface waves. Hence, SWATHs are generally expected to have better seakeeping
performance in waves than equivalent mono-hulls and catamarans. Earlier works on
determining the motions and wave loads of SWATHs showed that analytical methods
need to account for the effects of wave interaction between the two hulls. Soding
(1988) developed a strip theory method that takes this interaction into account and
the obtained results are more accurate than those obtained by solving merely the
boundary value problem involving two-dimensional twin cylinders. This is because it
15
considers the speed effects on the radiated and diffracted waves between the hulls.
Dallinga et. al. (1989) developed a seakeeping program to gain a physical understanding of SWATH motions and described improved prediction methods. Their
approach utilizes the 3D diffraction theory as well as empirical information on the
associated loads and this was compared with traditional strip theory method. Hullfins interaction and free surface effects was found to be significant and are difficult to
estimate using simple methods. In addition, they posited that accurate predictions of
the seakeeping of SWATHs will require a more detailed description of the dynamics
of the flow around the vessel.
A frequency domain based strip theory method developed by Papanikolaou and
Schellin (1991) accounts for the effects of forward speed on the hydrodynamic interaction between the hulls. This method gives motion predictions which agrees well the
model test measurements and three-dimensional diffraction theory calculations.
Brizzolara (2004) used an efficient differential evolution algorithm to implement
a parametric design of a SWATH vessel which is optimized for total resistance. The
optimized vessel is a variant of the original base vessel characterized by two humps at
the bow and stern area and an intermediate hollow shape. This method automatically
picks the optimized dimensions SWATH vessel with minimal total resistance. It was
demonstrated that the operation profile of a watercraft should play a major role in
shaping its hull during design.
More advancements in design of SWATHs have focused on reducing wave pattern
and added resistance. A major contribution to this development is achieved using the
parametric optimization method as demonstrated by Brizzolara and Chryssostomidis
(2012).
This approach involves the modification of the submerged hull shape to
maximize wave cancellation effects between both hulls.
Such optimization of the
SWATH ship hulls results in the reduction of total resistance by more than 25 percent
at high speeds with respect to a conventional design.
The computation of the hydrodynamics of multi-hull ships may or may not include
the interaction between objects piercing through the free surface at different locations
depending on the sophistication of the method employed. Although, it is important
16
11FOOPm-
to consider this hull interaction effects in SWATHs, its usually good practice to use
simple and computationally cheap methods which gives reasonably accurate results
at preliminary stages of design (Patrikalakis and Chryssostomidis, 1986).
1.2
Thesis Objectives
The aim of this thesis is to predict the motion of an unconventional SWATH USV in
waves using numerical and experimental approaches.
There exist three dimensional fully viscous time domain simulations to account
for non-linear free surface and viscosity effects but they can be computationally expensive. Hence, it's more efficient to develop quicker hybrid methods which account
for these effects especially at early design stages. However, these new methods require
sufficient experimental data for validation purposes and to boost the confidence in
their applications. Only a few experimental data on the hydrodynamics of SWATHs
are available in literature while some substantial numerical results can be found for
existing SWATH designs. Moreover, in order to further progress with this work, the
computationally predicted better performance of unconventional SWATHs in waves
must be validated using experiments. Hence, the need for more experimental results
such as those reported in this thesis.
17
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18
Chapter 2
Theoretical study of Motion of a
Floating Body in Waves
The motion of a body in random seas may be computed by the superposition of
the responses of the body to each of the constituent elementary waves.
Hence, it
is very useful to study this problem using regular exciting waves which are easy to
understand and apply the Wiener-Khinchin theorem to obtain the response of the
ship in irregular waves. Another very important simplification is the assumption of
potential flow and also the linearization of the free surface boundary condition in
terms of the velocity potential. The solution to this problem is achieved by solving
the diffraction and the radiation problems as would be discussed in following sections.
2.1
Theory of Ship Motions in Regular Waves
This section discusses the theory of ship motions in the context of input and output
into a linear time-invariant (LTI) system as assumed in studying small motions of a
ship in the six degrees of freedom. Furthermore, this guarantees that if the input,
that is- the exciting wave is sinusoidal, the response of the ship is also sinusoidal with
the same frequency but in general, a different amplitude and phase. The magnification factor is otherwise known as the Response Amplitude Operator (RAO). Here,
the RAO is a function of frequency but independent of the wave amplitude. This
19
concept as described is known as the linear theory of ship motions and it proves to
be accurate for estimating the responses of ships in regular waves and irregular waves
of considerable height. The regular wave elevation, q is given as follows:
q(x, y, z, t) = Re{Aew t - kx}
(2.1)
The response of the ship to sinusoidal exciting waves, (j can be expressed as:
(j(t) = Re{y (w)ewt}
(2.2)
The RAO which is the transfer function of the LTI system described above is the
ratio of the complex amplitude of the ship motion to the wave amplitude, A.
Hence,
RAO =
A
(2.3)
In model testing, the complex amplitude of the ship response is the measured amplitude of oscillation for a given frequency which can be measured using displacement
sensors or derived from readings obtained using an accelerometer.
Theoretically, the RAO is obtained from the solution of the equation of motion of
the ship-water system using the strip theory method described in the next section.
The equation of motion of a simple mass-spring system with a dashpot is a linear
second-order differential equation and is given as:
M
+ B
+C( = F (t)
20
(2.4)
2.1.1
Ship-Motion Strip Theory
The strip theory is a relatively simple method of computing the forces acting on a
submerged slender body by direct integration of the net pressure acting on a piece of
length, dx over the entire length of the body. This tranche is generated by "tearing
up" the body of the ship into sections also known as stations.
Generally, viscous effects are neglected and the flow is assumed to be irrotational.
Hence, the linear-ship motion strip theory is framed in the context of the potential
flow theory. Also, the ship is considered to be very long compared to its beam, that
is, the slender body theory is applied which allows surge motion to be ignored.
The slender body theory applied in strip theory makes it possible to compute
the exciting force and moment, hydrodynamic coefficients namely added mass and
damping of the entire ship from two-dimensional potential flow solution for oscillating
cylinders.
Also, it is assumed that the disturbance of the free surface is due to the incident
wave only. Hence, radiated waves can be ignored.
For a ship with forward speed, the encounter frequency is assumed to be high.
However, the strip theory gives good prediction for the heave and pitch motions even
at low encounter frequencies because the Froude-Krylov force is dominated by the
restoring force at low frequencies and is independent of the wave frequency.
The strip theory calculations yield the hydrodynamic coefficients, that is, added
mass and damping coefficient and the heave and pitch motion values with their corresponding phase relationship. These results are used as input to the formulation for
predicting the added drag and must be accurate to ensure reasonable prediction of
the added drag.
The hydrodynamic coefficients may be determined using close-fit conformal mapping methods or by considering time harmonic motions of small amplitude for ship
motions with the complex factor ei
applied to the periodic quantities as described
below.
The total fluid potential around a ship hull using strip theory method is given as:
21
6
<D (x, y, z, t)
=A(q5 0
+0~ 7 eiwt) +
Z
Cjojeiwt(2)
(2.5)
The diffracted and radiated wave potentials must satisfy the following conditions:
1. Laplace equation in the fluid domain:
a2<Dj
+y
0
(2.6)
ick~j + g 19) = 0
az
(2.7)
2
1Z2=
1. Linearized free surface condition:
z=0,
j=2,.....,6
(2.8)
1. Ship hull no flux condition:
=
iwnae
aOPo
an'
(j = 2,....,6)
(j = 7)
< radiatedwave potential >
< diffracted wave potential >
1. Far field radiation potential:
22
(2.9)
(2.10)
an
= 0,
Z = 00
(2.11)
According to the strip theory, the above complex amplitude of the wave potentials
in 3D, D, may be replaced by 2D time domain body linear potentials, IV, using the
relation below:
(2.12)
O' (t, y, z) = <P (x, y, z)eiwt
The 2D time domain potentials, Oj (t, y, z)are obtained for each strip ordered from
bow to stern.
The linear hydrodynamic pressure is obtain from the Bernoulli's equation and the
hydrodynamic force is obtained by integrating this pressure over the wetted section
of the ship. The hydrodynamic forces acting on the ship can be expressed in terms
of the 2D potentials as follows:
Hij =
P (x, y, z) nddS = -p
e-I
ezIt
'
ny dS
(2.13)
Hij is an analytic function which represents the radiation hydrodynamic coefficients and has the added mass, Aid and damping coefficients, Bid in its real and
imaginary terms respectively. This is written as:
Hij = w 2 Aij - iwBi
(2.14)
The strip theory method involves the integration of the force and moment per
unit length (sectional force) acting on each strip over the entire length of the ship.
This gives the total hydromechanics force and moments acting on the ship which is
23
added to the wave exciting force and moments respectively to obtain the equations
of motion based on the Newton's second law.
The hydromechanics force and moments consist of the hydrodynamic and hydrostatic components. For coupled heave and pitch motion, the equations of motion can
be written as:
In Heave:
(M + A 33 ) h3 + B3 3 h3 + C331
+ A 3 5 h5 + B 3 5h5 + C3 5 h5 = F3 eiwt
(2.15)
(155 + A 5 5 ) k5 + B5 5 h5 + C5 5 h5 + A 5 3 h 3 + B 53 h3 + C53 h3 = Fseiwt
(2.16)
3
In Pitch:
The harmonic motion can be expressed as a complex amplitude of the motion in
frequency domain:
hj (t) = Re {(e't}
(2.17)
For harmonic motions with small amplitudes, the equations of motion described
above can be expressed as a linear system in frequency domain given as:
[C 33 - (M + A 33 ) w 2 + iwB 33] (3 + [C 35 - A 35w 2 + iwB3 s]( 5 = F3
24
(2.18)
- --
-1--l
................
--
-1--1.-1---..1
[C55 - (M + A 55 ) w2 + iwBs] (5 + [C53
-
A 53w 2 + iBs]( 3
=
F5
(2.19)
For a ship with forward speed, the wave frequency, w which is the wave circular
frequency is replaced with the wave encountering frequency, We. The added mass and
damping are functions of the encountering frequency hence, are respectively calculated
for each encountering frequency. The expressions which accounts for forward speed
are given below (Salvesen, Tuck and Faltinsen, (1970):
In Heave:
eikxe-kds {pgb - w (wea33 - ib33 )} dx -
F3 = A
eikXAe-kdsw(wea A - ib
A
)
-
(2.20)
In Pitch:
F5 = - A
+
eikxe-kds
S1We
AU
x
[pgb - w (wea33 - ib3 3 )]
eikAe-kdswxA(weaA
-
Uw (wea33
-
ib3 3 ) dx
(2.21)
-
ib
)
.M
iWe
The accuracy of the RAO especially for heave and pitch motions is very important
since these motion results are the inputs to the added resistance formulation which
is highly sensitive to their accuracy.
2.1.2
Equations of Motion of a SWATH
The basic Newton's equation of motion as described in previous section is applicable
to the SWATH vessel being investigated in much similar way as it is applicable to
other floating bodies. However, the radiation and diffraction problem considers two
hulls which are symmetric about the centerline from port to starboard and this yields
the expressions below:
25
Zb
Xb
S
P
Yo
Yo
Figure 2-1: Transverse Section of a Theoretical SWATH Vessel
In Heave:
M - Fj = Fw
(2.22)
M -- F m= Fw
(2.23)
In Pitch:
The response amplitude operator of the SWATH in heave and pitch is obtained
from the above expressions. For a SWATH moving with forward speed, the Doppler
shift must be taken into consideration and in fact, the interaction between the hulls
becomes much more significant at low encountering frequencies. Also, because of the
shape of the unconventional SWATH hulls, viscous effect plays an important role and
cannot be neglected.
2.1.3
Radiation and Diffraction Problems of SWATH Vessels
in Regular Waves
The prediction of the motion response of a ship in waves based on the linear theory
requires that the hydromechanics coefficients and exciting wave forces and moments
26
.....................
be determined. This is achieved by solving two main problems:
* The Radiation Problem: This involves the excitation of the ship in calm water
to create waves which carry energy away from the ship. The aim is to determine
the added mass and damping coefficients which are expressed as the real and
imaginary part of an analytic function derived from radiated wave potential,
(Dr.
hj = Re
{(
e}wt}
(2.24)
41)r = (g i
(2.25)
As in the case of mono-hulls, the radiation force consists of the added mass and
damping terms.
It is important to note that these hydrodynamic coefficients are
frequency dependent. While the added mass component is in phase with acceleration,
the damping component is in phase with velocity. A more detailed proof can be found
in Marine Hydrodynamics by Newman.
The hydromechanics force acting on a twin hull swath oscillating in heave and
pitch in calm water is given as:
In heave:
-Fh=
(2a
+ 2b31 .f + 2c3 1x) + ( 2a 33 i + 2b33 z + 2c3 3 z) + (2a
35
+ 2b35 9 + 2c3 5 9)
(2.26)
The first term accounts for the heave-surge coupling, the second term is the pure
heave force contribution and the last term is the heave-pitch coupling term.
In pitch:
27
-Fa"m = (2an1; + 2b5l + 2c51 x) + ( 2a53 i + 2b53 + 2c5 3 z) + (2a55 0 + 2b55 0 + 2c 55 0)
(2.27)
The first term accounts for the pitch-surge coupling, the second term is the pitchheave force contribution and the last term is the pure pitch coupling term.
In both cases described above, the surge terms are neglected for this problem
because only the heave and pitch motion responses have significant contributions to
the added resistance of the vessel in head waves which is the end objective of this
project.
* The Diffraction Problem:
The goal here is to determine the wave exciting
forces and moments by fixing the ship and sending harmonic waves towards
the ship. The excitation wave forces and moments are computed starting from
the diffracted wave potential, FDd.
In the special case of a swath, the interaction between the parallel hulls must be
considered. The hydrodynamic interaction between the hulls comprises of two wave
systems:
* The radiated waves produced by the motion of one hull striking the other hull
* The incident waves acting on one of the hulls which are modified due to presence
of the other hulls (reflection and transmission).
However, the effects of the interaction between the hulls is more significant at
small encountering frequencies for ships with forward speed.
2.1.4
Wave Excitation Forces and Moments on SWATH Ships:
Strip Theory Method
The diffraction problem in which the floating body is constrained in the presence of
incident waves yields the exciting wave forces comprising of the Froude-Krylov and
28
diffraction force components. In the case of a Swath with zero speed in regular head
waves, the excitation forces can be expressed in terms of the 2D sectional potential
mass and damping components, the wave particle orbital velocities and accelerations
and the sectional Froude-Krylov force. The first order wave potential for deepwater
case is given as:
4D = Re
ei(wt-kbcos,6-kybsin8)+kzb
(2.28)
For the case of head waves, / is 180 degrees.
In the vertical direction, the following expressions for the wave particle orbital
velocity and acceleration for the portside (p) and starboard side (s) can be written
as:
=
nt =
Re
gkA et(wt+kxb)+kzb
}
Re {-igkAe(t+kXb)+kzb}
=
8
(2.29)
itS
(2.30)
The exciting wave force in heave and pitch is obtained by integrating the 2D
sectional components listed above along the length of the ship.
This is given as
follows:
In heave:
F3'=
j
[M
3 .(iP
+ its) + N3 3 . (UP + u3) + (FPFK + F(2FK)
In pitch:
29
(2.31)
F3=
[M' 3 .(h+hi)+Ni.(u +ui)+(FFK3+FFK)bb
(2.32)
The exciting forces in the remaining degrees of freedom namely yaw, roll, surge
and sway can be trivially obtained from the same concept described above.
2.2
Second-order Non-linear Forces Acting on Floating Bodies in Waves
The strip theory described so far is based on the linearized seakeeping theory. Using
this concept, the integration of the first order pressure force on a floating boding over
one wave period is effectively zero. However, any floating body in waves experience
a steady force which does not go to zero over the wave period. This steady force is a
second-order nonlinear force known as the drift force.
The contribution to the drift force may result from the non-zero component of
the time integral of the non-linear total wave potential but can be expressed in terms
of known first order terms (Salvesen, 1978).
Also, second order forces may result
from first order pressure terms integrated over the actual wetted surface and in irregular seas, second order terms may be due to wave coupling. The first-order terms
which contribute to the second-order drift force on a floating body in head waves
are the heave and pitch motions already discussed for general floating bodies and
specifically for the SWATH-type of vessel under investigation in this project. Hence,
it is very important to accurately predict the motion response of the vessel in order
to obtain a reasonable value for the second-order drift forces. The drift force could
act transversely, longitudinally or vertically with respect to the ship fixed coordinate
system. The expression below gives the theoretical representation of the second-order
longitudinal drift force according to Salvesen (Salvesen, 1978):
30
Fs =
2.2.1
2
p ]]
ff
(
an
an )(
VC* + 2-V()
0
B
ds
(2.33)
Added Resistance of Ships in Waves
The added resistance also referred to as the added drag is that part of a ship's total
drag which is due to its encountering of waves. The other part is known as the calm
water resistance resulting from its forward speed in still water. The added resistance
is typically in the range of 15% to 30% of the calm water resistance for a conventional
merchant ship. Some of the factors that increase the added drag of a ship in realistic
sea/weather conditions are wave reflection, ship motion in waves and wind effect on
ship's superstructure.
The added drag acts along the longitudinal axis of the ship hence, it creates
additional resistance to the motion of the ship. It can be deduced from Havelock's
formulation that the added resistance results from the phase relationship between
the ship motion and the exciting wave forces. Also, the energy loss associated with
the added resistance is dissipated mainly by the radiated waves generated by ship
motions and negligibly by fluid friction.
The added resistance consists of three components namely the drift force, diffraction effects and viscous effects. The drift force component results from the interference between the incident wave and the radiated waves generated due to ship motions
specifically in heave and pitch. The diffraction effect is the component which results
from the interaction between the diffracted wave and the radiated wave also due to
ship motion in heave and pitch. The viscous effect is due to the damping of the ship's
vertical motion.
It is important to note that all components of the added drag are additive and
each of them is proportional to the square of the wave amplitude which makes the
added drag a non-linear problem. Also, the viscous damping is very small compared
to the hydrodynamic damping of the entire ship. Hence, energy lost due to friction
is negligible. Therefore, in practical terms, added drag can be considered as a non31
viscous fluid flow phenomenon which makes it possible to apply potential flow theory
to this problem.
Added drag is important for choosing an accurate and reasonable value of weather
margin during ship propulsion design and also important for mapping out the routes
of an ocean going vessel to reduce fuel consumption, stress resulting from resistance
and maximize the efficiency of the ship. This involves using the calm water resistance
obtained from subtracting the predicted added resistance from the current ship's total
resistance.
2.2.2
Salvesen's Method for Predicting Added resistance in
Waves
The total potential of the fluid around the ship hull is given as:
# = 0 + #B
0 is the incoming wave potential,
#B
(2.34)
is the potential due to the body disturbance
and diffraction effects.
The added drag is the negative of the mean second-order wave force acting in the
longitudinal direction. There are three main contributing terms to the added drag
predicted by Salvesen as shown below:
R= E
{R+Rf } + R7
(2.35)
j=3,5
The first term, R is the incident wave contribution to the added drag. This term
is a function of the sum of the Froude-Krylov part of the excitation force in heave
and pitch.
The second term, Rf is the diffraction contribution to the added drag. It is a
function of the sum of the diffraction part of the excitation force in heave and pitch.
32
All
11111
The third term, R7 is a function of the sectional damping coefficients in heave and
sway.
The added drag can be further expressed in terms of sectional quantities which
are obtained from solving a two-dimensional problem of a cylinder oscillating in free
surface as follows:
R =-
k cos
(3 (Fj)* + Pf I + R7
)
(2.36)
j=3,5
(F/)* is the complex conjugate of the Froude-Krylov part of the exciting force.
F9 is the same as the diffraction part of the exciting force except that the incident
potential, 50 is replaced by its conjugate, <b*.
The conjugate of the Froude-Krylov force and moment is given as:
(FI)* =
f
ip
j
(FJ)* = -ip
< heave >
N3 J ds
xN 3 *ds < pitch >
(2.37)
(2.38)
The term F9 is closely related to the diffraction part of the exciting force and
moment and is expressed as:
P3
FPD=5
=
L
3 (x)
X+-
dx
i)h3(x)dx
W
33
< heave >
pitch>
(2.40)
(2.39)
h3
(x) = -pk
j4 (N + iN sin /)<dJdl
JC0
3
3
(2.41)
2
Salvesen's formula treats the ship as a slender body and a 'weak scatterer' justifying
the assumption that the body potential due to body-disturbance and diffraction effects
(damping),
iB's small compared to incident-wave potential,
#0.
Hence, quadratic
terms involving the body potential is neglected. This assumption is justified when
the wavelength is considerably longer than at least two main dimensions of the ship
as assumed in the derivation.
Typically, for navy ships, the maximum added drag is reached when the wavelength is about seven times larger than the beam of the ship when operating with
normal speed in head waves while the wavelength is only about three and half times
larger for zero speed ship.
Immediately, it is expected that Salvesen's formulation predicts the added drag
for a ship with normal operating speeds in head and bow waves more accurately than
for a ship with zero speed in beam and following waves.
2.2.3
Other Methods of Predicting Added Resistance of a Ship
in Waves
(1) The Momentum and Energy Method:
This method basically involves defining a control volume around the ship hull
and deriving the momentum balance in the control volume. The added resistance is
related to the linear momentum flow through the control volume. The fluid velocity
potential is divided into the incident wave potential, diffracted wave potential and
the radiated wave potential.
The solution to the added resistance problem involves finding the harmonic potential which satisfies three basic conditions namely: The linearized free surface potential,
the far field radiation potential and the ship hull no flux condition. These conditions
34
are simplified by considering a slender-body ship which its lengths much greater than
its beam and draft. Joosen (1966) using this approximations expanded Maruo's results in an asymptotic series of power of a slenderness parameter and considered only
first order terms. His formulation is given as:
Raw
=
1 W3
[B3 h2 + B5 h25 - 2B3 5 h3 h5 cos
3
2 g
(E3 -
65)]
(2.42)
The encounter frequency, We accounts for speed effects on the ship in waves.
(2) The Radiated Energy Method:
This method equates the energy loss by the ship during one period of oscillation
in regular waves to the added resistance of the ship. The most important parameter
in this approach is the average relative vertical velocity. The average relative vertical
velocity, V is the vertical velocity of the water particles relative to a point on the
ship. It can also be described as the velocity of the water particles expressed in the
ship's coordinate system.
Vz = W
Vh 5
h3 - Xbh5 +
-
*
(2.43)
We
=[1
Yb exp (kzb) dzbI
yw
(2.44)
-T
The radiated wave energy of the ship is equal to the work absorbed by the added
resistance. For one encountering period, the energy dissipated by the ship is given as:
Paw =
7r
We
jb
I
33 V
2~
dxb
(2.45)
The radiated wave energy during one period of oscillation is related to the added
35
resistance as follows:
Raw
=
b3 3V 2 dxb
7e
(2.46)
This method does not give accurate results for following waves with phase speed
approaching ship speed because the encountering frequency in the denominator tends
to zero. Hence, the added resistance results tends to infinity in this case which is not
practical. However, it gives good results in head to beam seas.
2.2.4
Comparison between the Integrated Pressure Method
and Salvesen's Method
Both methods compute the added drag using the products of first-order terms obtained from the linear ship-motion theory. In fact, all computational methods for
predicting added drag uses the linear ship responses as inputs although the added
drag is a highly nonlinear quantity (Salvesen, 1978).
Both methods do not consider the interaction between neighboring stations as they
employ transverse strip theory to compute the pressure on the submerged sections of
the ship.
The Salvesen's method considers diffraction and viscous effects contribution to
added resistance whereas the integrated pressure method neglects these effects. Hence,
for a bulky ship with high Cb, the Salvesen's method is expected to give better results
than the PDSTRIP method.
The integrated pressure method utilized in PDSTRIP gives more accurate results for oblique waves compared to other methods in theory including the Salvesen's
method.
The PDSTRIP approach over estimates the peak of the added resistance for high
Froude numbers and under estimates this value at lower Froude number compared to
other methods in theory (Arribas, 2006). Also, it is more reliable than the Salvesen's
method for predicting added resistance in short waves where diffraction contribution
36
MI
II |IliII
qiiIII
Iill~
ilN
P111111
11111
111
111
1111
nis|I
lli
T
11"
1111
111
1M
||||1|||91
lm
llipHp
slle
Iiff
111
"j
' 1,
In
is dominant (Matulja, 2011).
Generally, the PDSTRIP approach has more agreement with experiments than
the Salvesen's method especially for ship forms with low Cb.
2.2.5
Results of Added Resistance Computation for a Series
60 hull
The theoretical methods discussed above were implemented in different computer
programs. Results were obtained for both zero speed and forward speed case and
a brief discussion of the results are given based on theory. The drift force RAOs
presented are for Normand Ranger and Series 60 mono-hull ships.
24-
*
22-
20
CI
-
PDSTRIP
Ger&Beu
Salvesen
c
Experiment
18
"co 16
-
o 14
-h
Y 12
Cn
-
10
8
-
0.-
C]
In-
op-
6
0
4
2
2~
6
5
4
3
Encounter Frequency (We (/g) 1/2)
7
Figure 2-2: Added Drag Versus Encounter Frequency for Series60 with Cb=0.6 at
F,=0.283 in Head Waves
37
Cs-
0
WAMIT Pressure
WAMIT Momentum
PDSTRIP
-
*
*
*
2.8.
2.42-
c
41)
1.6-
0
-
1.20.8
0.4
0
0
0.2
0.6
0.8
0.4
Waveleng th (A/L)
1
1.2
Figure 2-3: Added Drag Versus Wavelength for Series60 with Cb=0.7 at Zero Speed
in Head Waves
38
2.2.6
Viscosity effects on the hydrodynamics of an unconventional SWATH
Radiation forces are the most influenced by viscosity effects especially for SWATHtype of vessels (Bonfigilo and Brizzolara, 2014).
The significance of the viscosity
effects on the radiation hydrodynamic force acting on a multi-hull ship is dependent
on the body and wave motions. Palaniswamy showed that the effect of non-linear
free surface and viscosity can be neglected for a multi-hull when body and wave
motions are small. Also, the kinematic boundary condition is unaffected by the effect
of viscosity (Ananthakrishnan, 2012).
However, for the kind of hull shape being
investigated in this project, viscosity cannot be neglected even with somewhat small
body and wave motions (Bonfigilo et. al., 2013a).
The struts of conventional SWATH ships is continuous from bow to stern which
allows for trapping of waves on the mean surface between the hulls. The modes of
motion undergone by these waves can be divided into two:
e Piston Mode of Motion: This mode of motion corresponds to the vertical oscillation of the mean surface between the hulls. This mode of motion is the most
affected by the viscous effect especially at resonance (Ananthakrishnan, 2012).
* Sloshing Mode of motion: This mode of wave motion is characterized by the
back and forth transverse oscillation of the standing waves between the hulls.
For the unconventional SWATH ship being investigated in this project, the struts
are not continuous from bow to stern. The water plane of the vessel comprise of four
struts piercing through the free surface. Since the radiation problem is conducted in
head waves, only the vertical modes of motion are deemed important. Hence, the
piston mode of motion also known as Helmholtz oscillation is expected to be the
dominant mode of wave motion in this case.
The fully viscous Navier-Stokes governing equation that accounts for viscosity of
the fluid is given as:
39
p(
( t
+V.VV
V ( + gZ)+LV2V
(2.47)
The Navier-Stokes equation must be solved for this kind of problem but this
approach is usually computationally expensive and hybrid methods which gives reasonably accurate results may be used at preliminary stages of design.
Since, the
frequency domain analysis of ship seakeeping problems is formulated in the context
of linear theory, a hybrid method must be developed to account for this non-linear
effect using a wholly linear analysis method (Bonfigilio et. al., 2013a).
The comparison between the potential flow based heave and pitch motion RAOs
and experiments at resonance frequency in the later part of this work shows clearly
that the viscous effects are non-negligible for this kind of hull topology.
2.2.7
Analytical Investigation of the effects of inertial characteristics on SWATH RAOs
The RAO of a floating body in the context of linear theory of seakeeping is independent of the wave amplitude. The RAO is characterized by its peak amplitude and
the frequency at which this peak occurs. The inertial characteristics under study are
the vertical center of gravity relative to the waterline, VCG and the pitch gyradius,
KY of the SWATH ship. This discussion is shaped to illustrate the effect of varying
the VCG and KY on the peak RAO and the wavelength at which it occurs.
The
wavelength is derived from the dispersion relation for deepwater as follows:
W2 = gk
Hence,
40
(2.48)
(2.49)
A = 27rg
2
w
The peak of the RAO of a damped linear system is related to the ratio as follows:
RAO m ax =
1
27V1 -
(2.50)
2
From the above relation, the peak RAO would increase as the damping ratio
reduces. Furthermore, the damping ratio can be expressed as:
actual damping, b
critical damping, br
The actual damping of a floating ship can be considered roughly to be proportional proportional to the waterplane area of the ship. SWATHs generally have small
water plane area which is responsible for their low damping characteristics. However,
according to the Haskind relation, the exciting force acting on a floating body is
proportional to the square root of the potential damping. Hence, water crafts with
small water plane areas like this unconventional SWATH experience smaller exciting
forces in waves which gives them a superior seakeeping performance over equivalent
mono-hulls.
From the above relation, increasing the critical damping reduces the damping
ratio. The critical damping is dependent on the mass (both inertia and virtual mass)
of the floating body and the restoring coefficient. The critical damping is expressed
as follows:
In heave only mode of motion:
bcr = 2
c33(m + a,,)
41
(2.52)
In pitch only mode of motion:
bc, = 2,/c 5 (m+ a,,)
(2.53)
The resonant frequency at which the peak RAO occurs is defined as follows:
Wn
c
(m a)
=
(2.54)
The restoring coefficient of the ship in the pitch mode of motion, c 55 is dependent
on the metacentric height which changes as the vertical center of gravity is altered.
The relation for the restoring coefficient in pitch is given as:
C55=
pgVGML
(2.55)
By definition, the VCG is zero on the waterline and is negative below the waterline.
Hence, lowering the VCG means increasing the GML which is favorable to the initial
stability of the ship.
The restoring coefficient in heave mode is given as:
C33=
pgAw
(2.56)
From the above relation, the restoring coefficient in heave does not depend on
the VCG and Ky. Hence, it may be inferred theoretically that the heave resonant
frequency is unaffected by the change in these inertial characteristics.
The mass moment of inertial is directly related to the gyradius in a corresponding
rotational mode of motion. The relation is given below as:
42
I=MK2
(2.57)
Increasing the pitch gyradius increases the moment of inertial by a square of the
factor of increase. The theoretical analysis presented here assumes that the SWATH
ship is excited by regular head waves at zero speed.
Based on the relations given above, the following theoretical deductions may be
presented:
* Lowering the VCG is expected to increase the pitch peak RAO
* Lowering the VCG would reduce the wavelength at which the pitch peak RAO
occurs.
e Increasing the Ky is expected to increase the pitch peak RAO
e Increasing the Ky would increase the wavelength at which the pitch peak RAO
occurs.
2.3
Summary
The discussions presented in this chapter shows clearly that the motion response of
an unconventional SWATH requires that other effects like the viscous forces be taken
into consideration especially in predicting the radiation forces acting on the vessel.
Also, the added resistance of a ship is shown to be theoretically related to its motion
response in waves.
Hence, a considerably accurate prediction of the contributing
RAOs is necessary to predict the added resistance within reasonable and acceptable
error limit. Finally, the numerical results presented in the next section shows a good
agreement with these analytical inferences.
43
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44
Chapter 3
Computational Study of the Motion
of a Floating Body in Waves
This section discusses the various numerical methods implemented in computer programs used for predicting the motion responses of floating bodies in waves. Some of
the computer programs presented can only solve problems involving mono-hulls while
some can solve problems involving multi-hulls. Additionally, 2D and 3D potential flow
solvers and computer programs which are capable of solving zero speed and forward
speed problems are also presented.
Some other sophisticated computational Fluid
Dynamics (CFD) techniques which solve the non-linear fully viscous problem exist
but these methods are computationally expensive and may only justify the enormous
task involved at later stage of the design. The accuracy in predicting the added resistance of floating vessels depends on the accuracy of the first order motion responses
of the ship in regular waves. In head waves, only the heave and pitch modes of motion
contribute significantly to the added resistance of the ship. Hence, it's important to
correctly predict the heave and pitch motion RAOs at least within some acceptable
error limit. In the results and discussion section, some results obtained for added
resistance of mono-hulls using the various computational methods are presented and
motion responses were compared to investigate their effect on the resistance of the
ship. Also, the results of the numerical study on the effects of inertial characteristics
of the unconventional SWATH on its motion responses in regular waves is presented.
45
3.1
2D Strip Theory Method (PDSTRIP and CAT
5D)
The 2D strip theory method is implemented in PDSTRIP (Public-domain strip).
PDSTRIP is a hydrodynamic strip code written in FORTRAN which computes the
seakeeping of mono-hull ships including sailing boats (Bertram et. al., 2006). It is
not applicable to structures with components piercing through the free surface more
than once. Hence, all appendages like fins must be submerged. PDSTRIP cannot
deal with multi-hulls like SWATHs because it does not consider the interactions between the hulls. PDSTRIP gives the responses of mono-hulls in regular waves as
RAOs. It follows the frequency domain approach hence, it is confined to predicting
linear responses but accounts for a few non-linear effects. The water depth being
investigated may be deep or shallow, but the water depth must be constant in space
and time. In irregular waves, the responses are given as significant amplitudes which
are essentially the average of the one-third largest positive maxima of the responses.
The significant amplitude is twice the standard deviation of the response. PDSTRIP
can accommodate unsymmetrical bodies including heeled ships and forces on sails
and fins can be taken into account. The patch method which is a modification of the
traditional panel method is followed in this program. The former computes the forces
more accurately than the latter. The potential which is the basic mathematical quantity that describes the flow in potential context is obtained from the superposition of
the point sources. Mathematically, this is expressed as:
I(y, z)
2 ln[(y - y4) +
=
(z - zi)]
(3.1)
i=1
Where:
ID(y, z, t) = I(y, z)ewt
46
(3.2)
The 2D section contour representing each strip of the ship is defined by a set of
offset points. A source is place near the midpoint of each segment between adjacent
offset points; however, the source is placed at distance of 1/20 of the segment length
inward from the surface. Additionally, grid points are automatically generated on the
free surface until a distance equal to 1/12 of the wavelength of the waves generated
by the ship motion is reached. A source is place between adjacent free surface grid
points near the midpoint, but at a vertical distance equal to the free surface segment
length above waterline. The sources are placed either above z=0 or within the section
contour to avoid having the discontinuity at the origin of the source in the fluid region.
The potential satisfies all boundary conditions in classical potential flow analysis.
The bottom condition for example is satisfied exactly using the method of images by
placing a source at a location (yi,2H-zi) for the shallow water case; for the deepwater
case, the bottom condition is satisfied automatically by the superposition equation.
It computes the added mass and damping matrix from solving the radiation problem
and computes the wave exciting forces by solving the diffraction problem.
More
details can be found in the PDSTRIP manual.
CAT 5D is also a strip theory code originally written in FORTRAN but translated
to MATLAB and can compute the ship motions of catamarans and SWATHs in five
degrees of freedom namely, heave, sway, pitch, roll and yaw. It is based on the linear
strip theory and can also account for viscous effects if externally computed viscous
related radiation forces, added mass and damping are provided (Brizzolara et. al,
2012). The code needs some more improvements to accurately predict the resonant
frequency of the response.
3.2
3D Potential Flow Solver (WAMIT)
The so called panel method is implemented in this computer program. It is based on
the boundary integral equation method (BIEM) and solves the 3D problem at zero
speed. The hull of the ship is represented using sources and if the source is of the
Rankine type, this method is appropriately termed the Rankine panel method.
47
Wave Analyses MIT (WAMIT) is a 3D potential flow solver which can solve problems involving fixed, constrained or neutrally buoyant bodies and these may be bottom mounted, submerged or surface piercing. It can also analyze problems involving
multi-body interactions including seakeeping of SWATHs and catamarans. WAMIT
includes options to use traditional low-order panel method or a more versatile higherorder panel method based on B-splines.
In WAMIT 3D approach, the hull surface is represented using quadrilateral panels
unlike segments in the 2D case. This allows more accurate prediction of the hydrodynamic forces than in the latter. Gaps are allowed between the panels and this does
not affect the accuracy of predicted, motions and forces significantly.
Based on the potential flow assumption including no separation and no lift effect,
the velocity of the fluid around a floating or submerged body can be described as the
gradient of a potential, <.Thepotentialmustsatisfythevariousboundaryconditionsasdescribedinearliej
V2<4)=0
(3.3)
The total velocity potential is divided into radiation potential and total diffraction
potential. Mathematically;
C,=3 + 4D
(3.4)
The time harmonic exponential is eliminated from the equation above because it
appears in all terms.
The radiation velocity potential is given as:
6
<DR = iWZE (j
j=1
The total diffraction potential can be expressed as follows:
48
(3.5)
4DD = 10
(3-6)
Cs
The integral equations satisfied by the radiation and total diffraction potential
is obtained by solving the boundary value problem using the Green's theorem. The
Integral equations are enforced at the collocation points located at the center of the
panels.
Applying the Green's theorem to the radiation potential gives:
27r4j (x) +
4
( ad
JfSb
ff nG (;, x) d
(3.7)
JSb
Similarly, applying the Green's theorem to the total diffraction velocity potential
gives:
The term G(c, x)is known as the wave source potential. It is the velocity potential
at a point x if a source of strength -47r is placed at point ;. More details can be found
in the WAMIT manual.
The figures below shows the computed motion of a Series 60 ship at zero speed
using WAMIT:
3.3
Numerical Investigation of the Effects of Inertial
Characteristics on motion RAOs
The numerical investigation was carried out using WAMIT. Two inertial characteristics namely VCG and K. where varied independently. That is, one was fixed while
the other was varied. The effects of these variables on the heave and pitch RAOs of
the SWATH vessel at zero speed was investigated and presented below.
49
0.45+
S+
0.4
PDSTRIP
WAMIT
0.35
0.3 1
0.25 1
0
0.2[F
cc)
0.15
0.1
0.05
0
0
0.2
0.8
1
0.4
0.6
Wavelength (Lambda/L)
1.2
1.4
Figure 3-1: Comparison of WAMIT and PDSTRIP Heave RAOs for Series 60 ship
with Cb=0.7 at Zero Speed
4
.
*
*
PDSTRIP
WAMIT
*
3.5
3
2.5[
0
4:
a:
C.)
0~
2
1.5
1
0.5[
A
00
0.2
1
0.8
0.4
0.6
Wavelength (Lambda/L)
1.2
1.4
Figure 3-2: Comparison of WAMIT and PDSTRIP Pitch RAOs for Series 60 ship
with Cb=0.7 at Zero Speed
50
1~r
K =0.75
y
o KY=1.15
101F
K =1.5
y
+
K =1.85
8
0
I
6
4
2
0
1
2
3
4
5
Wavelength (A/L)
6
7
8
Figure 3-3: Effect of Varying the pitch Gyradius on Heave RAO at Constant VCG
K y=0.75
o KY=1.15
8
[
K =1.5
+
K =1.85
-
7
6
5
0
a:
4
(L3
2
1
0'
0
1
2
3
4
5
6
7
8
Wavelength (AL)
Figure 3-4: Effect of Varying the pitch Gyradius on Pitch RAO at Constant VCG
51
VCG=-0.2
o
o
7
VCG=-0.3
VCG=-0.5
VCG=-0.6
+
6
5
0
<4
cc
CD
3
2
1
0
1
2
4
5
3
Wavelength (A/L)
6
7
8
Figure 3-5: Effect of Varying the VCG on Heave RAO at Constant Pitch Gyradius
35
30
o
VCG=-0.2
VCG=-0.3
o
+
VCG=-0.6
VCG=-0.5
25
U,
0
Er
-C
C)
20
15
a-
10
5
A
0
1
=and
2
3
4
5
Wavelength (JL)
6
7
8
Figure 3-6: Effect of Varying the VCG on Pitch RAO at Constant Pitch Gyradius
52
3.3.1
Discussion
Fig. 3-1 shows that WAMIT generally predicts less heave RAOs than PDSTRIP which
indicates the effect of diffraction contributions considered in WAMIT but neglected
in PDSTRIP. Also, because this range of wavelength depicts short waves, the results
from WAMIT are more numerically stable than those of PDSTRIP as expected.
Fig. 3-2 illustrates that at very small wavelengths, WAMIT gives more stable and
less values of pitch RAOs than PDSTRIP. However, both programs agree very well
at longer wavelengths close to the length of the ship.
Fig.
3-3 depicts that the added drag predicted by PDSTRIP is greater than
WAMIT predictions as expected since the added resistance depends on the first-order
motion responses of the ship in head waves. This agrees with the explanation given
in the theory section. Also, the diffraction contribution for short waves is significant
and cannot be neglected.
Fig.
3-4 shows that although, PDSTRIP neglects diffraction contribution, its
results agree very well with experiment and the Salvesen's method because the ship
considered here is not very bulky and hence, diffraction contribution is small. The
results obtained show larger discrepancies for PDSTRIP as CB increases.
Fig. 3-5 illustrates that the position of the peak of heave-RAO is unaffected by
the variation in the pitch gyradius since there is not direct relationship between the
parameters that determine the heave resonant frequency and the pitch gyradius as
predicted by theoretical investigation.
Fig. 3-6 shows that the peak of pitch-RAO increases and the position of the peak
moves to longer wavelength as the pitch gyradius is increased. This agrees very well
the theoretical study presented in the previous chapter.
Fig. 3-7 illustrates that the wavelength at which the peak of heave-RAO occurs
is unaffected by the change in VCG and this agrees with the theoretical study.
Fig. 3-8 depicts that the peak of pitch-RAO occurs at a shorter wavelength and
it increases as the VCG is lowered. This is because lowering the VCG increases the
pitch restoring coefficient, reduces the damping ratio and hence, the peak pitch RAO
53
increases as predicted by theoretical investigation.
3.4
Conclusions
The comparison of motion responses and added drag obtained using 2D and 3D panel
methods have been presented. The 2D panel method implemented in PDSTRIP gives
fairly accurate results for the motion responses of a floating body within acceptable
error limit for preliminary design but overestimates the second-order added drag
acting on a floating body because it does not consider diffraction contribution which
is non-negligible for waves in the short wavelength range.
54
Chapter 4
Experimental Study of the Responses
of SWATH USV in Regular Head
Waves
This section presents the experiments carried out in the MIT towing tank. The subsections detail the method employed in assembling the model, model characteristics,
towing tank dimensions, wave probe calibrations, experimental set up, instrumentation and procedure and processing of results.
4.1
The SWATH Model
The SWATH USV model has a scale ratio of 1:6 compared to the prototype.
It
is 1m long and designed to maximize wave cancelation effects using a parametric
optimization approach.
4.1.1
Assembling the Model
The components of the SWATH model were produced using a 3D printer and assembled by employing the process of additive manufacturing. Each of the parts printed
separately were joined carefully to make sure the model is water tight. The model is
55
made from Acrylonitrile Butadiene Styrene (ABS) plastic and lead weights were used
to ballast the model to achieve the required draft (DiMino, 2013).
4.2
Experimental Set-up
The wall and bottom effects on the experimental results are assumed negligible because the length of model compared to the width of the tank and the depth of water
in the tank satisfy the necessary requirements. To avoid these effects, the length of
the model must not be much in excess of the water depth and half the tank width.
The first table shows the main parameters of the SWATH model including its inertial
characteristics. The next table shows the tank dimensions and other testing conditions. The figures below show the profile, frontal and plan view of the model under
experimental condition
Towing Rig
La Sensor
Wave Probe
100 Cm
Figure 4-1: Profile View of Model during Experiment
56
Towing Rig
----Il-
.9
.7L
l
Figure 4-2: Frontal view of model during experiment
KZZ~
GI
0
KcZ~
Figure 4-3: Plan view of Model during experiment
57
The main parameters of the model including the inertial characteristics are presented in the table below:
Length Overall,LOA(m)
Breadth OverallBOA (m)
Test Draft, T (m)
1.000
0.907
0.187
LCG (m)
0.503
Displacement (kg)
19.000
VCG (m)
-0.250
Pitch Gyradius (m)
1.830
Table 4.1: Main Model Parameters
The table below shows the tank dimensions and factors which compare model size
to tank dimensions to show that wall and bottom effects are negligible.
Length,L(m)
Breadth,B (m)
Water depth,W.D (m)
LOA/B
LOA/W.D
30.48
2.44
1.22
0.41
0.82
Table 4.2: Tank Dimensions and Model Test Factors
The following configurations were used in setting up the experiment:
* Computer System-ADC-Wave-maker
The tow tank facility has a remotely operated pneumatic wave-maker which is connected to a computer system through an Analog to Digital Converter (ADC). In this
set-up, the ADC helps to convert digital signals from a computer program (LABVIEW) to analog signals which the wave-maker understands.
The power of the wave-maker was first switched-on, then pressure was supplied to
the wave-maker by turning a valve and the fan of the wave-maker box was turned on.
Waves with different frequencies and amplitudes can be generated on the computer
system.
58
* Wave probe-ADC-Computer System
This set up is similar to the one described above. However, the process is reversed as
the ADC in this case converts analog signals from the wave probe to digital signals
which the computer system understands.
The wave probe measure wave heights at a specific location and send these signals to the computer system as output voltages. Hence, the need for calibration to
determine the correlation between the output voltage and wave height.
* Laser sensors-converters-Computer System
A Micro-Epsilon displacement measurement laser sensor was used to acquire the bow
and stern displacement readings.
The signals from the sensors were conveyed to
a converter and the converter transmits the processed data via a USB cable to a
computer system which display the displacements in real time.
The table below shows the different cases of the model tests carried out in the first
batch batch of experiments.
Case No.
1
2
3
4
5
6
Amplitude [m]
0.00588
0.0161
0.0224
0.0245
0.0301
0.0203
A [m]
0.4836
0.9992
1.5136
2.1141
2.5581
3.1581
K [1/m]
12.9925
6.2882
4.1512
2.9720
2.4562
1.9895
F [Hz]
1.7968
1.2500
1.0155
0.8580
0.7775
0.6939
Slope, h/A [m]
0.0243
0.03223
0.02959
0.02318
0.02353
0.01286
7
0.0163
3.5406
1.7746
0.6502
0.0092
8
0.0248
3.9970
1.5720
0.6049
0.0124
9
10
0.0228
0.0143
4.5476
5.2205
1.3816
1.2036
0.5576
0.5084
0.01003
0.0055
11
0.0187
5.6144
1.1192
0.4830
0.0067
Table 4.3: Wave Parameters for Different Cases of Model Test
59
4.3
4.3.1
Instrumentation and Procedure
Laser Sensors
The displacement measurement devices used work by principle of Laser triangulation.
The optoNCDT 1302 sensors were used in the experiment. This laser diode of the
sensor projects a visible light on to the surface of the target object and the light
is reflected back to the sensor receiver at an angle. Shadowing must be avoided by
ensuring that the sensors are placed such that the reflected light does not encounter
an obstacle as it is being reflected back to the sensor's receiver. A complete set of the
device consist of a sensor and a converter with a USB cable connecting the converter
to a computer system. A 24V DC power supply was used to power the displacement
measurement instruments. One of the sensors is placed at the bow and the other at
the stern. Each of the sensors measures the distance from the target to the sensor.
4.3.2
Data Processing
The time series of the wave elevation captured using a calibrated wave probe was
processed using a Fast Fourier Transform (FFT) code in MATLAB to extract the
frequency of excitation for each of the runs carried out. The heave and pitch motion
with respect to the center of gravity was computed from the displacement measured
at the bow and stern. The complex motion amplitude with frequency closest to the
wave excitation frequency was extracted using an FFT code in MATLAB. The RAO
for heave and pitch is computed as the ratio of the complex motion amplitude and the
wave amplitude. For short wave length (first test with A = 0.5), no appreciable heave
and pitch motions were observed, so the measurements actually describe a noise. All
MATLAB scripts used are presented under appendices.
The heave and pitch motion of the SWATH USV can be derived from the bow
and stern displacements using the expressions below:
60
Heave Motion:
Zh =
l1df + l 2da
(4.1)
Pitch Motion:
Op = tan-
1
(d L da)
(4.2)
Where,
(4.3)
12
=(L - LCG)- 13
(4.4)
df = H - H1
(4.5)
da= HS - HS
(4.6)
The set of plots below are the time series of the heave and pitch motion obtained
when the expressions above are implemented in MATLAB.
61
Wave Test No. 1: A = 0.4836m; Motion Time History
Heave
-
1
0.8
--------------.
...
..-.
--
0.4
-
-- ----- -- --------------- -- -- - - ---- ----
-~
0.6
0.2
-
E
-
------
--
--
---------
0
-0.2
-0.4
...-. .------------
-0.6
-0.8
1
.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
42.5
45.0
47.5
50.0
52.5
47.5
50.0
I
52.5
0.2
-
0.15
-
-- ---- --
0.1
0.05
-
0
---
----- --------
-0.05
-0.1
-0.15
-0.2 "
0.0
2.5
-
5.0
- -
-7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
42.5
45.0
lime [s]
Figure 4-4: Heave and Pitch Motion Time History for Wave Test No. 1
62
Wave Test No. 2: A = 0.9992m; Motion Time History
'
Heave
--------
2
-----
-
-
-
-----
- --
-- --
--
-
---
- --
-
6
0.0
- --
- --- - -
2.5
5.0
-
Pitch
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
35.0
32.5
37.5
40.0
42.5
45.0
- - - - - - - - - - - - - - - - - - - - ----- ---- - - -- - - - - - - - - - -- - -- - - - - - - - - - - - - - - - - - - - - - - -- - - - - --- - - - - - - - - - - - - - - - - - - - - - - - -
0.8
7.5
-
- -
-- -
-- - ---- - ---
-
0.6
0.4
- - - - - - - - - - - -- - -~
~ ~-
-
- - - - - - -- - - - - ----- - - - - -
-
-2
-
2
0.2
0
-0.2
------ -------
-~--
-0.4
-0
0.0
.8 --
2.5
---
--
5.0
7.5
- --
10.0
-
12.5
--
---
15.0
17.5
20.0
22.5
Time [s]
25.0
27.5
30.0
32.5
-
- -
---
35.0
37.5
40.0
42.5
Figure 4-5: Heave and Pitch Motion Time History for Wave Test No. 2
63
45.0
-
-
-
-
-
-
-
-
-
-
-0.6
Wave Test No. 3: A
1.5136m; Motion Time History
Heave-
....
-- -..
-
-
- - --
-
4 - - -
- -.
---.
---..
..---.-.-.
2
.2
-- -----------
-
-4 -
..
.-------------------
-4
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
42.5
45.0
1.5
47.5
I
Pitch
----- ------
-- - ----
-------- -
50.0
52.5
T--
......
.
-6 -.
0.5
0
...... .....
-1.5
'
-1
.
VV
-0.5
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
42.5
45.0
47.5
Time [s]
Figure 4-6: Heave and Pitch Motion Time History for Wave Test No. 3
64
50.0
52.5
Wave Test No. 4: A = 2.1141m; Motion Time History
Heave
6
4
2
-
---- -- --
--
---------
0
-2
.4
-6
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
225
25.0
275
325
300
35.0
37.5
400
42.5
45.0
47.5
50.0
52.5
55.0
57,5
60.0
62.5
65.0
52.5
55.0
57.5
60.0
62.5
65.0
Pitch
15
.
- -- ...-.-.-.---------------..
....
..
..
0.5
0
-0.5
-1
0.0
2.5
5.0
7.5
10.0
12.5
-1.5
15.0 17.5
20.0
22.5
25.0
27.5
30.0
32.5 35.0
Time [s]
37.5
40.0
42.5
45.0
47.5
50.0
Figure 4-7: Heave and Pitch Motion Time History for Wave Test No. 4
65
Wave Test No. 5: A
2 1 1111
1
1
-
II
I
SHeav
I
I
I
I
--- -- --------0 - -- -- T -10-
-
--------------- ---
0
-30
-4.
-
-50
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
42.5
45.0
47.5
50.0
52.5
55.0
57.5
50.0
100
125
150
175
200
225
250
275
300
325
350
37.5
400
425
450
475
500
525
55.0
57.5
60.0
50.0
52.5 55.0
57.5
60.0
4
Pitch
-2-3-00
2
-5
25
1
50
75
-- - -
-
-
0.0
2.5
5.0
7.5
10.0
12.5
15.0
IT--
I
I--!I
-
17.5
20.0
- --
22.5
25.0
27.5
30.0
32.5
Time [s]
35.0
37.5
40.0
42.5
45.0
47.5
Figure 4-8: Heave and Pitch Motion Time History for Wave Test No. 5
66
lI,I
lfi|lm
iM
III
II
II'
01R
'
2.5581m; Motion Time History
lf|Ill
l1iIi'll |l'i~oil lop llllII|| III IIIf M1 .
111'
|
'
||
,
111 I 'l IIl"l
11|11
I
Wave Test No. 6: A = 3.1581m; Motion Time History
Heave
-
30
20
--------------- ---- ----10
0
-
-------
-
-
-
--
--- -
---
.------
-10
-~~
-. -..-..-----.
-.
-20
I
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
-
--
20.0
22.5
250
27.5
300
325
350
37.5
400
42.5
450
47.5
500
52.5
55.0
57.5
I
60.0
62.5
65.0
60.0
62.5
65.0
Pitch
4
-
-
-
27.5
30.0
32.5 35.0
Time [s]
---
- - -----
-------
2
0
-
--------------
-
-
-
-
12.5
15.0
17.5
20.0
-2
-4
0.0
2.5
5.0
7.5
10.0
22.5
25.0
37.5
40.0
42.5
45.0
47.5
50.0
52.5
55.0
57.5
Figure 4-9: Heave and Pitch Motion Time History for Wave Test No. 6
67
Wave Test No. 7: A = 3.5406m; Motion Time History
30
20
.-.
--
10
-
-
-
-
-
-
T0
-10
--
- ----
--
- -- ------
--
--
-- --
-20
-30
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
42.5
450
47.5
50.0
52.5
55.0
Pich
-
4
--
------
-
- ..
----
-..
..
-.
--.--.-.
2
---- ----
---- --0 --- --- ----b
----
-4
-2
-4
-6
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
Time [s]
32.5
35.0
37.5
40.0
42.5
45.0
47.5
50.0
52.5
Figure 4-10: Heave and Pitch Motion Time History for Wave Test No. 7
68
55.0
Wave Test No. 8: A = 3.9970m; Motion Time History
I
1 I C 1*
r
30
'*
Heave-
20
1
-- - --------
10
- --
-- ------
-------
--
-
---
------- -------
-
11 I
0
-
-----
-
-20
.30
2
0.0
5.0
2.5
7.5
10.0
-
-
12.5
17.5
15.0
20.0
22.5
25.0
30.0
27.5
32.5
35.0
37.5
41 .0
42.5
40.0
Pitch
--
- --
--
--
-
----- -
--
-
2
------- -...
-
- ----
-3-2-3
0.0
2.5
5.0
7.5
10.0
-
- -
-
12.5
-- - -
---
15.0
- --
17.5
-
- -
-- -
---
-
----- - ----
---- -
--
-5
-4
-
-- -
. -.
..-
-
20.0
22.5
Time [s)
25.0
27.5
30.0
-
----- -
---
-------
-
32.5
35.0
37.5
-
-2
-
-
- -
---
---
-----
--
1
-
------ ---------
--
-- -- ----------
-10
40.0
42.5
Figure 4-11: Heave and Pitch Motion Time History for Wave Test No. 8
69
45.0
Wave Test No. 9: A = 4.5476m; Motion Time History
40
Heave
30
- --
- --- ----- -
--
--
- --
-
-- - - -
20 ---
-- -- -------- --- -------- -------------10
E
--
0
--
-
----- --
---- -----
--
-10
------
----
------
-
-20
.30
-40
0.
0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
32.5
8
Pitch
6
- -- - - - - -
-
-
-
--
--
4
--
-
-----
-.
-.
----
-
2
0
--------
a.
-2
-----
-
--4
-6
-8
0.
0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
Time [s]
Figure 4-12: Heave and Pitch Motion Time History for Wave Test No. 9
70
32.5
-5. 27.5 -3-.- -3--- 5- -3 --. --37. -4-.--4--75--.----5-5.------20- 225
Wave Test No. 10: A = 5.2205m; Motion Time History
Heave-
-------- -- - - - - - - - - - - - - - - ---- ---- ----------------
15
- ------------- ---- ----------- --- - - --- - - - -------- - - -- - - -- - - - - - - - - - - --- - -- - - -- - -- ---- -- ------ -- --
-
20
10
0
-10
- - - - - -- - - - ----- -
- - -- - - -
-- - ---
- --
- -
-
-
- --
-
-
E
- - - - --- -- -- -------- - - --
------
--- ------ ---- -------------- - - ---------- - - - - - - - - - - - - - - - - - --
5
-15
---
-- ----
--
--
----
----- ---------------------------------
-20
-25
-30
0.
0
4
- - -
5.0
2.5
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
4, .5
40.0
Pitch
. .
. -
.
-.-.-- - - -
-
- -- -
-
~
.... -. .
- -..
-....--
- -
- -
-
-
-
-
- --
--
-
-
-
-
~
~
--
-----.
- -..
- ---
------------2
-
--
-
.
-.
3
0.
CL -2
-- -
-
25---
00
-4
-
-
-5 -6
0.0
--
5-0
-
--
--
----
----
--2
--- - -----
-
-- - -------
-
5.0
7.5
-
------ - -
12.5
15.0
17.5
20.0
-
. -
---
.
--
-.--
30.0
32.5
22.5
25.0
27.5
35.0
37.5
40.0
Time [s)
Figure 4-13: Heave and Pitch Motion Time History for Wave Test No. 10
71
- -
-.--
- --
-.. - ---10.0
- ------
---------
-
--
---- -- 2-5
-------
-------
42.5
-
-. .- .-
Wave Test No. 11: A = 5.6144m; Motion Time History
15
Hleave --
10
~~ ~ ~ ~ ~ ~ ~ - - -
- - - - - - -- - - - ---
-
- - -
- - --
- - -
-
-
- -
5
0
-- --- - --------
-10
-
-
- - - ----
--
- ---
-----
---
-
--..-.
- ----
--
-
-5
--
-.
-15
I
-A
-
20'0.0
2.5
5.0
7.5
10.0
12.5
-
-.
15.0
I17.5
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
42,5
45.0
47.5
50.0
|
52.5
3
PitchII
2
. - - ---- -- - -----
-
---
1
0
b-1
-
-
-------
-----
-2
-3
II-L
-.I ii jI-j
-1
0.0
1
5.0
2.5
1
7.5
10.0
1
12.5
1
I
15.0
17.5
I
20.0
J
22.5
25.0
27.5
Time [s]
30.0
32.5
35.0
37.5
40.0
42.5
45.0
47.5
50.0
Figure 4-14: Heave and Pitch Motion Time History for Wave Test No. 11
72
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52.5
Only a portion of the recorded of the recorded heave and pitch responses of the
SWATH USV is processed. These processed heave and pitch responses for test case
7 are present below and the same process is carried out on the responses recorded for
all the experiments.
-25
-30
11 .0
1
-
---
12.0
13.0
------
--
--- ---- -
-
-----14.0
15.0
16.0
17.0
18.0
19.0
20.0
21.0
---
22.0
~tch--
-
- ------
-
---------
--.
....-
0 - ------- . ....---1
----....----2 --------- - --------- -3
.
-
--------.
.
.
-c
---
-
.. ....-
-15
-20
4
3
2
----
.
U
CL
-.
5
0
-5
-10
-
-
>
10
H ea
-----
-
E
..
...-.
.. ----- -
-
E
25
20
15
-4
-5
12.5
15.0
17.5
Time [s]
20.0
Figure 4-15: Cut Heave and Pitch Responses of SWATH USV
73
22.5
4.4
Results and Discussion
The experimental and computational results are compared in this section. The discussion highlights quantitatively the significance of the viscous effects responsible for
the discrepancies between the potential flow solver results and experiments. The comparison of the heave RAOs obtained from experiment, CAT 5D (2D Potential flow
Method) and WAMIT (3D potential flow) method is presented below:
Heave RAO at VCG= -0.25m and K = 1.83m
y
3
*
o
+
2.5F
WAMIT
CAT5D
Exp
2
co,
0
1.5
ciz
-
1
0.5 k
0*
0
1
2
3
4
5
Wavelength (U/L)
6
7
8
Figure 4-16: Comparison of Experimental and Computational Predictions of Heave
Motions of SWATH USV
74
The comparison of the pitch RAOs obtained from experiment, CAT 5D (2D Potential flow Method) and WAMIT (3D potential flow) method is presented below:
12
Pitch RAO at VCG= -0.25m and K = 1.83m
y
*
o
+
10-
1
WAMIT
CAT5D
Exp
8LO
0
6-
4 --
2-
V0
1
2
3
4
5
Wavelength (L)
6
7
8
Figure 4-17: Comparison of Experimental and Computational Predictions of Pitch
Motions of SWATH USV
75
4.4.1
Discussion
Fig. 4-16 shows good agreement between the wavelength at which resonant heave
response occurs for both experiment and WAMIT. As expected, the potential flow
solution predicts greater peak response than experiment. This is mainly because the
viscous effect effect is important for this kind of hull shape. However, the a small
fraction of the discrepancy may also be due to friction on the rod on which the
model slides while heaving. The 2D potential solver gives a trend with agrees with
experiment but the wavelength at which resonance occurs is much greater than that
obtained by experiment. Improvement on this method is necessary to achieve better
agreement with others.
Fig. 4-17 illustrates a very good agreement between the pitch responses from experiment and WAMIT especially at wavelength below 2.5 times the length of the vessel.
The discrepancy in the peak RAO is however significant and is mainly due to viscous
effects unaccounted for in the potential flow based method.
The friction resistance on the sliding rod which anchors the model could influence
the response of the vessel, especially at higher frequencies. This also contributes to
the discrepancy between peak responses of the ship obtained through experiment and
numerical computation.
76
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Chapter 5
Conclusion
The research work presented here comprise of the analytical, numerical and experimental investigation of the hydrodynamics of an unconventional SWATH USV proposed to be used to monitor a team of underwater vehicles in a safer and more cost
effective manner.
The core of the work involves the assembly of a scaled model of the vessel and the
subsequent model testing at the MIT towing tank which involved the setting up of
specific instrumentation, data acquisition, data processing and comparison of experimental results with potential flow solvers predictions.
The accurate prediction of the motions of a ship in waves is very important in the
computation of the added resistance of the ship since the added resistance is dependent on the RAOs of the ship in heave and pitch especially when considering ships
in head waves.
Some added resistance results for mono-hulls were also presented
and comparison was made between the RAOs obtained from different methods. The
analyses confirms that the efficiency of a method in predicting the added resistance
depends on the accuracy of the input RAOs. This inference is also expected to be
true for the unconventional SWATH USV investigated in this work.
The investigation of the effects of inertia characteristics on the motion responses of
the SWATH USV showed that the vessel will have better responses when loading is
distributed about the center of gravity especially at resonance.
77
Summary of Results
5.1
The results presented achieve the thesis objectives of providing more experimental
data for validation of numerical results.
The discussions of the results provides insight into the behavior of the vessel in waves
under varying conditions and change in its inertia characteristics.
5.2
Recommendations
In order to better substantiate the agreement between experiment and WAMIT, more
test points are required especially at small wavelengths.
Additionally, the time at which the wave front reaches the wave probe must be noted
in order to allow for proper overlapping of the motion response and wave elevation
time series. This will improve the data processing accuracy by ensuring that the best
part of the motion history is cut for analysis.
Finally, installing a more effective beach in the tank would reduce the effect of interference resulting from reflected waves propagating back from the end of the tank.
This reduces the window of good data available for the cut motion time history.
78
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Bibliography
11] Ananthakrishnan, P. "Effects of viscosity and free surface non-linearity on wave
motion generated by an oscillating twin hull". Technical report, Department of
Ocean and Mechanical Engineering, Florida Atlantic University, 2012.
[21 Arribas, F.P (2006) "Some methods to obtain the added resistance of a ship
advancing in waves". Ocean Engineering, Science Direct Journal 34 (2007),
946-955
[3] Bertram V., Sdding, H. and Graf, K.(2006) "Program PDStrip: Public Domain
Strip Method", Hamburg, Manual
[4] Bertram V. PracticalShip Hydrodynamics. Butterworth-Heinemann, 2000.
[5] Bonfigilo, L., Brizzolara, S. and Seixas de Medeiros J. (2013a) "Influence of
viscous effects on numerical prediction of motions of swath vessels in waves".
Ocean Systems Engineering Int. Journal, Vol. 3, No. 3 (2013) 219-236. ISSN:
2093-6702 DOI: http://dx.doi.org/10.12989/ose.2013.3.3.219.
[6] Bonfiglio, L. Brizzolara, S. and Chryssostomidis, C. (2013b), "Viscous free surface
numerical simulations of oscillating SWATH ship sections". Proceedings of the
10th WSEAS InternationalConference on Fluid Mechanics. FLUIDS'13, Milano,
Italy, 9-11 January.
[7] Bonfigilo, L. and Brizzolara, S. (2014) "Unsteady viscous flow with non-linear
free surface around oscillating swath ship section". WSEAS Transactionon Fluid
Mechanics,Vol. 9, (2014). E-ISSN: 2224-347
[8] Brizzolara, S. "Parametric optimization of swath hull forms by a viscous-inviscid
free surface method driven by a differential evolution algorithm". Technical
report, University of Genoa Italy, August 2004.
19]
Brizzolara, S. and Chryssostomidis, C. (2012) "Design of an unconventional asv
for underwater vehicles recovery: Simulation of the motions for operations in
rough seas." ASNE International Conference on Launch Recovery, Linthicum
(MD), 14-15, November
[10] Chryssostomidis, C. and Patrikalakis, N.M. (1986), "Seakeeping calculations for
SWATH ships using a new modified version of 'CAT-5"', MIT Sea Grant Report
86-7TN.
79
[111 Dallinga R.P. et. al. "Development of design tools for the prediction of swath
motions". report, Maritime Research Institute Netherlands, DREA Canada,
June 1989.
[12] Faltinsen, O.M. "Sea loads on Ships and Offshore Structures", chapter 5. Cambridge University Press, 1990.
[13] Gerritsma, J. and Beukelmann, W. (1967) "Analysis of modified strip theory
for the calculation of ship motion and wave bending moments". International
Shipbuilding Progress 14(156), 319-337.
[14] Journee, J.M.J. (2001) Theoretical Manual of SEAWAY.
[15] Korvin-Kroukovsky, B.V. and Jacobs W.R. "Pitching and Heaving Motions of a
Ship in Regular Waves", TRANS.SNAME,1957.
[16] Mansour, A. and Choo, K.Y. (1973), "Motion and loads prediction of catamarans
in random seas", Report 73-6, MIT, Dept. of Ocean Engineering.
[17] Matulja, D., Sportelli, M., Guedes Soares, C. and Prpi-Orsid, J. "Estimation
of added resistance of a ship in waves" Journal BRODOGRADNJA, Vol. 62,
(2011), UDC: 629.5.015.24
[18] Newman J.N Marine Hydrodynamics: MIT Press, 1977
[19] Salvesen, N.
1978.
"Added resistance of ships in waves".
Journal of Hydronautics,
[20] Salvesen, N., Tuck, E.G and Faltinsen, O.M. "Ship motions and sea loads".
Transactions SNAME, Vol. 78, pp. 250-287, Jersey City, 1970.
[21] Schellin, T.E. and Papanikolaou, A. "Prediction of seakeeping performance of a
swath-ship and comparison with measurements". Technical report, Germanischer
Lloyd, National Technical University Athens, June 1991.
[22] Timman, N. and Newman J.N. " The coupled damping coefficients of a symmetric ship" Journal of Ship Research, 5:1 7 (1962)
[23] WAMIT Inc. Revised Manual of WAMIT Version 7.0, 2013.
[24] Young-Joong Kwon. "The effect of weather, particularly short sea waves on ship
performance". PhD thesis, November 1981.
80
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Appendices
The MATLAB script used to process wave signal is given below:
% Fourier analysis of wave probe signals from SWATH ASV experiments
% Script written by ABIODUN OLAOYE working with Dr. Luca Bonfigilo
% For MIT-iShip Lab headed by Professor Stefano Brizzolara
% December 17, 2014
clear all,
clc
g= 9.81; % Acceleration due gravity
t, = 0.025;%Timestepofsignal
Fs = 1/t,;%Samplingfrequency
NL = ll;%Numberofwavelengths
fid
=
f open('waveerbose','w');
fprintf (fid,'%15s%20s',' Wavelength[m]','WaveAmplitude[m]');
fidl = f open('wave',' w');
loadwavelength
WLR = wavelength(:, 1);%Requiredwavelength
f, = sqrt(g./(2 * pi. * WLR));%Requiredf requencyinhz
forLC = 1: NL%Lambdacount
prompt =' Enterfilename :';
str1 = input(prompt,'s');%Obtainresponsefromuser
M = load(stri);
%ExtractTimeseriesofWaightHeight
Time = M(:, 1);
81
Volt = M(:, 2);
h = (Volt - 1.7201)/0.1002;
figure(1)
plot(Time, h)
title('OriginalWaveHeightTimeSeries')
xlabel('Time(s)')
ylabel('WaveHeight(m)')
NUM = input('Enterstarttimeofsignal :');
NUM 2 = input('Entercuttimeofsignal :');%Endtimeofheavesignal
i = round(NUM1 /t,); j = round(NUM2/t,); hi = h(i j, 1);%Cutheavetimeseries
Time1 = Time(i j, 1);
figure(2)
)
plot(Time1, h1
title('abridgedwaveheighttimeseries')
xlabel ('Time (s')
ylabel('Waveheight(mm)')
L = length(h1 );
NFFT = 2nextpow2(L);%Nextpowerof 2fromlengthofy
H1 = f ft(h1, NFFT)/L;
f = Fs/2 * linspace(O, 1, NFFT/2 + 1);
H2 = 2 * abs(H1(1: NFFT/2 + 1));
[ ,idxl= min(abs(f - f,(LC, 1)));
WA = H2 (idx)/100;
w = 2 * pi * (f(1, idx));
WL = ((2 * pi * g)/w 2 );%Actualwavelengthofclosestwavetorequiredwave
WLWA = [WL; WA] ;%WLhererepresents(WL/Length)ofmodelsincemodelislmlong
fprintf (fid'%14.8f%20.8f', WLWA);
fprintf(fid1,'%12.8f%16.8f', WLWA);
end
82
Derivation of heave and Pitch Response
The MATLAB script used to derive the heave and pitch response from bow and stern
displacement is shown below:
% Derivation of Heave and Pitch Motion from Bow and Stern Displacement
% MATLAB script written by Abiodun Olaoye
clear
clc
N=11; % Number of Experiments
for NE=1:N
prompt= 'Enter file name:
str= input(prompt,'s');
M= load(str);
Time= M(:,1);
DispB. = M(:, 2);
Dispstern = M(:, 3);
H = (345.166 * DiSPBOW - 293.009 * Dispstern)/638.175;
Prad = atan((DispBn, - Dispstern)/638.175);
Pdeg = (Prad* 180/pi);
P = [Time'; H'; Pdeg'];
filename =
['Hp'retsprintf('%i', NE)];
f ilename2 = ['ModelTest'sprintf('%i',NE)];
f id = f open(f ilename,'w');
f id2 = f open(f ilename2,' w');
fprintf(fid,'%6s% 15s%24s',' Time',' Heave(mm)',' Pitch(deg)');
fprintf(fid,'%6.4f%15.8f%24.8f', P);
fprintf(fid2,'%6.4f%15.8f%24.8f', P);
end
83
Processing of Heave and Pitch Response
The MATLAB script used to process the heave and pitch response of the ship is
shown below: %Analyses of Experimental data obtained during SWATH ASV model
test to determine Heave and Pitch RAOs
% Script written by ABIODUN OLAOYE working with Dr. Luca Bonfigilo
% For MIT-iShip Lab headed *by Professor Stefano Brizzolara
% December 17, 2014
clear all;
clc
g= 9.81; % Acceleration due gravity
t, = 0.0013;%Timestepofsignal
Fs = 1/t,;%Samplingfrequency
NL
=
ll;%Numberofwavelengths
loadwave
%fid= fopen('RAOsverbose','w');
%fprintf(fid,'%15.8f%20.8f%S 24.8','Wavelength [m]','HeaveRA O[m/m]',' PitchRAO[radm/m]');
%fidl = fopen('RAOs','w');
for LC=1:NL % Lambda count prompt= 'Enter file name:
str1 = input(prompt,' s');%RObtainresponsefromuser
M = load(stri);
%SolvefortheheaveRAO
heavemotion
h = M(:, 2) ;%Originaltimeseriesof
Time = M(:, 1);%figure (1)
plot(Time,h)
title('Original heave motion')
xlabel('Time (s)')
ylabel('Heave (mm)')
NUM 1 = input('Enterstarttimeofsignal :');
84
NUM 2 = input('Ernternumberofperiods :');
WL = wave(LC, 1);%Wavelength
WA = wave(LC, 2);%WaveAmplitude
WP = sqrt((2 * pi * WL)/g);%WavePeriod
K = (2 * pi/WL);%Wavenumber
signalinthetimeve
i = round(NUM1 /t,);%Intergeridentificationofthetimethatcorrespondstostartof
NUM 3 = NUM1 +NUM2 *WP;%Endtimeofheavesignalj = round(NUM3 /t,);%Intergeridentificat
h1 = h(i: j, 1);%Cutheavetimeseries
Time, = Time(i: j, 1);
figure(2)
)
plot(Time1, h1
title('abridgedheavemotion')
xlabel ('Time(s)')
ylabel ('Heave(mm)')
L = length(hi);%S Length of signal
NFFT = 2nextpow2(L) ;%Nextpowerof 2fromlengthofy
H1 = f ft(hi, NFFT)/ L;
f = Fs/2 * linspace(O, 1, NFFT/2 + 1);
H2 = 2 * abs(H(1 : NFFT/2 + 1));
fe = sqrt(g/(2 * pi * WL));
[ ,idx]
= min(abs(f - fe));
Heave = H2 (idx)/1000;
HeaveRAO = Heave/WA;
% Process Pitch Signal p= M(:,3); % Time series of heave motion figure (3)
plot(Time,p)
title('Original pitch motion')
xlabel('Time (s)')
ylabel('Pitch (deg)') NUM 4 = input('Enterstarttimeofsignal:');
NUM 5 = input('Enternumberofperiods :');
signalinthetimeve,
i = round(NUM4 /ts);%intergeridentificationofthetimethatcorrespondstostartof
NUM 6 = NUM 4 + NUM 5 * WP;%Endtimeofheavesignal
85
signalinthetimeveci
j = round(NUM6/t,);%intergeridentificationofthetimethatcorrespondstoendof
pi = p(i : j, 1);
Time, = Time(i j, 1);
figure(2)
plot(Time1, pi)
title('abridgedpitchmotion')
xlabel('Time(s)')
ylabel('Pitch(deg)')
P1 = P1.* (pi/180);%pitchangleinrad
L = length(p1 );%Lengthof signal
NFFT = 2nextpow2(L);%Nextpowerof2fromlengthofy
P1 = f ft(p1, NFFT)/L;
P2 = 2 * abs(Pi(1 : NFFT/2 + 1));
f = Fs/2 * linspace(O, 1, NFFT/2 + 1);
, idx
=
min(abs(f - fe));
Pitch = P2 (idx);
PitchRAO = Pitch/(K * WA);
fprintf(fid,'%12.8f%17.8f%17.8f', RAOs);
end
86