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Bachelor of Engineering Thesis
Use of Initial Trim to Minimise Dynamic Under Keel
Clearance for Full Form Vessels.
Samuel Davey
27 March 2013
Supervisor: Jonathan Duffy
Thesis submitted in partial fulfilment of the requirements for the degree
of
Bachelor of Engineering (Naval Architecture)
National Centre for Maritime Engineering and Hydrodynamics
ii
DECLARATION
This project report contains no material which has been accepted for a degree or diploma by
AMC, University of Tasmania or any other institution, except by way of background
information and duly acknowledged in the report, and to the best of my knowledge and belief,
no material previously published or written by another person except where due
acknowledgement is made in the text of the report.
Signed:
Date:
STATEMENT 1
This project report is the result of my own investigation, except where otherwise stated. Other
sources are acknowledged in the text giving explicit references. A list of references is
appended.
Signed:
Dated:
STATEMENT 2
I hereby give consent for my project report to be available for photocopying, inter-library
loan, electronic access to AMC and UTAS staff and students via the UTAS Library, and for
the title and summary to be made available to outside organisations.
Signed:
Dated:
1
Table of Contents
Declaration ............................................................................................................................................. ii
List of Figures ......................................................................................................................................... 2
List of Tables .......................................................................................................................................... 2
1. Project Title ......................................................................................................................................... 3
2. Aim ..................................................................................................................................................... 3
3. Supervisors.......................................................................................................................................... 3
4. Scope ................................................................................................................................................... 3
5. Preliminary Literature Survey and References ................................................................................... 4
5.1 Reasoning behind survey .............................................................................................................. 4
5.2 Literature Review .......................................................................................................................... 4
5.3 Papers Still to be Read .................................................................................................................. 5
6. Schedule and Milestones..................................................................................................................... 6
6.1 Key Milestones ............................................................................................................................. 6
6.2 Meetings ........................................................................................................................................ 6
6.3 Test Program ................................................................................................................................. 7
7. Resources Required .......................................................................................................................... 11
7.1 Experimental Facility .................................................................................................................. 11
7.2 Model Particulars ........................................................................................................................ 11
7.3 Apparatus .................................................................................................................................... 13
References ............................................................................................................................................. 14
Appendix B – Gantt chart ..................................................................................................................... 15
2
List of Figures
Figure 1 - Model Vessel ........................................................................................................................ 12
Figure 2 - Model Vessel with Largest Beam Attachments on .............................................................. 12
Figure 3 - Setup Apparatus ................................................................................................................... 13
List of Tables
Table 1 - Key Milestones ........................................................................................................................ 6
Table 2 - Test Program Day 1 ................................................................................................................. 7
Table 3 - Test Program Day 2 ................................................................................................................. 7
Table 4 - Test Program Day 3 ................................................................................................................. 8
Table 5 - Test Program Day 4 ................................................................................................................. 8
Table 6 - Test Program Day 5 ................................................................................................................. 9
Table 7 - Test Program Day 6 ................................................................................................................. 9
Table 8 - Test Program Day 7 ............................................................................................................... 10
Table 9 - Test Program Day 8 ............................................................................................................... 10
Table 10 - Test Program Day 9 ............................................................................................................. 10
Table 11 - Test Program Day 10 ........................................................................................................... 11
Table 12 - Model Vessel Particulars ..................................................................................................... 11
Table 13 - Testing Equipment ............................................................................................................... 13
3
1. Project Title
Use of Initial Trim to Minimise Dynamic Under Keel Clearance for Full Form Vessels.
2. Aim
With the current demand for cargo vessels to be larger and travel faster the effects of shallow water
have become more significant to the shipping industry. When travelling in shallow water full form
vessels will undergo sinkage and trim due to pressure effects on the hull. Therefore the aim of this
research is to determine whether an initial trim on a full form vessel will reduce the amount of vessel
squat and trim when underway.
3. Supervisors
The primary supervisor is Jonathan Duffy from the Australian Maritime College (AMC). Chris Hens
and Giles Lesser from O'Brien Maritime Consultants International (OMCI) are industry participants
and will co-supervise the project.
4. Scope
The scope of this project is to investigate the effect initial trim has on a full form generic bulk carrier
with regards to ship squat in shallow waterways. This will be achieved through physical scale model
tests in calm water in the Model Test Basin (MTB) at the AMC. A 2.385 metre long ship model will
be tested at a number of different speeds and depth to draft ratios for a range of initial bow down
trims. The heave and pitch will be measured using the motion capturing system Qualisys. A detailed
test program can be seen in 6.3.
If time permits the effect of changing the vessel beam will also be investigated. The model will be
tested at 3 different beams. Only two initial trims will be tested for the different beams, as well as zero
trim due to the limited testing time available. After the completion of the first beam test, if insufficient
time remains this part of the research may be removed from the scope.
As part of another first semester subject a CFD analysis may also be conducted. If this project is
completed it will also be used in this thesis. Results will be compared to experimental data and an
empirical formula and the correlation between the three methods will be analysed.
The data obtained will be compared to zero trim tests to determine if there are any benefits of loading
vessels with set trim by the bow.
4
5. Preliminary Literature Survey and References
5.1 Reasoning Behind Survey
The literature survey will be conducted to achieve the following:
To see if any other research has been undertaken in this field.
To gain a wide range of knowledge about the subject.
To determine what methods should be used.
To decide upon what parameters to investigate.
To determine what results are to be expected.
5.2 Literature Review
Dietze et al (1997) investigated the problem of squat and compared current methods to predict vessel
squat. Work they compared and discussed included Huuska (1976), Icorels (1980), Millward (1990)
and Barrass (1971). Three common hull forms were chosen to compare these techniques to predict
vessel squat in shallow waters. The tests were conducted in straight channels with no ship interaction
and no sudden changes in channel configuration. From the tests they concluded that it was not
possible to recommend one single squat estimation method as each method had different answers,
with the largest variation came at a Froude Depth Number of 2. However, from the tests they were
able to develop guidelines and graphs to recommend the most accurate method depending on different
situations.
Millward (1990) conducted experiments to investigate the influence of hull-form on vessel squat. His
results showed that the largest instances of squat were usually located at a Froude Depth Number of
0.9, while at supercritical speeds the squat would usually become negative, hence the vessel would
rise in the water rather than squat. He concluded that the squat curve will follow the same shape
regardless of the hullform with the only difference being in the values of squat. He took an empirical
approach to find a family of curves for the prediction of ship squat. Millward used Gibbing’s curve
fitting technique to find formulas to fit the curves. He found 4 constants, 3 of which were similar for
any hull form and one which was based on hull parameters. For the constant based on hull parameters
he used the extreme points from his results so his squat predictions would be larger to err on the side
of caution. From his results he deduced that the squat of the vessel was always greater at the bow
compared to amidships, especially for fuller formed vessels. He also concluded that the way the vessel
will trim is to do with where the centre of buoyancy (COB) is compared to amidships. For example if
the COB was forward of midships the vessel was more likely to trim towards the bow. In conclusion
Millward’s formulae showed good agreement to current squat formulae, in almost every case gave a
better agreement than the simple formulae previously used.
Eryuzlu (1994) conducted experiments into the under keel requirements for large vessels in shallow
waterways for the Canadian Coast Guard. They tested 5 models at a scale of 1/100, with varying
depth to draft ratios and speeds while keeping level trim in an unrestricted waterway. Over 1000 tests
were conducted and through multiple regression analyses two dimensionless equations were derived.
He stated that these equations can be applied with confidence to certain vessels (19000-227000DWT
with depth to draft ratios of 1.1 – 2.5). Extra tests were also conducted to determine the effect of
channel width. From these tests a channel width factor was derived and included in the equations. The
new equations were then compared to field data and other squat formulae, including Tuner, Barrass
and Simard’s equations. These equations were more accurate in both unrestricted and restricted
5
waterways than previous formulae, nothing that accuracy in restricted waterways was only high when
the width factor was used. The results from the tests concluded that for vessels with no trim, bow
squat was predominant. He made special mention as to the limits of applicability of their equations
and not to use them outside these bounds.
Dand (1971) investigated the effect of bulbous bows on ship squat. Again he did this as most of the
methods in use did not take into account any hull shape when predicting squat. He conducted model
tests in shallow water with two different hulls, one with a hemispherical bulb and the other having an
S-type protruding ram bow. The models were self-propelled and without rudders. The expected results
were that both sinkage and trim would increase at the same rate with a decreasing depth to draft,
however the results showed that only the mean sinkage increased. Dand was unsure of the reasons but
speculated that it would likely be to do with how close the bulb is to the free surface and the suction
effects a bulb would experience at different speeds. He concluded that certain ships with bulbous
bows and high block coefficients will have an increased value of mean sinkage, but they will not trim
by the head when h/T ratio is decreased, as many normal full form vessels would.
Dand (1972) also conducted research into the effect of squat in shallow water. The main difference in
his work to previous papers was that his research was based on predicting squat in waters with
unrestricted width while most previous papers predicted squat in restricted waterways. Dand
investigated three methods, a simple one-dimensional method, the Sogreah Method which is an
empirical method and a theoretical method by Tuck (1970). According to Dand these methods have
merit but they all disregard the effect of viscosity, propeller and squat due to the hull wave system,
therefore under predicting squat. From this investigation and model tests, Dand developed a
prediction method using the one dimensional theory with some modifications. Through his model
tests he calculated an effective channel width factor using the fluid disturbance around the hull as an
effective channel width. He then provided correction factors for the viscosity and the effect of the
propeller directly from model tests.
5.3 Papers Still to be Read
The first list of papers is directly relevant to this thesis. These papers will all be read for the
completion of this thesis as well as others that are found to be relevant. The second lists papers that
have some relevance to this thesis and therefore will only be read if time permits.
List 1:
Tuck (1966)
Hooft (1974)
ICORELS (1980)
Millward (1992)
Barrass I (1979)
Barrass II (1979, 81)
Eryuzlu and Hausser (1978)
List 1 Cont:
Romisch (1989)
Harting et al (2009)
Briggs et al (2009)
Gourlay (2007, 08)
Stocks et al (2004)
Harting et al (1999, 2002)
Lataire et al (2012)
Briggs (2010)
List 2:
Huuska (1976)
Ankudinov et al. (2006)
Barras (2004)
Beaulieu et al (2009)
Dunker (2002)
Delefortrie et al (2010)
Eloot et al (2008
6
6. Schedule and Milestones
6.1 Key Milestones
The final year thesis subject for 2013 has 6 submissions. Table 1 shows the dates each submission is
due. A Gantt chart with every milestone has been included in Appendix B – Gantt
Table 1 - Key Milestones
Date Milestone
19/04/2013 Project Plan
21/06/2013 Interim Report
21/06/2013 JEE418 Completed
TBD Testing
03/10/2013 Journal Article Submission
03/10/2013 Executive Summary Submission
03/10/2013 Supporting Documents
25/10/2013 Oral Presentation
29/11/2013 JEE419 Completed
6.2 Meetings
Meetings will be held every Wednesday at 11am in the Towing Tank conference room with
Jonathon Duffy, Shaun Denehy, Tim Vaughan, Rohan Langford and Mitchell Todd.
Additional meetings will be held with Jonathon Duffy when deemed appropriate.
Minutes will be taken at every meeting and will be discussed at the following meeting to keep
track of progress.
Additional meetings between the four students will take place when deemed appropriate.
7
6.3 Test Program
The testing will be conducted in a group basis with Timothy Vaughan, Rohan Langford and Mitchell
Todd. The testing program is shown in Table 2 to Table 11. Each table represents one days’ worth of
testing. The test runs that will be of interest for my research are highlighted in grey. The testing has
been estimated to take 10 full days. This includes re runs, change over times, time needed for the tank
to settle and time needed to alter the model for different conditions. Set up time has not been included.
A length of time for setup will need to be decided upon discussing with technicians.
Preliminary testing has already taken place. This testing was done to confirm whether the Qualisys
system would be accurate enough to determine the small changes in heave and pitch needed for this
investigation. Another reason for the preliminary testing was to see whether the capture area for the
Qualisys program was large enough to record both the initial condition and a steady state in a single
run. Results from the initial testing indicate that Qualisys will have sufficient capture period to record
an initial value and a steady state period.
Table 2 - Test Program Day 1
Run
No.
h/T Water
Depth (m)
Draft
(m)
Beam
(m)
Trim
(deg)
Speed
(m/s)
Time
1 2 0.18 0.09 0.473 0 0.23 900
2 2 0.18 0.09 0.473 0 0.23 930
3 2 0.18 0.09 0.473 0 0.46 1000
4 2 0.18 0.09 0.473 0 0.69 1030
5 2 0.18 0.09 0.473 0 0.92 1030
6 2 0.18 0.09 0.473 0.5 0.23 1100
7 2 0.18 0.09 0.473 0.5 0.46 1130
8 2 0.18 0.09 0.473 0.5 0.69 1200
9 2 0.18 0.09 0.473 1 0.23 1230
10 2 0.18 0.09 0.473 1 0.23 1300
11 2 0.18 0.09 0.473 1 0.46 1330
12 2 0.18 0.09 0.473 1 0.69 1400
13 2 0.18 0.09 0.473 2 0.23 1430
14 2 0.18 0.09 0.473 2 0.46 1500
15 2 0.18 0.09 0.473 2 0.69 1530
16 1.4 0.18 0.129 0.473 0 0.23 1600
17 1.4 0.18 0.129 0.473 0 0.23 1630
18 1.4 0.18 0.129 0.473 0 0.46 1700
Table 3 - Test Program Day 2
19 1.4 0.18 0.129 0.473 0 0.69 900
20 1.4 0.18 0.129 0.473 0 0.92 930
21 1.4 0.18 0.129 0.473 0.5 0.23 1000
22 1.4 0.18 0.129 0.473 0.5 0.46 1030
23 1.4 0.18 0.129 0.473 0.5 0.69 1030
24 1.4 0.18 0.129 0.473 1 0.23 1100
25 1.4 0.18 0.129 0.473 1 0.23 1130
8
26 1.4 0.18 0.129 0.473 1 0.46 1200
27 1.4 0.18 0.129 0.473 1 0.69 1230
28 1.4 0.18 0.129 0.473 2 0.23 1300
29 1.4 0.18 0.129 0.473 2 0.46 1330
30 1.4 0.18 0.129 0.473 2 0.69 1400
31 1.2 0.18 0.15 0.473 0 0.23 1430
32 1.2 0.18 0.15 0.473 0 0.23 1500
33 1.2 0.18 0.15 0.473 0 0.46 1530
34 1.2 0.18 0.15 0.473 0 0.69 1600
35 1.2 0.18 0.15 0.473 0 0.92 1630
36 1.2 0.18 0.15 0.473 0.5 0.23 1700
Table 4 - Test Program Day 3
37 1.2 0.18 0.15 0.473 0.5 0.46 900
38 1.2 0.18 0.15 0.473 0.5 0.69 930
39 1.2 0.18 0.15 0.473 1 0.23 1000
40 1.2 0.18 0.15 0.473 1 0.23 1030
41 1.2 0.18 0.15 0.473 1 0.46 1030
42 1.2 0.18 0.15 0.473 1 0.69 1100
43 1.2 0.18 0.15 0.473 2 0.23 1130
44 1.2 0.18 0.15 0.473 2 0.46 1200
45 1.2 0.18 0.15 0.473 2 0.69 1230
46 1.1 0.18 0.164 0.473 0 0.23 1300
47 1.1 0.18 0.164 0.473 0 0.23 1330
48 1.1 0.18 0.164 0.473 0 0.46 1400
49 1.1 0.18 0.164 0.473 0 0.69 1430
50 1.1 0.18 0.164 0.473 0 0.92 1500
51 1.1 0.18 0.164 0.473 0.5 0.23 1530
52 1.1 0.18 0.164 0.473 0.5 0.46 1600
53 1.1 0.18 0.164 0.473 0.5 0.69 1630
54 1.1 0.18 0.164 0.473 1 0.23 1700
Table 5 - Test Program Day 4
55 1.1 0.18 0.164 0.473 1 0.23 900
56 1.1 0.18 0.164 0.473 1 0.46 930
57 1.1 0.18 0.164 0.473 1 0.69 930
58 1.1 0.18 0.164 0.473 2 0.23 1000
59 1.1 0.18 0.164 0.473 2 0.46 1030
60 1.1 0.18 0.164 0.473 2 0.69 1100
61 1.05 0.18 0.171 0.473 0 0.69 1130
62 1.05 0.18 0.171 0.473 0 0.23 1200
63 1.05 0.18 0.171 0.473 0 0.46 1230
64 1.05 0.18 0.171 0.473 0 0.69 1300
9
The afternoon of day 4 will be spent changing the beam ready for the beginning of day 5.
Table 6 - Test Program Day 5
Run
No.
h/T Water
Depth (m)
Draft
(m)
Beam
(m)
Trim
(deg)
Speed
(m/s)
Time
65 2 0.18 0.09 0.433 0 0.23 900
66 2 0.18 0.09 0.433 0 0.23 930
67 2 0.18 0.09 0.433 0 0.46 1000
68 2 0.18 0.09 0.433 0 0.69 1030
69 2 0.18 0.09 0.433 0 0.92 1030
70 2 0.18 0.09 0.433 0.5 0.23 1100
71 2 0.18 0.09 0.433 0.5 0.23 1130
72 2 0.18 0.09 0.433 0.5 0.46 1200
73 2 0.18 0.09 0.433 1 0.23 1230
74 2 0.18 0.09 0.433 1 0.46 1300
75 1.4 0.18 0.129 0.433 0 0.23 1330
76 1.4 0.18 0.129 0.433 0 0.23 1400
77 1.4 0.18 0.129 0.433 0 0.46 1430
78 1.4 0.18 0.129 0.433 0 0.69 1500
79 1.4 0.18 0.129 0.433 0 0.92 1530
80 1.4 0.18 0.129 0.433 0.5 0.23 1600
81 1.4 0.18 0.129 0.433 0.5 0.23 1630
82 1.4 0.18 0.129 0.433 0.5 0.46 1700
Table 7 - Test Program Day 6
83 1.4 0.18 0.129 0.433 1 0.23 900
84 1.4 0.18 0.129 0.433 1 0.46 930
85 1.2 0.18 0.15 0.433 0 0.23 1000
86 1.2 0.18 0.15 0.433 0 0.23 1030
87 1.2 0.18 0.15 0.433 0 0.46 1030
88 1.2 0.18 0.15 0.433 0 0.69 1100
89 1.2 0.18 0.15 0.433 0 0.92 1130
90 1.2 0.18 0.15 0.433 0.5 0.23 1200
91 1.2 0.18 0.15 0.433 0.5 0.23 1230
92 1.2 0.18 0.15 0.433 0.5 0.46 1300
93 1.2 0.18 0.15 0.433 1 0.23 1330
94 1.2 0.18 0.15 0.433 1 0.46 1400
95 1.1 0.18 0.64 0.433 0 0.23 1430
96 1.1 0.18 0.64 0.433 0 0.23 1500
97 1.1 0.18 0.64 0.433 0 0.46 1530
98 1.1 0.18 0.64 0.433 0 0.69 1600
99 1.1 0.18 0.64 0.433 0 0.92 1630
100 1.1 0.18 0.64 0.433 0.5 0.23 1700
10
Table 8 - Test Program Day 7
101 1.1 0.18 0.64 0.433 0.5 0.23 900
102 1.1 0.18 0.64 0.433 0.5 0.46 930
103 1.1 0.18 0.64 0.433 1 0.23 1000
104 1.1 0.18 0.64 0.433 1 0.46 1030
105 1.05 0.18 0.171 0.433 0 0.23 1030
106 1.05 0.18 0.171 0.433 0 0.23 1100
107 1.05 0.18 0.171 0.433 0 0.46 1130
108 1.05 0.18 0.171 0.433 0 0.69 1200
The afternoon of day 7 will be spent changing the beam ready for testing at the start of Day 8.
Table 9 - Test Program Day 8
Run
No.
h/T Water
Depth
(m)
Draft
(m)
Beam
(m)
Trim
(deg)
Speed
(m/s)
Time
109 2 0.18 0.09 0.393 0 0.23 900
110 2 0.18 0.09 0.393 0 0.23 930
111 2 0.18 0.09 0.393 0 0.46 1000
112 2 0.18 0.09 0.393 0 0.69 1030
113 2 0.18 0.09 0.393 0 0.92 1030
114 2 0.18 0.09 0.393 0.5 0.23 1100
115 2 0.18 0.09 0.393 0.5 0.23 1130
116 2 0.18 0.09 0.393 0.5 0.46 1200
117 2 0.18 0.09 0.393 1 0.23 1230
118 2 0.18 0.09 0.393 1 0.46 1300
119 1.4 0.18 0.129 0.393 0 0.23 1330
120 1.4 0.18 0.129 0.393 0 0.23 1400
121 1.4 0.18 0.129 0.393 0 0.46 1430
122 1.4 0.18 0.129 0.393 0 0.69 1500
123 1.4 0.18 0.129 0.393 0 0.92 1530
124 1.4 0.18 0.129 0.393 0.5 0.23 1600
125 1.4 0.18 0.129 0.393 0.5 0.23 1630
126 1.4 0.18 0.129 0.393 0.5 0.46 1700
Table 10 - Test Program Day 9
127 1.4 0.18 0.129 0.393 1 0.23 900
128 1.4 0.18 0.129 0.393 1 0.46 930
129 1.2 0.18 0.15 0.393 0 0.23 1000
130 1.2 0.18 0.15 0.393 0 0.23 1030
131 1.2 0.18 0.15 0.393 0 0.46 1030
132 1.2 0.18 0.15 0.393 0 0.69 1100
133 1.2 0.18 0.15 0.393 0 0.92 1130
134 1.2 0.18 0.15 0.393 0.5 0.23 1200
135 1.2 0.18 0.15 0.393 0.5 0.23 1230
11
136 1.2 0.18 0.15 0.393 0.5 0.46 1300
137 1.2 0.18 0.15 0.393 1 0.23 1330
138 1.2 0.18 0.15 0.393 1 0.46 1400
139 1.1 0.18 0.64 0.393 0 0.23 1430
140 1.1 0.18 0.64 0.393 0 0.23 1500
141 1.1 0.18 0.64 0.393 0 0.46 1530
142 1.1 0.18 0.64 0.393 0 0.69 1600
143 1.1 0.18 0.64 0.393 0 0.92 1630
144 1.1 0.18 0.64 0.393 0.5 0.23 1700
Table 11 - Test Program Day 10
145 1.1 0.18 0.64 0.393 0.5 0.23 900
146 1.1 0.18 0.64 0.393 0.5 0.46 930
147 1.1 0.18 0.64 0.393 1 0.23 1000
148 1.1 0.18 0.64 0.393 1 0.46 1030
149 1.05 0.18 0.171 0.393 0 0.23 1030
150 1.05 0.18 0.171 0.393 0 0.23 1100
151 1.05 0.18 0.171 0.393 0 0.46 1130
152 1.05 0.18 0.171 0.393 0 0.69 1200
7. Resources Required
7.1 Experimental Facility
Testing will be conducted in the MTB at the AMC.
7.2 Model Particulars
The model to be used in the study can be seen in Figure 1and Figure 2. Particulars of the model ship
are shown in Table 12. The plasticine seen at the bow of the vessel in Figure 1 shows how the vessel
is faired at the bow when the beam extensions are attached. In Figure 2 the widths of the extension
pieces can be seen, taking the beam from 393mm at its skinniest to 473mm as depicted
Table 12 - Model Vessel Particulars
Length (m) 2.385
Beam A (m) 0.393
Beam B (m) 0.433
Beam C (m) 0.473
Draft (m) 0.09 – 0.171
12
Figure 1 - Model Vessel
Figure 2 - Model Vessel with Largest Beam Segments Attached
Beam Segments
13
7.3 Apparatus
The model test has a fairly simple set up. Figure 3 depicts how the model will be set up when testing
is conducted. The equipment needed for the setup is shown in Table 13.
Table 13 - Testing Equipment
Equipment Use
Qualisys - Motion Capture System
To record all the motions of the vessel including heave, pitch and to verify the speed of the model.
4 x Qualisys Capture Markers and Stands
To act as markers on the model for the Qualisys program.
Qualisys Battery Pack and Radio System
To provide power to the Qualisys marker system and to transmit signals back to the computer.
Ball Joint An attachment at the forward end of the model which has 3 DOF in each rotation.
Slide/Ball Joint An attachment at the aft end of the model which has 5 DOF (not sway).
Winch Cart Setup To attach the model to the propulsion system.
Winch Cables and Motor
To provide a means of propulsion for the model.
Mass Ballast for the model.
Poles To extend the capture width of the model for higher accuracy.
Winch Speed Capture System
To record the vessel speed.
Posts & Post Bearings Attaches mode to cart & allows vertical displacement.
Figure 3 - Setup Apparatus
14
References
Dand, I. W. (1971). Squat Measurements: Bulbous Bow Ships in Shallow Water. London, National
Physical Laboratory.
Dand, I. W. (1972). Full Form Ships In Shallow Water: Some Methods for Prediction of Squat in
Subcritical Flows. London, National Physical Laboratory.
Eryuzlu, N. E., Y. L. Cao, et al. (1994). Under Keel Requirements for Large Crafts in Shallow
Waterways. 28th International Navigation Congress. PIANC, PIANC: 17-25.
Dietze, W., Rekonen, T., van Toorenburg, J.C.K., Vantorre, M., Wijinstra, R. (1995)
Joint PIANC-IAPH Working Group II-30., International Maritime Pilots Association., et al.
Approach channels, preliminary guidelines. Brussels, Belgium
Millward, A. (1990). "A Preliminary Design Method for the Prediction of Squat in Shallow-Water."
27: 10-19.
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