Validation of the Strip Theory Code SEAWAY
by Model Tests in Very Shallow Water
Marc Vantorre 1 and Johan Journée 2
Introduction
For a sea-going vessel it is of utmost importance to respond in an adequate way to the
governing wave climate. The survivability of a ship in severe sea conditions depends,
among other factors, on the resistance of the ship construction to vertical and
horizontal shear forces, longitudinal bending moments and torsion moments, excited
in the ship's hull by wind waves and swell. For the ship designer, it is therefore crucial
to dispose of reliable estimation methods for these loads.
Besides the strength aspect, also the ship motions induced by the waves deserve full
attention. Large roll motions may cause capsizing and can therefore jeopardise ship
safety. If the bow leaves the water due to extreme motions relative to the water
surface, slamming can occur, causing additional vibration loads. Alternatively,
relative vertical motions may cause shipping of water by the bow, resulting into
damage to deck and superstructures and affecting safety.
The navigating officer has only limited means to influence the ship's response to
waves. In high waves, steering the ship towards the dominant wave direction can
reduce roll motions. In these conditions, reduction of speed in general results into a
decrease of the relative vertical motions of the bow and, therefore, of the frequency of
occurrence of slamming and/or shipping of water. However, such measures, if applied
frequently, are detrimental for fulfilling the ship's operational mission. For a container
ship, as an example, it is of great importance that the speed can be maintained in
severe seas, without the occurrence of excessive motions forcing the master to speed
reduction.
A thorough knowledge of ship behaviour in waves is therefore essential for ship
design. Consequently, it is not surprising that the first applications of numerical
methods that were developed since the middle of the last century for calculating a
ship's response to waves were concentrating on design conditions. This were waves
with considerable height and a wave length in the order of magnitude of the ship's
length – the latter causing the largest values for pitch motion and bending moments –
in full ocean conditions, viz. in deep water.
A ship approaching a harbour, on the other hand, is confronted with completely
different environmental conditions. The shipping traffic is guided through access
channels, which mainly affects control in the horizontal plane. In open sea course
keeping is most important; in channels, a specific trajectory has to be followed. The
1
2
Professor, University of Ghent, Maritime Technology Department
Associate Professor, Delft University of Technology, Ship Hydromechanics Laboratory
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MARC VANTORRE & JOHAN JOURNEE
VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
effect of disturbances – such as wind, current, other ships and bank effects – causing
deviations from the desired track becomes more important. In many navigation areas
the command is in fact taken over by a pilot, who is informed about the local
conditions. As an additional complication, the limitations of the depth of the access
channel may significantly affect the manoeuvring behaviour of a ship.
A ship's behaviour in waves is affected by water depth restrictions as well, for many
reasons. In the first place, the presence of the bottom yields a new boundary
condition, affecting the kinematics of and the pressure distribution in the wave. Also
the hydrodynamic characteristics – damping and inertia – are strongly dependent on
water depth. Due to the combinations of these influences, response functions (RAO’s
or Response Amplitude Operators) obtained in shallow water may deviate
considerably from those obtained in deep water conditions.
Furthermore, ship behaviour in shallow navigation areas such as access channels to
harbours usually does not belong to the scope of interest of ship designers, as in
general wave conditions are less severe in coastal areas compared to open sea. On the
other hand, ship characteristics in shallow water are of interest for channel designers,
waterway administrations, shipping traffic control services: their concern is to reduce
the probability of contact between the ship's keel and the bottom of the channel to an
acceptable minimum. For this purpose, a prediction of the vertical ship motions due to
squat effects and wave excitation with a rather high level of accuracy is required. As
a matter of fact, compared to phenomena such as slamming and shipping of water, the
(acceptable) probability of bottom touch is in general extremely small. Furthermore,
the wave characteristics can deviate significantly from ship design conditions. The
wave height is relatively limited, as most shipping channels are closed in (too) severe
wave conditions. Moreover, the dominating wavelengths are not always of the same
order of magnitude compared to the ship length.
Although in both ship and channel design a similar question is raised – how does the
ship respond to the governing wave climate? –, the boundary conditions are
essentially different. Obviously, techniques developed for ship design are not
immediately applicable for waterway design. In this paper, consideration will be
given to the oldest, but certainly not the least numerical technique that has been
developed for calculating responses of ships in waves: strip theory. Within this
approach, the ship is divided into a number of cross sections; the hydrodynamic
coefficients and exciting wave forces are calculated for each section and integrated
over the ship's length. The computer program SEAWAY, developed by the second
author, is based on this principle. More specifically, the feasibility of applying this
program for shallow water problems will be discussed.
Numerical calculation methods can only be considered as reliable if they have been
validated by comparison with experimental results. For deep water conditions
sufficient data are available for validation of the numerical results. For shallow water
conditions, on the other hand, the number of available data is much more restricted.
In the period 1996-1999, comprehensive series of sea-keeping tests in shallow water
conditions were executed in the Towing Tank for Manoeuvres in Shallow Water at
Flanders Hydraulics Research, Antwerp, Belgium, resulting into a considerable
amount of validation data. This experimental program was carried out in co-operation
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with the Division of Maritime Technology of Ghent University in the frame of a
research project on behalf on the Coastal Waterways Division (Ostend). This was one
step in the development of a probabilistic admittance policy for the Scheurpas and the
Pas van het Zand, channels giving access to the Western Scheldt and the harbour of
Zeebrugge. Use will be made here of a selection of test results to discuss the
opportunities and restrictions of strip theory.
History of the Strip Theory and SEAWAY
The six ship motions of and about its centre of gravity G have been defined in the
Figure 1.
Figure 1: Definition of motions
According to Newton’s second law, the equations of motion for six degrees of
freedom of an oscillating ship in waves in a earth-bound axes system have to be
written as follows:
∑ {M
6
j =1
ij
⋅ &x& j } = sum of all forces or moments in direction i
for: i = 1,...6
Because a linear system has been considered here, the forces and moments in the right
hand side of these equations consist of a superposition of:
• so-called hydromechanic forces and moments, caused by harmonic oscillations of
the rigid body in the undisturbed surface of a fluid being previously at rest, and
• so-called exciting wave forces and moments on the restrained body, caused by the
incoming harmonic waves.
With this, the system of a with six degrees of freedom moving ship in waves can be
considered as a linear mass-damping-spring system with frequency dependent
coefficients and linear exciting wave forces and moments:
∑ {(M
6
j =1
ij
+ aij ) ⋅ &x& j + bij ⋅ x& j + cij ⋅ x j } = Fi
for: i = 1,...6
In here, x i with indices i = 1,2,3 are the displacements of G (surge, sway and heave)
and x i with indices i = 4,5,6 are the rotations about the axes through G (roll, pitch
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VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
and yaw). The terms with indices ij (with i ≠ j ) present for motion i the coupling
with motion j .
The masses in these equations of motion consist of:
• solid masses or solid mass moments of inertia of the ship ( M ij ) and
• “added” masses or “added” mass moments of inertia caused by the disturbed
water, called hydrodynamic masses or mass moments of inertia ( a ij ).
An oscillating ship generates waves it self too, since energy will be radiated from the
ship. The hydrodynamic damping terms ( bij ⋅ x& i ) account for this.
For the heave, roll and pitch motions, hydrostatic spring terms (c ij ⋅ x i ) have to be
added.
The right hand sides of these equations of motion consist of the exciting wave forces
and moments ( Fi ).
In the so-called strip theory, the ship will be divided in 20 to 30 cross sections, of
which the two-dimensional hydromechanic coefficients and exciting wave loads will
be calculated. To obtain the three-dimensional values, these values will be integrated
over the ship length numerically. Finally, the differential equations have to be solved
to obtain the motions. All calculations will be performed in the frequency domain.
It was in 1949 that Ursell published his 2-D potential theory for determining the
hydrodynamic coefficients of semicircular cross sections, oscillating in the frequency
domain in the surface of a fluid with infinite depth. Using this, for the first time a
rough estimation could be made of the motions of a ship in regular waves at zero
forward speed.
Shortly after that, Tasai, Grim, Gerritsma and many other scientists have used various
already existing conformal mapping techniques (to transform ship-like cross sections
to a semicircle) together with Ursell’s potential theory, in such a way that the motions
in regular waves of more realistic hull forms could be calculated too. Most popular
was (and still is) the 2-parameter Lewis conformal mapping technique.
The exciting wave loads were found from the loads in undisturbed waves – the socalled Froude-Krilov forces or moments – completed with diffraction terms,
accounting for the presence of the ship in these waves. When calculating these
diffraction terms, use had been made of – by Haskind in 1957 for zero forward speed
derived and by Timman and Newman in 1962 generally confirmed - relations between
the diffraction potentials and the radiation potentials, the so-called Haskind relations.
Borrowed from the broadcasting technology, Denis en Pierson published in 1953 a
superposition method to describe the irregular waves too. The sea was considered to
be the sum of many simple harmonic waves, each wave with its own frequency,
amplitude, direction and random phase angles. By calculating the responses of the
ship on each of these individual harmonic waves and adding up these responses, the
energy distribution of the ship’s behaviour in irregular waves could be found. These
irregular responses are characterised by significant amplitudes and average periods.
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VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
However, these potential theories provided the motions at zero forward speed only. In
1957, Korvin-Kroukovski en Jacobs published a method - which was improved in the
sixties - to account for the effect of forward ship speed too.
So at the end of the fifties, all components for an elementary ship motions computer
program for deep water were already available.
Fukuda published in 1962 a calculation technique for the internal sheer forces and
bending moments in a cross section of a ship.
Frank published in 1967 his pulsating source theory to calculate the hydrodynamic
coefficients of a cross section of a ship in deep water directly, without using a
mapping technique. The potential coefficients of fully submerged cross sections (at
bulbous bows) and sections with a very low area coefficient (often present in the aft
body) could be calculated now too.
Using the Lewis conformal mapping technique, Keil [6] published in 1974 his theory
for obtaining the potential coefficients in very shallow water.
Very useful theories to calculate the added resistance of a ship due to waves were
given by Boese (integrated pressure method) in 1970 and Gerritsma and Beukelman
(radiated energy method) in 1972.
So far, all hydrodynamic coefficients had been determined with the potential theory.
In this context, references to the large amount of work of Tasai in the sixties and the
well-known 1970 report of Salvesen, Tuck and Faltinsen should not be absent here.
However, in particular roll required a viscous correction. Ikeda, Himeno and Tanaka
published in 1978 a very useful semi-empirical method for determining viscous roll
damping components.
The introduction of PC’s in the early eighties increased the accessibility for carrying
out ship motion calculations considerably; even non-specialists could become users
too. From then on, computer capacity and computing speed increased very fast. Also
three-dimensional theories could be developed much easier and cheaper now.
Because of the complex problem of forward speed in 3-D theories however, the 2-D
approach (strip theory) is still very favourable when calculating the behaviour of a
ship at forward speed. Faltinsen and Svensen have discussed in 1990 the many
advantages and few disadvantages of 2-D, when comparing with 3-D, very clearly.
As a consequence of the work of the researchers mentioned above, the Delft
University of Technology had completed at the end of the eighties a PC strip theory
code, called SEAWAY. This DOS program has been verified and validated
extensively in the nineties by the second author [2], many students and a large number
of commercial users. Many improvements and modifications have been implemented
in the course of years.
A Windows version of SEAWAY has been created recently by AMARCON; see web
site http://www.shipmotions.nl or http://www.amarcon.com for more detailed
information [3].
An extensive theoretical manual [1] can be downloaded from the first web site too.
Also, this manual contains the references to the great many authors who had provided
us the present basic hydromechanic knowledge.
Figure 2 shows an example of the very advanced input data options of AMARCON’s
program “SEAWAY for Windows”.
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VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
Figure 2: Selection of input data for “SEAWAY for Windows”
Based on the linear strip theory, SEAWAY calculates for six degrees of freedom in
the frequency domain the hydromechanic loads, wave loads, absolute and relative
motions, added resistance and internal loads of displacement ships, barges and yachts
in regular and irregular waves. When ignoring interaction effects between the two
individual hulls, the behaviour of catamarans and semi-submersibles can be calculated
too. The program is suitable for deep water as well as for very shallow water. Viscous
roll damping, bilge keels, free-surface anti-roll tanks, external moments and (linear)
mooring springs can be added.
Because free accessible results of model experiments on ship motions in very shallow
water are occasional, the validations mentioned above were mainly related to the
behaviour of the ship in deep water. Only one report on experiments in very shallow
water could be used for validation - see Chapter 7 of reference [2] - with very
promising results, however.
However for a short time now, this program - and particularly Keil’s potential theory
[6] in it - could be validated well with results of extensive model tests in very shallow
water [4]. These experiments and the validation of the theory will be described here.
Experiments in Shallow Water
These tests were carried out by the first author in the Towing Tank for Manoeuvres in
Shallow Water at Flanders Hydraulics Research in Antwerp (see Figure 3) in the
frame of a co-operation contract with the Division of Maritime Technology of Ghent
University. This towing tank has a total length of 88 m, of which 67 m can be used
effectively during the tests. The width is 7 m and a maximum water depth of 0.50 m
can be attained. This water depth is sufficient for research on the behaviour of ships in
shallow waters, which are typical for harbours, access channels and canals. The tank
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VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
is equipped with a computer controlled towing carriage with a "planar motion
mechanism", capable of imposing any trajectory in the horizontal plane to a ship
model. In the vertical modes (heave, pitch and roll) the ship model is free to move. A
wave generator can produce both regular and irregular waves for investigating the
vertical motions of ship models due to waves. Tests can be performed in an
unmanned, completely automated way.
Figure 3: Towing tank at Flanders Hydraulics Research in Antwerp, Belgium
In the scope of the project "Ship motions in the Scheurpas" [4] the seakeeping
characteristics of four ship models were investigated in shallow water: two normative
ships (D, E) and two critical ships (F, G), see Table 1.
Model
Scale
D
E
F
G
(-)
1/75
1/85
1/50
1/50
Length over all
Loa
(m)
300.00
343.00
200.00
190.00
Length between perpendiculars
L pp
B
T
(m)
291.13
325.00
190.00
180.00
(m)
40.25
53.00
32.00
33.00
(m)
15.00
21.79
11.60
13.00
Moulded breadth
Maximum draught
(-)
0.60
0.85
0.60
0.85
CB
Table 1: Ship models examined on behalf of project "Ship motions in the Scheurpas"
(Flander Hydraulics – Ghent University)
Block coefficient at max. draught
The purpose of this experimental program consisted of generating a database for the
development of a calculation procedure for estimating the probability of bottom touch
by a ship during her passage through the Scheurpas and/or the Pas van het Zand (see
Figure 4).
This calculation procedure takes account for the following parameters: ship
characteristics, tide, waves (including swell) and ship speed. The project was executed
for supporting a probabilistic access policy for shipping traffic in these channels, [5].
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VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
Figure 4: Access channels to the mouth of the Western Scheldt and to Zeebrugge
Figure 5: Typical wave spectra in the Flemish Banks region
The normative ships are the largest ones in their category expected to frequent the
harbours of Antwerp, Ghent and Zeebrugge in long term. As ship shape parameters
influence the motion responses, two normative ship types were considered: a slender
ship type (model D, container carrier) and a full one (model E, tanker / bulk carrier).
Taking account for the wave characteristics in the southern part of the North Sea - see
Figure 5 - it can be expected that these normative ships will not be subject to the
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VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
largest motions; smaller types may have a larger probability of bottom touch. For this
critical category, two ship models were selected as well. A slender (model F,
panamax container carrier) and a full ship (model G, panamax bulk carrier) were
selected.
The four ship models were examined at a number of loading conditions, water depths,
ship speeds and incident wave directions, as shown in Table 2. The frequency range
of the regular waves varied between 0.3 and 1.2 rad/s, with special attention to the
values between 0.6 and 1.2 rad/s. The models were free to heave, pitch and roll, and
fixed in the other modes of motion.
Table 2: Overview of the entire experimental program
(m)
(m)
32.00
33.00
(m)
11.60
11.60
Trim
L pp
B
T
t
Ship F518
Series FA
190.00
(m)
0.00
0.00
Block coefficient
CB
(-)
0.60
0.84
KG
T0 φ
(m)
11.60
11.60
(s)
15.70
24.20
(-)
0.250
0.250
no
no
13.60
13.60
Length between perpendiculars
Breadth (moulded)
Amidships draught
Centre of gravity
G
Mean natural roll period
Longitudinal radius of inertia
k yy / L pp
Bilge keels
Water depth
h
(m)
Ship G518
Series GA
180.00
Table 3: Test conditions for the considered experiments
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VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
In 2001 these model test results became available for validation of the seakeeping
program SEAWAY. The validation described in the next sections is based on two
specific series of experiments, viz. FA and GA (see Table 2), carried out at a water
depth – draft ratio h T ≈ 1.17 .
The principal ship characteristics during these test series are listed in Table 3; the
frames of the two ship models are represented in Figure 6. The mean experimental
natural roll period has been used to determine the radius of inertia for roll of the solid
mass of the model.
Figure 6: Body plans of ship models F and G
Verification of the Theory of Keil in SEAWAY
The general transformation formula of a ship-like cross section to the unit semicircle
of Ursell as used by Tasai et al for deep water – and later with N = 2 used by Keil for
deep to very shallow water - is given by:
N
{
z = M s ⋅ ∑ a2 n −1 ⋅ ζ − (2 n −1)
n= 0
}
where:
z = x + iy
plane of cross section contour
ζ = ieα ⋅ e − iθ
Ms
a −1
a 2 n −1
N
plane of unit semicircle
scale factor
= +1.0
conformal mapping coefficients ( n = 1,... N )
number of parameters ( 2 ≤ N ≤ 10 )
Figure 7 shows an example of this unit semicircle with a - by N = 2 (Lewis
transformation) created - contour of a cross section.
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VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
Figure 7: Conformal mapping of the section contour to the unit semicircle
For a specific Lewis transformation ( N = 2 ), the scale factor M s and the coefficients
a1 and a3 will be determined such that the breadth B , the draught T and the area As
of the re-mapped and the actual cross section are similar.
For the more general N -parameter transformation, the wetted length of the cross
section contour will be divided in 32 line elements of equal length. Then, M s and a1
until a2 N −1 will be determined such that the sum of the squares of the deviations of the
33 offsets on the contour are minimal. De RMS (Root Mean Square) value of the
deviations is an indication of the accuracy of the performed transformation.
An example of the deviations of a rectangular cross section from his re-mapped
contour is given in Figure 8.
Figure 8: Conformal mapping of a rectangle ( B T = 2.80 )
These transformation techniques provide for cross sectional area coefficients in the
range of ± 0.5 ≤ As ( B ⋅ T ) ≤ ±1.2 - see the white region in Figure 9 – very acceptable
results, where the accuracy increases with an increasing N . In case of a cross section
with an area coefficient in the grey region, this area coefficient will be shifted to the
boundary of the white and valid region, for the determination of the Lewis
coefficients in SEAWAY.
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VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
Figure 9: Valid regions for Lewis conformal mapping
The deep to very shallow water potential theory of Keil [6] uses this Lewis
transformation technique of cross sections. Compared with 10-parameter close-fit
conformal mapping, this transformation technique provides for both models rather
large RMS values of the contour deviations. Besides this, the bulbous sections in the
fore body ( As (B ⋅T ) > ±1.2 ) en wedge-shaped sections in the aft body
( As ( B ⋅ T ) < ±0.5 ), actually require a straightforward approach with a distribution
pulsating sources on the contour, as done by Frank.
Figure 10 presents the longitudinal distribution of the RMS values of the calculated
deviations for the two ships, given in Figure 6.
Ship F5 18-F A
RMS of deviations (m)
2 .0
1 .5
2-param eter L ewis t ansfo rm ation
10-param ete r co nfo rm e tran sforma tion
1 .0
0 .5
0
0
APP
50
100
15 0
2 00
FPP
Distrib ution over ship le ngth
(m )
Ship G5 18-GA
RMS of deviations (m)
2 .0
1 .5
1 .0
0 .5
0
0
APP
50
100
Distrib ution over ship le ngth
15 0
F PP
2 00
(m )
Figure 10: RMS values of cross sectional contour deviations
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VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
Given the relatively large contour deviations in Figure 10, the question arises whether
the Lewis transformation in Keil’s theory suits for these ships. This can be
investigated with program SEAWAY for deep-water cases only.
For both ships travelling in deep water, Figure 11 and Figure 12 show, using the
classical strip theory and:
• Ursell’s theory and 2-parameter Lewis conformal mapping (solid line),
• Ursell’s theory and 10-parameter conformal mapping (dashed line) and
• ditto, with – where required - locally Frank’s theory (dotted line),
a mutual comparison of the calculated heave, pitch and roll motions.
Starting point here is the harmonic incoming wave, defined in the centre of gravity G
of the ship by:
ζ = ζ a ⋅ cos(ωe ⋅t )
The resulting harmonic heave, roll and pitch motions of the ship are:
x3 = x3 a ⋅ cos ωe ⋅ t + εx3ζ
x 4 = x4 a
x5 = x5 a
(
)
⋅ cos (ω ⋅ t + ε )
⋅ cos (ω ⋅ t + ε )
e
x4ζ
e
x5ζ
where ωe is the circular frequency of encounter of the waves, t is the time and εxiζ is
the phase shift between the (angular) displacement xi and the wave elevation ζ .
The three response amplitudes have been presented here in a non-dimensional format:
x5 a
x3 a
x4 a
,
en
.
ζa 2⋅ζa B
2 ⋅ ζ a L pp
8 kn / 180
2 par. Lewis
10 par. Conf.
10 par. Conf. +Frank
1.5
Amplitude pitch (-)
Amplitude heave (-)
2.0
0
1.0
0.5
0
0 kn / 090
2.5
5
2.0
4
Amplitude roll (-)
8 kn / 180
1.5
1.0
0.5
0
3
2
1
Deep water
0
0
0.25
0.50
0.75
1.00
0
0
0.25
0.50
0.75
1.00
0
360
360
360
270
270
270
180
90
0
180
90
0.25
0.50
0.75
W av e frequenc y (rad/s)
1.00
0.50
0.75
1.00
0.25
0.50
0.75
1.00
180
90
0
0
0.25
0
Fase roll ( )
0
Phase pitch ( )
0
Phase heave ( )
0
0
0
0.25
0.50
0.75
W av e frequenc y (rad/s)
1.00
0
W av e frequenc y (rad/s)
Figure 11: Effect of mapping on computed behaviour of F518-FA in deep water
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MARC VANTORRE & JOHAN JOURNEE
VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
8 kn / 180
2 par. Lewis
10 par. Conf.
10 par. Conf. +Frank
1.5
Amplitude pitch (-)
Amplitude heave (-)
2.0
0
1.0
0.5
0
0 kn / 090
2.5
5
2.0
4
Amplitude roll (-)
8 kn / 180
1.5
1.0
0.5
0
3
2
1
Deep water
0
0
0.25
0.50
0.75
1.00
0
0
0.25
0.50
0.75
1.00
0
360
360
270
270
270
180
90
0
180
90
0
0
0.25
0.50
0.75
W av e frequenc y (rad/s)
1.00
0.25
0.50
0.75
1.00
0.25
0.50
0.75
1.00
0
0
Phase roll ( )
360
Phase pitch ( )
0
Phase heave ( )
0
180
90
0
0
0.25
0.50
0.75
W av e frequenc y (rad/s)
1.00
0
W av e frequenc y (rad/s)
Figure 12: Effect of mapping on computed behaviour of G518-GA in deep water
The figures show generally rather small mutual differences between the calculated
motions in deep water. Only roll shows with a Lewis transformation a somewhat
higher peak at the natural frequency, which could be expected because of the lower
potential damping (see Figure 8, for the extreme case of a rectangle). Assuming a
comparable behaviour in shallow water, it can be concluded that the conformal
transformation in Keil’s theory does not require an extension of 2 to 10 parameters, at
least from a practical point of view. Also, only a very small the effect of a local use of
Frank’s theory has been found for these ships.
These conclusions apply for phenomena following from total – so over the full ship
length integrated - values. It does not necessarily apply for local values like local
pressures, sheer forces and bending moments.
Besides this, the method of Keil has been verified for the deep-water case with the
Ursell-Lewis method; see Figure 13 en Figure 14. Both methods provide almost fully
identical results (solid line and the almost invisible dotted line behind this), which
confirms that both methods within the strip theory have been implemented in
SEAWAY correctly.
FLANDERS HYDRAULICS RESEARCH , N UMERICAL MODELLING COLLOQUIUM, ANTWERP , BELGIUM, 23-24 O CTOBER 2003
DUT-SHL REPORT 1373-E
14
MARC VANTORRE & JOHAN JOURNEE
VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
8 kn / 180
Lew is deep water
Keil deep water
Keil shallow water
1.5
Amplitude pitch (-)
1.0
0.5
0
0.25
0.50
0.75
2.5
5
2.0
4
1.5
1.0
0.5
1.00
3
2
1
0.25
0.50
0.75
1.00
0
360
360
360
270
270
270
90
0
180
0.25
0.50
0.75
0.50
0.75
1.00
90
0.25
0.50
0.75
1.00
180
90
0
0
0.25
0
0
180
0
0
0
Phase pitch ( )
0
0 kn / 090
0
0
Phase heave ( )
0
Phase roll ( )
Amplitude heave (-)
2.0
0
Amplitude roll (-)
8 kn / 180
1.00
0
0
W av e frequenc y (rad/s)
0.25
0.50
0.75
1.00
0
W av e frequenc y (rad/s)
W av e frequenc y (rad/s)
Figure 13: Effect of water depth on computed behaviour of F518-FA
Amplitude pitch (-)
1.0
0.5
0
0.25
0.50
0.75
1.00
0
0 kn / 090
2.5
5
2.0
4
1.5
1.0
0.5
0
0
0.25
0.50
0.75
1.00
2
1
0
360
270
270
270
Phase roll ( )
360
180
90
0
0.25
0.50
0.75
W av e frequenc y (rad/s)
1.00
0
0.25
0.50
0.75
1.00
0.25
0.50
0.75
1.00
0
180
90
0
0
0
3
360
0
0
Lewis deep water
Keil deep water
Keil shallow water
1.5
0
Phase heave ( )
8 kn / 180
Phase pitch ( )
Amplitude heave (-)
2.0
0
Amplitude roll (-)
8 kn / 180
180
90
0
0
0.25
0.50
0.75
W av e frequenc y (rad/s)
1.00
0
W av e frequenc y (rad/s)
Figure 14: Effect of water depth on computed behaviour of G518-GA
From the foregoing can be concluded that Keil’s potential theory has been
implemented in SEAWAY correctly and that – at least from a practical point of view Keil’s theory with a 2-parameter Lewis transformation does not require an extension
FLANDERS HYDRAULICS RESEARCH , N UMERICAL MODELLING COLLOQUIUM, ANTWERP , BELGIUM, 23-24 O CTOBER 2003
DUT-SHL REPORT 1373-E
15
MARC VANTORRE & JOHAN JOURNEE
VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
with the more accurate N -parameter conformal transformation with for instance
N = 10 .
Validation of SEAWAY with Experiments in Shallow Water
At the same time, Figure 13 and Figure 14 show the calculated results for very
shallow water (dashed line) too. These figures show here for the two extreme cases
( h T = 1.17 en h T = ∞ ) a significant influence of water depth on the ship motions.
The wave forces and moments on the ship consist of forces and moments in the
undisturbed wave (the so-called Froude-Krilov contribution) and a correction that
accounts for the disturbance of the wave by the presence of the ship (the diffraction
contribution).
This diffraction part can be calculated (estimated should be a better word for the strip
theory) in two different ways by the computer code SEAWAY:
• The classical strip theory method.
With this, the 2-D diffraction part has been found from the hydrodynamic mass
and damping of the cross section and the velocity and acceleration of the water
particles. However, the amplitudes of these water motions are not constant in a
wave; they decrease downwards exponentially. Required equivalent or average
values have been found here from the Froude-Krilov part.
• An approximating 2-D diffraction method.
With this, the amplitudes of the 2-D wave loads will be obtained from the
diffracted energy of the incoming wave. Using theories in principle valid for the
low and high frequency part of longitudinal end beam waves only has
approximated the phase shifts. A practical and satisfying solution has been found
for the transition zones.
Both methods have been described in detail in reference [1].
Comprehensive validation studies, carried out in the past for the deep-water case [2],
show that – generally - the approximating 2-D diffraction method provides somewhat
better results than the classical strip theory method. But, is this true for the shallow
water case too?
Using both methods for a water depth to draught ratio of h T = 1.17 , the by Keil’s
potential theory in SEAWAY calculated motions have been compared here with the
experimental data of the first author. The viscous roll damping has been estimated by
using the semi-empirical method of Ikeda, Himeno en Tanaka.
Figure 15 and Figure 16 compare the heave behaviour of both ships in head waves at
three forward speeds and Figure 17 and Figure 18 show this for the pitch behaviour.
Figure 19 and Figure 20 show the heave, pitch and roll behaviour in beam waves at
zero forward speed.
Excluding the phase shifts in the (practically often less important) region with
response amplitude operators decreasing to a minimum value, the differences between
the results of both methods are very small.
Comparing all experimental data of the FA and GA series in Table 2 with calculated
results, hardly any preference for one of the two theoretical wave load approaches
FLANDERS HYDRAULICS RESEARCH , N UMERICAL MODELLING COLLOQUIUM, ANTWERP , BELGIUM, 23-24 O CTOBER 2003
DUT-SHL REPORT 1373-E
16
MARC VANTORRE & JOHAN JOURNEE
VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
could be found. Only a slight trend has been found that amplitudes will be calculated
somewhat better by the classical strip theory approach (especially at the higher
frequencies) and phase shifts somewhat better with the approximating 2-D diffraction
method.
Overall, it may be concluded that the results of both calculation methods agree “quite
well to good” with the experimental data.
Especially in the framework of the project “Ship motions in the Scheur channel”
however, preference would be given to the classical strip theory. Somewhat better
agreements have been found here for frequencies above 0.6 rad/s. This means for
h T = 1.17 waves with a length not exceeding 110 m, so less than about half the
ship’s length. Nevertheless the small responses at these frequencies, the spreading in
the experimental data is rather small. Larger spreading has been found for frequencies
less than 0.6 rad/s, which can be explained by tank wall interference. At a full-scale
frequency of 0.4 rad/s, the FA and GA tests had a wavelength to tank width ratio of
about 0.5.
Conclusions
Based on the results of the investigations described here, the following conclusions
may be drawn for ship motions at shallow water:
1. The two-dimensional potential theory of Keil has been implemented correctly in
the computer code SEAWAY.
2. An extension of the Lewis transformation of the section contour in Keil’s theory
to a 10-parameter close-fit transformation is not required.
3. When using the classical wave loads and the 2-D diffraction wave loads, the
mutual differences between the computed motions of both ships are rather small.
4. The results of both computational methods agree “quite well to good” with the
experimental data. However, in the higher frequency region at shallow water, a
slight preference for the classical method has been found.
5. Taking this into account and the slight preference found in the past for the
approximating 2-D diffraction method in deep-water cases, a general preference
for this method can be maintained; this with a reservation for the shorter
wavelengths at very shallow water.
Summarised, based on these computations and experiments, it may be concluded that
the strip theory with the potential theory of Keil (read: the computer code SEAWAY)
provides a “quite well to good” prediction of the motions of ships at deep and shallow
water.
Acknowledgements
The model tests have been carried out at Flanders Hydraulics Research at Antwerp on
behalf of the “Ministerie van de Vlaamse Gemeenschap, Departement Leefmilieu en
Infrastructuur, Administratie Waterwegen en Zeewezen, Afdeling Waterwegen Kust”.
The authors are very grateful for the permission using these test data for this
validation study.
FLANDERS HYDRAULICS RESEARCH , N UMERICAL MODELLING COLLOQUIUM, ANTWERP , BELGIUM, 23-24 O CTOBER 2003
DUT-SHL REPORT 1373-E
17
MARC VANTORRE & JOHAN JOURNEE
VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
0
0 kn / 180
8 kn / 180
Amplitude heave (-)
2.0
2.0
1.5
1.5
1.5
1.0
1.0
1.0
0.5
0.5
0.5
0
0
0
0
12 kn / 180
2.0
0
Phase heave ( )
0
0.25
0.50
0.75
1.00
0
0
0.25
0.50
0.75
1.00
0
360
360
360
270
270
270
180
180
180
90
90
90
0
0
0
0.25
0.50
0.75
1.00
2-D diffraction
Class ic
Exp.
0.25
0.50
0.25
0.50
0.75
1.00
0.75
1.00
0
0
W av e frequency (rad/s)
0.25
0.50
0.75
1.00
0
Wave frequenc y (rad/s )
Wave frequency
(rad/s)
Figure 15: Heave in head waves of F518-FA in very shallow water
0
Amplitude heave (-)
0 kn / 180
8 kn / 180
2.0
2.0
1.5
1.5
1.5
1.0
1.0
1.0
0.5
0.5
0.5
0
0
0
0
10 kn / 180
2.0
0
Phase heave ( )
0
0.25
0.50
0.75
1.00
0
0
0.25
0.50
0.75
1.00
0
360
360
360
270
270
270
180
180
180
90
90
90
0
0
0
0.25
0.50
0.75
W av e frequency (rad/s)
1.00
2-D diffracton
Clss ic
Exp.
0.25
0.50
0.25
0.50
0.75
1.00
0.75
1.00
0
0
0.25
0.50
0.75
Wave frequenc y (rad/s )
1.00
0
Wave frequency
(rad/s)
Figure 16: Heave in head waves of G518-GA in very shallow water
FLANDERS HYDRAULICS RESEARCH , N UMERICAL MODELLING COLLOQUIUM, ANTWERP , BELGIUM, 23-24 O CTOBER 2003
DUT-SHL REPORT 1373-E
18
MARC VANTORRE & JOHAN JOURNEE
VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
0
Amplitude pitch (-)
0 kn / 180
8 kn / 180
0
12 kn / 180
2.5
2.5
2.5
2.0
2.0
2.0
1.5
1.5
1.5
1.0
1.0
1.0
0.5
0.5
0.5
0
0
0
0.25
0.50
0.75
1.00
2-D diffraction
Classic
Exp.
0
0
0.25
0.50
0.75
1.00
0
360
360
360
270
270
270
180
180
180
90
90
90
0.25
0.50
0.25
0.50
0.75
1.00
0.75
1.00
0
Phase pitch ( )
0
0
0
0
0.25
0.50
0.75
1.00
0
0
W av e frequency (rad/s)
0.25
0.50
0.75
1.00
0
Wave frequenc y (rad/s )
Wave frequency
(rad/s)
Figure 17: Pitch in head waves of F518-FA in very shallow water
0
Amplitude pitch (-)
0 kn / 180
8 kn / 180
0
10 kn / 180
2.5
2.5
2.5
2.0
2.0
2.0
1.5
1.5
1.5
1.0
1.0
1.0
0.5
0.5
0.5
0
0
0
0.25
0.50
0.75
1.00
2-D diffraction
Classic
Exp.
0
0
0.25
0.50
0.75
1.00
0
360
360
360
270
270
270
180
180
180
90
90
90
0.25
0.50
0.25
0.50
0.75
1.00
0.75
1.00
0
Phase pitch ( )
0
0
0
0
0.25
0.50
0.75
W av e frequency (rad/s)
1.00
0
0
0.25
0.50
0.75
Wave frequenc y (rad/s )
1.00
0
Wave frequency
(rad/s)
Figure 18: Pitch in head waves of G518-GA in very shallow water
FLANDERS HYDRAULICS RESEARCH , N UMERICAL MODELLING COLLOQUIUM, ANTWERP , BELGIUM, 23-24 O CTOBER 2003
DUT-SHL REPORT 1373-E
19
MARC VANTORRE & JOHAN JOURNEE
VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
0
0 kn / 090
0 k n / 090
1.0
0.5
0.25
0.50
0.75
4
1.5
1.0
0.5
1.00
3
2
1
0
0
0.25
0.50
0.75
1.00
0
360
360
270
270
270
180
90
0
180
0.25
0.50
0.75
0.50
0.25
0.50
0.75
1.00
90
0.75
1.00
180
90
0
0
0.25
0
0
Phase roll ( )
360
Phase pitch ( )
0
2.0
0
0
0
5
2-D diffrac tion
C lass ic
Exp.
Amplitude roll (-)
1.5
0
Phase heave ( )
0 k n / 090
2.5
Amplitude pitch (-)
Amplitude heave (-)
2.0
0
1.00
0
0
W av e frequency (rad/s)
0.25
0.50
0.75
1.00
0
W ave frequency (rad/s)
Wave frequency
(rad/s)
Figure 19: Motions in beam waves of F518-FA in very shallow water
0
0 kn / 090
0 k n / 090
1.0
0.5
0.25
0.50
0.75
4
1.5
1.0
0.5
1.00
3
2
1
0
0
0.25
0.50
0.75
1.00
0
360
360
270
270
270
180
90
0
180
90
0
0
0.25
0.50
0.75
W av e frequency (rad/s)
1.00
0.25
0.50
0.25
0.50
0.75
1.00
0.75
1.00
0
0
Phase roll ( )
360
Phase pitch ( )
0
2.0
0
0
0
5
2-D diffrac tion
Class ic
Exp.
Amplitude roll (-)
1.5
0
Phase heave ( )
0 k n / 090
2.5
Amplitude pitch (-)
Amplitude heave (-)
2.0
0
180
90
0
0
0.25
0.50
0.75
W ave frequency (rad/s)
1.00
0
Wave frequency
(rad/s)
Figure 20: Motions in beam waves of G518-GA in very shallow water
FLANDERS HYDRAULICS RESEARCH , N UMERICAL MODELLING COLLOQUIUM, ANTWERP , BELGIUM, 23-24 O CTOBER 2003
DUT-SHL REPORT 1373-E
20
MARC VANTORRE & JOHAN JOURNEE
VALIDATION OF THE STRIP THEORY CODE SEAWAY BY MODEL TESTS IN VERY SHALLOW WATER
References
[1]
Johan Journée and Leon Adegeest
Theoretical Manual of "SEAWAY for Windows"
Ship Hydromechanics Laboratory, Delft University of Technology, Report
1370, September 2003, Internet: http://www.shipmotions.nl.
[2]
Johan Journée
Verification and Validation of Ship Motions Program SEAWAY
Ship Hydromechanics Laboratory, Delft University of Technology, Report
1213a, February 2001, Internet: http://www.shipmotions.nl.
[3]
Leon Adegeest and Johan Journée
Brochure of “SEAWAY for Windows”
AMARCON and Delft University of Technology, October 2003, Internet:
http://www.amarcon.com or http://www.shipmotions.nl.
[4]
Marc Vantorre, Bart Wackenier, et al
Schip Motions in the Scheur Channel (in Flemish)
Internal reports on research project UGent 174P1295 – WLH Mod. 518,
Antwerp/Ghent, 1996-2000.
[5]
Marc Va ntorre, Erik Laforce, Guido Dumon and Bart Wackenier
Development of a Probabilistic Admittance Policy for the Flemish Harbours
30th PIANC Congress, Sydney, September 2002.
[6]
Harald Keil
Die Hydrodynamische Kräfte bei der periodischen Bewegung zweidimensionaler Körper an der Oberfläche flacher Gewässer
Institut für Schiffbau der Universität Hamburg, Bericht Nr. 305, Februar 1974.
FLANDERS HYDRAULICS RESEARCH , N UMERICAL MODELLING COLLOQUIUM, ANTWERP , BELGIUM, 23-24 O CTOBER 2003
DUT-SHL REPORT 1373-E
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