WAVE DRIFT FORCES IN SHALLOi
J.A. Pinkster
R.H.M. Huijsmans
%
S
Technical University of Delft
Maritime Research Institute Netherlands
SUMMARY
Wave drift forces on a tanker in shallow water have been measured in regular and irregular waves. The results of measurements are compared with computations based on three-dimensional potential theory. Results of computations and measurements show the effect of the second order part of the wave
potential associated with the 'set-down' phenomenon. In order
to measure the low frequency part of the second order wave
forces, a special system for restraining the model of the
vessel was developed. This system incorporated active position control devices working partly on the basis of 'feedback' of the position error of the model and partly on the
basis of the application of a 'wave-feed-forward' control
signal generated from real-time measurements of the relative
wave elevation around the model. The time records of the mean
and low frequency forces on the vessel were analysed by bispectral analysis techniques to obtain frequency domain quadratic transfer functions of the second order forces. Attention was also paid to low frequency phenomena in the undisturbed irregular waves such as wave grouping and wave setdown. Measured data on both phenomena were compared with theoretical predictions based on the random wave model and on
potential theory respectively.
1
INTRODUCTION
Wave drift forces have been the subject of many studies
throughout the last 20 years, see for instance, Pinkster [l],
Dalzell [2], Van Oortmerssen [3], Bowers [4] and Wichers [S].
It is generally agreed that, at least for large monohull vessels, the mean and low frequency wave drift forces are dominated by pressure forces which may be predicted reasonably
accurately using computation methods based on potential theory. Results of comparisons of computations and measurements
of the mean second order forces on vessels of various shapes
ranging from monohull forms such as tankers and barges to
more slender hull forms such as semi-submersibles have been
given, among others by Faltinsen and Michelsen [6], Pinkster
[l], Pinkster and Huijsmans [ 7 ] ,
Van Oortmerssen [3] and
Wichers [5].
From theoretical considerations, as pointed out for instance
by Bowers [4], and from results of computations based on
three-dimensional potential theory it can be shown that mean
and low frequency forces of vessels moored in shallow water
will be higher than in deeper water. These forces will also
contain
significant effects from pressure contributions
which, although in principle also present in deeper water,
can in such cases generally be neglected. The increase in the
mean forces in shallow water relative to the forces in deeper
water can in part be explained by the decrease in the wave
length in shallow water for the same wave frequency and in
part by the modification of the vessel motions when considering low draft/water depth ratios.
In shallow water the irregular incoming waves exhibit the
wave set-down phenomenon. This non-linear effect appears as
long waves bound to the incoming short waves. Set-down wave
elevations are related to second order pressures in the wave
field which in shallow water is dominated by second order
potential effects. The incoming irregular waves are characterized by wave grouping which is a term describing the fact
that the waves contained in the train display amplitudes
which are relatively slowly varying in time and space, thus
giving the impression that waves progress in almost distinct
groups. The long waves associated with wave set-down and
bound to the short waves generally exhibit wave troughs where
the wave group amplitudes are largest and wave crests where
wave group amplitudes are smallest. See for instance Fig. 1.
Based on potential theory, it can be shown that the set-down
effect is strongly increased in shallow water. It can also be
shown that the set-down phenomenon does not contribute to the
mean value of the second order forces but only to the slowly
varying components.
Figure 1 . Set-down components in an irregular wave in shallow
water.
The computations of the low frequency forces only take into
account the bound second order potential effects and not the
effects due to the free long waves which are a basin-bound
effect, see [ 4 ] . In the test program described in this paper,
attention was paid to these effects by straightforward spectral analysis of the wave elevation records in order to detect the amount of low frequency activity in the wave elevation record and by analysing the measured undisturbed irregular wave trains using bi-spectral analysis methods in order
to obtain the quadratic transfer function of the second order
wave elevation record. This could be compared with corresponding results of potential theory computations.
The measured undisturbed wave records were analysed to yield
spectra and distributions of the wave envelope square process
which is related to the wave group phenomena; the results
could be compared with predictions based on random wave the-
ory. By and large, such predictions of wave grouping seem to
conform well to full scale data.
As stated above, such effects on the low frequency second
order wave drift forces are predicted to be large in shallow
water but little experimental evidence has been gathered to
substantiate such predictions. One of the reasons for this
lack of data is the fact that experiments which have to be
carried out to measure the slowly varying second order wave
exciting forces on a moored vessel require sophisticated measuring systems in order to measure the forces as correctly as
possible. Also special signal analyses have to be carried out
to identify the quadratic transfer function associated with
the wave forces.
In order to measure the low frequency second order wave
forces on a model in irregular waves it is necessary to emp1oy.a dynamic system of restraint which allows the vessel to
carry out freely the wave frequency motion components associated with the short waves in the spectrum, while at the same
time restricting fully motions at low motion frequencies associated with the slowly varying second order wave forces.
The first requirement must be met since the second order wave
force itself is influenced by the wave frequency motion components. The second requirement is due to the fact that model
motions in the frequency range of the force of interest will
result in dynamic amplification effects in the measured
force. Both requirements have been taken into account in selecting the system of restraint of the model.
For tests in which the mean force in regular or irregular
waves is the only component of interest, a simple soft spring
restraining system may suffice. For tests in irregular waves
where the non-zero frequency low frequency components were of
interest an elaborate dynamic system of restraint using both
feed-back and feed-forward control signals was developed.
Results of tests in irregular waves are the time records of
the restraining forces of the model and of the associated
irregular wave elevation.
In order to extract the frequency domain quadratic transfer
function of the mean and'slowly varying forces use was made
of bi-spectral analysis techniques. In this case a program
was developed based on the original work by Dalzell [ 2 ] . In
order to be able to identify, with sufficiently reliable results, the quadratic transfer function for the low frequency
forces, relatively long time records of the forces measured
under stationary irregular wave conditions are required. Results of calculations of the quadratic transfer functions of
the forces based on the use of a three-dimensional diffraction program DIFFRAC developed at MARIN were compared with
results obtained from model tests in regular and irregular
waves.
The program DIFFRAC was developed by Van Oortmerssen [3] and
subsequently extended by Pinkster [l] for the computation of
the quadratic transfer function for the mean and slowly vary-
ing drift forces using the pressure integration method. The
computational procedure makes use of an approximation for the
force contributions associated with the wave set-down phenomenon, also called the second order potential contribution,
which takes into account the so-called diffraction terms but
neglects terms arising from interactions of incoming, diffracted and radiated first order waves. The method has recently been compared with more complete formulations and
other approximations by Kim and Yue [ E ] with respect to the
second order wave forces on a vertical cylinder and previously by Benschop, Huijsmans and Hermans [g] with respect to the
forces on the same tanker. The method, although incomplete
insofar as that a number of contributions are not included,
appeared to be adequate for determining the low frequency
force components due to the second order potential term. It
was pointed out by Huijsmans, [l01 and [Ill, that the approximation, which is based on the transformation of the first
order force in a wave with the same wave number as that associated with the set-down wave, is exact with respect to the
Froude-Kryloff and diffraction components in the limiting
case of small water depth.
In this paper, after a brief introduction to the mathematical
description of the wave drift forces, the model tests will be
treated. This part includes discussion regarding the adjusted
regular and irregular waves, the model test set-up, the test
procedure and the measurements. Following this the bi-spectral analysis method will be discussed. Results of quadratic
transfer functions for the second order wave set-down obtained using bi-spectral analysis will be compared with predictions based on potential theory. Wave envelope spectra and
distribution functions will be compared to random wave theory
results. Finally, the results of three-dimensional potential
theory computations regarding the quadratic transfer function
for the mean and low frequency second order horizontal drift
forces will be compared with the measured data.
2
THEORY ON SECOND ORDER WAVE DRIFT FORCES: THE PRESSURE
INTEGRATION METHOD
In the following a brief summary of the main points of the
computation method for mean and low frequency horizontal
drift forces is given. A more complete review of the theory
may be found in ref. [ l ] or in [5]. The theory is developed
in accordance with perturbation theory methods. All quantities related to the flow, i.e. the potential, the fluid pressure, the body motions, etc. are expanded in a power series
of a small parameter, in this case typically the wave slope.
By grouping all components of quantities such as wave forces,
etc. which are found by integration of the hydrodynamic and
hydrostatic pressure over the actual wetted surface, into
terms which are the coefficients of the powers of the chosen
small parameter, the expressions for first and second order
wave forces can be derived. The results for the first order
forces correspond with the well-known expressions and need
not be repeated here. The expression for the second order
force is as follows:
in which:
<L1) = first order relative wave elevation at the waterline
dl
= lenqth
element waterline
+(l) = total first order potential including incoming wave,
diffracted wave and radiated wave effects
dS
= surface element of the mean wetted surface of the body
-n
= unit normal vector positive into the fluid domain
R(')
= first order rotational vector of the body
-X
= first order acceleration vector of the body
+(2) = second order velocity potential including effects from
incoming waves, diffracted and radiated waves and all
interactions of first order potential contributions.
-
-
..
Based on eq. (l), the low frequency second order drift forces
in irregular waves can be derived. We assume that the irregular wave elevation can be described as follows:
in which:
= i-th frequency
1
A:l.
= amplitude of i-th frequency component derived from the
wave spectrum
E.
= random phase angle
ztt) = wave elevation in a point.
W.
Based on eqs. (1) and (2), it can be shown that
order wave force is of the following form:
the second
N
~ ( ~ ' ( t=)
N
C
i=1 j=1
cos
+
(E.-E.)
1
3
l
+
in which:
Pij = quadratic transfer function of the part of the force
which is in phase with the wave envelope square process
Qij = quadratic transfer function of the out-of-phase part of
the force.
As can be seen from eq. (3), the quadratic transfer function
is a function of two wave frequencies and physically represents the in-phase and out-of-phase parts of the second order
force in a regular wave group consisting of two regular waves
with frequencies wi and w. respectively. The frequency of the
wave group corresponds to7wi-w
j'
Eq. (1) is used as the basis for evaluating the components of
the quadratic transfer function for the drift forces using
three-dimensional potential theory based computational methods, see for instance ref. [l]. When applied to conventional
hull forms, such as tankers, it is generally found that the
mean and low frequency horizontal drift force components are
dominated by the first term of eq. (1) which is due to the
relative wave elevation. The second term generally counteracts the first term. The remaining terms involving products
of first order quantities may be large at certain wave frequencies if motions are large. The last term involving the
second order potential requires special treatment if it is to
be solved correctly, see ref. [8]. A number of approximations
have been developed in order to evaluate this term which, except in special cases, such as in very shallow water, or for
deep floating structures, is generally not large. The reader
is referred to the relevant literature for more details regarding the theoretical and computational procedures.
3
THE MODEL BASIN
All model tests were carried out in the Wave and Current Basin of MARIN at Wageningen. This basin measures 60 X 40 m.
The water depth is adjustable from zero to 1.1 m. For the
model tests, which were carried out at a scale of 1 to 82.5 ,
water depths were adjusted corresponding to 22.68 m and 30.20
m. The basin is equipped with paddle-type wavemakers along
two sides of the basin and opposite to the wavemakers with
wave damping beaches with adjustable slope in order to minimize wave reflections at all water depths. With a view to
generating sufficiently long time records of the measured
forces, all model tests in irregular waves were carried out
with a duration of 6 hours full scale. All results given with
respect to the adjusted waves were also derived on the basis
of 6 hour records.
3.1 The Ship Model
The model tests were carried out with an 1 to 82.5 scale model of a 200 kDWT tanker in fully loaded condition. The main
particulars of the vessel are gi;en in Table 1. A body plan
is given in Fig. 2. Prior to testing the displacement, position of the centre of gravity and the radii of gyration in
air for the yaw and pitch motion were adjusted. The roll period was adjusted with the model in the water.
Table 1. Main particulars of the 200 kDWT tanker
Symbol
Designation
Length between perpendiculars
Breadth
Draft
Displacement weight
Centre of buoyancy forward of
section 10
Centre of gravity above keel
Metacentric height
Longitudinal radius of gyration
Transverse radius of gyration
:PP
T
A
FB
KG
GM
k
kYY
XX
Unit
Magnitude
100% T
m
m
m
tons
m
m
m
m
309.90
47.17
18.90
240,697
6.61
13.32
5.78
77.47
17 .OO
m
General arrangement
A.P.
St. 10
r
F.P.
B
J
A.P.
S t . 10
Body p l a n
E
Figure 2. The 200 kDWT tanker.
F.P.
3.2 Restraininq System
Two systems of restraint were used for the Dresent model test
program. For tests in regular waves use was made of a simple
soft-spring system which consisted of long steel wires incorporating linear springs attached at one end to fixed points
in the basin and at the model to force transducers located on
the deck of the vessel. The set-up is shown in Fig. 3.
Figure 3. The soft-spring mooring system and the location of
force transducers.
For tests in irregular waves the model was restrained by
means of a dynamic mooring system. The purpose of this type
of mooring system is to allow the vessel to carry out freely
the motions at wave frequency while restraining, as fully as
possible, motion components at the frequency of the second
order wave drift forces as indicated in the introduction.
The restraining system consisted of controlled servo-units
which applied forces to the vessel in surge, sway and yaw
direction through steel wires attached to force transducers
located on the deck of the vessel. The servo-units were activated by a feed-back control loop based on the measured instantaneous position of the vessel. The position of the vessel was measured using an optical tracking system based on a
point light source midships and on a heading gyroscope located in the model. The characteristics of the feed-back
control were of the proportional-differential type. In order
to augment the position-keeping capability of the system of
restraint, additional servo-units fed by a feed-forward control loop, based on a real-time estimate of the low frequency
longitudinal and transverse drift forces and yaw drift moment
were also applied.
Based on an analysis of the components of the wave drift
forces on the same tanker using the direct pressure integration method, Pinkster [l] showed that the major part of the
second order horizontal forces was due to the contribution
associated with the square of the relative wave elevation
around the waterline of the vessel. By arraying eight wave
probes around the vessel, this contribution to the mean and
low frequency second order force could be evaluated in real
time during the model tests. By applying appropriate gain
factors determined by trial and error methods, the feedforward and the feed-back loops could be optimized to reduce
as much as possible the low frequency horizontal motions of
the model in irregular waves. This ensured that the measured
force signals were as little affected by dynamic magnification effects as possible. In Fig. 4 a sketch is given of the
principle of the dynamic system of restraint. A sketch of the
set-up of the model is given in Fig. 5.
Waves
Wave-feed-forward
control system
o
Drift
forces
Force - Vessel
Force
-
Horizontal
motions
l
Feed-back
control
system
Figure 4. Block diagram of the dynamic system of restraint.
Feed-forward control
St. 14
St. 1 9 i
ion c o n t r o l l e r (feed-back)
Force transducer
Wave probes f o r position
control system
Universal j o i n t
Force transducer
Position c o n t r o l l e r (feed-back)
Feed-forward control
Dimensions in m ( f u l l s c a l e )
Figure 5. Set-up of the dynamic system of restraint.
The low frequency motions that still were present, despite
the control algorithm, are used for the corrections of the
low frequency wave drift forces as measured by the control
algorithm. This was established using Newton's law with estimated added mass and damping. This correction procedure has
been used with several settings of the control system and
gave the same answers. In this way the low frequency residue
motions are incorporated into the wave drift forces. For the
higher sea states this procedure may lead to inaccurate estimates due to the presence of wave drift force damping effects.
4
ANALYSIS OF MEASURED FORCE TIME HISTORIES
The measured tine traces of the low frequency wave drift
forces requires an analysis that reflects the nature of the
wave drift forces; i.e. the transfer function of the wave
drift forces is quadratic in nature, so the analysis should
be able to identify this quadratic relationship.
In the analysis of linear processes it is custom to use spectral analysis type of techniques. A natural extension to quadratic type of systems is the Cross-Bi-Spectral techniques
(CBS), which originate from the Volterra series expansion
techniques to describe non-linear systems.
full review of these techniques is given by Brillinger
1121. In this section we will explain the details of analysis
for the wave drift force time trace analysis.
A
In quadratic theory the CBS representation is used:
)
As can be seen the Quadratic Transfer Function H ( w ~ , w ~(QTF)
and the CBS now depend on frequency pairs instead of one frequency as in.the linear case. The process of identifying the
QTF is based on identifying the CBS. In order to use this
spectral analysis technique we consider the fact that in essence the QTF of the wave drift force is of low frequency
type. This essentially means that the axis on which the QTF
of the wave drift forces is tiescribed is dependent on the sum
and difference frequencies.
The CBS is estimated using the Fourier transform of the cross
) . The cross covariance is determined from:
covariance (R
XXY
The CBS is then:
Possible analysis errors when identifying the QTF from the
CBS lies in the fact that the quadratic nature is not fully
reflected. This means that to ensure a quadratic relationship
we impose the following extra condition:
In this equation the denominator reflects the QTF of the input-input squared relation. In theory this QTF should be
identical to one. However, in determining this QTF of the
input-input squared relation from measured wave records one
often sees a deviation from one.
Another feature of these analysis techniques has to be reviewed. Since the Nyquist frequency from the wave drift force
time traces is high NYQQ = NYQQ = n/bt, we are faced with a
e
l
L
resolution problem in the frequency range of interest. The
number of lags for the spectral analysis must be very high to
ensure enough resolution. This however imposes a restriction
on the statistical confidence bands of the estimated QTF at
the varies frequencies.
In order to overcome this dilemma a reduction of the Nyquist
frequency along the difference frequency axis must be obtained. Fortunately this is possible by selection of a filter
scheme on products of the input in the estimation of the
cross covariance function Rxxy.
5
WAVE S
5.1 Wave Set-Down
Examples of regular waves adjusted at water depths corresponding to 2 2 . 6 8 m and 3 0 . 2 0 m are shown in Fig. 6 . It will
be noted that, as may be expected, the lower wave frequencies
contain larger higher harmonic components at the lower water
depths. Spectra of the irregular waves are shown in Fig. 7.
These are based on time records with a duration of 6 hours
full scale. Excerpts from the time records of the wave elevations are given in Fig. 8 . The time records also show the
low-pass filtered records showing the wave set-down. The setdown is seen in the low frequency tail of the wave spectra.
In the figures showing the wave spectra, the theoretically
predicted spectrum of the wave set-down in the low frequency
range is also shown. The prediction is based on the following
expression:
in which:
W
P
T(w,w+p)
sc(W)
SSD(p)
wave frequency
frequency shift = set-down wave frequency
= quadratic transfer function for wave set-down
= wave spectrum (high frequency part)
= spectrum of wave set-down.
=
=
Water d e p t h 30.20 m
Water depth 22.68 m
r~
Seconds
0
I
I
I
I
I
I
50
7
100
Figure 6. Regular waves in shallow water.
For the derivation of eq. ( 1 0 ) , it is necessary to invoke the
random wave assumption, i.e. it must be assumed that the component waves in the wave train are independent random processes. Correspondence of measured and computed spectra
therefore also has a bearing on the validity of the random
wave assumption with respect to the adjusted waves. The quadratic transfer function expresses the amplitude of the setdown wave elevation in a wave field consisting of two regular
waves with frequencies w and w+p. This is also called a regu-
lar wave group. p expresses the frequency difference between
the two regular wave components of the regular wave group and
this corresponds with the frequency of the envelope process
of the group. w+p/2 expresses the mean frequency of the regular wave group. The quadratic transfer function is derived
using potential theory based on the following expression for
the second order wave elevation:
in which:
(2) = second order potential
+ ( l ) = first order potential
<(l) = first order wave elevation.
*
The comparison of the computed and measured low frequency
parts of the wave spectra given in Fig. 7 shows that these
are predicted reasonably well based on knowledge of the high
frequency part of the wave spectrum and the quadratic transfer function based on potential theory. The low frequency
tail of the measured wave spectra do not show any significant
peaking; the presence of which might suggest seiches or
standing waves in the basin.
By considering the wave set-down as a quadratic function of
the high frequency part of the wave spectrum, the quadratic
transfer function of the set-down may also be obtained by
cross-bi-spectral analysis methods considering the wave frequency components of the measured wave record as input and
the low frequency part of the measured wave record as output.
This separation of the time record of the measured wave elevation is affected by low-pass filtering to obtain the low
frequency part. Subtracting this part from the time record of
the total wave elevation record yields the high frequency
part. The principle of this is illustrated in Fig. 9. Results
of cross-bi-spectral analysis of the undisturbed wave elevation records are shown in Figs. 10 and 11. In these figures
the amplitude of the quadratic transfer function is shown to
a base of the frequency difference between the two component
waves of a regular wave group. Each of the small figures applies to the stated value of the mean frequency of the component waves. The lines given in the figures correspond to
results obtained from tests in irregular waves. The dots
given in the figures correspond to the potential theory based
values of I~(w,w+p)l.Comparison of the results shows that
potential theory results correspond well with results of
measurements and that the wave set-down phenomena is significantly affected by changes in the water depth when considering low water depth values. From the comparison indications
were also obtained regarding the applicability of the crossbi-spectral analysis technique. Good comparisons were obtained for mean frequencies which are close to the peak frequency of the wave spectra considered. At mean frequencies
further away from the peak frequency of the considered wave
spectrum the comparison becomes worse. The limits, with re-
spect to the mean frequency at which cross-bi-spectral analysis could still give reasonable results, were established
from the analysis of the wave set-down for the various wave
spectra. These limits were also used when analysing the wave
drift force records from tests in the same waves.
------ Measured
Theoretical
Water d e p t h 30.20 m
WAVE SPECTRUM
Measured:
H, = 5.89 m, T1 = 11.17
s
SET-DOWN
10.
Water d e p t h 22.68 m
-
WAVE SPECTRUM
Measured :
HS = 5.89 m, T 1 = 11.45 s
SET-DOWN
0.25'--,
10.
-
\
3
L?
v,
.
0.
LP-.-.L
0.5
w i n rad/s
1 .o
Figure 7. Spectra of waves and set-down.
1172
w in r a d l s
1
Water depth 30.20 m
H, = 2.77 m, T, = 9.73 s
Water depth 22.68 m
Hs = 2.76 m, T, = 9.74 s
Seconds
1111lrlllr1
0
50
100
Figure 8. Irregular
ter.
wave trains and set-down
in shallow wa-
Wave
W
Figure 9. Set-down separated from the high frequency part of
the wave spectrum.
Water depth 30.20 m
------
Theoretical
I r r e g u l a r waves: H
H:
= 2.77 m, T
= 9.73 5
= 5.89 m, T,l = 11.17 s
Figure 10. Quadratic transfer function of the
at 30.20 m water depth.
wave set-down
Water depth 22.68 m
------
Theoretical
I r r e g u l a r waves: H
H:
m, T1
=
5.89 m, T1
=
= 2.76
=
9.74 s
11.45 s
Figure 11. Quadratic transfer function of the
at 22.68 m water depth.
wave set-down .
------
Derived t h e o r e t i c a l l y based on spectrum of measured wave
Derived from low frequency part of squared wave record
Water depth 3 0 . 2 0 m
Irregular waves:
H, = 2.77 m, T, = 9 . 7 3 S
0
0.25
0.50
Irregular waves:
0.75
0
0.25
Figure 1 2 . Spectra of wave envelope square.
0.50
0.75
Figure 13. Distribution function of wave envelope square.
Grouping in irregular waves can be evaluated through analysis
of the wave envelope. Since wave drift forces are a quadratic
function of the wave elevations it is appropriate to analyse
the properties of the square of the wave elevation. In particular, since we are interested in the low frequencies of the
drift forces we restrict ourselves to the low frequency part
of the square of the wave elevation. This is equivalent to
the square of the wave envelope and thus contains information
on the wave grouping. By squaring and Low-pass filtering of
the wave elevation records, time records of the wave envelope
square are obtained which may be subjected to spectral analysis and of which the distribution may be determined. Conversely, based on knowledge of the wave spectrum (from the
measured elevation record) and the random wave assumption,
expressions can also be given for the expectation of the wave
envelope square spectrum and the distribution of the wave
envelope square process. These are as follows:
P(A
2
=
1
. exp 2m0
- A ~
0
in which:
= wave frequency
S (W) = spectral density of high frequency waves
= frequency shift = envelope square frequency.
P<
W
In Fig. 12
as derived
as derived
quency wave
comparisons are given of envelope square spectra
directly from the wave elevation time records and
from eq. (10) using the spectrum of the high frecomponents.
Comparisons of the distributions of the envelope square records with the results of eq. (12) are given in Fig. 13. The
good comparison indicates that the adjusted irregular waves
conformed well with the random wave assumption.
6
RESULTS OF MODEL TESTS IN REGULAR WAVES
6.1 Mean Drift Forces in Head Waves
The results on the mean surge drift force in regular head
waves are shown in Fig. 14 for both water depths tested. The
results of computations are also given. In general the correlation between measurements and computations is satisfactory.
In shallower water the scatter in the experimental data is
somewhat larger than is usually found when testing in deeper
water. An unusual feature in the mean longitudinal drift
force is the high peak at a wave frequency of about 0.5
rad/s. This peak increases considerably when the water depth
is decreased from 30.20 m to 22.68 m. This behaviour is also
followed by the computed data.
Figure 14. Mean surge drift force in head waves forthe fully
loaded condition.
6.2 Low Frequency Surqe Forces
These results are shown in Fias. 15 and 16 in the form of the
base of the
amplitude of the quadratic transfer function to
frequency difference w -w which corresponds to the frequency
of the force componen&. Sach sub-figure is valid for a constant value of the mean frequency of the components of a regular wave group. The values for zero difference frequency
correspond to the mean drift force in regular waves. Two
tests were carried out in irregular waves at each of the two
water depths. In Fig. 15 the results of the test in irregular
waves with the lowest wave height.were omitted at a mean frequency of 0.43 rad/s. Results from the test in the highest
wave have been omitted for a mean frequency of 0.68 rad/s.
These results were considered unreliable since the wave spectra did not contain sufficient energy at these frequencies.
This leads to large inaccuracies in the cross-bi-spectral
analysis results.
a
Results of computations show fair agreement with the results
of cross-bi-spectral analysis of the measured data from the
model tests. The computed and the measured low frequency
surge force show a large peak at a difference frequency of
about 0.1 to 0.15 rad/s at the lower mean frequencies. Inspection of the computed results shows that this peak is due
to the second order potential contribution to the drift
force. As indicated previously this is associated with the
phenomenon of wave set-down. Comparison of the results for
22.68 m and 30.20 m shows that the force amplitudes are about
a factor two higher at the lower water depth. This is fully
in agreement with the sensitivity of the set-down for decreasing water depth as seen in Figs. 10 and 11.
Water depth 30.20 m
Theoretical
Irregular waves: H,
W-----
H,
=
=
2.77 m, T 1 = 9.73 s
5.89 m, T, = 11.17 s
Figure 15. Quadratic transfer function of the surge drift
force in head waves for the fully loaded condition
at 30.20 m water depth.
Water depth 22.68 m
Theoretical
Irregular waves: H~ = 2.76
HS = 5.89
&(wl+w2) = 0.49
0.25
Figure 16. Quadratic transfer function of the surge drift
force in head waves for the fully loaded condition
at 22.68 m water depth.
The correlation between the measured data for the different
tests and the results of computations is such that it is
concluded that this contribution is significant. It is also
noted that the quadratic transfer function for the low frequency force with a (difference) frequency of about 0.1 rad/s
is considerably higher than thevalue for a difference frequency of zero at the same mean frequency. This indicates
that when low frequency force components at such frequencies
are of interest, as could be the case with the vessel moored
to a jetty which results in relatively high natural frequencies for the horizontal modes of motion, the mean drift force
transfer function severely underestimates the true value.
This means that approximations such as proposed by Newman
[ l 3 1 or Pinkster [l] must be viewed very carefully before
being applied for the analysis of relatively stiff shallow
water mooring systems which induce high natural frequencies
in the horizontal motions of the moored vessel combined with
sea conditions with relatively long wave periods.
7
FINAL REMARKS
In this paper attention has been paid to mean and low frequency horizontal wave drift forces on a loaded tanker in
shallow water conditions. It has been shown on the basis of
both the results of computations and the results of model
tests in irregular waves that decreasing the water depth
results in an increase in the magnitude of the wave drift
forces. The effect on the low frequency drift force components of the contribution associated with the wave set-down
phenomenon has also been demonstrated both from the results
of model tests and computations.
It has been found that although drift forces are predicted
reasonably well based on potential theory methods, the
scatter in the measured data on the drift forces tends to be
larger in shallow water than in deeper water. This may be
associated with the occurrence of larger higher harmonic components in the incoming undisturbed waves which are present
in shallow water. It has been found that in the present test
program no undue effect of free long waves on the measured
low frequency forces could be discerned even though these
waves are predicted by potential theory to be of the same
order as the bound long waves (set-down waves). This may be
due to the fact that in the basin under consideration the
wave paddles, where the free long waves are supposed to
originate in order to compensate the bound long waves, are in
fact, situated in an area where the water depth is considerably larger than in the test basin proper where the measurements took place.
The model tests aimed at measuring the low frequency force
components were carried out using a novel dynamic system of
restraint aimed at minimizing the low frequency motion response of the otherwise soft-moored vessel. In this system
use was made of a feed-back loop and a feed-forward loop in
order to control the dynamic system of restraint. The feed-
forward loop was based on the application of real-time evaluation of the relative wave elevation contribution to the
drift force. The results of measurements and the subsequently
applied cross-bi-spectral analysis technique are encouraging
and will be developed further as standard tools for experimental investigation of low frequency wave drift forces.
ACKNOWLEDGEMENT
The data presented in this paper have been made available
from an extensive Joint Industry Research Program carried out
by MARIN on behalf of a group of companies engaged in offshore activities.
REFERENCES
[l]
[2]
[3]
[4]
[S]
[6]
[7]
[E]
[g]
[l01
[l11
[l23
Pinkster, J.A., Low frequency second order wave exciting
forces on floating structures, PhD thesis, Technical
University of Delft, 1980.
Dalzell, D.F., Application of the fundamental polynomial
model to the ship added resistance problem, 11th Symposium on Naval Hydrodynamics, University College, London,
1976.
Oortmerssen, G. van, The motions of a moored ship in
waves, PhD thesis, Technical University of Delft, 1976.
Bowers, E.C., Long period oscillation of moored ships
subject to short seas, Proceedings of the Royal Institute of Naval Architecture, 1975.
Wichers, J.E.W.,
A simulation model for a single point
moored tanker, PhD thesis, Technical University of Delft, 1988.
Faltinsen, O.M. and Michelsen, F.C., Motions of large
structures in waves at zero Froude number, International
Symposium on the Dynamics of Marine Vehicles and Structures in Waves, London, 1974.
Pinkster, J.A. and Huijsmans, R.H.M., Low freauencv motions of semi-submersibies, 2nd BOSS conference, 19i2.
Kim, M.H. and Yue, D.K.P:, . Slowly varying wave drift
forces in short-crested Irregular seas, Applied Ocean
Research, 1989, Vol. 11, No. 1, pp. 2-9.
Benschop, A., Hermans, A.J. and Huijsmans, R.H.M., Second order diffraction forces on a ship in irregular
waves, Journal of Applied Ocean Research, Vol. 9, 1987.
Huijsmans, R.H.M.
and Dallinga, R.P., Non-linear ship
motions in shallow water, Symposium on Ship and Platform
Motion, 1983.
Dallinga, R.P.! Elzinga, Th. and Huijsmans, R.H.M., Simulation of shlp motions: an adeauate aid to harbour design, 19th ICCE-Conference, oust on, 1984.
Brillinger, D.R., Time Series: Data Analysis and Theory,
Holden Dav. 1981.
Newman, J.N., Second order slowly varying forces on vessels in irregular waves, International Symposium on the
Dynamics of Marine Vehicles and Structures in Waves,
London, 1974.
a .
[l31