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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 385
Frequency Domain Numerical and Experimental Investigation of
Forward Speed Radiation by Ships
M.Guilbaud,1,2 J.Boin,1,2 M.Ba3
(1Laboratoire d'Etudes Aérodynamiques UMR CNRS, 2CEAT-Université de Poitiers, 3ENSMA, France)
ABSTRACT
In this paper we present an experimental and numerical investigation of the radiation effect of a ship with forward
speed in the frequency domain. The tests were performed in a recirculating water channel. Forces, moments and wave-
elevation were measured on series 60 CB=0.6 and 0.8 ship models in forced heave and pitch oscillations. A velocity based
first order Boundary Element Method was developed using the forward speed diffraction radiation Green function. The
calculations of this function and its derivatives as well as its integration on flat panels were performed by controlling both
the accuracy and the computational time. The Fourier integration was done using an Adaptative Simpson method with a
prescribed error. In what concerns the surface integrations, a mixing numerical technique (Gauss method with a number
of points which are function of the distance between the field point and the source panel) and an analytical integration
(based on the Stokes theorem to transform the boundary integral into a contour one, the remaining Fourier integrals over
the complex exponential function are then computed with the same Adaptative method) was used. For the wave pattern
calculations, an extrapolation technique was used to obtain improved numerical results for a field point located on the free
surface.
INTRODUCTION
As pointed out by Okhusu and Wen (1996) and Okhusu (1998), the comparison of measured and calculated global
forces (or motions) on ships running in waves is not an efficient check for the validity and the quality of a numerical
method. This is due to the fact that this kind of data represents an integrated effect involving plenty of factors and not
only fluid mechanics but also mechanics. So the test results are not always clear concerning the quality of the prediction
given by numerical methods. Seakeeping experiments are generally ship motion measurements or global force
measurements for forced motion tests. Thus, they only give results which are not accurate enough to investigate the
validity of the modelling of flow by numerical methods. Consequently, it is also necessary to measure local data such as
pressure distribution or free surface elevation around the ships in order to have a better understanding of the method of
computation. Few experimental data are available on this subject. Some experimental results obtained from free models in
waves or on fixed models in forced motion can be found, for example, Okhusu and Wen (1996) or Okhusu (1998). They
describe some diffraction and radiation wave patterns for the OHS form or Series 60 CB=0.8 ship models running in
waves, using several probes located on a path parallel to the displacement of the model and then extract the linear
component of the first harmonic by data processing. Iwashita et al. (1993), see also Okhusu (1998), also present pressure
measurements of the diffraction problem for a VLCC ship running in waves. Furthermore, for free models in waves, test
errors are present both on motion and force and moment measurements. Finally, radiation and diffraction are not easy to
separate, even if it is well known that diffraction waves vanish more rapidly than the radiation ones.
The experimental work presented here tries to give both global and local data on hydrodynamic radiation flow in
order to compare them with the numerical method in the frequency domain under development. For the experimental
work we use the forced motions for heave and/or pitch motions. A first experimental planar motion system, Guyot (1995),
Guyot and Guilbaud (1995) was built to study a Series 60 model with a block coefficient CB=0.6 and with a length
L=0.6m. It is an improved version of the device used by Delhommeau et al. (1992). Some difficulties were encountered
when trying to obtain an accurate wave pattern map due, in particular, to the weak amplitudes measured Therefore, the set-
up has been modified to perform tests on a model with a length of L=1.2m. Forces and moments and also wave pattern
measurements have been done with two models, one in which CB=0.6 and the other with 0.8, in order to study the
influence of the block coefficient. Influences of
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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 386
frequency, free-stream velocity were investigated in the recirculating water channel of the Ecole Centrale de Nantes. The
test apparatus enables to separate the radiation waves, which are not directly visible in a towing tank, from the mean
steady wave pattern.
The numerical method developed is a velocity based panel method using the diffraction-radiation with forward speed
Green function, satisfying a linearized free-surface boundary condition. The main advantages with respect to the Rankine
methods (described in a large review by Sclavounos, 1996) is the reduction of the size of the linear system to be solved,
the automatic satisfactory of the radiation condition (particularly difficult to insure in the Rankine methods whatever are
the values of the frequency and of the forward speed) and of course, of the free surface boundary condition. The use of
the corresponding Green function prevents any problem related to the existence of boundaries of the computational
domain on the free surface, responsible for wave reflections difficult to suppress in the Rankine methods. Furthermore,
due to the fact that no grid is present on the free surface, there is no filtering of the smaller wavelengths. Although the
corresponding Green function for seakeeping calculations in the frequency domain around bodies with forward speed is
quite difficult to compute and relatively time consuming, the progress of computers during the last years as well as the
improvements of the algorithms of computation enable us to develop numerical codes running on a cheap workstation or
PC in less than 2 hours for the computation of pressure distribution, forces and moments. The fastest and more accurate
techniques of calculation are the steepest descent method, Iwashita and Okhusu (1989, 1992), Brument et Delhommeau
(1997) or Brument (1998) using the Steepest descent method for the function and its derivatives or Iwashita (1992) for
boundary integrations of this function, the method of the Super Green function developed by Chen and Noblesse (1998),
or the Adaptative Simpson method for the function, Nontakaew et al. (1997), or for surface integrations on panels, Boin et
al. (2000). But to have an accurate method to compute free surface flows, it is necessary to accurately calculate not only
the Green function but also the boundary integrals on panels and the line integrals on the waterline. We have developed a
mixed technique for the surface integration using both a numerical Gauss method (with a number of points which are
function of the distance between the field point and the source element) and an analytical method of integration, derived
from the Stokes theorem, closer to the source element, Boin et al. (2000). All these methods give accurate results in
moderate computational times. We present here such a velocity-based method for non-lifting flows. Nevertheless, it is
well known that these computations are very difficult and it is quite important to check the results with test measurements.
EXPERIMENTAL STUDY
Experimental set-up
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Figure 1 Planar motion generator

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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 387
The experiments were performed in the test-section (2m wide and 1m high) of the recirculating water channel of the
Ecole Centrale of Nantes where the maximal velocity is 1.7m/s. The photograph in figure 1 shows the plane motion
generator. It is made of two cam-crankshaft systems driven by an electric motor (1) with variable rotation speed. These
systems give a sinusoidal controlled motion to the ship model (2), which can be either a pure heave movement through
the rod (4) (the pitch rod (7) being decoupled), or a pure pitch movement with the rod (7) (the heave rod (4) being
decoupled). With the combination of two eccentrics, it is also possible to obtain a heave-pitch combined motion with a
variable phase lag between heave and pitch motions. The model position is recorded with the help of a heave linear
transducer (5) or a pitch linear transducer (6). The model is fixed to the motion device through either a 3 component
dynamometer or a rigid modulus (3) used for the wave pattern measurements. The centre of rotation, which also
corresponds to the centre of moment, is in the plane of the undisturbed water level and is located at 0.608 and 0.560 m
from the forepart of the model for the CB=0.6 and 0.8 respectively. The pitch motion is also transmitted at the waterline
level of the model (pitch rod (7)). The maximal amplitudes available are 0 to 20mm and 0 to 6° for heave and pitch
respectively.
The position of the plane motion generator can be moved by 90°, i.e. located horizontally (instead of vertically for
the heave and pitch motions, figure 1) and thus is able to produce sway and yaw motions, Nontakaew et al. (1996). In this
case, a new dynamometer is used.
Models and test conditions
Two series 60 CB=0.6 and 0.8 ship models (L=1.2m) were built. Their characteristics are given in table 1; the
dimensions are based on the original methodical series, Todd (1963). The models are made of composite materials
(carbon fibre) in order to minimise the inertia components in the force measurements. Their weights are 0.8 and 1kg for
the CB=0.6 and 0.8 models respectively. The tests were carried out for both models, for pure heave and pure pitch motions
with 10.8mm and 1.8° as amplitude respectively. Global forces and moments have been recorded for frequencies ranging
from 2.75 to 3.25Hz if F=0.04, 2.5 to 6Hz at F=0.2 and 0.3. For the wave-elevation, the measurements were performed
for f=0.89 and 1.06Hz at F=0.12 (τ=0.22 and 0.27) and f=3 and 3.9Hz at F=0.2 and 0.3 (the test conditions are
summarised in table 2 and 3). In order to reduce the reflection of the waves on the side walls of the test section at very
low velocity, the flow velocity around τ=1/4 was increased for the global measurements. The Reynolds number of the
tests were R=1.68.105 (F=0.04), 0.84 and 1.2.106 (F=0.2 and 0.3).
Table 1: Ship sizes
CB 0.6 0.8
Lm 1.2 1.2
Bm 0.1574 0.1816
Hm 0.0629 0.0726
S m2 0.238 0.324
∇ 0.007 0.012
Table 2: Test conditions for free surface elevation measurements
U m/s F Hz τ F
0.4 0.87 0.22 0.12
0.4 1.07 0.27 0.12
0.7 3 1.35 0.2
0.7 3.92 1.76 0.2
1 2.97 1.90 0.3
1 3.85 2.47 0.3
Table 3: Test conditions for force measurements
U(m/s) f(Hz) τ F
0.13 2.75 to 3.25 0.22–0.27 0.04
0.7 2.5 to 6 1.25 to 2.7 0.2
1.0 2.5 to 6 1.75 to 3.8 0.3
Measurements of forces and moments
The dynamometer used is composed of 3 miniature force transducers; forces and moments are uncoupled by the use
of needles. Before being connected to the analog to digital converter, electric signals were amplified using band pass
filters. Static weights were used for calibration to determine the calibration matrix for the data reduction.
Linear displacement transducers record the motions. They are not used to measure the motion amplitudes but to give
the reference lag for the motion. However, the lags have been corrected from parasite lags introduced by the linear
transducers and by the data acquisition system by the measured phase lags during the inertia measurements with the
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known value of 180°.
For the force measurements, tests had to be performed twice, with the model oscillating in air (inertia forces) or in
water (total forces) at the same frequency. Then, the inertia forces were subtracted from the total ones to obtain the
hydrodynamic ones. Guyot and Guilbaud (1995) have shown that the results are equivalent if the calibration matrix is
applied before the signal analysis (then applied to the forces or moment) or after. In this last case, the signal analysis is
applied directly to the electric signal. We used this last solution here. Once global forces were determined, added mass
and damping coefficients were calculated. It was also necessary to measure the hydrodynamic restoring to obtain the
added-mass coefficients. This was done at the corresponding Froude number to take into account the true shape of the
mean free surface by measuring the differences between the forces and moment on the model located at the extreme
positions of the motion.

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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 388
Wave patterns measurements
Free-surface elevations were measured using a resistive probe which consists of two parallel chromel wires of
0.2mm in diameter and 150mm in length which were 10mm apart and held by a Plexiglas frame to avoid electric
perturbations. These two wires were mounted as one branch of a Wheastone bridge supplied by a 3kHz alternative current
to prevent water electrolysis. After demodulation and amplification, a signal with voltage directly related to the depth of
immersion was obtained. As for the force measurements, band pass filtering processed the signal. In order to obtain a
good accuracy in the measurements of wave-amplitudes, a probe calibration was done for each flow velocity. For each
one, the curve voltage-immersion was approximated by a five-order polynomial, in order to determine the probe
immersion. A slight influence of flow velocity was also found, justifying, thus, this procedure. The free-surface mesh was
obtained over one side of the model (the flow being symmetrical) each 40mm for x and y coordinates close to the hull,
figure 2. This mesh was finer than that used in a former work (Guyot, 1995). The measurement area was 1880mm
streamwise and 360mm crosswise. During these tests, we took care to have a significant measurement domain upstream
of the bow in order to highlight the upstream wave phenomena close to τ=1/4. In short, the wave-elevation contour
cartography included 350 measurement points, see for example figure 2 for the CB=0.6 model.
Figure 2 Part of the free surface where the elevations was measured (L=1.2m)
Acquisition system and signal analysis
The experimental set-up included a Pentium 100MHz personal computer, with an acquisition card Keithley
DAS1600 (with an internal clock of 10MHz) and a sample and hold SSH-4/A module (thus the time lag between the
channels during the acquisition did not exceed 40ns). Four channels were used for the global measurements (3 for forces
and moment and one for the motion) and two channels only for the wave pattern measurements (one for the model
position recorder and the other for the free-surface probe). The probe motion was semi-automatic: automatic along the x
co-ordinate with the help of a stepper motor driven by the computer, and manual along the y direction. During the tests,
an optical device located close to the motor adjusted the frequency of motion (motor rotation). As already shown (Guyot,
1995), data reduction depends on the number of acquisition points: 6000 for the force measurements and for the wave
patterns, 1024 samples were recorded for each channel. The data treatment procedure was as follows:
a) Rough determination of the motion frequency fini of the model by a Fourier analysis. This frequency was the
initial value for the following calculations;
b) Probe-signal's Fourier analysis around this frequency value for the range fini−0.1Hz≤f≤fini+0.1Hz in order to
determine the accurate value of the frequency, corresponding to the maximal energy (spectrum); 100 points
of computation were distributed in this range and the Fourier analysis step was reduced when the studied
frequency came closer to fini;
c) Finally, for the probe signal: determination of the amplitude and the phase lag with respect to the forced hull
motion.
During the acquisition, the free surface elevation showed 2 peaks on the spectra, one with the frequency of motion
and a second one at about f=0.15Hz. This last oscillation corresponded to a variation of the recirculating water free
surface height during the tests; this level oscillation could reach 2.5 to 3mm and is removed by the analysis.
TEST MEASUREMENTS
Added-mass and damping coefficients
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Figure 3 Added mass coefficients for the heave motion

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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 389
Figure 5 Added mass coefficients for the pitch motion
Figure 4 Damping coefficients for the heave motion
Figure 6 Damping coefficients for the pitch motion
The added-mass coefficient CMij=Mij/(r Ln) and the damping one CAij=Aij/(rwLn) for j=3 (n=5), or j=5 (n=3) where
w=2pf, versus the frequency f (in Hz) for the 2 ship models and the two values of the Froude number are plotted in figures
3 to 6.
Figures 3 and 4 correspond to the heave motion. The effect of the Froude number is relatively weak, except at lower
frequencies for the added-mass CM33. In figure 3 the model shape has a stronger effect, CM33 increasing with the block
coefficient. It must be noticed than when the frequency decreases, CM33 is very sensitive to errors in the hydrodynamic
restoring coefficients. Concerning the damping coefficients, figure 4, these two parameters show weak influences, except
at the lower values of the frequency, leading to high values of CA33.
Figures 5 and 6 are for the pitch motion, CM55 and CA55. Conclusions are similar for the heave motion, the effect of
the kind of model (value of CB) being greater than the Froude one, particularly on CM55, figure 5. No important
coefficient variations were observed close to τ=1/4.
Wave pattern measurements
Flow close to τ=1/4
Figure 7 Wave pattern close to τ=1/4 (heave) Figure 8 Wave pattern close to τ=1/4 (pitch)
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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 390
One of the first aims of this work was to investigate the wave-elevation contours close to τc=0.25 for both CB=0.6
and 0.8 models. It is known that below this critical value of τc one part of the waves field propagates upstream of the
model. Beyond τc=0.25 a wedge appears at the bow and stops this propagation, thus the waves are exclusively convected
downstream inside this V-shape pattern (the wave group velocity is lower than the ship forward velocity). Figures 7 and 8
show the wave-amplitude contours for both heave and pitch motions for the CB=0.6 model. The upper half and lower
parts of each figure correspond to τ=0.22 and 0.27 respectively. The form of the waterline of the CB=0.6 model is also
plotted and the white area around the hull corresponds to a non-investigated zone (this domain was not accessible to the
free-surface probe). For the heave at τ=0.22 (figure 7), two waves can be seen close to both the bow and the stern, and
their crests are roughly parallel to the hull axis. The relatively large height of these waves is probably linked to a
reflection phenomenon on the channel lateral walls. When the motion frequency increases (τ=0.27), the upstream
perturbations vanish and the more pronounced wave-elevations are located at the downstream end of the model. An
increase of τ leads to the formation of the V-shape pattern, as will be shown in the next paragraph. Figure 8 shows the
wave-elevation contours for the pitch motion. At τ=0.22 stronger perturbations are convected upstream of the model (in
comparison with the heave motion); the wave pattern stays relatively homogeneous along the hull downstream. The
pattern at τ=0.27 shows the emergence of a new flow state: upstream, the wave-elevations decrease strongly. However,
the wave-amplitudes pattern remains slightly rough; downstream, a perturbed wave field is observed, nevertheless the
heights are slightly smaller in comparison with the upstream measurements.
The results corresponding to the CB=0.8 model close to τ=1/4 are not reported in the present paper. Indeed, this
model generated larger wave-elevations, which were reflected on the channel walls. The obtained wave pattern was
strongly disturbed and difficult to analyse. These measurements are less accurate than those obtained at higher values of
the Brard parameter due to the low flow velocity enabling wall reflection of waves close to the model and to the weak
amplitudes.
Flow at τ≫1/4
Figures 9 to 14 show the different wave-elevation patterns for both CB=0.6 and 0.8 models. The upper half and lower
parts of each figure refer to the CB=0.6 and 0.8 models respectively for the same test configuration. The wave-elevation
measurements highlight the effect of four parameters on the free-surface waves: the motion frequency imposed to the
model, the flow velocity (Froude number), the type of hull movement (heave or pitch) and the ship block coefficient.
Figure 9 Wave amplitudes (heave motion; F=0.2; f=3Hz)
The recorded wave patterns have the same characteristics whatever the test configurations and the model motion:
two zones in V-shape with the tip in the upstream direction are visible with strong amplitude values, at the bow and the
stern (these amplitude values being stronger at the stern). The whole wave field is contained in this V-shape pattern.
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Figure 11 Wave amplitudes (heave motion; F=0.2; f=4Hz)
Figure 10 Wave amplitudes (pitch motion; F=0.2; f=3Hz)
The opening angle of this wedge and the wave-amplitudes are decreasing functions of the motion frequency: the
figures 9 and 11 for heave motion, for the CB=0.8 model for instance, show a decrease of the wave amplitudes just

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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 391
behind the bow and downstream from the stern when the frequency increases from 3 to 4Hz. Moreover, the frequency
increase prevents the propagation of strong wave-amplitudes in the flow field as shown for example in figures 10 and 12
for CB=0.6 and 0.8, in the case of pitch motion. The wave-amplitudes are quickly damped far from the model. The highest
waves are located close to the stern (particularly for the pitch motion). A diminution of the two V-shape zone angles is
also observed.
Figure 13 Wave amplitudes (heave motion; F=0.3; f=3Hz)
Figure 12 Wave amplitudes (pitch motion; F=0.2; f=4Hz)
Figure 14 Wave amplitudes (pitch motion; F=0.3; f=3Hz)
The increase of the flow velocity induces the same behaviour: for the CB=0.6 and 0.8 models, figures 9 and 13
clearly show that the wave-amplitudes decrease and the front V-shape pattern is less visible when the Froude number
increases; the bow waves are reduced and the region of strong amplitude moves from nearly x/L=0.2 at Fr=0.2 to about
0.4 for Fr=0.3 for the CB=0.8 model. A similar observation can be done for pitch motion, figures 10 and 14. However it
should be pointed out that the frequency effect seems more pronounced in comparison with Froude number one; the wave
amplitude variation is weaker in this last case.
Figures 9 and 10, both for CB=0.6 and 0.8, emphasize that on pitch motion, the back V-shape pattern presents higher
wave amplitudes in comparison with the heave one, but the areas of strong wave amplitudes are reduced. This fact can
also be observed at a higher frequency, figures 12, F=0.2 and f=4Hz.
For the same test configuration, the block coefficient effect is quite pronounced: figure 9 for instance provides
stronger wave amplitudes for CB=0.8, but the wave angle seems to have the same value. For the CB=0.8 model, just
behind the bow wave, a local area of low wave amplitudes appears, and three areas with high amplitude are clearly seen
close to the hull (see also the CB=0.8 figure 11, for a frequency increase). The bow wave and front V-shape pattern vanish
with the increase of the flow speed (figure 13, CB=0.8), only the stern wave is present, with the same amplitude but a
smaller area. Results are similar for pitch motion, figures 10, 12 and 14.
Analysis of flow
Unsteady wave motion
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Figure 15 Model positions during motion
The advantage of this kind of experiments is to underline the evolution of the free-surface unsteady part, which is
defined by z=Asin(ωt+φ), where A is the amplitude, by removing the steady component which, because of its high values,
hides the unsteady phenomena. The time variation is quite similar for both models, but the CB=0.8 model results show a
more pronounced wave-amplitude pattern in comparison with the other model. These results are therefore presented in
this paper in order to make the understanding easier.

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CB=0.8 model in heave motion (F=0.3 and f=3Hz)
Figure 16 Unsteady wave pattern during a period for the
CB=0.8 model in pitch motion (F=0.3 and f=3Hz)
FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS
Figure 17 Unsteady wave pattern during a period for the
392

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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 393
The model positions during a period are given in figure 15: for heave at t/T=1/4, the model is in the highest position;
the lowest one corresponds to t/T=3/4. For the heave motion, the figure 16 shows the free-surface unsteady component
during a period T, for eight positions every t/T=1/8 (from t/T=0 to 7/8), at F=0.29 and τ=1.90 (the flow comes from the
right).
As it can be observed, the waves spread out parallel to the hull axis and move away downstream and slightly
sideways from the ship during a period (follow the wave development from t/T=1/2 to 7/8). For 5/8

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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 394
and
The limits of integration for G2 are:
The modified complex integral functions, g1, g2, and g3, are defined by:
The complex integral function is given by:
The poles Ki and Zi of eq. (4) are given by:
With the previous assumptions and by neglecting the boundary integrals on the free surface as a first calculation.
Brument et al. (1998) have shown that the influence of the waterline is weak for this problem,. Then equation (4) for
on S can be transformed into:
(5)
where is the non-dimensional circular frequency.
To deal with a non-lifting problem, by a choice concerning the arbitrary potential in the inner domain defined by the
hull, it is possible to use only a source distribution and the previous equation can be rewritten as:
(6).
By applying the ∂/∂n operator to equation (6), the following integral equation enables us to compute the source
distribution on the body by using the body condition (1) and leads to the integral equation:
(7).
Once the source distribution is known from equation (7), equation (6) enables the calculation of the potential. Then,
the pressure on the body can be easily computed by:
(8),
and by integration of the pressure, forces and moments are obtained.
Finally, the unsteady free surface elevation can be computed by:
(9)
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Derivatives in equations (8) and (9) have to be computed by differentiating with respect to x the potential given by
equation (6).

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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 395
Numerical method
The body is divided into nc columns of np panels giving N=nc np panels on the body. To take advantage of the
symmetry of the boat for non-lifting flow, only one half of the hull is discretised. Consequently, the waterline is also
divided into np segments. The method developed is a constant panel method Equation (7) leads to the following equation
with unknown sj and si for the calculation:
(10).
Coefficients aij and bij to be computed are respectively given by boundary and line integrals on the derivatives of the
Green function. Integrals on the function also have to be performed to compute using eq. (5) the velocity potential
needed in eqs. (8) or (9). This Green function is computed as in Ba and Guilbaud (1995); the only difference is that the
Fourier integration is performed here by an adaptive quadrature method proposed by Lyness (1970) and Malcom and
Simpson (1975), where the integration step decreases as the integrand becomes more oscillating with a prescribed error,
instead of a fourth order Runge-Kutta method. Each interval is divided into 2 parts, the integral on the whole interval and
the sum of the 2 integrals on the 2 sub domains are computed with a 5 points Simpson method and the results compared,
Nontakaew et al. (1997). The procedure is pursued until convergence is obtained. It is easy to relate the error on the
whole domain and the corresponding error on one of the sub-domains, Guttman (1983). This method reduces the CPU
times and gives accurate results for any value of the parameters. Nevertheless, not only do the Green function and its first
derivatives have to be computed accurately, but also the boundary and contour integrals involved in the equation (7). For
the boundary integrals on panels (coefficients aij in equation (10)), the accuracy of the integration by a Gauss method has
been controlled, Boin et al. (2000) by comparing with the results of an analytical integration based on the Stokes theorem
transforming the surface integrals into contour ones, following Bougis (1981):
(11).
Coefficients Ak in equation (11) are computed using the following properties of the complex integral function
concerning the derivatives:
(·) and (¨) are the first and second derivatives of the functions with respect to the parameter ξ. For the integrations
(j=1 to 3):
The term G0, often studied in aerodynamics or hydrodynamics using Rankine's singularities, will not be studied here.
For G1 and G2, the following expressions are obtained:
(12)
for i=1, 2 or 3 and:
Mk(xk, yk, zk) is the node k of a panel with Mm+1=M1; the outer unit normal to the panel is and:
the authoritative version for attribution.
For G2, we obtain:
The terms Ii in the above expression are very similar to equation (12) with different coefficients instead of
involving other values of c or the modified complex functions. Details can be found in Boin et al., 2000. Quantities ()' are
deduced from the () ones by replacing χ by χ', with, for I=1, 2:

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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 396
Concerning the Green function, coefficients are obtained with similar expressions by replacing the unique integrals
for by double integration with respect to ξ Furthermore, we have the following relations:
For the integration on a segment, a similar expression to equation (11) can be obtained:
(13).
In equations (11) and (13), the Fourier integrations are also performed with the Simpson Adapative method already
used for the Green function. Finally a technique to calculate the boundary integrals was used. It mixes a numerical Gauss
method of integration with an analytical one. The number of Gauss points required to have a good accuracy ranges from a
unique one, far from the source panel and particularly upstream, to 224 Gauss points close to it. As the computational
time for the analytical method of integration is more or less equivalent to the numerical one with 4 Gauss points, the
analytical method is more efficient close to the panel. Therefore, we have decided to use the numerical method (with the
number of Gauss points ranging from 1 to 4) for field points out of a vertical cylinder centred on the source panel with a
radius equal to 8 times the panel length, except in a zone limited by angles amin and amax, defined in figure 18, where it is
not possible to compute accurately the integral over G2, even with a large lumber of Gauss points. In the cylinder, both
integrals on G1 and G2 are integrated analytically. Details of the method are explained in Boin et al. (2000).
The values of the limiting angles in the figure 18 have been determined from numerical tests on single panel and a
field point describing the whole fluid domain by varying both the Froude number and the motion frequency.
Figure 18 Limiting angles for the definition of the zone where the integrals on G2 have to be calculated analytically
They have shown that amin is a function of the Brard parameter given by
(14).
It is more difficult to find an expression for amax, but we have chosen:
Nevertheless, if this mixed technique of boundary integration used for these kinds of calculations is efficient when
the field points are not to close to the free surface, numerical difficulties arise when calculating the wave pattern from
equation (9), the field points being in this case on the free surface (z=0). This is quite evident for the first row of panels
closest to the free surface, none of the 2 integration techniques (numerical or analytical) gives correct results. Better
integration is made using the analytical method by reducing the prescribed error to 10−4 and by increasing the number of
steps during the adaptative integration to 10000, but only if z/L≤−0.1. We then decided to compute the integrals for this
value of z and to extrapolate the values to z=0. Results have been shown to be nearly independent of the method of
extrapolation used. We are still working on the improvement of the calculation of these integrals. Similar developments
can be also done for the integrals over the waterline segments.
Finally, equation (10) leads to a linear system of equations that can be inverted giving the source intensities on the
panels. After having obtained these intensities, the pressure is computed by equation (8) and by integration on the body
surface, the forces and moments on the body. Such a computer code has been developed without incoming waves to
compare the results with the present test results and with the numerical results obtained by Nontakaew et al. (1997) for a
flat plate in sway and yaw motions using a vortex lattice method. Numerical results and the comparison with the test
measurements will be presented and discussed in the next paragraph.
Computational times for pressure, forces and moments are typically 2 hours on a PC for 400 panels and 7.5 hours for
the computations of 200 points on the free surface for one value of the frequency and of the Froude number.
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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 397
NUMERICAL RESULTS AND COMPARISON WITH TEST MEASUREMENTS
Series 60 hulls
Figure 19b Damping coefficients for Series 60 CB=0.8
Figure 19a Added-mass coefficients for Series 60 CB=0.8 hull in heave motion (F=0.2; a/L=0.009; 490 panels)
hull in heave motion (F=0.2; a/L=0.009; 490 panels)
Figure 19 and 20 plots the added mass and damping coefficients versus the non dimensional frequency
for the Series CB=0.8 hull for the two motions with 490 panels at F=0.2. The heave amplitude was a/
L=0.009 and the pitch one ψ=1.8°. The dashed lines are for the numerical results (without the waterline integral) and the
symbols are for the test measurements. These results show oscillations probably due to the existence of irregular
frequencies. To remedy this problem, we have added a surface of flat horizontal and slightly immersed panels (−0.5% of
the total body length) inside the free-surface where a zero velocity condition is satisfied. The results obtained using this
technique for suppressing the irregular frequencies are shown by the full line. The first irregular frequencies have been
effectively removed showing an improvement of the results, but there are still irregular frequencies at higher
values. Work is in progress to improve these results. Nevertheless, we can observe a good agreement between
computations and measurements for particularly for the heave motion. Results are also good for CA55 but
calculated results over predict the test results. The agreement is less good for CM55 but the variation of this coefficient
with is qualitatively predicted and this coefficient has very weak values.
Figure 20a Added-mass coefficients for Series 60 CB=0.8
Figure 20b Damping coefficients for Series 60 CB=0.8
hull in pitch motion (F=0.2; ψ=1.8°; 490 panels)
hull in pitch motion (F=0.2; ψ=1.8°; 490 panels)
Figures 21 depict the plots of the unsteady wave amplitude patterns around the Serie-60 CB=0.8 at F=0.2 and τ=1.75
the authoritative version for attribution.
(f=3.9Hz) for heave motion (top graph) and for pitch

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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 398
one (bottom one). The upper part of each graph shows the numerical results and the lower one the test measurements; the
higher plot represents the heave motion and the lower one the pitch one. The motion amplitudes are the same than
previously. The same general wave patterns can be observed with the V shape already mentioned and some regions with
high amplitudes at the bow, middle part and end. The calculations show particularly a zone with high wave amplitude
behind the hull. This fact has not been explained yet, but it has already been mentioned, Brument et al. (1998).
Figure 21 Comparison of wave height amplitudes for a series-60 CB=0.8 model in forced heave or pitch motion (F=0.2;
f=3.9Hz; τ=1.75; a/L=0.009 or ψ=1.8°)
Figure 22 Comparison of wave height amplitude for a
Figure 23 Comparison of wave height amplitude for a
series-60 CB=0.8 model in forced heave motion at y/
series-60 CB=0.8 model in forced heave motion at y/
L=0.166 (F=0.2; f=3Hz; τ=1.35; a/L=0.009)
L=0.166 (F=0.2; f=3.9Hz; τ=1.75; a/L=0.009)
For a quantitative comparison, figures 22 and 23 plot the longitudinal relative wave amplitude (h/L) profile for y/
L=0.166 for the Series 60 CB=0.8 hull in heave motion at respectively f=3 and 3.9Hz and F=0.2. Full line is for 245
panels and the dashed line for 400 panels. The symbols are for the test measurements. The increase of the number of
panels lead to weak variation of the wave profile but with a smoother curve. In figure 22 (f=3Hz), some discrepancies
appear between computations and measurements but this frequency is close to an irregular one. At the contrary, in
figure 23 (f=3.9Hz), the agreement is better except at the hull end where, as already mentioned, the calculations
overpredict the wave amplitude; this frequency is located between two irregular frequencies, so the results are better and
the agreement with the test results is more fair, except behind the hull, as already mentioned.
Flat plate in forced sway motion
Figures 24 and 25 plot the added-mass and damping coefficients CM22=2M22/(r SL) and CA22=−2A22/(r SU) versus
the non-dimensional circular frequency V for a surface-piercing flat plate of aspect ratio AR=0.5 at Froude number
F=0.32 in forced sway motion. The shape of the plate is a Wigley hull defined by:
the authoritative version for attribution.
with L=1m, T=0.5m and b=0.02m. The mean amplitude is α=0°. The present results (full lines) are compared with
the calculations of Nontakaew et al. (1997) using a vortex lattice method based on the same diffraction-radiation with
forward

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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 399
speed Green function as in the present study for a zero thickness flat plate (dashed line). Agreement is relatively good
when it is considered this assumption for these last calculations, particularly for high values of Nevertheless, even at
high values of curves have similar shapes.
Figure 25 Damping coefficients for a Wigley-shaped flat
Figure 24 Added mass coefficients for a Wigley-shaped
plate in sway motion (F=0.32; AR=0.5; 400 panels)
flat plate in sway motion (F=0.32; AR=0.5; 400 panels)
CONCLUSION
We have presented some experimental and numerical results in the frequency domain concerning the radiation flow
around ship models of Series 60 with block coefficient CB=0.6 and 0.8 in forced oscillations of heave or pitch motions.
Added-mass and damping coefficients were measured as well as the unsteady free-surface elevation around the hull. The
measurements show a strong influence of the block coefficient when compared with the Froude number both, on global
forces and also free surface elevations. Furthermore, the unsteady radiation wave pattern evolution during a period has
been analysed by removing the mean free surface elevation from the measurements. The graphs show waves, oriented
along the longitudinal axis, travelling downstream and sideways from the model, both starting from the fore and back
parts of the hull. These waves moved in V-shape areas that can be observed on the wave amplitude plots. For heave
motion, the upstream and downstream waves have the about same phase lag, while the phase lag is about 180° for the
pitch motion. Close to the singular value of the Brard parameter τc=1/4, if the wave pattern changes from waves travelling
in the whole domain if τ1/4, no sharp variation of the forces coefficients are then
observed.
Calculations have been performed with a constant panel method using the diffraction radiation with forward speed
Green function. Both the accuracy of the Green function and derivatives but also of the integration on flat panels have
been controlled. The comparison of added mass and damping coefficients for Series 60 hulls shows relatively good
agreement between measured and calculated values. The results obtained with the code developed in the present study
have also been compared with other numerical results available for a flat plate in forced sway motion and are in relatively
good agreement. These calculations show also the presence of irregular frequencies at high values of the reduced
frequencies. Only the first irregular frequencies have been suppressed and the technique used has to be improved. Some
difficulties appear for the free surface elevation calculations, particularly in the wake of the model whereas the wave
amplitudes are overpredicted by the computations. Nevertheless, the wave pattern is qualitatively correctly represented by
the computations. Furthermore, the free surface amplitude are not accurate when the frequency is too close of an irregular
one.
Finally, work is in progress first to improve the technique used to suppress the irregular frequency, to study the
seakeeping of ships in regular waves and to introduce the waterline integral computational code controlling the accuracy
of the integrals on the segments of the waterline. The influence of the latter will be investigated for several kinds of boats.
The lifting effects will be also introduced to deal with boats in yawed flows as sailing boats or manoeuvring ships. In this
case, integrals dealing with the second derivatives of the Green functions have to be considered. More comparison are to
be done both on global data and local ones. A study of the calculation of the wave amplitude behind the hull must be
pursed in order to understand the overprediction of the free surface elevation in this zone.
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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 400
AKNOWLEDGEMENTS
The authors gratefully thank the DGA-DRET (French Ministry of Defence) for its grant n°95–068 used to finance
the experimental part of this study. The authors thank also G.Delhommeau and its group of LMF-DHN (UMR CNRS n°
6598) of Ecole Centrale de Nantes (France) for many exchange and discussion about this study.
REFERENCES
Ba M. and Guilbaud M., “A fast method of evaluation for the translating and pulsating Green's function,” Ship Technology Res., Vol. 42, April 1995,
pp. 68–80.
Boin J.P., Ba M. and Guilbaud M., “Sea-keeping computations using the ship motion Green's function,” Proceedings of ISOPE2000 Conference, Vol.
IV, Seattle (USA), May 2000.
Bougis J., “Etude de la diffraction-radiation dans le cas d'un flotteur indéformable animé d'une vitesse moyenne constante et sollicité par une houle
sinusoïdale de faible amplitude,” Thèse de doctorat, Université de Nantes (France), 1981.
Brument A. et Delhommeau, G., “Evaluation numérique de la fonction de Green de la tenue à la mer avec vitesse davance”, Proceedings of the 6th
Journées de l'Hydrodynamique, Nantes (France), 1997, pp. 147–160.
Brument A., “Evaluation numérique de la fonction de Green de la tenue à la mer”, Thèse de Doctorat, École Centrale de Nantes (France), 1998.
Brument A., Delhommeau G., Gaillard L. and Guilbaud M., “Comparison between numerical computations and experiments for seakeeping on
ship's models with forward speed”, Proc. of Euromech374, Poitiers (France), 1998, pp. 241–248.
Chen X.B. and Noblesse F., “Super Green functions”, Proceedings of the 22nd Symposium on Naval Hydrodynamics, Washington (USA), 1998, pp.
860–74.
Delhommeau G., Ferrant P. and Guilbaud M., “Calculation and measurement of forces on a high speed vehicle in forced pitch and heave,” Applied
Ocean Research, No. 14, 1992, pp. 119–126.
Guével P. and Bougis J., “Ship motions with forward speed in infinite depth,” Int. Ship. Progress, Vol. 29, 1982, pp. 103–117.
Guttman C., “Etude théorique et numérique du problème de Neumann-Kelvin tridimensionnel pour un corps totalement immergé,” Rapport de
recherche No. 177, 1983, ENSTA, Paris (France).
Guyot F., “Etude expérimentale de la résistance ajoutée d'une maquette de navire soumise à des oscillations harmoniques. Etude du champ de vagues
instationnaires associé,” Thèse de doctorat, Université de Poitiers, 1995.
Guyot F. and Guilbaud M., “Force and free surface elevation measurements on a series 60 CB=0.6 ship model in forced oscillations,” Proceedings of
the 5th ISOPE Conference, The Hague (Netherlands), Vol. IV, 1995, pp. 507–514.
Iwashita H. and Okhusu M., “Hydrodynamic Forces on a Ship Moving at Forward Speed in Waves”, J.S.N.A. Japan, Vol. 166, 1989, pp 87–109.
Iwashita H. and Okhusu M., “Green function method for ship motions at forward speed,” Ship Technology Research, Vol. 39, pp. 87–109, 1992.
Iwashita H., “Evaluation of the Added-Wave-Resistance Green Function Distributing on a Panel”, Mem. Fac. Eng. Hiroshima Univ., Vol. 11, N°2,
1992, pp. 21–39.
Ishawita H., Ito A., Okada T., Okhusu M., Takaki M. and Mizoguchi S., “Waves forces acting on a blunt ship with forward speed in oblique sea
(2),” T. Soc. Naval Arch. Japan, Vol. 173, 1993, pp. 195–208.
Lyness J.N., “Simpson quadrature used adaptively-noise killed: algorithm 379,” Comm. ACM, No. 13, 1970, pp. 260–63.
Malcom M.A. and Simpson R.B., “Local versus global strategies for adaptative quadrature,” ACM Transactions on Mathematical Software, Vol. 1,
No. 2, 1975, pp. 129–146.
Nontakaew U., Guilbaud M. and Ba M., “Experimental and numerical study of the wave radiation by a surface-piercing oscillating and translating
plate using Green's function,” Proc. Of the 2nd Symposium on Hydrodynamics, Hong Kong, Vol. I, 1996, pp. 171–176.
Nontakaew U., Ba M. and Guilbaud M., “Solving a radiation problem with forward speed using a lifting surface method with a Green's function,”
Aerospace Science and Technology, No. 8, 1997, pp. 533–43.
Okhusu M. and Wen G., “Radiation and diffraction waves of a ship at forward speed,” Proceedings of the 20th Symposium on Naval Hydrodynamics,
Trondheim (Norway), 1996, pp. 29–44.
Okhusu M., “Validation of Theoretical methods for ship motions by means of experiments,” Proceedings of the 21st Symposium on Naval
Hydrodynamics, Washington D.C. (USA), 1998, pp. 1–18.
Sclavounos P.D., “Computation of wave ship interaction,” in Advances in Marine Hydrodynamics, Okhusu M. editor, Computational Mechanics
Publication, U.K., 1996.
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Taylor Model Basin. Washington D.C.
the authoritative version for attribution.

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FREQUENCY DOMAIN NUMERICAL AND EXPERIMENTAL INVESTIGATION OF FORWARD SPEED RADIATION BY SHIPS 401
DISCUSSION
V.Bertram
Hamburg Ship Model Basin, Germany
The authors present detailed measurements to validate advanced seakeeping computations for a publicly available
hull form. Together with recent experiments by Prof. Iwashita in Japan, these are the first such benchmark data available
to the general public and I am sure that the world-wide community of researchers will be grateful for this.
Iwashita [1] showed in transversal wave cuts for diffraction waves near the bow for the Series 60 CB= 0.8 that even a
“fully 3-d” Rankine singularity method is not able to reproduce the local unsteady wave field in this region, although it
improved results over GFM computations. I wonder if the authors have tried similar transverse cuts for radiation waves
and found similar results.
Have results been compared with Iwashita's experiments for the same geometries to obtain a feeling for the accuracy
of these very difficult measurements?
Both Rankine singularity methods and Green function methods for ships at considerable forward speed are still not at
a stage where we can be satisfied with the results when we look at local effects like waves which indicate also to some
extent how local pressures are likely to differ. How will we overcome these shortcomings in the future? Will sophisticated
potential flow solvers, perhaps RSM in the near field coupled to GFM in the far field, in the frequency domain ever be
sufficient to get local pressures with sufficient accuracy?
1. Iwashita, H. “Prediction of Diffraction Waves of a Blunt Ship with Forward Speed Taking account of the Steady Nonlinear Wave Field”, 2nd
Numerical Towing Tank Symposium NuTTS'99, Rome, 1999.
AUTHOR'S REPLY
Unfortunately, we have not obtained the paper from Iwashita [1] mentioned by Professor Bertram in spite of asking
it from library, so no comparison has been yet made. Nevertheless, we are interested to make such comparison.
We have tried also to compare the transverse cuts computed and measured. The figure 1 show such cuts for the
Series 60 CB=0.6 hull in heave motion at F=0.2 and f=4Hz, far from irregular frequencies. The agreement seems to be
correct close to the stern but some discrepancies can be shown closer to the bow for low values of Y/L. Nevertheless, it is
quite difficult with our probe to make measurements very close of the model. Other comparisons are being done in
different cases and for different values of the flow parameters.
Concerning the last point evoked in the comments of Prof. Bertram, more comparisons with local measurements are
needed to validate the various codes, Rankine or Green, as wave pattern or pressure distribution. But, we think that
probably close to the hull these kinds of inviscid methods are not very accurate and that a coupling between viscous
method close to the body and inviscid one, able to represent precisely the far field may be a track to follow to improve the
numerical methods.
REFERENCE:
1. Iwashita, H. “Prediction of Diffraction Waves of a Blunt Ship with Forward Speed Taking account of the Steady Nonlinear Wave Field”, 2nd
Numerical Towing Tank Symposium NuTTS'99, Rome, 1999.
Figure 1
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