Proceedings of The 13-th (2003) Conference
Proceedings of The 13-th (2003) Conference
Proceedings of The 13-th (2003) Conference
571
of modeling is satisfactory in many practical cases. calculated through direct integration of the dynamic and hydrostatic
pressure over the instantaneously wetted surface Sw(t) of the body,
Second-order effects, like drift forces and related motions, are defined by the undisturbed incoming wave and the instant position of
encountered by freely floating bodies, when in a seaway. Also other the body.
nonlinear phenomena, like the effect of flare of the body shape near the FFK (t ) = ∫∫ pn ds (2)
calm waterline, the body motion and wave run-up in extremely large Sw (t )
amplitude waves are few of nonlinear phenomena of special interest to where the pressure p comprises hydrostatic and dynamic terms,
designers and operators. according to the next formula.
∂Φ 1
In this paper a comprehensive review of the developed time domain p = − ρgz − ρ − ρ ∇Φ 2 (3)
simulation method by Spanos (2002) is provided, and its validation by ∂t 2
comparison of relevant numerical results for two standard type Radiation forces are herein calculated by use of the added mass and
cylinders with corresponding ones derived by exact, small amplitude damping coefficients calculated in frequency domain and properly
second-order potential theories in the frequency domain and model transformed into the time domain applying the impulse response
experiments is presented [Molin (1983); Mavrakos, Bardis and function concept by Cummins (1962).
Balaskas (1987); Zaraphonitis (1990)]. ∞
FR ,i (t ) = − Aij (∞ )UD Gj − ∫ K ij (τ )U Gj (t − τ ) dτ , i, j=1,…, 6 (4)
THE TIME DOMAIN SOLUTION 0
where Aij are the added masses at infinite frequency of oscillation and
A mathematical model accounting for the motion of floating bodies the kernel functions Kij are calculated by the cosine Fourier transform
under the excitation of sea waves and the influence of other external of the forced motion damping coefficients calculated in the frequency
forces or induced dynamic phenomena, like mooring forces or even the domain.
effect of accumulating water on deck or other body spaces, has been
developed at the Ship Design Laboratory of NTUA, and it is For irregular seaway excitation, elementary diffraction forces
implemented in the computer code CAPSIM. It provides an efficient corresponding to the elementary wave frequencies of an assumed
way to investigate the motions of ships or floating structures in general irregular seaway spectrum are taken to be directly proportional to the
coupled with other dynamic phenomena. The model allows the corresponding elementary diffraction forces calculated in the frequency
consideration of large amplitude motions and establishes a motion domain. Finally, nonlinear viscous effects are taken into account for the
simulation base upon which the dynamic performance of floating roll motions using a semi-empirical quadratic roll velocity model.
bodies can be analyzed.
Radiation and diffraction forces in the frequency domain were herein
The floating body is considered as a rigid one of arbitrary shape calculated by applying the computer code NEWDRIFT, Papanikolaou
moving in six degrees of freedom. The equations of motion of the body (1989). It is a six degrees of freedom, three-dimensional panel code
in the three dimensional space can be derived by application of the program for the calculation of motions and wave induced loads,
momentum conservation theorem. The relevant equations are of the including drift force effects acting on arbitrarily shaped bodies in
following general form: regular waves. The code is based on a zero-speed Green function
D pulsating source distribution method and employs triangular or
mU G + ω ∧ U G = F (1a) quadrilateral panels for the modeling of the wetted ship surface.
[I ]ωD + ω ∧ [I ]ω = M (1b) The solution of the above formulated set of differential equations of
motion (Equ. 1a and 1b) is accomplished by applying an integration
where m denotes the mass of body, [I] is the matrix of body’s moments method based on the extrapolation scheme, which permits high
of inertia and the vectors UG is the linear velocity of center of mass G, accuracy results at reduced computational effort, Spanos (2002).
ω the angular velocity, F and M the external forces and moments
respectively. Dot over vector denotes differentiation with respect to One particular problem of interest in the framework of the validation of
time. Both equations are expressed with respect to the body fixed the present time domain solution is the effect of higher-order wave-
coordinate system GXYZ that has its origin at the mass center G. induced effects, like second-order diffraction and drift forces, on the
body motion responses. Second-order diffraction forces are not
These equations supplemented by a set of equations relating the included in the presently used hydrodynamic modeling, whereas the
velocities to the time rate of change of relevant position vectors define quasi-second order drift forces, depending in the frequency domain on
the full set of the governing twelve ordinary nonlinear differential first-order quantities, are indirectly included in the present time domain
equations of body motions. When the external forces, acting on the formulation. Systematic comparisons of the resulting numerical drift
mass, are specified, then the motion of the body can be calculated by forces with experimentally measured drift forces acting on shiplike
integration of the set system of differential equations. bodies indicated that the formulated time domain solution captures
satisfactorily the dominant part of the drift forces effects, Spanos,
The external to the body acting forces and moments appearing on the Maron and Papanikolaou (2002).
right hand side of equations (1) express the entire set of forces acting
on the present inertia system and causing its motion. These forces In particular, the second order wave-induced forces acting on a floating
comprise gravity, hydrostatic and hydrodynamic components; others body can be derived by a direct integration method in the frequency-
might be added in a straightforward way (wind & current forces, etc). domain according to Zaraphonitis (1990) (see equ. (5)). This
Wave exciting forces, as the dominant dynamic part, are further expression results by further elaboration of the originally introduced
analyzed into Froude-Krylov or undisturbed incident wave, radiation formulation by Pinkster and Van Oortmerssen (1977).
and diffraction forces, following linear potential theory.
1 2 1 2
F ( 2) = ρgAw d x 4(1) + x5(1) i3 + D (1) F (1) + ρ ∫∫ ∇Φ (1) n ds +
2
Incident wave forces together with the hydrostatic ones are herein 2 2 Swo
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+
1
(
) 1
ρ ∫∫ x (1) ∇Φ t(1) n ds − ρg ∫ ζ R(1)
2 Swo 2 WL
2 n
cosθ
dl (5)
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In next Figure 6 the horizontal drift force for the cylinder is presented.
The zero frequency force component resulting in the spectral analysis
of the time series of horizontal force due to waves is the drift force
presented in the diagram. The other methods’ drift force results as the
mean values of the horizontal force over one period.
The general behavior of the drift forces over the entire frequency range,
Figure 4. Heave motion of cylinder (T/R=1.0) described above, fits to the experimental observations made during
model tests with ship models in regular beam seas. As shown in next
Figure 7 the mean drift velocity of a model of a passenger/RoRo ferry
(L = 5.333m, B = 1.014m) that has been recently tested in a model tank
[Spanos, Maron and Papanikolaou (2002)], in regular beam waves and
zero speed, is properly predicted by the simulation method for higher
values of kB, where B is the model’s breadth. For the lower values of
kB the measurements and numerical predictions by the simulation
method diverge, likewise the observed behavior for the horizontal drift
force on the cylinder in Figure 6. For the model test shown in Figure 7,
the horizontal drift force seems to be underestimated compared to the
experimental values, as indicated by the higher drift speed resulting
from the numerical simulations.
Experimental
Numerical
The overall first order results for the coupled motion of surge, heave Figure 7. Mean drift velocity of a passenger/RoRo ferry model in beam
and pitch of the floating cylinder in regular waves indicate the regular waves
successful implementation of the time domain solution of the CAPSIM
code.
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response.
It remains to be clarified why the present simulation tends to
overestimate the drift force effects at lower frequencies, where The cylindrical radius equals 4.0 m and its draft equals 7.90 m. The
diffraction effects are expected to anyway die out, hence any related center of gravity is located 3.60 m above keel and the radius of gyration
simplifying assumptions in the present theory would be more valid than equals 4.0 m. This cylinder has been also studied by Molin-Marion
at higher frequencies. (1985) and Zaraphonitis (1990, 1993), and corresponding results are
reproduced with the simulation method and compared with those
For a better illustration of the time domain simulation responses, a published above.
sample of the surge motion of the cylinder for one wave frequency is
given in the next Figure 8. This is the surge motion time series for The cylinder’s geometry is modeled with a mesh similar to that
kR=1.00. depicted in Figure 1, consisting of 360 surface elements (240 on side
and 120 on bottom). The cylinder has not any restoring force in the
surge direction like the cylinder studied in the previous paragraph, and
is free to drag downstream under the action of the drift force and the
induced resistance because of the body’s steady downstream speed.
The cylinder coupled motion in surge, heave and pitch in regular waves
of constant steepness (H/λ)=1/100 has been calculated with the
simulation code CASPIM. The steady state response of the cylinder has
been analyzed in the frequency domain and the two components
corresponding to the wave frequency (ω) and that of the double
frequency (2ω) are depicted in Figures 10, 11 and 12, together with the
corresponding numerical results of the other methods. The NEWDRIFT
numerical results correspond to the second order theory calculations by
Zaraphonitis, and the MOLIN results to the relevant second order
calculations by Molin. In these diagrams the thick lines correspond to
the (ω) components whereas the thin lines to the (2ω) components.
In the next Figure 9 the spectral analysis of the above surge motion
results to a zero frequency component that corresponds to the mean
surge displacement and the motion corresponding to the wave
frequency of 3.13 rad/sec. Note that the motion spectrum has been
herein analyzed for the motion 150 sec after initialization of the
simulation and a small amount of transient motion energy is still
present at about 0.15 rad/sec.
Figure 10. Surge motion of cylinder (T/R=1.975)
A second cylinder having a draft to breadth ratio equal to 1.975 was Figure 11. Heave motion of cylinder (T/R=1.975)
additionally studied. The prediction of the second order responses of
the cylinder in regular waves is herein of interest and particularly the
double frequency (2ω) components included in the cylinder’s motion
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Figure 12. Pitch motion of cylinder (T/R=1.975) Figure 13. Asymptotic behavior of cylinder surge motion (T/R=1.975)
The comparison for the pitch motion of double frequency is better for For the examination of the non-linearity effect of wave height on the
the higher frequencies whereas for the lower frequencies correlation cylinder’s motion the responses were simulated for various wavelength
worsens. It is observed as substantial discrepancy mainly in the range to wave height ratios. Figure 15 presents the non-dimensional
of pitch resonance. This behavior is presumably related to the large responses for the three degrees of freedom and for both components of
amplitude pitch motion that the cylinder performs at the resonance. The (ω) and (2ω) for a wave excitation corresponding to kR=0.50. As
first peak met at the left side of the diagram corresponds to the observed the responses become practically independent of the wave
superharmonic resonance of pitch motion as the double frequency of height for (λ/H) ratios greater than 60. The corresponding wave height
this response coincides to the natural pitch period of the cylinder. A to cylinder’s draught ration (Η/Τ) below of which the linear regime is
divergence between the three methods as regards the (ω) response at recognized is about 10%. Based on the above, it is evident that the non-
lower frequencies is also observed, similar to the surge behavior. This linearity effect of the wave height on the motion responses is small,
result could be explained by the assumed absolute magnitude of the thus the above discussed results could be expressed as well defined
wave amplitude used in the simulation. For constant steepness waves non-dimensional Response Amplitude Operators (RAOs) with respect
studied herein large absolute wave amplitudes compared to the cylinder to wave amplitude. A possible nonlinear effect of wave amplitude on
dimensions result for the lower frequencies. When the wave amplitude the motion responses would obviously have required an alternative
used in the simulation method is reduced then the (ω) motion converges approach of presentation, as the RAOs concept is linear.
to that predicted by NEWDRIFT.
In the next two Figures 13 and 14 the asymptotic responses for surge
and pitch respectively at lower frequencies are shown. The square
symbols in these diagrams denote the response according to CAPSIM
code, and the lines correspond to the results of the previous Figures 10
and 10. The asymptotic surge motion converges to the asymptotic value
1.0, similar to the values obtained by finite wave steepness simulation,
whereas the asymptotic pitch motion converges to the NEWDRIFT
results.
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and Calculation of Motion and Load of Arbitrarily Shaped 3D
CONCLUSIONS Bodies in Waves", Journal of Marine Structures.
REFERENCES
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