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OCR for page 67
A Coupled Time and Frequency Approach
for Nonlinear Wave Radiation
P. Ferrant (Laboratoire d'Hydrodynamique Navale, France)
ABSTRACT
In this paper we report on computations using a semi
nonlinear time domain formulation for the threedimensional
wave radiation problem with a free surface. The body
boundary condition is applied at the actual timedepending
body surface, and the free surface conditions are linearized.
An initial value problem is solved for the potential on the
moving body, using a boundary elements method. The
method allows the hydrodynamic forces on bodies of arbitrary
geometry undergoing large amplitude forced motions in six
degrees of freedom to be determined, as well as the unsteady
wave field generated by the body motions.
Two different applications are presented. The first
one refers to the problem of a sphere undergoing large
amplitude periodic motions below the free surface. A
thorough parametrical study (amplitude  frequency) has
been completed in this first case, and the influence of the
body nonlinearity is clearly highlighted both on the
hydrodynamic forces and on the structure of the radiated
wave field, which is investigated using specific frequency
domain Green functions associated to the bodynonlinear
problem. Some results on a submerged spheroid starting
from rest with a constant velocity parallel to the free surface
are also presented, as an unsteady approach to the wave
resistance problem.
INTRODUCTION
Time domain modelization of freesurface
hydrodynamics is all but a new subject. Some time before the
publication of the commonly quoted paper by Finkelstein
(1957) on the timedomain Green function, Brard (1948) gave
the expression of the time depending freesurface potential
generated by a submerged source of arbitrary path and
strength. The formulation of the integral equations of the
linearized wavebody interaction problem in the time domain
has been presented by many authors, including Stoker (1957)
and Wehausen (1967), the latter with a clear presentation of
the connection between frequency and time domain solutions
for fully linearized problems.
With the increasing power of computers, practical
numerical solutions of linearized time domain formulations
have become available, starting with 2D problems with for
example Adachi & Ohmatsu (1980) or Young (1982) with
boundary elements methods (BEM). Jami (1982) solved the
2D problem using a mixed formulation associating finite
elements and integral representation, and formulated the
solution of the 3D problem, giving practical expressions for
(1) now with SIREHNA SA, 2 quad de la Joneliere. 44300 NANTESFRANCE.
67
the computation of the 3D time domain Green function.
Newman (1985a) solved the problem of the impulsive
heaving motion of a floating cylinder, using time depending
ring sources. Some time later, the 3D problem was solved
numerically, both for forced and free motions of a floating
sphere by Jami & Pot ( 1985), whereas Liapis ( 1985)
presented the first results of a timedomain BEM
formulation for the radiation problem with forward speed.
Specific algorithms for the computation of the 3D time
domain Green function were given. Newman (1985b)
presented his own algorithms for the Green function
computation. The solution of the forward speed problem was
extended to diffraction by King & al (1988).
The accurate numerical schemes for the computation
of the timedomain Green function, developed at that time,
were a prerequisite for the reliable solution of the time
domain wave body interaction problem, and besides academic
applications, computations on realistic bodies such as Tlp's
became possible and demonstrated the interest of time
domain methods for industrial applications, compared with
more conventional frequency domain methods (Korsmeyer &
al 1988). Nevertheless, the Cpu requirements of the model,
mainly due to Green function evaluations, restricted the use
of the corresponding codes to vector computers, with Cpu
times much higher than for linear frequency domain
analysis, on equivalent cases. A considerable speedup was
obtained by Ferrant ( 1988a), using a tabulation
interpolation procedure for the evaluation of the timedomain
Green function in infinite depth. The power of the method
was demonstrated on the linear time domain analysis of the
ISSC Tlp, discretized by 1200 panels. The results were
obtained on a scalar computer (Vax8700) with moderate Cpu
times.
Besides its interest for linear timedomain analysis,
the tabulationinterpolation procedure puts Cpu
requirements at a sufficiently low level to allow the socalled
bodynonlinear problem to be solved in the time domain. In
such a formulation, a linearized free surface condition is
maintained, but the body boundary condition is applied at
the exact timedepending body surface. This leads to an
integral formulation very similar to the fully linearized one,
except for a line integral appearing in certain cases for
surfacepiercing bodies. The additional difficulty mainly lies
in the numerical implementation, for the Green function
terms in the convolution integrals have to be entirely
recomputed at each time step, due to the changing position of
the body surface on which the integral equations are solved.
This results in O(Nt2) Cpu times, where Nt is the number of
time steps, to be compared with O(Nt) when the body
boundary condition is linearized.
OCR for page 67
A first experience on the solution of the body and:
nonlinear problem in the frequency domain was reported in
Clement & Ferrant (1987), where partial results were given
on a submerged sphere with forced heaving motion. Although
results were successfully compared with the experiments of
Dassonville (1987), the formulation was very heavy and the
resulting code was not considered to be fully reliable, due to
problems of convergence of the influence coefficients at large
amplitude. Furthermore, the extension of the formulation to
arbitrary motions was not possible. The only alternative was
to solve the problem in the timedomain, but intensive
computations with the basic code based on conventional
schemes for the Green function were practically impossible,
with about 12 Cpu hours on a Vax 8700 to reach steadystate
with sufficiently fine time and space discretizations. This was
in fact our main motivation for the development of the
tabulationinterpolation procedure for the evaluation of the
timedomain Green function. With simple arrangements in
the convolution computations leading to O(Nt) Cpu times, the
resulting code is considerably faster, and a typical run on the
heaving sphere as presented in this paper requires now
about 10 minutes.
Intensive runs of the program being possible, a
complete parametrical study (amplitudefrequency) in the
case of a submerged heaving sphere has been undercome, the
results of which are presented in this paper. Timedepending
forces and wave elevation, as well as the results of the
harmonic analysis of steadystate are given for various
values of the amplitude, over the whole significant frequency
range. A method for the fast analysis of the steadystate
radiated wave field, based on frequency domain Green
functions for the bodynonlinear problem is also presented.
A few additional results are given on the time domain
approach of the wave resistance problem, in order to
demonstrate the versatility of the timedomain body
nonlinear formulation, which is basically able to cope with
any freesurface linear problem.
TIME DOMAIN FORMULATION
Basic Assumptions
The fluid domain D(t) is bounded by a free surface
Silt), the body surface Sb(t), and is unbounded in horizontal
directions. The fluid depth is infinite. A fixed coordinate
system is chosen so that the Taxis points upwards, and the
origin lies in the mean free surface. An ideal fluid is
assumed, with irrotational flow, so that the fluid velocity
derives from a potential satisfying Laplace's equation:
^~)( x,y, z,t) = 0 in D(t)
(1)
U (x, y,z, t) = V4)(x,y,z, t) in D(t) (2)
The body boundary condition is applied at the actual
time depending body surface, while the perturbation at the
free surface is assumed to remain sufficiently small for a
linearized condition to be valid, so that:
an
 =V.n onSb(t)
an
(3)
68
at an
+g =0 onSf(z=O)
at az
with:
infinity:
(4)
n
V
unit normal on Sb(t) pointing out of the fluid
domain D(t)
local velocity of the body surface
Additionally, the fluid velocity must vanish at spatial
VO(x,y,z,~)  0 for (x2+y2) moo or zoo (5)
and the fluid is supposed to be initially at rest:
A, Ot =0 fort=0
Integral Equation
(6)
The fluid problem being now completely defined,
various integrodifferential representations of the solution
may be derived, either for the potential on the moving body
(distribution of sources and normal dipoles), or for the source
density on the body (sources only), using Green's theorem
and the threedimensional time domain Green function
G(M,P,t) (see Appendix 1). In the first case, we obtain after
some transformations the following integral equation to be
solved for the velocity potential O(M,t) on the moving body
surface:
fr
Q(M)~(M,t)   O(P,t)aa Go(M,P)dSp
Sb(t)
=  Ji Go(M,P)O(P) dSp
Sit) P
+: dl :: [me l )~F(P(I),M(t),tl)F(P, M,tl)~(P,7 )] dSp
O Sal)
+ J do  [~(P,7)F(P(I),M(t),tl)F(P,M,tl)h(P,~)] (n Adl).V c(P,I)
(7)
a
onp
u Cal)
where Cb(t) is the the closed line defined by the intersection
between the instantaneous body surface Sb(t) and the XY
plane, Vc is the velocity of a point on Cb and Q(M) is the
solid angle under which the fluid domain D(t) is seen from M.
Go and F (see Appendix 1) are defined by:
(8)
G(M,P,t) = Go(M,P) . S(t) + H(t) . F(M,P,t)
The actual occurence of the line integral in (7) is
governed by Vc. For example, this term is zero for the
linearized problem without forward speed (see e.g. Ferrant
OCR for page 67
1988), where Cb is time invariant. In the applications
presented in this paper, the body is fully submerged, and the
line integral obviously vanishes.
Hydrodynamic Forces.
The unsteady hydrodynamic pressure (without
hydrostatics) in the fluid domain is given by:
p(M,t) 3~(M,t) I (My) ~ (9)
It is more convenient for the present study to
introduce the total derivative of the potential on the body,
yielding for M on Sb(t):
P(=  ~[~(M, t)]  ~ ~ VO(M,t) ~ + V. VO (10)
The first term in the right hand side of (10) is directly
obtained by finitedifferencing in time the potential on the
body. The two other terms require the computation of the
fluid velocity on the body. The method used for this obtained from:
computation depends on the space discretization scheme and
will be discussed when describing the numerics.
Force computations follows by simply integrating (10)
over the discretized body surface.
Free Surface Elevation
According to the linearized condition (4), the free
surface elevation is given by linearized Bernoulli's equation:
1 am(~t)
(it)= g at (11)
~ may be computed by finite differencing in time the
velocity potential at the free surface, or directly from an
integral representation of Ot which for a submerged body (no
line integral) and M on the free surface shrinks to:
d
~(~ t) =
at
J= Adam
o she T)
[F,(P, M,t~)~P(P,~)  O(P ~)an F. (P. TV)] dSp
(12)
STEADYSTATE COMPONENTS FOR PERIODIC FORCED
MOTIONS
In the bodynonlin ear problem of the forced
oscillations of a submerged body about a fixed mean position,
the influence of the body boundary condition nonlinearity on
forces and freesurface elevation is investigated.
First, the time depending forces on the body are
straightforwardly computed frorr' (10), after solution of the
transient integral problem. These forces tend rapidly to a
periodic steadystate which is Fourieranalyzed for a
quantification of nonlinearities.
On the contrary, the transient wave field, if computed
at some distance from the body to eliminate near field
components, needs a long simulation to reach the periodic
steadystate, mainly because of the low group velocity of the
higher harmonics. This point offers an opportunity to use
special Green functions already developped for the solution of
the bodynonlinear problem in the frequency domain
(Clement & Ferrant 1985, 1987). These Green functions are
shortly described in appendix 2. Thus, for a direct and
economical computation of the steady state wave field, we
first extract the harmonic components of the singularity
distribution on the moving body, obtained from the time
domain solution after a few cycles of motion. That is, for
example in the case of sources only and periodic forced
motion:
n2 o an(M) c06 n ~ + an ( M) sin n cat ( 13)
for M on Sb(~) and t  so
The steadystate potential in the fluid is then
~s(P t) = ~ Ii band G~n(M P t) + on (wG2n(~P t) ~ dsM
sb
(14)
and the corresponding freesurface elevation, for P on the
freesurface:
(15)
rls(P, I) = g ~ J: [c~n(M) ~Gln(M P t) + an (M) iG2,,(M,P,t) ] dSM
sb
For harmonic heave motion, the expressions of Gin
are simplified, and for fixed points P on the freesurface, the
harmonic components of Us and Us are directly calculated by
eliminating the time variable from the expressions of Gin
(A2.10), (A2.11).
When only the far field is to be computed, which is
sufficient to study the structure of the radiated wave field
(amplitudes of harmonics, dispersion of energy on the
components of A), asymptotic expressions of Gin are used.
Again, the computation is drastically simplified in the
present case of the heave motion, Gin reducing to the very
simple expressions (A2. 12), (A2. 13). No numerical
integration, but only computations of modified Bessel
functions are involved.
NUMERICAL IMPLEMENTATION
The major part of the numerical results given in this
paper have been obtained using a first version of the
computer code completed in 1988. In this version, a very
classical discretization scheme is used. The body surface is
discretized into plane polygonal panels over which
69
OCR for page 67
singularity distributions are assumed to be constant. The
impulsive part of the Green function (Go) is integrated
analytically over the panels, and the memory part F is
integrated using a variable order Gaussian quadrature.
Numerical tests have proved that, at least for submerged
bodies, one single point of integration per panel is sufficient
for a good accuracy, the local flow being dominated by the
singular part of Go. Thus, wave terms are treated as
monopoles situated at panel centroids, which greatly reduces
computational requirements. Note that such a mixed
procedure for the space integration of a Green function as
also been proved to be a valuable compromise for the solution
of the steady wave resistance problem (Doctors & Beck,
1987). The time variable is discretized into constant time
steps and the convolutions integrals are evaluated using a
trapezoidal rule.
At each time step the convolution terms at the right
hand side of the linear system of equations are actualized by
computing and assembling the corresponding wave terms,
and a new kernel is obtained by computing the motion
dependent part of Go. The linear system is then solved using
a standard Gauss solver. Faster solvers are obviously
available, but in fact the computing time is dominated by the
evaluation of the convolution terms.
Actually, two integral equations are solved at each
time step. First a mixed distribution of sources and dipoles is
used for a direct computation of the potential on the body.
Then the integral equation for sources only is solved, and the
result is used for the computation of the fluid velocity at
panels centroids. Although increasing the computer time, the
method allows the fluid velocities, and thus the full
hydrodynamic pressure to be obtained without having to
calculate the second spatial derivatives of the Green function.
At the end of the simulation, the different terms in
the hydrodynamic pressure are computed at panels centroids.
Ot is obtained by finite differencing the time depending
potential, and the quadratic terms are computed from the
previously computed fluid velocity. Forces are then obtained
by integrating the pressure which is assumed to be constant
on each panel. The time depending wave elevation at
prescribed points is also available. For simplicity, this
computation is based on the source solution.
In the case of a periodic forced motion, the solution of
the timedomain problem is followed by the computation of
the harmonic components of the steady state part of the
response. For reasons initially related to an economical
computation of the convolution terms, the time step is
adjusted to obtain an integer number of steps per period, and
the harmonics can be accurately computed by simple
trapezoidal rule over one single period, typically the last, in
order to provide the best approximation of the steady state.
The harmonic analysis is applied first to the time depending
forces, and then to the source solution. The latter results are
used for the computation of the steadystate periodic wave
field using the Gip functions described in appendix. In
consistency with the method used for the solution of the
timedepending problem, this computation is based on a
monopole approximation of the source distribution on the
body. In the case of the heave motion, the computation of the
harmonic components of the far field wave system is
straightforward and of negligible CPU cost.
At last, for a quantification of the influence of the
body boundary condition nonlinearity, the linear solution is
systematically computed for comparison.
REDUCTION OF COMPUTING TIME
Apart from the use of frequency domain Green
functions, the numerical implementation as described in the
preceding paragraph is very classical. In fact, the main
difficulty is related to the extensive Cpu and mass storage
requirements of the bodynonlinear timedomain
formulation. These requirements may be lowered first by
accelerating the Green function computations, and secundly
by reducing the number of Green function computations
necessary to the evaluation of the convolutions. These two
points have been addressed in the present study.
TabulationInterpolation of the Green Function
The wellknown time domain Green function for a
source of impusive strength in infinite depth is given in
Appendix, with the following notation:
G(M, Pi,) = 6(~) . Go(~P) + Ho) · F(~ Pit) (16)
The memory part F of that Green function can be
easily put under the following form:
r°°
gl/2r. 3/2 1 It2 21/2 _>, 1/2
F(M,P,t) =  2~ J sin(x A) Jo[~(l~~ ) ] e ~ do
o
or:
F(M P. t) = gl/2r 3t2~(~' 0
with ~ = (z+z')/r' and ,B = t(g/r')V2
(17)
(18)
The first parameter is linked to the relative positions
of points M and P. and varies from 0 to 1., whereas ~ is an
essentially positive time parameter.
Thus, the only nontrivial terms to be evaluated
during the computation of the convolution integrals are
reduced to the bivariate function F and its first derivatives.
This fact can be exploited for deriving a very fast
procedure for the evaluation of the wave terms. This
procedure, based on very simple principles, has been already
been described in Ferrant (1988). The 2D domain described
by ,u and ~ is truncated at a large value Oman, and the
remaining bounded domain is mapped by a discrete set of
equispaced points for which F and its first derivatives are
computed by numerical schemes very similar to the ones
described by Newman( 1985) or King & al ( 1988). This
computations are performed once for all, and the results are
stored on permanent disk files. When a simulation has to be
performed, the resulting evaluations of the memory part of
the Green function are based on linear bivariate
interpolations of the stored data. Note that the content of the
file is read once for all at the beginning of the simulation, so
that no disk access is necessary during the timestepping
procedure. The tabulated part of the (,u,0) domain is
sufficiently extended to allow the use of simple largetime
asymptotic expressions when ,B>pmax. In a very thin layer
near 11=0 where the function presents large oscillations the
~ a, ,
precision of the interpolation may be unsuff;cient and we
simply return to the original numerical schemes. However,
70
OCR for page 67
for the forced motions of a submerged body about a fixed
mean position, computations never occur in this part of the
domain and the interpolation is used throughout the
simulation. The use of a regular tabulation grid allows for a
very quick search in the tables, and the resulting code is very
fast: less than 30 As are necessary for one evaluation of the
Green function and its first space derivatives, on a Vax 8700
computer, from and back to physical variables. This is about
the time needed for one evaluation of the sine function. The
grid is composed of 200x1200 points in the rectangular
domain defined by O < ~ < 1. and 0 < ~ < 30. For the Green
function and its gradient, 3 tables have to be stored, resulting
in about 720Kwords or 2.9 Mbytes on a 32 bit computer. Note
that we do not pretend to have an optimized set of tabulation
parameters. Such an optimization could be necessary to
reduce memory requirements for a given accuracy, but the
present size of the tabulation files is not a problem on the
computer we use.
When the problem to be treated leads to large
horizontal displacements, as it happens with forward speed,
a substantial number of computations may occur near ~ = 0,
and a special scheme is needed to maintain both precision
and low Cpu in this portion of the computational domain.
Such a refined procedure has been subsequently
derived by Magee & Beck (1989), exploiting an analytical
approximation of the Green function in the vicinity of the p=0
axis, and a higher order interpolation scheme for the
remaining part of the Green function on various subdomains.
Although more precise for a given number of tabulated
points, their method is certainly a bit less efficient in terms
of computing time, but comparisons are delicate between
runs on scalar and vector computers. The use of an analytical
approximation is a very clever idea for reducing the
oscillations of the function near ~ = 0, and we intend to adopt
a similar scheme, but in our opinion, bilinear interpolation
should be preferred for the interpolation of the remaining
smooth function, whenever a very high precision is not
necessary. We are not sure that the precision of 108 obtained
by Magee & Beck is necessary when the overall accuracy of
the computations is dominated by space & time discretization
errors. A precision of about 105, easily obtained by a simple
bilinear interpolation method in the major part of the
computational domain is certainly sufficient for most
applications.
Reduction of the Number of Green Function Evaluations
Basically, during one run of the program, Np2(It1)
computations of the Green function are required at time step
It for the evaluation of the convolution. This results in a total
of Np2.Nt(Nt1) computations for a simulation over Nt time
steps, if Np is the number of panels. A crude method for
reducing this number is to truncate the convolution integrals,
considering that the influence of the past history of the
solution tends to zero for large time delays. However, such an
approach is not safe, for it is very difficult to estimate the
influence of the truncation on the final results.
In many particular problems of forced motions, an
adequate choice of the time step allows a substantial
reduction of the number of Green function evaluations
without affecting the numerical results. The idea is as
follows:
The Green function terms to be computed in the
convolution depend essentially on MP and tt, where M(t) is
a point on the body at time t (field point), P(T)IS the position
of a point on the body at time T (source point), and tT.iS the
time delay. The idea is to Just the time step At, i.e. the
positions of the body at times iAt to reduce the number of
combinations of MP and tT to be considered in the
computation of the discretized convolution.
For clarity, let us illustrate the procedure on the case
of forced motions of period T about a fixed mean position, for
which the appropriate choice is At = T/Nper. Consider the
discretized convolution to be computed at time step it, it 2
Nper. The Green function terms to be evaluated may be
schematically written G[MP(it,iT),(itiT)At], with IT < it. For
iT 2 Nper, we have obviously:
G [MP(it,i~),( it iT)] = G [MP( it Np~r ,i~Nper),(itNper)(*Nper)]
(19)
The term at the righthand side has already been
necessary for the computation of the convolution at time step
itNper, and thus has already been computed and stored.
Thus the only new terms to be computed correspond to the
influence at time step it of the Nper time steps of the first
period. The total number of NpxNp sets of terms to be
computed is now:
N,(N,1)
NCaI! = 2 for N. ~ NPer
N. (N. 1)
N. In= per 2 per +(NtNpe,)Npe,.
N. N. Nper(Nper~ 1) forN' 2 Nper
(20)
(21)
The resulting total number of Green function
evaluations is now a linear function of the number of time
steps, as opposed to the quadratic behaviour obtained when
At is arbitrary. The ratio of reduction, after a few periods of
simulation is approximately:
N. NT
G 2 N per ~
(22)
where NT is the number of simulated periods. For the
computations presented in this paper, 4 to 8 periods are
simulated, and the Green function computations are reduced
by a factor 2 to 4 using this artifice.
The procedure that we have illustrated in the case of
periodic motions is also applicable when a constant forward
speed is superimposed to the oscillations. In the case of a
smooth starting of duration NStart, the gain is lower, but a
linear law is maintained:
NCal~ = N.' . (Now+ N ta ) (Nper NStart)(Nper + Nstart+ 1) (23)
2
For problems with constant forward speed without
superimposed oscillations (unsteady approach of the wave
resistance problem), simulations with an abrupt start lead to
the same number of computation as for a linear code. If a
smooth start is chosen, Ncall is given by (23), with Nper = 0.
In this case, the introduction of a smooth start of duration
N Start multiplies the number of Green function evaluations
by the factor Nstart.
71
OCR for page 67
NUMERICAL RESULTS
Submerged Heaving Sphere
We give in this section the results obtained on the
problem of a submerged heaving sphere, with a mean depth
of submergence equal to the diameter (Zo/R = 2., Figure 1).
Starting from rest, the body is given a purely harmonic
motion, with frequency co.
~,~ Sc(t
Figure 1
Time Domain Results
Kl=~2R/g
Z = Zo + A cos cot
We first give in figures 2 to 9 a sample of the
unsteady (vertical) force signals computed form the time
domain formulation. The results of the fully linearized
formulation (dashed line) have been systematically
computed for comparison with the bodynonlinear
formulation. The difference of the two results is also plotted.
Results are given for two values of the amplitude,
A/R = 0.50 and A/R = 0.70, and four values of the
fundamental wavenumber K1 = ce2R/g=0.1, 0.25, 0.50, 2.0. A
periodic steadystate is very rapidly obtained, and a
significant influence of the body boundary condition
nonlinearity is observed, mainly for K1 = 0.25, the
difference between the two results being weaker at low and
high frequency. This difference mainly appear in the form of
higher harmonics in the response, the fundamental being
apparently not much modified. At high and low frequency,
the superharmonics tend to be in phase with the
fundamental. The analysis will be easier on the harmonic
components of the steady state response, given in a following
section.
Then, in figures 10 to 17, for the same two values of
the amplitude, and for K1 = 0.2, 0.4, 0.7, 1.0, we give the
transient wave elevation computed at a distance 20R from
the sphere. Again, the linearized solution in dashed line is
given for comparison. At low frequency, the influence of the
nonlinearity is much stronger than on forces, but the
difference between linear and bodynonlinear solutions tends
to zero at high frequency, at least for the smaller amplitude,
A/R = 0.50. For A/R = 0.7 and K1 = 1.0, a small second
harmonic component remains sensible. At low frequency, the
influence of the body boundary condition nonlinearity is
already very sensible for a moderate amplitude, as
demonstrated by figure 18 giving the wave elevations for
A/R = 0.30. For such an amplitude, a linear analysis is
commonly considered to give correct results. The present
results prove that at least for wave generation, a linear
analysis is inadequate, the body nonlinearity having a very
strong influence on the structure of the radiated wave field.
This phenomenon, already highlighted using a frequency
domain analysis of the problem by Clement & Ferrant
(1987), has been fully confirmed by the experiments of
Dassonville (1987). Comparisons were made over a wide
frequency range, but we simply give in figure 19 a
comparison between experiments and time domain body
nonlinear analysis for A/R = 0.50, and K1 = 0.28, i.e. in the
vicinity of the maximum nonlinear behaviour given by the
numerical model. Due to experimental constraints, the
distance from the sphere is here equal to 16.5R. A small
difference in the starting transients is observed, due to the
imposed smooth start of the experimental apparatus. During
the steadystate part of the response, the wave elevation is
slightly overestimated by the numerical model, but
harmonic content as well as phases are perfectly recovered.
At large time, the comparison is not significant, the
experiments being affected by tank wall reflexions.
Harmonic Components
Concerning the hydrodynamic forces exerted on the
moving body, the harmonic analysis is performed after
obtention of the steadystate, and the results are given
under the form:
00
F(t) = of, F cosn~t+ F. sonnet (24)
pgKlR2A n=0 n n
Full results may be found in table 1, where the
harmonic components of the vertical force on the spheres
from the constant term to the fifth harmonic, are given for
three values of the amplitude, over the whole frequency
range of interest, together with the purely harmonic
frequency domain linear results. A part of these results is
graphically represented in figures 20 to 27, for the two
largest amplitudes, and up to the fourth harmonic. The
results being scaled in the usual manner of frequency
domain analysis, the first harmonic components tend to the
usual linear added mass and damping coefficients when the
amplitude tends to zero. In figures 22 and 23, the linear
hydrodynamic coefficients are plotted together with the first
harmonic components of the bodynonlinear results. The
stronger relative difference lies in the damping coefficient,
especially between 0.2 and 0.9, in the ascending part of the
curve.
At low and high frequency, F** components tend to
zero. At high frequency, F* components tend to a constant.
This behaviour cannot be observed at low frequency, the
frequency range being insufficient. These observations agree
with the asymptotic analysis of the problem, which gives
predominant added mass effects at low and high frequency,
where the steadystate flow is in phase with the body
motion.
The steadystate farfield wave elevation harmonic
components are computed from equation (16), after an
harmonic analysis of the singularity distribution on the
body. This procedure avoids the computation of the
transient wave field, and is of negligible Cpu cost. The
results are given in the form of the distribution of the total
radiated energy over the different harmonics of the wave
field, in figures 28 and 29, for A/R = 0.50 and A/R = 0.70, as
a function of frequency. The very strong deviation from the
OCR for page 67
non
K1 = 0.25
KRIL3D+
 Iineaire
0.6  Differellce ,
0 10 20 30 40
Figure 3 T*sqrt(g/R)
A/R = 0 70 Zo/R = 2.00
A/R = 0.50 Zo /R = 2.00
o
Figure 6
0.4
0.2
0.0
0.2
0.4
fiO 60 70 0
Figure 7
~A/ R = 0.50 Zo/R = 2.0C
v." 
0.4
0 0
0.4
0.8
20 30 40 50 0
T*sqrt(g/R) Figure 8
1.0
0 10
Figure 4
AL 

co
*
lo:
*
~ o
o
\
_p .
K1 = 2.00
KR~3D+
 Idneaire
 Difference
4
0 5 1
Figure 5
on 40
60
T*sqrt(g/R)
A/R = 0.50 Zo/R = 2.00
. . .
10 20 30 40
T*sqrt(g/R)
10 20 30
T*sqrt(g/R)
A/R= 0.70 Zo/R= 2.00
* 1 
* 0 
o
~ 1 
2 
3
. .
10 15 20 25 0
T*sqrt(g/R) Figure
73
O6
70
40 50
A R = 0.50 Zo/R = 2.00
. .
5 10 15
T*sqrt(g/R)

20 25
OCR for page 67
*10 3
12 .
RSL = 20*R A/R = 0.50 Zo/R= 2.00
12 ~
~ n  .
K1 = 0.20
 sol. Iineeire
KRll:3D+
. l l l l l
0 20 40 60 80 100 120 140 160 180 200
Figure 10 T*SQRT(g/R)
RSL = 20*R A/R = 0.50 Zo/R = 2.00
3.0
*10 2
K1 = 0.70
  sol. linea~re
ER~D+
K1 = 0.40 1
  sol. linea~re 
.KRI~D+ 
0 20 40 60 80
Figure 11 T*SQRT(g/R)
RSL= 20*R A,/R = 0.50 Zo/R = 2.00
100 120 140
. .
0 20
Figure 12
$10 2
F.n RSL = 20*R A'R = 0.50 Zc,/R = 2nn
5.0
20 RSL= 20*R A/R = 0.70 Zo/R = 2.00
5
0
~: 5
E3 O
5
10
15  .
1 1 1 1 1 1 1
0 20 40 60 80 100 120 140 160 180 200
Figure 14 T*SQRT(g/R)
*10 2
RSL = 20*R A/R = 0.70 Z0/R = 2.00
. . . .
0 20 40 60 80
Figure 15 T*SQRT(g/R)
 6 RSL= 20*R A/R = 0.70 Zo/R = 2.00
5_
2
1
c,3 0~
1
2
3
_4
5 
40 60 80 100 120
T*SQRT(g/R)
. . .
100 120 140
K1 = 1.00  .
 sol. linesire
KRE.3D+
_ I I ~
0 20 40 BO
T*SQRT(g/R)
Figure 13
6 .
4
a, ~ ~ I
3
4
5
.~..
80 100 ° 20
Figure 17
7~1
1 1 1 · . .
~20 40 60 80 100 120
*lo_2 Figure 16 T*SQRT(g/R)
RSL= 20*R A/R = 0.70 Zo/R = 2.00
11:
11
K1 = 1.00 v I
 sol. 1iIleeire Y
KRIL3D+
40 60 80 100
T*SQRT(g/R)
120
OCR for page 67
*10 3
al
6
4
Ad; 2
EM
o
2
4
RSL = 20*R A/R = 0.30 Zo/R = 2.00
i,
0 40
Figure 18
80 120 160 200
T*SQRT(g/R)
linear analysis already observed on time domain results is
confirmed, with a maximum about co.sqrt(R/g) = 0.2 (K1 =
0.4) where for A/R = 0.50 only 60% of the total energy input
by the moving body is recovered on the first harmonic of the
wave field. This phenomenon appears approximately in the
same frequency range as the maximum deviation from
linear results of the damping coefficient F1**. For
confirmation, we give in figures 30 and 31 the damping
coefficient computed from the body nonlinear formulation,
together with linear results. The difference is also plotted, as
a function of frequency.
Results on the sphere have been obtained with a
discretization of 200 panels on the entire body.
Time Domain Wave Resistance of a Submerged Ellipsoid.
Results are now given on the problem of a
submerged ellipsoid of beam/length ratio B/L = 0.2, and
submergence Zo/L = 0.16. Two different discretizations are
used, with respectively 60 and 168 panels on the halfbody,
and the simulations were run over 150 time steps. Starting
from rest, the body is abruptly given a constant velocity in
the positive X direction, parallel to the free surface. We give
in figures 32 to 34 the transient horizontal force on the
body, for three values of the Froude number, Fr = 0.4, 0.45,
0.5. After some oscillations, the force tends to a constant,
which actually is the wave resistance of the body. We give in
table 2 the Cw wave resistance coefficient estimated from
the ultimate value of the transient force, compared with the
results of a conventional NeumannKelvin wave resistance
code (Delhommeau 1987), obtained with a 192 panels
discretization. The results of a semianalytical formulation,
deduced from curves given in Farell (1973), are also
presented. With the finest discretization, Npan=168, the
agreement between timedomain analysis and steadystate
results is correct. The influence of the timestep has not been
investigated. A smaller time step would certainly lead to a
better agreement, especially for larger Froude numbers.
Fr Cw (time domain)
(Np=60) (Np=168)
0.01226 0.01267 0.01175
0.45 0.01592 0.01667 0.01734
0.50 0.01619 0.01671 0.01775
Table 2
Cw (NK) Cw (Farell)
(Np=192)
0.01320
0.01713
0.01822
Note that the oscillations are greater for smaller
Froude numbers. At Fr = 0.8 (figure 35), only a small
*10 ~3
RSL=16.5*R A/R = 0.50 z0m = 2.00
0 10 TO 30 40 50 60
T*sqrt(g/R)
Figure 19
70 80 90 100
overshoot is observed during the transients, a constant
resistance being very rapidly obtained.
Similar results on the transient approach of the wave
resistance problem have already been published by Jami &
Gelebart (i987), and more recently by Magee & Beck (1990),
with comments on the oscillatory behaviour of the time
depending wave resistance and its connection with the
critical parameter ~ = 0.25 of the forward speed seakeeping
problem. Present computations simply intend to demonstrate
the versatility of the bodynonlinear time domain
formulation.
CONCLUSION
A systematic attention on computational efficiency
has led to a time domain bodynonlinear code with Cpu
requirements sufficiently low for intensive computations to
become possible, as demonstrated on the problem of the
heaving sphere. Strong nonlinear effects, both on forces and
wave field, already detected using a frequency domain
approach (Clement & Ferrant 1987) have been confirmed and
their validity extended to the whole frequency range and to
larger amplitudes. Furthermore, the time domain
formulation can cope with arbitrary motions and is
absolutely robust. Future computations will be undercome
with a refined version of the code based on a linear
representation of the velocity potential on the body.
Other researchers are also working on the time
domain bodynonlinear problem, and for example a
considerable experience has been gained on the time domain
seakeeping problem with forward speed by Professor Beck's
team. The variety of possible applications will certainly
motivate many interesting studies in the future. The
extension to finite depth depends on the development of fast
algorithms for the computation of the corresponding Green
function, on which preliminary results have been obtained by
Newman (1990).
75
OCR for page 67
Harmonic Components
K1 0.100 1
0~= 0.316
Fo
I A=O. 00 1
K1 0.025 1 AS0.30 1 0.0104
on. 0. 15B I AO.5O 1 0.0194
1 AO.70 1 0.0324
1 A0. 00 1
K1 O.OSO 1 A0.30 1 0.012S
0 ~0.224 1 A~0.50 1 0.0237
1 A0.70 1 0.0411
I A~0.00 1
1 A.0.30 1 0.01Si7
1 A~O .50 1 0.0305
1 A=O.70 1 O .0554
I AsO. 00
K1= 0.2001 A~0.30
on. 0.4471 AeO. 50
1 A=O.70
I A O . 00
AO. 30
A=O. SO
1 A=O. 70
I AsO. 00
AO. 30
A=O. SO
1 A=O. 70
I A~0.00
1 A=O.30
1 A. O . 50
1 A=O. 70
I AO.00
K1 0.600 1 A0.30
0~= 0.775 1 Ar O. SO
I A0.7a
I A=O.OG
1 Aa0.30
I AO. 50
1 A=0.70
I A=O. oa
AO.30
A=O. SO
1 AO.70
I A=O.00
1 AO. 30
I A=O. SO
1 A0.70
I AO. 00
1 A=O.30
I A ~ O . SO
1 A0.70
I A ~ O . 00
1 AsO. 30
I A=O. SO
1 AO. 70
I A=O. 00 1
1 Aa0. 30 10.0312
1 A=O. 50 10.0535
1 A=O . 70 1 O . 077S
I A=O. 00 1
1 Au0.30 10.0327
1 AeO.SO lO.OS70
1 A=0.70 10.0843
I AO. 00 1
K 1 = 2. 000 1 A'0. 30 10 .0323
on ~1. 414 1 As O. SO 10. OSb8
1 A0.70 10.08S6
I A=O. 00 1
K 1 2. SOO 1 AO.30 10 .0279
0~= 1.581 1 AO.SO 10.0500
1 A*O . 70 1 O .0778
I AO. 00 1
K1. 3.000 1 A=0.30 10.023b
0~. 1.732 1 AO.SO 10.042b
1 AO. 70 10.0674
K1 0.300 1
0~2 O. 548 1
K1. 0.400 1
0 ~O. 632 1
Kl= 0.500 1
0~ O.707 1
Kl  0.700
0~ O. 837
K1 0.800 1
0~= 0.894 1
K1 1.000 1
0~= 1.000 1
K1 ~1.200 1
OM. 1.095 1
K1 1.400 1
0~= 1.183 1
K1 l.bOO I
0~= 1.26S 1
K1C 1.800 1
0~ 1 . 34 2 1
1 0.0223
1 0.0433
1 0.0770
1 0.0281
1 0.0512
1 0.0829
1 0.0310
1 O.0530
1 0.0781
1 0.0305
I O . OSOO
1 0.0686
1 O . 026q
1 O. 043S
1 O .0576
1 0.0208
1 0.0337
1 0.0444
1 0.0131
1 0.0215
1 0.0287
I O.0034
1 O "0 0 2
1 O .0066
1 0.0172
10 . 02as
1 0.0391
1 O .0264
1 O . 044S
IO . Ob30
F1.F1#.
2.12900.0003
2. 13SS0.0003
2.14890.0003
2.17390.0004
2.1478
2. 1S68
2. 17S2
2.21 1S
2.1807
2.1941
2.2222
2.2782
2.2319
2. 246S
2.2757
2.3286
2.2533 O. 1 624
2. 2S87 O. 1 830
2.2696 0.2234
2.2888 O. 29~0
2.2287 O. 2570
2.2263 0.27 44
2.22 1 9 O.3084
2.2147 0.367S
2. ~ S67 O.3296
2.1520 0.3411
2. 1 422 0.3640
2.1240 0.4046
2.0784 O.3730
2.0750 O. 3805
2.0666 O. 39S7
2.0486 O. 4227
1. 99?6 O. 38b5
1. 9959 0.3925
~ .9905 O.404 1
1.9769 0.4241
1.9247 O.3763
1.9238 0.3821
1.9204 O.3930
1.9106 0.4107
1.8190 0.3148
1.8168 0.3214
1.8121 0.3333
1.8033 O.3~20
1. 76S9 O. 2361
1.7604 0.2429
1. 750S 0. 2S53
1.7356 0.2749
1.7503 O. 1 648
1.741S 0.1711
1.7257 O. 1326
1.7023 0.20 12
O. 1 092
O. ll4S
0.1245
O. 1 4 07
O.0694
O.0737
0.0818
O. O9S 1
1.7563
1.7449
1.7244
1.6933
1.7723
1. 7S94
1.7357
1. 69eq
1.7914 0.0427
1.7777 0.0460
1.7524 0.0522
1.7121 0.0626
1.8333
1.8202
1.7950
1.7533
1.8603
1.8487
1.8261
1. 787S
F2t
O .0364 0 .0005
O . Ob68 0 . OOO'
0.1093 0.001S
0.0020
0.0022 ' O .0524
O .0026 0 .097 3
0.0039 O. 1631
0.0130
0.0156
0.0217
O.0364
o .07 1 1
0.0857
O . 1 1 S?
O. 1747
0.0110
O . 0 1 2S O .0508
O . 0 1 53 O .09 1 B
0.0201 0.1443
0.002 1
O. 0026
O.0037
0.0056
O .074 9 0 .037 2
O. 1383 0.0717
O .228 1 0 . 1 292
O .0354 0 . 1 062
O .0606 0 . 1 94 8
0.0853 0.3186
O . 02b 7 0 . 102 ~
0.0490 O. 1857
O .082 1 0 .2939
O.OS12 0.0701
O .0927 0 . 1 287
O . 1 477 0 .2060
O .0489 0.0444
0.0906 0.0823
0.1471 0.1334
O .0397 0 .0350
O . 07S3 0 . Ob33
O. 12S7 O. 1002
O .0332 0 .034 7
O .063 1 0.0604
O. 1062 0.0912
O .03 1 4 0.0376
O . OS83 0 .0642
O . O9S8 0. 093B
O .0370 0.0420
O .0650 0 .07 1 7
0.0991 0.1040
O .0460 0.0403
O . 079S O .0697
0.1172 0.1033
O .053 1 0 .034 2
0.0918 O.ObO2
O. 13S2 0.0917
O .0568 0.0266
O .0990 0 .0477
0.1475 0.0748
0.0576 0.0194
O. IO1S 0.035S
O . 1 S33 0 . OS73
O.OS65 0.0134
O . 1 OOS O .0252
0.1539 0.0419
O .0454 0 .00 1 1
O .0826 0. 0026
O. 1316 0.00SS
Table 1
F2. ~f3.
0.0052 0.0129
0.0100 0.0399
O .0 1 82 0.0936
O .004 S O.0075
O .009 1 0.0223
O .0 1 65 O .04 8 1
76
F3~. F45
O .008 1O .0004
0.02490.0013
O.0575O.003 1
0.01220.01S8
O . 03bB0.0497
0.07880.1209
0.00290.0137
0.01070.056S
0.0325O. 1254
O . 00350.0 1 53
O. 026b0.0447
O.0624O.0933
0.01230.01 1S
0.0363O.0336
O .07 aoo . 069q
0.01270.0062
0.03730.0188
0.07880.041D
O . 0 1 1 50. 0030
O.0343O. OO9S
O.07 370.0224
0.00960.0014
0.0291O.0045
0.06390.0114
O .00 800.0009
0.02420.0029
O. OS39O. Q070
O.0062O.00 17
O .0 1 85O .0046
O. 0408O.0090
O. OObO0.0026
O . 0 1 7S0.0072
O .0370O .0 1 45
0.0065O. 0029
O .0 1 870. 008S
0.03BS0.0179
O . 007 10.0028
O.02040.0084
0.04190.0183
O. 007S0.0024
O . 02 1 8O.0073
0.04500.0165
O.0077O. 00 19
O.02250.0058
0.04710.0137
O.0070O.000 1
0.02 1 1O. ooOB
O. 0462O.0026
o .00370.002 1
0.01180.0109
O.0306O. O3S3
O.00070.0008
O. 002bO .0041
O .0068O .0 1 2 1
FSt F5~'
0.00140.0002
O . 007SO .00 1 2
O . 024BO .0044
0.0013
O .0073
O .0270
O .00080 .003 ~
0.00330.0164
O .00520.0549
0.00160.0021
0.00840.0102
O .02930.0286
0.001S0.0010
0.00790.0048
0.02470.0118
O .00 1 40 .0009
0.00680.0040
O.Ol960.0101
0.00130.0005
O. OObOO .0027
0.01660.0076
0.00130.0003
0.00630.0018
0.01770.006t
0.00120.0001
0.00610.0009
0.01770.0038
O .00 1 10 .0000
O . O OSSO .0002
0.01670.0017
O .0009 O .000 1
0.0043 0.0002
O .0 1 35 O .000 1
O .0007O .0000
O .00360.000 1
0.01100.0004
O .0007O .000 1
0.00330.0006
o.ooq?0.0017
O .00070.0002
O .00340.0009
O.00990.0027
O .00080.0002
0.00360.0010
0.01040.0032
O .0 0080 .0002
O .0038O .0009
0.01 1 10.0031
O.0000
O .0005
O .0020
O .0008O .000 1
O .0040O .000 1
0.01210.0008
O .0002O .000 1
0.0019Q.0007
O.0090O.0040
0.0002O. 0003
0.00190.00~8
O .0074O.0 1 49
O .000 10. 0004
O.00 1 10.0036
. ooe40. 01 5b
0.00020.0001
O. 0021O. 0007
O. OO9b0.0010
0.0002O.0000
0.00140.0000
O .0054O . 00 1 2
O .000 1O.0000
0.0008O.0000
O.0030O.0004
O.000 1O.0000
O.0004O. 0001
O.0011O. OOOb
O. 0001O. 0000
O.0005O.0003
O.OOlb0.0015
O.000 1O.0000
O. 0006O.0003
0.0023O.00 17
O .000 1O.0000
0.0007o . 0OO?
0.0029O. 00 1S
0.0001O.0000
0.00070.0001
0.00320.0008
O .000 1O.0000
O. 0006O. 0000
0.0028O. 0003
O .000 1O .0000
O.0005O.0000
0.0024O.0003
O.000 1O.0000
O .00050.000 1
O. 0022O. 0005
O. 0001O. 0000
O. 00050.0002
0.0021O.0006
O .000 1O . 0000
O. OOOS0.0002
0.0022O. 0007
O. 000 1O.0000
O.00060.0001
O. 002S0.0006
O .000 1O . 000 1
~.0006O.0000
O.00260.0002
OCR for page 67
n: O,OC~
cy
6
y
t:) 0.0
*
\
O0.05 
L~
0.1
21
.
0.1
o.o 02
Figure 20
2.5
2 ~o
~r
6 15
O 10
CY
\
05
QO ~
0.0 02
Figure 22
A/R = 0.50 ZO/R = 2.00
0.4
~ o.o ~ ~
z
o.o o'
Figure 24
~4
19 1'
A/R = 050 ZO/R = 2.00
12t4
.
.
n`L
0.1
CY
~ 0.05
Y ~ ~
O ~
0.0
L~
0.05
0.0 _
Figure 26
1F 1f
~~
~ zo
6 1.5
y
O 10
~ 05
QO I ~
16 ~20 0.0 02
Figure 23
14 ~
A/R = 050 ZO/R = 2.00
_ oO ~ ~ C
~ ~ C1 ~ ~ E
0 4 0.8 12 16
2~0
HA~'t. 3
F~ ~) 1
F* (~) I
A/~ = O70 7n/R = 70n
v.e
tt0.05
*
~0.0
o
O  .05
0.1

0.0 02 0.4
Figure 2
0.6 0 ~ 10 12 t4 t6 ~0
PULSATION W*SQRT(R/G)
A/~ = n7n
70/~ = 700
~ C . ., .. ~
0.4
02
Y
_' 0.0
L~
02
0.0 02
Figure 25
~o
mr~
1
013 10 ~ 14
PULSATION W*SQRT(R/G)
.
.
0.4 C
A/R = 0.70 ZO/R = ~co
o~ ~
i
011
~* 1
~ o.c~ 1
Y 1
oo 1
~ 1
\o~o5 1
o.1 1
0.0 0.4
Figure 27
t6 ~20
A/R = 0.70 ZO/R = 2.00
~1 1 1
0~o ~
~ tO [1 C] m
1m C1 1 1
1 1 1
0.6 0.8 10 12 t4 16 ~0
PULSATION W*SQRT(R/G)
~1
i I I I
1
~1
1
1
1
I c~ (5 c
1
D ~C1 [!1 C
1 c~
1
G [~
1
.
o ~t2 16 Zo
PULSATK N W*SQRT(R/G)
77
OCR for page 67
0.02
~_
CY
0.01
6
*
y
*
0.0
z
LL
0.01,
1.0
C.8
., O.B
0.4
0.21
0.0
ns
0.4
(f  ) o3.
02
C)
1i
LU
8 0.1
_
C
. _
. _
o.o 02
0.4 0.6
~o ~
t4 t6
A' R = 0.~0 m 2~0/p,R c,i.00 c
. =~
0.0 0.2 0.4 O.B 0.8 1.0 t
Figure 28
o.o ~ ~
o.o 02
Figure 30
1.4
A/R =
1
c
Q ~
,.. 050 ZO/R = 2.00
___
~ OW
~O [ ~
~ _
(!) ,t, Q Q Q ~ Q ~ ~ ~ ~
1
0.4 0.6 o ~to 12
1.O
1.8 2.0
O.oB
CY
* 0.04
6
y
CY o.o
0.04
1.0
0.8
O.B
=0.4
0.2
0.0
05
0.4
L1J
03
~ 0.2
C)
0.1
0.0
A/R = 0.70 ZO/R = ZOQ
o:~ J~ooo m~
o.o o~ Q4 0.6 o~ to t2 t4 t6 tB 21 0
PULSATION W*SQRT(R/G)
~ j~ ~ j
0.0 0.2 0.4 0.6 0.6 1.0 1.2 1.4
Pulsation w*sqrt(R/g)
Figure 29
WAVE ENERGY DISPERSION
A/R = 0.70 ZO/R = 200
1.6 1.6 2.0
_ =
o ~
2 0 o 0 o. 2 0.4 0;6 o~ to 12 t4
PULSATION W*SQRT(R/G)
Figure 31
DAMPING COEFFICIENT
78
LU
A ~
t6 t8 zo
OCR for page 67
*103 Submerged Ellitsoid Fr=0.40 *109
0.2
co
~0o.4
*
~
o.~
~
0.8
1.0
1 Npan=168
~. Npan= 601
1 1 1 1 1 1 1
0 10 20 30 40 50 60 70
t*sqrt(g/L)
Figure 32
0.2
CO
to0.4
O ~
0.8
1.0
80 °
9 Submerged Elli o d Fr=0.50 ,,, 9
0 B/L ~ 0.20 PZojL  0.16 10
0.2
 ~0.4
o
0.6
0.8
1.0
Submerged Ellipsoid Fr=0.45
B/L = 0.20 Zo/I~ = 0.16
.
1
 Npan168 
 Npan= 60
1 1 1 1
0 20 30 40 50 60 70 80
t*sqrt(g/L)
Figure 33
Submerged Ellipsoid Fr=0.80
B/L = 0.20 OWL = 0.16
.
0.2
 Npan=168 
 Npan~ 60
CO _
=0.4
o
0.~
~
0.8
1.0
1 1 1 1 1 1 1 1.U 1 1
0 10 20 30 40 50 60 70 80 0 10 20
t*sqrt(g/L)
Figure 34
79
30 40 50 60 70 80
t*sqrt(g/L)
Npan= 601
Figure 35
OCR for page 67
REFERENCES:
Abramowitz M., Stegun I.A., "Handbook of mathematical
functions". Dover Publications, New York, 1965.
Adachi H., Ohmatsu S., "On the timedependent potential
and its application to wave problems". Proc. 13th ONR
Symposium on Naval Hydrodynamics. Tokyo, 1980.
Brard. R., "Introduction a ltetude theorique du tangage en
marche" Bulletin de l'ATMA, 1948.
Clement A., Ferrant P., "Free surface potential of a pulsating
singularity in harmonic heave motion". Proc. Fourth Int.
Conf. on Num. Ship Hydrodynamics. Washington, 1985.
Clement. A, Ferrant. P., "Superharmonic waves generated by
the large amplitude motion of a submerged body". IUTAM
Symposium on Nonlinear Waves. Tokyo, 1987.
Dassonville B., "Etude experimentale des effete nonlineaires
du mouvement de grande amplitude d'un corps faiblement
immerge", these de DocteurIngenieur. Universite de Nantes,
1987.
Delhommeau G., "Les problemes de diffractionradiation et
de resistance de vagues: Resolution numerique par la
methode des singularites". These de Doctorat es Sciences,
Nantes, 1987.
Doctors L.J., Beck R.F., "Convergence properties of the
NeumannKelvin problem for a submerged body". Journal of
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Farell C., "On the wave resistance of a submerged spheroid".
Journal of Ship Research, Vol. 17, n° 1, pp. 111, 1973.
Ferrant P., "An fast computational method for transient 3D
wavebody interactions". Proc. Int. Conference on Computer
Modelling in Ocean Engineering. Balkema Publishers, 1988.
Ferrant P., "Radiation d'ondes de gravite par les
deplacements de grande amplitude d'un corps immerge:
Comparaison des approches frequentielle et instationnaire",
these de Doctorat, Universite de Nantes, 1988.
Finkelstein A.B., "The initial value problem for transient
water waves". Comm. on Pure and Applied Maths., vol. 10,
pp 5 1 1522, 1957.
Gradshteyn I.S., Ryzhyk I.W., "Tables of integrals, series and
products", Academic Press, New York, 1965.
Jami A., "Etude theorique et numerique de phenomenes
transitoires en hydrodynamique navale." these de Doctorat es
Sciences. ENSTA, 1982.
Jami A., Gelebart L., "Une approche transitoire des
mouvements de translation de corps immerges". Premieres
Journees de l'Hydrodynamique, Nantes, 1987.
Jami A., Pot G., "Finite element solution for the transient
flow past a freely floating body". Proc. Fourth Int. Conf. on
Num. Ship Hydrodynamics. Washington, 1985.
King B.K, Beck R.F., Magee A.R., "Seakeeping calculations
with forward speed using timedomain analysis". Proc. 17th
ONR Symposium on Naval Hydrodynamics. The Hague,
1988.
Korsmeyer F.T., Lee C.H., Newman J.N.,, Sclavounos P.D.
"The analysis of wave effects on tensionleg platforms".
OMAE Conference. Houston, 1988.
Liapis S., Beck R.F.,"Seakeeping computation using time
domain analysis". Proc. Fourth Int. Conf. on Num. Ship
Hydrodynamics. Washington, 1985.
Magee A.R., Beck R.F., "Vectorized computation of the time
domain Green function". Fourth Int. Workshop on Water
Waves and Floating Bodies. Oystesee, Norway, 1989.
Magee A.R., Beck R.F., "Time domain analysis for predicting
ship motions". Iutam Symposium. London, 1990.
Newman J.N., "Transient axisymmetric motion of a floating
cylinder". J.F.M., Vol. 167, pp. 1733, 1985.
Newman J.N., "The evaluation of free surface Green
functions". Proc. Fourth Int. Conf. on Num. Ship
Hydrodynamics. Washington, 1985.
Newman J.N., "The approximation of freesurface Green
functions". F. Ursell Retirement Meeting, Manchester, 1990.
Pot G., "Etude theorique et numerique des mouvements
libres de corps flottants ou immerges. Extension au cas des
grands mouvements de corps immerges". These de Docteur
Ingenieur, ENSTA, 1986.
Stoker J.J., "Water Waves", Interscience Publishers, New
York, 1957.
Webster W.C., "The flow about arbitrary, threedimensional
smooth bodies". Journal of Ship Research, Vol. 19, n° 4, pp.
206218, 1975.
Wehausen J.V., Laitone E.V., "Surface Waves" in Handbuch
der Physik., Springer Verlag Ed., Berlin, 1960.
Yeung R.W., "The transient heaving motion of floating
cylinders". Journal of Engineering Mathematics, Vol. 16, pp.
97119, 1982.
80
OCR for page 67
APPENI)IX 1
INTEGRAL REPRESENTATION OF THE VELOCITY
POTENTIAL
1. Green function
Let us first introduce the threedimensional Green
function for a point source of impulsive strength:
G(M9 P,t) = {i(~) . Go(M9P) + H(t) . F(M, P,t)
with the notations:
Go(M9P) = + (A1.2)
Or Or'
r
F(M,P,t) = J (gk)~ sin[(gk)~nt] Jo(kR) ek(Z+Z )dk
2~
o
M= (x9y,z)
P = (x',y',z )
r= [(xx') +(y_yt)2+(z z,)2]ln
r' = [(Xxt)2+(yy.)2+(z+z.)2l1/2
R = [(xx' )1/2+(y y, )I/2]
JJ
S(T)
Q(M)
8(t~) · O(M9 1) = (A1. 10)
41`
a a
[~P'~) a (p) (M9P,t I)  ~(p)(~>(P,7) G(M,P,t I)] dSp
where Q(M) is the solid angle under which D is seen from M,
when M is in D or S. (1.10) is then integrated with respect to
from 0 to t, giving, due to the fundamental property of o:
Q(M) (A1. 11)
O(M,t) =
4~
J dame [~)(P't) Pa C;(M,P,t~) ~ Pa O(p,:) G(M,P,t~)] dSp
O S(T)
~ being bounded at infinity (finite energy), and
accounting for (A1.8), the integral over SO in the right hand
side of last equation vanishes. Next step is to transform the
integral over Sf:
Isf = J damp [~(P,~)G(M,P,t I)  a GAP I) G (M9P,t~ )] dSp
O Sf(T)
and where o(t) and H(t) are respectively Dirac's delta
function and Heaviside's step function. surface condition, so that:
G is the potential induced at point P by a point source
of strength 6(t) located at point M, and can be shown to have
the following properties:
AG(M,P,t) = (or) . 6(t) (A1.4)
(A1.5)
2
G(M,P,t) AVG(M9P,t) O fops Sl(z=O)
et2 Liz.
G(M9P,O ) = G`(M9P,O ) = 0 (A1.6)
G(M,P,t) = G(P,M,t) (A1. 7)
G = 0(r~2); VG = 0(r~3) when r ~ +oo (A1.8)
2. Green's identity
Applying Green's identity in the fluid domain at time
to O(P,:) and to G(M(t),P(~),t~) yields:
J: ~ POP t) AG(M(t).P(~). to)  A~P.~) G(M(t).P(~).t~)l dV~ =
S(T)
v u
[~(P9~ ) a ups (M9P9t I)  ~ (p)O (Pa) G (M9P9t I)] d Sp
with S(T) = Sb(~) + Sf(~) + SO
Left side of equation (A1.9) is a Stieltjes integral
which can be reduced accounting for (A1.4) and (1), to give:
(A1. 12)
On Sf, ~ and G both satisfy the linearized free
~n(M; Pig) =  g 'I'=(M,P,~)
1 (A1. 13)
Gn(M,P,t~)= g GTT(M9P9tl)
[At = _ _ J ~ IJ [A GTT G Tl] P
= _
g I
[A GT T T P
(A1. 14)
then, introducing the total derivative in ~ of the integral over
Sf:
(A1.9) 1C J J [
APPENI)IX 2
FREQUENCY DOMAIN GREEN FUNCTIONS FOR
1~ BODYNONLINEAR PROBLEM
+
Isf = (A1. 16)
1
J1
d1
o
J do f [(Pa), 7) F~(M(t),P(~),t~)  FO ] (np~dl p).Vc(P)
O Cb~)
[~(P(~),~)G`(~P, A)  G(M,P,t~)OT(P,~)] dSp
The first term is zero, due to the initial conditions for
G and A, and gathering terms, we finally obtain the following
integral representation of the velocity potential in the fluid
domain D(t), in terms of a mixed distribution of sources and
normal dipoles on Sb. (Go terms in the integral over Sb have
been extracted accounting for the integral property of 6(t),
and eliminated from the line integral, for Go is identically
zero at the free surface)
Q(1~
~ O(M,t) = (A1. 17)
Ji [~(P,t)~ Go(M,P)~Go(M,P)~(P,t)]dSp
Sb(t)
+ tdz(t
+ g J do f [~P(~) to F'(M(t),P(~),t~)  FO ] (np~dl p.V~P)
O Cb(~)
a ~
[~(P,7) ~(p)F(M P. t`c) (Pa,) F(M, Pity)] dSp
When M is on Sb, Q(M) = 2~ (if the normal is
continuous in M) and (A1.17) gives the integral equation to
be solved for ~ on Sb, considering amen to be known (forced
motions). After solution, (A1.17) may be used to compute
everywhere else in the fluid domain D, where Q(M) = 4~.
When the body is fully submerged or when Cb is time
invariant, the line integral in (A1. 17) simply vanishes.
At last, note that the solution may also be
represented by a distribution of sources only. The derivation
of the corresponding integral representation follows
comparable steps and, for brevity, will not be given.
We are interested here in the steadystate periodic
freesurface potential generated by a submerged point source
following an arbitrary periodic motion of frequency lo, with a
source strength of frequency pal, p 2 0, i.e. solution of the
following problem:
AGip(M(~),p,[) = arm)) mS pet (i=l)
sin pot (i=2) (A2. 1)
32C2,p + g amp = 0 for z. = 0 (A2.2)
VGip ~ O for r > ~(A2.3)
+ ra liadon condition (A2.4)
for M(t) = [x(t),y(t),z(t)] prescribed modon of frequency
P = [x ,y',z' ] fixed point
r(t) = [(x(t)x ) + (y(t)y') + (z(t) z')2]~/2 (A2.5)
An exhaustive study of the solutions of this problem
may be found in Ferrant (1988), where these Green functions
were initially developed for the solution of the bodynonlinear
problem in the frequency domain. Only the results liable to
be useful for the present paper will be repeated in this
appendix.
1. General case  Arbitrary periodic motion
The preceding problem may be solved using different
methods (Fourier transform, time asymptotic limit of
unsteady solution,...), the details of which being given in
Ferrant (1988). Although involving some tedious algebra, the
results may be expressed in a very concise form, involving
the frequency domain Green function for a fixed point source.
In the general case of an arbitrary periodic motion, G1p and
G2p are given by:
~ .
Glp( M(t), P. t) = 2~'1 costar [ G`30(0, M(~),P)
00
+ 2Re2, G~,o(lt0,M(~l~),P) e ( ) ] do
2n
r
G2p( M4t),P, t) = ~J S~Pt [ G  (0,M(~),P)
00
+ 2 Re A, Go,o(lc~, M(~/cd)7P) e ( ) ] do
(A2.6)
(A2.7)
where G`~o(~7M,P) is the complex Green function for the
frequency domain diffractionradiation problem. G`,,o(O,M7P)
is the zero frequency limit of this function, i.e. for infinite
depth:
G`~,o(O,M?P) =  Cur ~ Or' (A2.8)
82
OCR for page 67
Nota: Although initially derived for the infinite depth case,
expressions (A2.6), (A2.7) can be easily shown to hold for
finite depth too, provided the finite depth diffraction
radiation Green function is substituted to Gino.
2. Sine heave motion
2.1 Basic formulations
Explicit forms of the bodynonlinear frequency
domain Green functions may be derived by first imposing the
analytic form of the source path, and then introducing
appropriate formulations of Gino (NearField, Haskind, ...). In
the case of a timeharmonic heaving motion, we have:
M(t)=(xO.yO. zO+acoscx) (A2.9)
Then, substituting for example the NearField
formulation of Glaxo in (A2.6), (A2.7), we get, after some
transformations:
cos pot r 1 1 ~
G~p[M(t).P, t] = ~ 4~c Lr(~) + r'(t) (A2. 10)
I, 12 kn ~ c" pa c" near ~ ~ em ME [a (/)] don do . cos nix
+ ~ e n ° [Ip+n(akn)+Ip n(akn) ] . [Ho(knR) cOs not  Jo(knR) sin not]
and:
sin pot ~ 1 1
G2p[M(t),P,t] ~~ 47c r(t) r (t) (A2.11)
I ~3 J sin p~·sinn~ Re IT e n E~[~(il~l))] do d~.sinnC0t
+7en ° [Ipn(akn)~Ip+n(akn)] [Ho(knR) sinn()t+Jo(knR)cosnot]
where Ho and JO are respectively the Struve and Bessel
functions of order 0, Ip is the modified Bessel function of
order p, E1 is the exponentialintegral function, and:
kn = (no) 2/g
R=[(xox) +(YOY)]
(n(t) = kn [z(t) + z' + j R cos 0]
These expressions are mostly appropriate for the
computation of Gip when R is small. For larger radial
distances, formulations based on the modified Haskind form
of Gino are to be preferred; see details in Ferrant (1988).
2.2 Farfield behaviour
Accounting for the asymptotic behaviour of the
special functions Ho, JO and E1, the following farfield
expressions for Gip are easily derived, for a source motion
being given by (A2.9):
G,p(M(~).P.~) = (A2.12)
n~27TR)I e [Ip+n(akn)+9n(akl,)].sin(k,,R )7/4)
+o(R~) for R woo
00 . ~/2
G2p(M(~).P.~) = (A2.13)
t2~R] e [Ip n(akn)Ip+n(akn) ] cos(knRncot~/4)
+ o(R~~) for R   oo
As can be deduced from these expressions, the far
field is a superposition of regular circular waves of velocities
Cn = gnu, 1 < n < oo., the amplitudes of the harmonics being
explicitly given by the coefficients in front of the sine
functions, involving modified Bessel functions of arguments
akn.
83
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