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OCR for page 239
Nonlinear Simulation of Transient Free Surface Flows
R. Cointe
Institut Frangais du Petrole
RueilMalma~son, France
1 Abstract
The application of the Mixed EulerianLagrangian method
to the simulation of transient free surface flows in the vicin
ity of a free surface piercing structure is considered. A par
ticular attention is given to the validation of the numerical
procedure.
Several applications are studied. Comparisons between
the results of the numerical scheme and those of approxi
mate theories and/or experiments are shown. They demon
strate the accuracy and versatility of the simulation that
can be used as a "standard" to check the applicability of
approximate theories.
The main limitation of the method is that it cannot
account for viscous effects, in particular in the vicinity of
the free surface. Approximate ways to simulate dissipative
phenomena associated to breaking would be most useful.
2 Introduction
The direct numerical simulation of unsteady twodimen
sional potential free surface flows using a Mixed Eulerian
Lagrangian method (MEL) has received considerable at
tention since the pioneering work of LonguetHiggins and
Cokelet t224. If many codes exist now that use this method,
their suitability for the study of nonlinear fluidstructure
interaction problems has only been demonstrated for some
particular applications (e.g., water impact t15], simulation
of breaking in a tank t12], forced heaving of a cylinder t18],
etc...~.
Compared to the initial application of the MEL to the
study of steep periodic waves (e.g., t22~), several new dif
ficulties appear when a structure is present, especially if
it pierces the free surface. The first one is related to the
proper description of the flow in the vicinity of intersection
points between the body and the free surface. A second
difficulty is to be able to control the incident wave train
that interacts with the body.
The main difficulty, however, is probably that the accu
racy of the simulation is difficult to establish for transient
flows in the vicinity of free surface piercing bodies, because
of a lack of reference cases. Checking such requirements
as conservation of fluid or energy might not be sufficient.
Surprisingly enough, the linear solution is not always com
puted and quantitative validations of the numerical proce
dure are almost inexistant, even for weal;ly nonlinear flows.
These considerations have partly motivated the present
work which associates numerical and analytical studies.
A code based on the MEL Sindbad has been de
velopped that has for purpose the simulation of a two
dimensional wave tank using potential flow theory. A par
ticular attention is given to the proper validation of the
numerical scheme by comparison of its results to those of
experiments and of asymptotic studies. To make this com
parison easier, an option of the code allows the linear prob
lem to be solved. Such an approach appeared necessary in
order to gain confidence in the code and eventually extend
its range of applicability.
A direct simulation of experiments that can be carried
out in a wave tank is performed. For this purpose, waves
are generated in a rectangular tank by use of a piston
type wavemaker. These waves can then interact with sub
merged or free surface piercing cylinders in forced or free
motion. Reflection from the wall opposite to the wave
. . . . .
maker is avoided by the use of a damping zone see fig
ure 1
The results of the classical first and secondorder dif
fraction radiation theories can be recovered using the sim
ulation. A good agreement with experimental results is
obtained in cases where these classical theories fail.
The main limitation of the method is that viscous and
turbulent effects cannot be accounted for. This problem
is crucial when viscous effects occur in the near vicinity of
the free surface, in particular during breaking.
This work has been reported elsewhere while in progress
(hi, Pi, [8], 9. If we focus here on the results of the
simulation, a more detailed description of the numerics and
of some of the approximate theories referred to here can be
found in t74.
239
OCR for page 239
Y ~Dnu~nerico/u~or~a Bank l g1
o
EM
3 Numerical Scheme
3.1 Outline of the Method
Since there exists now quite a large number of codes based
on the MEL, the numerical method used will only be briefly
outlined. The attention will focus in the next sections on
the main difficulties that have been encountered and on the
methods used to overcome them.
The main idea of the numerical procedure is to choose
markers initially at the free surface and to follow them in
their motion.
We use a coordinate system (x, y). The xaxis coincides
with the reference position of the free surface and the yaxis
is upward vertical  see figure 1 for geometric definitions.
The fluid is assumed to be incompressible and the flow
irrotational so that the velocity field v is given by:
v = V¢, (1)
with:
A¢' = 0. (2)
The computation is performed in a bounded domain.
Along rigid boundaries (x ~ En~t)~, the normal velocity in
the fluid is equal to the normal velocity of the boundary.
We have, therefore, a Neumann boundary condition.
Along the free surface, we use Bernoulli equation and
the fact that the free surface is a material surface. The
corresponding equations are written for a marker x on the
free surface (x ~ I`~(t)) and the associated value of the
potential, ¢(x). This yields):
Do _y _ ~ ~ _ <' ¢2 + ~ ¢2 _ p + cats (3)
Do _ ~ ~ ~ 1 ~ ~ `4)
where D is used to indicate a material derivative, s and n
are vectors tangent and normal to the free surface, respec
tively, and ~ is an arbitrary constant.

~ For the sake of simplicity, we use units such that the acceleration
of gravity, 9, the specific mass of water, p, and the depth of the tank,
h, are equal to 1.
Pistontype Tested body
wave maker
Figure 1: Sindbad geometric definitions
240
hl
"Beach"
'(absorbing, zone)
This constant specifies the tangential motion of the
markers: ~ = 1 identifies markers and particles while ~ = 0
yields a zero tangential motion of the markers. This last
choice allows a current to be simulated in the tank. For the
applications discussed here, we will however always take
= 1.
We will assume that the pressure is constant along the
free surface. It can therefore be included in the function
of time city. With an appropriate choice of the velocity
potential, this function can be taken equal to zero.
At a given instant t, if o and A', are known along the
free surface Fatty, then the righthand sides of (3~~4) can
be evaluated. The fact that ¢> is harmonic in the fluid
domain allows the value of ¢'n along I`~(t) to be computed
from the values of o along I,~(t) and of ¢>72 along I`ntt).
The kinematical constraint /~¢ = 0, associated with the
boundary condition on rtn' permits therefore to express
the free surface boundary conditions (3~~4) as an evolution
equation for (¢>, x). This evolution equation can be solved
numerically using standard timestepping procedures, such
as a fourthorder RungeKutta algorithm.
The main numerical difficulty is to be able, at each
timestep, to solve for the harmonic function ~ knowing
¢, along Lotte and ¢)n along En~t). We use the integral
equation:
~ ~ ~ ~ ~r`~+rn ~ ~ ~ ~ Q
= .,lr +r An (Q ~ G(P' Q ~ dog, (5)
where P is a point on the boundary, G is the Green func
tion, 8~) the angle between two tangents of the boundary
at P (equal to ~ for a smooth curve) and s a curvilinear
abscissa along I`. Equation (5) is discretized using a stan
dard collocation method. The boundary of the domain is
approximated by segments and ~ and ¢>r' are assumed to
vary linearly along each segment. This allows an analyti
cal integration of the Green function, its normal derivative
and their products by the curvilinear abscissa so that the
calculation of the matrix elements is rather simple (and
vectorizes well).
OCR for page 239
3.2 Numerical Treatment at the Intersections
At each timestep, we have to solve a Neumann [along
[nit)]  Dirichlet talon" l`~(t)] boundary value problem.
It is well known that the solution of such a "mixed" prob
lem is singular at the intersection points, i.e., at points
belonging both to I,n~t) and Pelt) e.g., [16;.
Let us consider, as an example, the wavemaker problem.
For a piston type wavemaker and an horizontal free surface
(90° intersection angle), the complex potential ~ = ~ + i ~
solution of such a problem behaves like z log z near an
intersection located at z — 0. This gives the expected
behavior of the problem discretized in time.
Discretizing in time is similar to performing a small
time expansion, i.e., to write:
¢(x, t) = jO(X) + t ¢~(x) + · · · (6)
Not surprisingly, performing such an expansion also leads
to a z log z behavior for ¢~ (e.g., t27i, t21~. Does this,
however, necessarily imply that this singular behavior has
to be expected for the solution of the transient problem?
The answer is no, just because the small time expansion
is not regular near the intersection point. It is, therefore,
improper for a local analysis.
A regular expansion can be found in the weakly nor~lin
ear regime, i.e. here, for a small acceleration of the wave
maker (relative to that of gravity) see A. It appears that
in this case the first approximation (in an asymptotic sense)
is provided by the classical linear solution. The boundary
condition for this solution is not a Dirichlet boundary con
dition; it is given by:
¢~+¢y = 0. (7)
If regularity in time is assumed, the local behavior of the
complex potential for this solution can be shown to be in
z2 logz. Not surprisingly, this singularity is the same
as that appearing for the harmonic problem (for which
= ~ exptiwt)) which was studied by I
OCR for page 239
3.4 Validation
As indicated previously, the proper validation of the nu
merical simulation has been one of our main objectives.
Solving the fully nonlinear problem can only be interesting
if the accuracy of the numerical scheme is well established.
Several strategies exist for this validation.
Consistency checks should of course be performed. For
that purpose, the volume and the rate of change of en
ergy are computed. When no damping zone is present, this
last quantity is compared to the power input from exterior
loads. The pressure exerted by the fluid on rigid bound
aries is computed by two different methods. Both of them
use Bernoulli's equation, but they diner by the evaluation
of the ¢' term. In one case, this term is evaluated by fi
nite differences in time. In the other case, it is obtained by
solving the boundary integral equation for hi.
In our opinion, consistency checks are not sufficient and
comparisons with results from approximate methods are
also needed. In the weakly nonlinear regime, the reference
solution is the classical linear solution. However, a direct
comparison with this linear solution is difficult, for two
main reasons:
* comparing the nonlinear simulation to a linear result
can be confusing since discrepancies might be due to non
linear phenomena. For that reason, a linear version of the
code has been derived that only differs from the nonlinear
version by the boundary conditions that are satisfied;
* usually linear results are for the steadystate response
while the simulation is transient. Discrepancies might be
due to longlasting transient phenomena, to the wave gen
eration or absorption mechanism used, etc... A first step
has therefore been to make proper comparisons with linear
results for the transient problem in a tank of finite length.
The main interest of the method being to be able to ac
count for nonlinear phenomena, a proper validation of the
nonlinear response should be made. For that purpose, a
comparison is performed with approximate nonlinear the
ories, such as secondorder theory or shallow water theory.
A final test is provided by comparisons with experi
ments. In our opinion, however, these comparisons should
only be made last if it is the accuracy of the numerical
Figure 2: Sloshing free surface profiles
scheme that has to be evaluated. In any event, a compar
ison with a linear result appears necessary in order to be
sure that nonlinear phenomena are important. It should
also be kept in mind that viscous effects can play an im
portant role and are not accounted for in the simulation so
that discrepancies might not be due to the inaccuracy of
the numerical scheme but to an improper physical model
ing.
4 Numerical Results
The purpose of this section is to show some numerical re
sults that can appropriately be compared to other theories
or to experiments. Numerical instabilities encountered
by many similar methods—may appear for waves of large
steepness. In this case, a 5 points smoothing algorithm
similar to that used in t22] is employed.
4.1 Sloshing in a Tank
The study of sloshing provides a first simple test for the
accuracy of the simulation. We consider a rectangular tank
and the free motion of a fluid initially out of equilibrium.
One of the main advantages of this configuration is that
it allows quasianalytical solutions of the transient problem
to be derived at first and secondorder A. The efficiency
of the numerical scheme for the solution of the linear and
nonlinear problems can therefore be directly evaluated.
We consider the case of a constant initial slope of the
free surface, i.e., an initial elevation given by:
y = ~ d (d—0.5), (11)
where d is the length of the tank. We show on figure 2
the free surface profiles for d = 1 (i.e., a depth equal to
the length) and ~ = 0.35. With such an initial amplitude,
breaking occurs in the tank.
In order to evaluate the accuracy of the simulation, we
compare on figure 3 the perturbation elevation,
¢
achieved, indicating that both the linear and secondorder
component of the wave elevation are accurately computed.
For ~ = 0.35, breaking occurs and the agreement deterio
rates, suggesting that higher order effects become impor
tant. More results on this topic can be found in A.
4.2 Harmonic Motion of a PistonType Wave
maker
The wavemaker problem is interesting because it provides
a simple configuration to study most of the difficulties as
sociated with the simulation. In particular, it involves a
moving free surface piercing body (the wavemaker itself)
that is absent in the sloshing problem.
In this section, the wavemaker motion is given by:
Arty = 0 t<0 (13)
tots =  2e corset) t > 0. (14)
The length of the tank is d. Note that even though the
motion is harmonic for t > 0, transient phenomena are
expected.
4.2.1 Linear Solution in a Tank of Finite Length
For a tank of finite length, it is possible to derive a quasi
analytic solution for the linear transient problem that can
be evaluated quite easily. This solution can be found in t124;
an alternative solution that seems to have better conver
gence properties has also been derived A.
Since this problem involves a moving rigid boundary
that pierces the free surface, it allows the numerical treat
ment of the intersection point to be evaluated. Here, we
compare the result of the linear simulation and the quasi
analytic result. This ensures that discrepancies are only
due to numerical errors.
As an example, we consider a wavemaker motion at the
frequency ~ = ~r/2. The corresponding wave period and
wave length are 4 and 2.5, respectively. The accuracy of
the simulation has been evaluated at t = 5 in a tank of
length 5. We show on table 1 the root mean square of the
error (the reference 100 is taken as the error for the coarsest
grid used, equal to 0.0903 at) as a function of the number
of nodes per wave length, NX, and the number of time
steps per wave period, NT3.
VT ~ / NX
4
16
32
64
128
256
5
100
61.3
63.7
64.0
64.0
64.0
64.0
10
oo
26.2
29.3
29.9
29.9
29.9
29.9
20
oo
0.6
11.5
13.7
14.4
14.4
14.4
Table 1: Convergence table
40 .
oo
.
4.2
.
5.3
.
5.4
5.4
5.4
oo 1
1
1.8 1
1
2.5 1
160
on
oo
oo
1.3
1.4
.4
.4
3A uniform grid is used and' thanks to the symmetry, the bottom
is not d~scretized. No smoothing is applied. However, using the 5
points smoothing procedure introduced in [22], even a each tinestep,
does not significantly alter the accuracy of the method.
243
From this table, it appears that:
* a sufficiently small timestep has to be used in order
for the numerical scheme not to blow up. The influence of
the timestep is otherwise rather small;
* for a given spatial discretization, results converge as
the timestep goes to zero. Note, however, that the numer
ical results do not converge to the exact solution;
* as the spatial discretization increases, results seem to
converge to the exact solution. However, the convergence
rate is not secondorder in the grid size. This is very likely
due to the numerical treatment at the intersection that is
not secondorder4.
4.2.2 Linear Solution in a Tank of Infinite Length
In order to test the efficiency of the damping zone, we con
sider now a tank of infinite length. For a piston type wave
maker, the solution far from the wavemaker and for large
times is a progressive wave of amplitude:
e k + sigh k cosh ~ (15)
where k is the wavenumber associated to ~ (~2 = k tanh k).
Note that in deep water (k ~ oo), a ~ at.
We perform the simulation for the same wavemaker mo
tion as in 4.2.1 but in a tank of length 10 equipped at its end
of an absorbing zone. We compare on figure 4 the steady
state linear solution and the result of the linear simulation
with and without an absorbing zone for the wave elevation
at the middle of the tank (distance 5 from the wavemaker).
The damping coefficient is given by (10) with cat = 1 and
= 1. It appears clearly that the damping zone provides
a simple and efficient way to avoid any reflection.
1.0
z 0.5
o
0.0
STEADY STATE LINEAR SOLUTION
_LINEAR SIMULATION WITH ABSORPTION
....LINEAR SIMULATION WITHOUT ABSORPTION
0.5 1 1 , , , , 1 ,
0 x0 Em ~~. Kim ~~ 6~ ]0
TIME
Figure 4: Wave elevation with and without damping zone
~
4 this indicates that numerical errors are mainly due to the numer
ical treatment at the intersection. A convergence test for the problem
with periodic boundary conditions is therefore not relevant to assess
the accuracy of the simulation applied to the wavemaker problem.
OCR for page 239
of 
o
to
LLI
—O 1 , . , , , . , . 1
·4C t.0 60.0 80.0 1 00.0 1 20.0 140.0
TIME
Figure 5: Damping zone relative error vs. time
In order to evaluate more precisely the efficiency of the
damping zone, we show on figure 5 the difference between
the steady state linear solution and the result of the linear
simulation for the relative wave elevation (wave elevation
divided by the wave amplitude ~ at) at the same point. If
longlasting low frequency oscillations appear, the relative
error is only of a few percents and the absorption mecha
nism appears to be quite satisfactory.
4.2.3 Nonlinear Solution in Shallow Water
In order to estimate the accuracy of the method for the
nonlinear computation, we first consider the case of a shal
low water swell.
Mel and Unluata t24] have explained how such a swell
can experience very drastic nonlinear deformations when
it propagates. Very recently, Chapalain t2] has performed
experiments at the Institut de Mecanique de Grenoble that
neatly confirm this biharmonic resonant behavior. It ap
peared therefore interesting to try and reproduce them with
the simulation.
The only data for the numerical simulation are the ge
ometry of the tank, the law of motion of the wavemaker
and the friction coefficient f,,, used to model dissipations.
The experiment was performed in a 40 cm deep tank with
a pistontype wavemaker the motion of which was given
by (14) with an = 15.9 cm and ~ = 2.5 rad/s. The total
length of the tank, 36 m, is simulated using 300 nodes on
the free surface. The simulation on 30 s took approximately
7 minutes on a CRAYXMP.
sin order to model dissipative effects in a way similar to that
used by Chapalain for Boussinesq equations we substitute to Bernoulli
equation:
with
1 _ _
go+ 2 vet v¢+y+~=o
4 a' w
v = 3 fill 2
We took the value of Jo, used by Chapalain, fw = 0.1. Note that this
modeling of dissipative effects, already used in [14] for the study of
sloshing, is similar to the modeling of the damping zone.
_
l
~ n 1 n n 15 n 20.0 2s.0 30.0
1
1
To 1 o.o 1 5.0 20.0 2s.0 30.0
t ~ A A ~ A A A A A Al i ~ A A A ~ A A A A I
Ji , ~ ~
0.0 5.0 1 0.0 1 5.0 20.0 2s.0 30.0
a,/ ~~; A ~~ ~ ~<~
· ~ I. lo. 3°. be. 'I 0.0 5.0 1 0.0 1 S.0 20.0 2s.0 30.0
~/~/\\,AJ\J\J\J\J\Vl\J\Jl ~~
·~° I I 1 1 1 ~ , , , , I
a. 1~. 20. I. "~ 0.0 5.0 1 0.0 1 5.0 20.0 2s.0 30.0
. .
A ~ /\ /\ ~ ~ ~ /~\ 1\ ~ f\ ~ A ~ A A ~ A ~
. \/ \J \J \J \J A \J V V \) A i A/ \/ ~ V ~ \J A ~ ,
Figure 6: Shallow water swell measured (left) vs. com
puted (right) wave elevations at several points in the tank
We compare on figure 6 the measured and computed
wave elevations at several points in the tank. An excellent
agreement is achieved. This agreement is confirmed by a
Fourier analysis performed once a steady state is reached.
The amplitudes of the first three harmonics is plotted as a
function of the distance along the tank on figure 7. It ap
pears that the simulation is very efficient to model shallow
water waves and their generation by a wavemaker.
70.0
60.0

~ 50.0
_'
c, 40.0
c' 30.0
20.0
10.0
_SINDBAD
EXPERIMENTS (CHAPALAIN)
0.0
0.0 5.0 10.0 15.0 20.0 25.0
DISTANCE FROM WAVEMAKER (M)
Figure 7: Shallow water swell measured vs. computed
244
OCR for page 239
4.3 Arbitrary Motion of a PistonType Wave
maker
In deep water, nonlinear phenomena in the propagation
of a regular swell are rather long to develop. Numerical
methods with periodic boundary conditions, such as t10],
are probably more suited to the study of this problem than
the direct simulation of a wave tank.
An appropriate choice for the motion of the wavemaker
can however lead to wave focusing that results in break
ing. Experiments based on this principle were performed
at MIT and a comparison with a numerical simulation sim
ilar to ours made see t12] where the law of motion of the
wavemaker is given. The main drawback of this test case
is that it involves a large amount of computer time. We
have run Sindbad on the same case, but using an absorb
ing zone and a somehow coarse grid in order to minimize
computational effort. The calculation has been performed
on a CRAYXMP and demanded "only" 30 minutes with
250 nodes on the free surface (compared to 30 hours with
500 nodes on a C RAY 1 for the simulation performed at
MIT).
._,
0.25
0.00~
 0 . 25  ,
0 10 20 30 40 50 ut
0.25 
0.00 ~ , ,
0.25
0 10 20 30 40 50 60
0.25 
0.00' ~
0.25
a
0.25  _
0.00  _
.O,,l, , , , , , 1
~ ~ ~ 20 30 40 50 60
0.25 
0.00 _ ~
0.25  _
o
I I ~ I I 1
1 0 20 30 40 50 60
O.25
' ! ` ~ O DO   ~
0 . 25  ,
0 10 20 30 40 50 60
Figure 8: Steep deep water waves measured (left, dashed
line) vs. computed (solid line, left: MIT, right: Sindbad)
wave elevations at several points in the tank
60ur own experience tends to suggest that "numerical" overturning
is very sensitive to the discretization used and more particularly to
the node distribution along the free surface. Here again the validity of
the simulation is difficult to establish. Our interest has been mainly
to perform the simulation up to the point where breaking occurs.
The results of our simulation are in good agreement
with both experimental and numerical results obtained at
MIT up to breaking see figure 8. If overturning develops
at the same time, our computation fails sooner than their
(before the closing of the tube)6.
This confirms that the numerical simulation can repro
duce accurately nonlinear phenomena observed experimen
taly.
4.4 Wave Diffraction on a Submerged Cyl~n
der
The case of the wave diffraction on a submerged cylinder al
lows a first study of wavestructure interaction. This prob
lem has been studied extensively in the past. In partic
ular, Ursell t31] used linear theory and showed that, for
a circular cylinder, there is no reflection. Ogilvie t25] ex
tended Ursell's results and computed the secondorder ver
tical drift force. Very recently, Vada t32] computed the
secondorder potential and calculated the diffraction loads
and the diffracted waves to secondorder.
Chaplin t3] measured diffraction loads in the labora
tory while Grue and Granlund [17] measured the diffracted
waves. These experiments have partly confirmed the re
sults of first and secondorder theories. They have also
exhibited some important nonlinear phenomena not ac
counted for by these theories. Such nonlinear phenomena
can either be due to nonlinear free surface effects (of third
order or higher) or to viscous effects.
Consequently, the present study has two main objec
tives:
* to recover the results of first and secondorder theo
ries in order to assess the numerical accuracy of the method;
* to compare with experimental results in order to de
termine the relative importance of viscous and free surface
effects for the nonlinear phenomena observed.
Fully nonlinear simulations similar to ours have been
performed by several authors (e.g., t29], [11~. In general,
periodic boundary conditions were used. To our knowledge,
however, comparisons with secondorder theory were not
achieved.
4.4.1 Diffraction Loads
We consider a circular cylinder of radius r = 0.06. The
coordinates of its center are xc = 3.5 and Yc = 0.12.
Waves are generated by a pistontype wavemaker moving
at the frequency w = 1.85. The simulation is made in a
tank of length 10 with 200 markers at the free surface and
60 timesteps per period.
The forces acting on the cylinder are computed by di
rect integration of the pressure.
In order to compare the results with those of experi
ments or of first and secondorder theory, we use a Fourier
series expansion of the transient signal (once a steadystate
is reached). This yields:
3 2 = F(°) + ~ F(n) cos(n`~,t + 8(n)) (16)
r ~ n>1
3Y2 = F(°) + ~F(n) cos(ncot + 8(n))' (17)
24s
OCR for page 239
where the subscript x and y denote the horizontal and ver
tical components of the force, respectively.
ka Kc
0.05 0.50
0.07 0.75
0.10 1.00
0.12 1.25
0.14 1.47
F(°) F(°) F(1) F(1) F(2) F(2)
0.00 0.04 1.07 1.07 0.07 0.07
0.00 0.10 1.58 1.58 0.15 0.15
0.01 0.16 2.07 2.06 0.24 0.25
0.02 0.24 2.52 2.51 0.33 0.34
0.03 0.30 2.88 2.87 0.41 0.41
3.00 r
2.50
_ 2.00
1 .50
I) 1.00
Table 2: Diffraction loads Sindbad O 50
As in t3] we introduce the KeuleganCarpenter number,
I{c. For a linear deepwater wave, Kc is given by:
Kc =—exp~kyc). (18)
Table 2 gives the values of F(n) and F(n) for n = 0, 1, 2 vs.
Kc in the case just described (that corresponds to Chaplin
case E t34~.
Following Chaplin, we write:
F(n) = ~ CxnmKc,
F(n) =
m>1
~ CynmI{c
m>1
The classical inertia coefficients are equal to Call et
Cysts According to linear theory, these are the only non
zero coefficients and they are equal. Ogilvie [25] calculated
them; they go to 2 as the immersion depth goes to infinity.
We give on figure 9 the horizontal and vertical inertia
coefficients vs. Kc. For a small value of the Keulegan
Carpenter number, both experimental and numerical re
sults go the value predicted by linear theory, 2.25. As I(C
increases, however, the sharp decrease of the inertia coeffi
cient observed experimentally is not predicted by the sim
ulation. As argued by Chaplin, this is very likely due to
viscous effects (creation of a circulation around the cylin
der).
Figures 10 and 11 give the secondorder vertical drift
force and the force at the double frequency, respectively.
Here, a good agreement appears between experiments, sec
ondorder theory and the present simulation. It should
be stressed that recovering the results from secondorder
steady theory with the simulation is by no means obvi
ous: it demands a good accuracy and a proper control of
the wave generation and absorption mechanism. It can be
argued that the results of the simulation are better than
those of secondorder theory for large values of Kc.
4.4.2 Diffracted Waves
Grue and Granlund t17] performed experiments related to
incoming deepwater Stokes waves passing over a restrained
submerged circular cylinder. For a small cylinder submer
gence, a strong local nonlinearity is introduced at the free
surface above the cylinder and free higher order harmonic
waves are generated.
We remind of the observations of Grue and Granlund quency
246
0.00
° `0~ x
00 t~
dO R
oEXPERIMENTS (CHAPLIN)
xSINDBAD
_UNEAR THEORY (OGILVIE)
.
0.00 0.50
x x
 do
0 to 0
e8 0
1 .00
Kc
=
1.50 2.00
Figure 9: Diffraction loads inertia coefficients
(19)
(20)  1 .O
~—2.0
I_
' 3.0
—4
oEXPERIMENTS (CHAPLIN)
xSINDBAD '/
_ SECOND—ORDER (OGILVIE) ~~ x
d.
Fox
Otto
0/
~0
~4
ye
mix
_ _ . . . . . . .
—2.00  1.50  1.00  0.50 0.00 0.50 1.00
In(Kc)
Figure 10: Diffraction loads vertical drift force
—1 .0
_ 
c~
~ 2.0
cod
x—3.0
_'
4.0
5.0
oEXPERIMENTS (CHAPLIN) /
xSI NDBAD /
_SECOND—ORDER (VADA) ~ Coax
a'
<~ don
Ox
y
—2~00  1.50  1.00  0.50 0.00 0.50 1.00
In(Kc)
Figure 11: Diffraction loads— response at double fre
OCR for page 239
0.06 
0.05 
0.04 
cat 0.03
c,
0.02
0.01
~~ ° x x
Fax
0.00 .k . . .
0.00 0.05
/ oEXPERIMENTS (GRUE)
/ XSINDBAD
/ _SECOND—ORDER (VADA)
o o
.
. . . . . . .
0.10 0.15 0.20
~ k
Figure 12: Diffracted waves amplitude of secondorder
free wave
concerning the wavefield far away from the cylinder:
* upstream of the cylinder: an incoming Stokes wave
train. No reflected waves, even to higher order;
* downstream of the cylinder: shorter free second har
monic waves of considerable amplitude are riding on the
transmitted Stokes wave.
If these trends are well predicted by secondorder the
ory [32i, the quantitative agreement with experiments is
rather disappointing. The amplitude of the second har
monic free wave, as, only increases as the square of the
amplitude of the incoming wave, a, for very small values of
a. A "saturation" rapidly appears; thereafter a2 remains
almost constant see figure 12.
These findings suggest that the range of validity of
secondorder theory is quite narrow in this case. Observ
ing that this theory predicts amplitudes of the secondorder
free wave as large as that of the incoming wave, this should
not appear as totally unexpected. In order to see if non
linear free surface effects and not viscous effect are,
indeed, responsible for this deficiency, it appeared interest
ing to run Sindbad on this case.
In order to compare our results with those of Grue and
Granlund, we write (once a steady state is reached):
* for the incident wave:
Hi = a costed—fit+ 8)
+ at cos 2(kx—cat + 8) + · · ·, (21)
where at is the amplitude of the secondorder locked wave;
* for the diffracted wave:
y~ = al costed—cot + 8~)
+ ad cos 2(kx  At + 8~ ~
+ a2 cos(4kx—Act+ 82~+ ·, (22)
where ad and a2 are the amplitudes of the secondorder
locked and free wave, respectively.
In order to exhibit the secondorder free wave, it is nec
essary to take a rather fine grid. The wavenumber of this
free wave is, indeed, four times larger than the wavenum
ber of the incoming wave. The wavenumber of the incoming
wave is chosen equal to ~ = 3.42 (so that we are in deep
3.0 ~
....SINDBAD: LINEAR CALCULATION
_SINDBAD: ok = 0.03
_SINDBAD: ok  0.09
2.0
c,
1.0 i
0.0 
1 .0
—2.0 .
0.0
x
3 C
On
Figure 13: Diffracted waves free surface profiles
5.0
water). With these choices, the results shown by Grue
and Granlund correspond to a cylinder radius r = 0.117
and a depth of immersion of the center of the cylinder
Yc = 0.1755.
The simulation was perfomed in a tank of length 8,
with a damping zone of length 3. The cylinder center was
located at a distance xc = 2 from the wavemaker. 340
nodes were distributed on the free surface and 60 time steps
used per period of the incoming wave.
On figure 13, free surface profiles after 7 periods are
shown for several amplitudes of the incident wave. The
apparition of a perturbation of wavenumber 4k is obvious.
However, its amplitude does not increase as the square of
the incident wave amplitude.
A Fourier analysis of the diffracted wave confirms this
trend. We show on figure 12 the comparison between Grue
and Granlund's experiments, Vada's secondorder theory
and the present calculation. The agreement between the
numerical simulation and the experiments is very good, in
dicating that the "saturation" is, indeed, a nonlinear free
surface phenomenon not accounted for by secondorder the
ory.
Grue and Granlund observed breaking for ah ~ 0.085,
while we were able to perform the numerical simulation up
to ah = 0.12. It is rather interesting to note that this does
not seem to affect the amplitude of the secondorder free
wave.
A little more surprising is the reason for which the nu
merical computation fails for ah = 0.12 after 3.2 periods.
The computation does not blow up because of the overturn
ing of the crest, as would have been expected, but because
of a concentration of particles just aft the cylinder. Phys
ically, it seems that particles flow very rapidly over the
cylinder and are then decelerated. Here again, the validity
of the simulation is difficult to establish.
It is rather interesting to note that for waves passing
over a submerged cylinder, nonlinear free surface effects
are important for the diffracted waves but do not seem to
affect very much the forces.
247
OCR for page 239
0.01 
CY
o
Lo
.$ 0.00
C~
F
>
0.01  at_ T ' I
0 0 2.0 4.0 6.0 8.0 10.0 12.0
TIME
Figure 14: Forced heaving transient force
4.5 Wave Radiation by a Free Surface Pierc
ing Cylinder
As a last example, we consider the case of a free surface
piercing body in forced or free motion.
4.5.1 Forced Motion
Forced motions of a free surface piercing circular cylin
der have been studied quite extensively using linear and
secondorder theory (e.g., t26i, t19~. Fully nonlinear cal
culations have also been attempted for forced heaving of
a circular cylinder by a few authors. Initially (~13i, [333)
only the starting phase was considered. Recently, Hwang
et al. t18] calculated the steady state response and made
comparisons with first and secondorder theories and with
experiments.
As an example, we consider the case of a halfimmerged
circular cylinder with kr = 1. The simulation is performed
in a tank of length 4 with a forcing frequency ~ = 3.16 (so
that k = 10~. The cylinder center has for elevation:
Yc = 0 t < 0
Yc = ac sin(wt) t > 0.
(23)
(24)
Because there is no wavemaker in this case, absorbing
beaches (with cat = ~ = 1) are located at each end of the
tank. We used 200 nodes on the free surface. In order to
avoid too small or too large segment sizes near the inter
section, a regridding procedure similar to that introduced
in t11] was used when a node came too close or too far from
the intersection.
We show on figure 14 the transient vertical force on
the cylinder as a function of time for ac = 0.5 r. If a
steady state is rapidly reached, the signal is obviously not
monochromatic and harmonics are present.
The free surface profiles corresponding to this case are
shown on figure 15.
Once a steady state is reached, we write the force as:
Fy + 2r Yc
0.51r r2 ac w2
y t .
x
~ ' it/
f rr _/
f ~~ an,
Figure 15: Forced heaving free surface profiles
The second term on the lefthand side is the linear hydro
static contribution. The accelerationphase and velocity
phase components of the force at the forcing frequency
would correspond to the added mass and damping coef
ficients for the linear problems.
LIT
F(°) F(l a) F(l b) Fy(2) F(3) F;,(4)
0.l 0.02 0.61 0.39 0.06
0.2 0.03 0.62 0.38 0.12 0.02
0.3 0.05 0.63 0.36 0.18 0.04 0.02
0.4 0.07 0.65 0.35 0.23 0.08 0.05
0.5 0.10 0.66 0.33 0.30 0.14 0.09
Table 3: Radiation loads Sindbad
We give on table 3 the amplitude of the different har
monics as a function of ~ = aC/r. The Fourier analysis is
performed on the four last periods of the signal. For small
values of c, the results from linear and secondorder theory
are recovered (e.g., t26i, t19~. As the amplitude of the mo
tion increase, the added mass increases while the damping
coefficient decreases. This behavior is in agreement with
available experimental observations t19] and other numer
ical results t184.
The importance of relatively highorder harmonics is
quite remarkable. For ~ = 0.5, the ratio of the amplitude
of the fourth harmonic to that of the first is almost equal to
15~o. This behavior, that obviously cannot be accounted
for using secondorder theory, is quite different from that
observed for the diffraction over a submerged cylinder. It
shows the interest of a fully nonlinear simulation, in partic
ular in order to assess the range of validity of approximate
theories. If these results for forced motions are very promis
ing, more experimental data would be needed to make pro
prer comparisons.
Note that for ~ = 0.6 the numerical computation breaks
down before a steady state is reached, apparently because
the cylinder goes out of the water.
=
F(°) + F(1a) sin(wt)—F(1v) cos(wt)
oo
~ F(n) sin(nc`;t + 8(n)).
rl=2
(25)
7 Note that for the nonlinear problem the distinction between
addedmass and hydrostatic components is somehow arbitrary.
248
OCR for page 239
2.0 1
1.0
0.0
_SINDBAD (YO = 0.9 r)
LINEAR THEORY (URSELL)
_ 1
—1 .0
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
T
Figure 16: Free heaving vertical displacement
4.5.2 Free Motion
The free motion of a surface piercing circular cylinder was
calculated using linear theory in [23~. A fully nonlinear
computation was performed in [33] using periodic boundary
conditions. A good agreement between these two theories
was achieved.
Here, we performed again the same calculation. The
body motion is calculated in a way similar to that used in
t33] (see t7] for details on the numerical implementation).
At t = 0, the fluid is at rest and the cylinder, which is
halfimmerged at equilibrium, has an elevation:
Yc(0) = Yo. (26)
We show on figure 16 the subsequent vertical displace
ment of the cylindre, tic, for yo = 0.9 r. The simulation is
performed exactly in the same conditions as in the preced
ing section. The absence of any nonlinear behavior for the
cylinder elevation is rather striking and contrasts with the
forced motion case.
5 Conclusions
The MEL has been applied to the numerical simulation of
a wave tank equipped at one end with a wave maker and at
the other end with a damping zone. This configuration al
lows twodimensional wavestructure interaction problems
to be studied.
Results have been presented that include a wide range
of applications, such as wave generation and absorption,
wave diffraction and wave radiation by submerged or sur
face piercing bodies in forced or free motion. Results from
approximate theories (linear, secondorder or shallow wa
ter theory) can be recovered, not only for short transient
evolutions but also for steady state responses. Some non
linear phenomena observed experimentally and that have
not been otherwise explained are also accounted for. This
seems to indicate that the simulation can be used as a
"standard" that allows the validity of approximate theo
ries to be assessed. Applications of the method in that
direction suggest themselves. In particular, the roll motion
of barges is being presently studied.
249
If the MEL provides an efficient and versatile way to
study twodimensional free surface problems, it cannot ac
count for viscous effects (except in a very crude way for
instance for the modeling of bottom friction). This is par
ticularly a problem for viscous effects occurring in the near
vicinity of the free surface, i.e., viscous effects associated
to breaking. Breaking is a major limitation for the simu
lation because the calculation has to stop whenever a local
breaking event occurs.
It would therefore be most useful to be able to simulate
breaking and associated dissipative effects, even in a crude
way. Some hope exists (e.g., [30i, t44) for spilling break
ers, but there is an obvious need for more theoretical and
experimental work on the subject.
6 Acknowleclgements
This work is a result of research sponsored in pant by the
"Ministere de la Defense", DRET, under convention num
ber 88/073. This support is gratefully acknowledged.
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