|
Items in cart [0] |
Questions? Call 888-624-8373 |
| Copyright © 2009. National Academy of Sciences. All rights reserved. Terms of Use and Privacy Statement |
Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 239
Nonlinear Simulation of Transient Free Surface Flows
R. Cointe
Institut Frangais du Petrole
Rueil-Malma~son, France
1 Abstract
The application of the Mixed Eulerian-Lagrangian 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 two-dimen-
sional potential free surface flows using a Mixed Eulerian-
Lagrangian method (MEL) has received considerable at-
tention since the pioneering work of Longuet-Higgins and
Cokelet t224. If many codes exist now that use this method,
their suitability for the study of nonlinear fluid-structure
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 second-order 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 240
Y ~Dnu~nerico/u~or~a Bank l g-1
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 x-axis coincides
with the reference position of the free surface and the y-axis
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.
Piston-type Tested body
wave maker
Figure 1: Sindbad geometric definitions
240
h-l
"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 right-hand 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 time-stepping procedures, such
as a fourth-order Runge-Kutta algorithm.
The main numerical difficulty is to be able, at each
time-step, 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 241
OCR for page 242
OCR for page 244
OCR for page 245
OCR for page 246
OCR for page 247
OCR for page 248
OCR for page 249
OCR for page 250
Representative terms from entire chapter:
linear solution
3.2 Numerical Treatment at the Intersections
At each time-step, 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
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 steady-state response
while the simulation is transient. Discrepancies might be
due to long-lasting 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 second-order 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 quasi-analytical solutions of the transient problem
to be derived at first- and second-order 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 second-order
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 Piston-Type 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 tine-step,
does not significantly alter the accuracy of the method.
243
From this table, it appears that:
* a sufficiently small time-step has to be used in order
for the numerical scheme not to blow up. The influence of
the time-step is otherwise rather small;
* for a given spatial discretization, results converge as
the time-step 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 second-order in the grid size. This is very likely
due to the numerical treatment at the intersection that is
not second-order4.
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.
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
long-lasting 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 bi-harmonic 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 piston-type 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 CRAY-XMP.
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
4.3 Arbitrary Motion of a Piston-Type 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 CRAY-XMP 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 wave-structure 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 second-order ver-
tical drift force. Very recently, Vada t32] computed the
second-order potential and calculated the diffraction loads
and the diffracted waves to second-order.
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 second-order 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 second-order 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 second-order 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 piston-type 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 time-steps 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 second-order theory, we use a Fourier
series expansion of the transient signal (once a steady-state
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
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 Keulegan-Carpenter number,
I{c. For a linear deep-water 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 second-order vertical drift
force and the force at the double frequency, respectively.
Here, a good agreement appears between experiments, sec-
ond-order theory and the present simulation. It should
be stressed that recovering the results from second-order
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 second-order 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-
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 second-order
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 second-order 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
second-order theory is quite narrow in this case. Observ-
ing that this theory predicts amplitudes of the second-order
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 second-order 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 second-order
locked and free wave, respectively.
In order to exhibit the second-order 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 second-order 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 second-order 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 second-order 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
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
second-order 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 second-order theories and with
experiments.
As an example, we consider the case of a half-immerged
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 left-hand side is the linear hydro-
static contribution. The acceleration-phase 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 second-order 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 high-order 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 second-order 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
added-mass and hydrostatic components is somehow arbitrary.
248
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
half-immerged 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 two-dimensional wave-structure 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, second-order 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 two-dimensional 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.
References
t1] BAKER, G.R., MEIRON, D.I., and ORSZAG, A.,
1981, "Applications of a generalized vortex method
to nonlinear free-surface flows," Third International
Conference on Numerical Ship Hydrodynamics, Paris,
pp. 179-191.
t2] CHAPALAIN, G., 1988, "Etude hydrodynamique et
sedimentaire des environnements littoraux domines
par la houle," These de Doctorat de l'Universite
Joseph Fourier, Grenoble (in French).
t3] CHAPLIN, J.R., 1984, "Nonlinear forces on a horizon-
tal cylinder beneath waves," J. Fluid Mech., Vol. 147,
pp. 449-464.
t4] COINTE, R., 1987, "A theory of breakers and break-
ing waves," Ph. D. Dissertation, University of Califor-
nia, Santa Barbara.
t5] COINTE, R., 1988, "Remarks on the numerical treat-
ment of the intersection point between a rigid body
and a free surface," Third International Workshop on
Water Waves and Floating Bodies, Woods Hole, Mass.
t6] COINTE, R., 1989, "Calcul des efforts hydrody-
namiques sur un cylindre horizontal en theorie
potentielle non-lineaire", Deuxiemes Journees de
l'Hydrodynamique, Nantes, pp. 89-104 (in French).
t7] COINTE, R., 1989, "Quelques aspects de la simulation
numerique d'un canal a houle", These de Doctorat de
l'Ecole Nationale des Ponts et Chaussees, Paris (in
French).
t8] COINTE, R., JAMI, A., and MOLIN, B., 1987,
"Nonlinear impulsive problems," Second International
Workshop on Water Waves and Floating Bodies, Bris-
tol.
[9] COINTE, R., MOLIN, B., and NAYS, P., 1988, "Non-
linear and second-order transient waves in a rectangu-
lar tank," BOSS'88, Trondheim.
t10] DOLD, J.W., and PEREGRINE, D.H., 1986, "Water-
wave modulation," Bristol University, Report AM-86-
03.
t11] DOMMERMUTH, D.G., 1987, "Numerical methods
for solving nonlinear water-wave problems in the time
domain," Ph.D. Dissertation, MIT.
t12] DOMMERMUTH, D.G., YUE, D.K.P., RAPP, R.J.,
CHAN, E.S., and MELVILLE, W.K., 1988, "Deep-
water plunging breakers: a comparison between poten-
tial theory and experiments," J. Fluid Mech., Vol. 189,
pp. 423-442.
t13] FALTINSEN, O.M., 1977, "Numerical solutions of
transient nonlinear free surface motion outside or in-
side moving bodies," 2nd International Conference on
Numerical Ship Hydrodynamics, Berkeley, pp. 347-
357.
t14] FALTINSEN, O.M., 1978, "A numerical method of
sloshing in tanks with two-dimensional flow," Journal
of Ship Research, Vol. 22, pp. 193-202.
t15] GREENHOW, M., 1987, "Wedge entry into initially
calm water," Applied Ocean Research, Vol. 9, pp. 214-
223.
[16] GRISVARD, P., 1984, "Problemes aux limites darts
les polygones mode d'emploi", Prepublication du
laboratoire de mathematiques de l'Universite de Nice
(in French).
[17] GRUE, J., and GRANLUND, K., 1988, "Impact of
nonlinearity upon waves traveling over a submerged
cylinder," Third International Workshop on Water
Waves and Floating Bodies (Woods Hole).
t18] HWANG, J.H., KIM, Y.J., and KIM, S.Y., 1988,
"Nonlinear Hydrodynamic Forces Due to Two-
dimensional Forced Oscillation," Proceedings, IU-
TAM Symposium on Nonlinear Water Waves (Tokyo),
Springer-Verlag, pp. 231-238.
[19] KYOZUKA, Y., 1982, "Experimental study of second-
order forces acting on a cylindrical body in waves,"
Proceedings, 14th Symposium Naval Hydrodynamics,
Ann Arbor.
t20] KRAVTCHENKO, J., 1954, "Remarques sur le Calcul
des Amplitudes de la Houle Lineaire Engendree par un
Batteur", 5th Conf. Coastal Eng., Grenoble, Fiance,
pp. 50-61 (in French).
t21] LIN, W.M., 1984, "Nonlinear motion of the free sur-
face near a moving body," Ph. D. Dissertation, MIT,
Cambridge, Mass.
[22] LONGUET-HIGGINS, M.S., and COKELET, E.D.,
1976, "The deformation of steep surface waves
on water. 1. A numerical method of comptation,"
Proc. R. Soc. London, Vol. A 364, pp.1-26.
t23] MASKELL, S.J., and URSELL, F., 1970, "The tran-
sient motion of a floating body," J. Fluid Mech.,
Vol. 44, pp. 303-313.
t24] MEI, C.C., and UNLUATA, U., 1972, Harmonic gen-
eration in shallow water waves," Waves on beaches,
R.E. Meyer, Ed., Academic Press, pp. 181-202.
t25] OGILVIE, T.F., 1963, "First- and second-order forces
on a cylinder submerged under a free surface," J. Fluid
Mech., Vol. 16, pp. 451-472.
t26] PAPANIKOLAOU, A., and NOWACKI, H., 1980,
"Second-order theory of oscillating cylinders in a reg-
ular steep wave," Proceedings, 13th Symposium Naval
Hydrodynamics, Tokyo.
t27] PEREGRINE, D.H., 1972, Unpublished note.
t28] ROBERTS, A.J., 1987, "Transient Free Surface Flows
Generated by a Moving Vertical Plate," Q. J. Mech.
Appl. Math., Vol. 40, Pt. 1.
[29] STANSBY, P.K., and SLAOUTI, A., 1984, "On non-
linear wave interaction with cylindrical bodies: a vor-
tex sheet approach," Appl. Ocean Res., Vol. 6, pp.
108- 1 15.
[30] TULIN, M.P., and COINTE, R., 1987, "Steady and
unsteady spilling breakers: theory," Proceedings, IU-
TAM Symposium on Nonlinear Water Waves, Tokyo,
pp. 159-165.
[31] URSELL, F., 1950, "Surface waves in the presence of
a submerged circular cylinder, I and II," Proc. Camb.
Phil. Soc., Vol. 46, pp. 141-158.
t32] VADA, T., 1987, "A numerical solution of the second-
order wave-diffraction problem for a submerged cylin-
der of arbitrary shape," J. Fluid Mech., Vol. 174, pp.
23-37.
t33] VINJE, T., and BREVIG, P., 1981, "Nonlinear, two-
dimensional ship motions," Norwegian Institute of
Technology, Rapport R-112.81.
250
