Ship Hydromechanics laboratory
Library
Mekelweg 2 26282 CD DeIft Phone: +31 (0)15 2786873 E-mail: p.w.deheertudelft.nlTime-Domain Calculations
of
Ship Motions
Time-Domain Calculations
of
Ship Motions
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Deift,
op gezag van de Rector Magnificus Prof. ir. K.F. Wakker, in het openbaar te verdedigen ten overstaari van een commissie,
door het College voor Promoties aangewezen, op dinsdag 17 februari 1998 te 16.00 uur
door
Louisette Merete SIEREVOGEL,
wiskundig ingenieur,
Prof. dr. ir. A.J. Herrnans
Samenstelling promotiecomrnissie:
Rector Magnificus,
Prof. dr. ir. A.J. Hermans,
Prof. D.V. Evans,
Prof. dr. ir. G. Kuiper, Prof. dr. H.G. Meijer, Prof. dr. ir. LA. Pinkster, Prof. dr. ir. P.J. Zandbergen,
Dr. ir. J,E.W. Wichers,
Financia1 support by the Maritime Research Institute, the Netherlands, is gratefully acknowledged.
Copyright © by Lisette Sierevogel, 1997
ISBN 90-9011329-O
voorzitter
Technische Universiteit Deift, promotor
University of Bristol
Technische Umversiteit Deift Technische Universiteit Deift Technische Uruversiteit Deift Universiteit Twente
Summary
Time-Domain Calculations of Ship Motions
When designing a new ship, naval architects are interested in the forces acting on the ship and its motions. To reduce the number of expensive experimental techniques, engineers and mathematicians try to calculate these forces and
mo-tions. In this thesis, a numerical model is extended. The model computes in the time domain the forces due to speed, current and waves on ships with
small forward speed. The physical problem is linearized under the assumption that the motions and waves are small. The model is able to compute the hy-drodynamic coefficients, the first-order forces, the motions and the wave-drift forces and damping, and gives good results in comparison to results shown in
the literature.
The numerical algorithm requires an artificial boundary. In this thesis, a new absorbing boundary condition is developed. This condition, based on the semi-discrete DtN method, is independent of the wave frequency and the artificial boundary can be placed about one wavelength away from the object. Because
of the frequency independence it is possible to compute the hydrodynamic
coef-ficients using general time signals. This reduces the computing time
consider-ably.
To test the model, both a two- and a three-dimensional simple problem is
considered: a cross section of a circular cylinder of infinite length and a float-ing hemisphere. We test, among others, the influence on the motions of water
depth and of small forward speed, using a harmonic or general time signal. The results agree well with other computations or measurements, as far as these
have been carried out. In two dimensions, we verify the assumption that the wave-drift damping can be computed by the derivative of the drift forces to the velocity. And in three dimensions, we verify the engineering view to calculate the wave-drift damping.
The stability analysis for the two-dimensional test problem shows that the grid
and time-step size are chosen sufficiently accurate and that instabilities will
ap-pear for a combination of high speed, small panel Froude number and central
discretization of the x-derivatives. Using upwind discretization instead will
make the instabilities disappear. The two-dimensional test problem confirms this analysis.
Because of the satisfying results of the test problems, we also apply our model to real vessels, to a supertanker and to an LNG carrier. For the first one it is interesting to look at the moored situation. That means current with a low ve-locity and waves from different directions. For the LNG carrier it is interesting to look at situation of service speed. We study the influences of water depth and different angles of incoming waves on the forces arid motions of the su-pertanker. The results are satisfactory, as far as we are able to compare them with other computations or measurements. The results for oblique incoming waves are sometimes less accurate, probably because 01: the viscous effects of
the roll damping. The engineering view to compute the wave-drift damping does not give satisfying results for the tanker.
We studied the influence of the bulbous bow for an LNG carrier. Smoothing this bulbous bow influences the drift forces especially near the peak.
Further-more we can conclude that the results for forward speed are Further-more accurate when we use a Dawson-hull instead of interpolating the steady potential on
another mesh. When extending ta higher speed, the model seems to give good results, but the derivatives on the hull have to be computed very accurately, in particular for the second-order forces. The derivatives are required to be more accurate than can be done by using our mesh with flat panels.
Five suggestions for further research are the following. Firstly, the influence
of oblique incoming current can be computed with a little extension of the
computer code. lt is recommended to check the viscous roll damping again. Secondly, the use of B-splines instead of a finer grid is recommended to obtain more accurate results. B-splines have to be used ori the hull and maybe on the free surface as well. Thirdly, it is not recommended to extend this model to get more accurate results for higher speeds. For the higher speed approach, raised panels in combination with a nonlinear steady potential have to be used as a basis for the diffraction problem. Fourthly, to compute the yaw moment the
additional terms have to be studied more accurately. Bui: we have to remember that for the yaw moment the viscous effects are the most important, so that the necessity of computing the yaw moment has to be questioned. Fifthly,
an extension to nonlinearity can better be done by an extension of this linear model than by trying to compute the problem in a fully nonlinear way, because of the large increase in computing time in order ta get accurate and interesting results.
Contents
V Summary 111 List of symbols vu Introductioni
Backgroundi
Solving the problem 3
Outline of this work 6
I General theory 7
1.1 Overview 7
1.2 Mathematical model 12
1.3 Numerical model 20
2 Absorbing boundary condition 23
2.1 introduction 23
2.2 Overview 24
2.3 Senm-discrete DtN method (orthogonal functions) 26
2.4 2D Semi-discrete DtN method (boundary-integral) 31
2.5 3D Semi-discrete DtN method (boundary-integral) 37
3 Two- and three-dimensional test problems 43
3.1 Introduction 43
3.2 Cylinder of infinite length 44
3.3 Floating hemisphere 55
4 Stability analysis 65
4.1 Introduction 65
4.2 Continuous Fourier Transform 67
4.3 Discrete Fourier Transform 69
4.4 Evaluation of the discretization schemes 72
4.5 Discussion on temporal stability 82
5 Results 85 5.1 Introduction 85 5.2 200kDWT tanker 87 5.3 LN4G carrier 103 Conclusions 111 A Green functions 115
A.1 Inñffltedepthin3D
115A.2 Finite depth in 3D 117
B Discrete Fourier Transforms 119
B.1 TheGreenfunction 119
B.2 The derivatives 121
Bibliography 123
Samenvatting (Summary in Dutch) 127
Curriculum vitae (In Dutch) 131
List of symbols
The list of symbols gives an explanation of symbols which are used in different places in this thesis. Symbols only used once are explained close to the equation where they are used in.
(i) k X 2D 3D U ß
(
Ça ¡2 pr
(2) time derivativemean or asymptotic value continuous Fourier transform (in chapter 4 only)
discrete Fourier transform
i-th order in direction k partial derivative two dimensional three dimensional total displacement angle of incident wave small parameter surface elevation
wave height incoming wave wavelength constant in exterior (2/(g(At)2)) second coordinates second coordinates fluid density non-dimension parameter (Uw/g) unsteady potential
unsteady part of unsteady potential q5 (2) Wo W' Wi W2 WC Wd W
çI
aki bkf
h steady potentialsteady part of unsteady potential frequency of encounter frequency scaled frequency of encounter scaled frequency
frequency of slow oscillation
continuous frequency discrete frequency
non-dimensional parameter (wAt/ (27r))
displacement of vessel panel size on free surface panel size on hull
time step panel size in 2D velocity potential rotation relative to numerical damping numerical dispersion added-mass coefficient damping coefficient free surface water depth V!I
v)
coefficients of difference schemeD
matrixE
matrixF
force Fm Froude number(U/v"Jor U/../Lg)
Fn
panel Froude numberG Green function
fl
hull L length of vessel M momentM
mass matrix O order ;ymbo1R radius of cylinder or sphere
RHS right hand side
S fluid domain
T draught of vessel
T period of forced oscillation U forward speed
W free-surface operator
X translation relative to OE9
(Ï,)
g gravitational acceleration
k wave number
k continuous wave number kd discrete wave number k non-dimensional parameter
(kx/(27r))
n normal vector p pressurer
distance t time Zg centre of gravity X coordinates2yz coordinates
A added mass matrix
B damping matrix B breadth of vessel
B1 wave-drift damping 13 absorbing boundary
C
restoring force matrix CI numerical dampingCR numerical dispersion
Introduction
This introduction explains the contents of this thesis, why the forces on a vessel are calcu-lated, what kind of forces there are, and how they are calculated. Readers familiar with this subject are advised to omit this introduction, except the outline. On the other hand readers not familiar with this subject or with applied mathematics are advised to read only this chapter and omit the rest.
Background
Figure 1.1: A model test at the MARIN.
When designing a new ship, naval architects are interested in the forces acting on the ship and its motions. Therefore, in large basins, as those at the M
ari-time Research Institute Netherlands in Wagemngen, the Netherlands, scale
models of ships are used to measure the forces and motions, see figure 1.1.
These experimental techniques are expensive. Therefore engineers and math-ematicians started to calculate the forces and motions mathematically a century ago. From the 1960's the calculations have been done with the help of digital computers. Nowadays computer codes are still not able to replace the model tests, but they can give a first indication of the behaviour of a ship.
Before making a computer code you have to know what kind of problem you are
going to solve, in this case what kind of ship you are going to study, and what the physics of your problem is. Firstly, we mention tha: we restrict our
prob-lem to large vessels, as oil- or gas-tankers. Consequently we do not handle
sailing boats, catamarans or speedboats. The vessel's velocity in proportion to its length is very low. Secondly, we have to determine what kind of forces are
acting on such a vessel, and what kind of forces we are interested in. When such
a large vessel sails on the ocean, several forces are acting on it. For instance forces due to the wind, to the engine, to the forward speed, to the current and to the waves.
Because of the complexity it is not possible to solve the 'problem with ail those
different forces. Therefore we assume that we can split the problem into
in-dependent parts. In this thesis, we look at the last three mentioned forces.
And because a ship sailing with forward speed in one direction is exactly the same as a ship with no forward speed in a current in the other direction, there are only two forces left: due to the current and due to the waves. The waves generated by these forces are shown in figure 1.2.
Figure 1.2: Waves due to different forces on the ship (vertically scaled).
The waves due to the current are stationary waves, because they are independ-ent of time. This means that if the currindepend-ent stays the same, the waves due to
this current do not change in time. The waves due to the incoming waves are instationary, thus dependent on time.
The incoming waves cause first-order wave forces, these: are forces of the same
order as the waves, which means that they are proportional to the wave height
and have the same frequency. This can be imagined as the motion of a ship together with the waves. On the other hand the waves cause second-order
forces, the so called wave-drift forces. These forces cause a ship to drift away. lt is easy to imagine that wind or current cause a ship to drift away, but how can waves? Pinkster (1980) illustrates the drift forces due to waves in one of
his theses, stellung IV:
'Playing kids throwing a baH in the water often use second-order wave-drift forces to float the ball within the range again. This ¡s, till now on, the only useful use of this
phenomenon, which is known?
Solving the problem 3
Prins (1995) explains that the stones thrown at the ball generate waves which cause the ball to drift to the shore. Prins also gives another example of wave-drift forces; a ship lying at anchor into waves. Due to the waves the ship will rotate until it lies in head waves.
These wave-drift forces are second-order forces because they are proportional
to the square of the wave height, which is smaller if the wave height is
as-sumed to be small. And these wave-drift forces oscillate with low frequencies.
This is important to moored constructions, like tankers used for exploration
and production. The natural frequency of those constructions is low, and the damping of the tankers is low as well, because most of the tankers are designed for sailing. Combining this with the natural frequency of the wave-drift forces, leads to large amplitude resonant behaviour of the motions, see ñgure 1.3.
5m
o
5m
20m a. Record of an irregular sea.
o
Pn'
rJj
2Onb 100
Time insec. 200 300 b. Record of the horizontal surge motion.
Figure 1.3: Low frequency surge motions of amooredvessel, a Liquid Natural Gas carrier, in irregular head waves, from Pinkster (1980).
lt turns out that the effect of the water depth and the direction of the incoming wave is important for the drift forces, and that the combination of current and
waves also influences the drift forces. In this thesis, we will study the influences
of all these aspects.
Solving the problem
To calculate the physical problem described above, i.e. the forces due to
cur-rent and waves, we have to translate the physics into mathematics.
Unfor-tunately, the problem is still too complex to solve analytically. Therefore we have to make assumptions and simplifications.
Firstly, we linearize the problem. This means that we assume the waves to be small due to both the current and the incoming waves, and that we assume the waves to be harmonic, like a sine curve. Therefore we cannot handle breaking waves. Secondly, we solve our problem numerically. Therefore we discretize,
that means divide into panels, the water surface and the hull, see figure 1.4.
On each flat panel we assume the quantities, like pressure and forces, to be constant.
a. The water surface, b. The hull (under the water line). Figure 1.4: Panel distributions (different scales).
We also discretize the time, thus we do not assume the time to be continuous, but we use time steps. We compute the situation on every next time step on
the basis of the situation on the previous time steps. The calculations with
time steps are called calculations in the time domain. lt is dear that the accur-acy of the model is dependent on the panel size and time-step size; the smaller
both are, the more accurate, but also the more time consuming the model is.
Our main goal is to compute the forces acting on the hull. We do this by adding
the forces on every panel of the hull. The force on a panel is computed with the help of the pressure on that panel, which is computed on the basis of the velocity potential. This potential has no physical meaning, but is a
mathem-atical resource to compute the pressure. The derivative of the potential to a
direction is the velocity in that direction.
We compute the potential using a boundary-integral method, called Green's second theorem. Instead of solving the three dimensional problem, a
bound-ary-integral method only uses the boundaries, thus the solver is reduced to
two dimensions. Green's second theorem needs a Green function and a closed contour, then the potential can be computed on every panel. The Green
func-tion has to satisfy some physical condifunc-tions, says something over the influence of one panel to another and is dependent on the distance between two panels. Different Green functions can be used, for instance one over 4ir times the dis-tance between two panels, 1/(4irr).
The closed contour can be realised by taking the hull, the water surface, the bottom and a wall on a finite distance of the hull, see the cross section ¡n fig-ure 1.5. The problem with this closed boundary is the wall, called the artificial boundary, 5.
This boundary does not exist in the real problem, or it is infinitely far away. If the boundary in the mathematical model is also infinitely far away, we need very many panels on the water surface and that takes a lot of computer time and memory. On the other hand the use of a wall dose to the hull means less panels, but will cause reflections and is therefore not representing the real sea
Solving the problem 5
like a sponge. This boundary is called an absorbing boundary. In this thesis, we develop a new absorbing boundary, which is efficient and independent of the wave frequency.
Using the Green function and the
closed contour in the boundary-in- water surface t
tegral method, we compute the ve-
)
locity potential. To compute the iu11
forces and motions, we first
corn-pute the hydrodynarnic coefficients. water
We do this by forcing the vessel to
b
oscillate in the three different trans- Ottern
lations and the three different ro-
-- Figure 1.5: The closed boundary
tations. Then we can compute the
added mass and damping due to the forced motions. Secondly, we fix the
vessel and compute the first-order forces on the vessel due to the incoming
waves. Combining the hydrodynamic coefficients and the first-order forces we compute the motions and second-order forces.
Alter developing the mathematical model and writing the computer code, we
firstly tested the model on two simple problems. The advantage of a simple shape is that the hull is described mathematically and therefore no errors are made in the description of the hull. We first test the model on a two-dimensional test problem, a cross section of a circular cylinder of infinite length,
see figure I.6.a. This model only needs a few panels and can therefore be com-puted very quickly. And subsequently on a three-dimensional test problem, a floating hemisphere, see figure l.6.b.
a. Two dimensional.
Figure1.6: The test problems.
b. Three dimensional.
Because of the satisfying results of the test problems, we finally applied our model to two real vessels. We chose a super tanker and a Liquid Natural Gas carrier, because we have a great number of measurements and other
calcula-tions of both vessels to compare with. For the super tanker it is interesting to look at the moored situation. That means current with a low velocity and
max-imum 2m/s, this is Froude number equals 0.035. The Froude number is a
non-dimensional quantity representing the ship velocity in proportion to its
length. lt is also interesting to look at the wave-drift damping. On the other hand, for the LNG carrier it is interesting to look at the service speed. The
usual speed of an LNG carrier it is lOm/s or Froude number equals 0.2. This means that we have to extend our model to higher speeds.
Outline of this work
Chapter 1 first gives an overview of different methods describing ship hydro-dynamics. In the second section, the mathematical model used in our method
is given. In the third section, the numerical tools are given which we use to solve the mathematical model. The mathematical model and the numerical
algorithm are based on the ones given by Prins (1995).
Chapter 2 gives, after an introduction to absorbing boundary conditions, an overview of different methods which are used to absorb free-surface waves. Subsequently the semi-discrete DtN method is described. This method is given
in Givoli (1992). On this method we based our semi-discrete DtN method, which is, both in two and three dimensions, described in the final sections of chapter 2.
In chapter 3, we test our model on a two- and three-dìmensionai problem,
respectively a cross section of a circular cylinder of infinite length and a
hemi-sphere. We compute the hydrodynamic coefficients, first- and second-order forces and motions, and we study the influence of water depth and the
velo-city. We also test the formulation of the wave-drift damping.
In chapter 4, a theoretical model for the stability analysis is developed. Firstly, the importance of this analysis is determined. Secondly, the continuous
Four-ier Transform is applied. Subsequently the discrete FourFour-ier Transform is ap-plied. Then the numerical algorithm is evaluated using t1e FourierTransforms.
In the final section, our conclusions are given. For this analysis, we extended
the works of Raven (1996) and Nakos (1990).
In chapter 5, the model is applied to two real vessels, a super tanker and an
LNG carrier. For the super tanker we compute the zero and low speed
coeffi-cients, with different angles for the incoming waves. For the LNG carrier, we compute the forces for higher speed.
i
General theory
1.1
Overview
This overview presents the way mathematicians and engineers try to calculate
the ship hydrodynamics, as described in the introduction, for more than one
century. It explains the simplifications and approximations which are necessary to overcome the complexities of the equations. This overview in hydrodynam-ics is confined to problems with an interaction of the water and a body. And, of course, this overview will unfortunately not cover the complete research done in this topic.
Theoretical work until the 1960's
7
This chapter first gives an overview of different methods describing ship hydrodynamics. n the second section, the mathematical model used in our method is given. In the third section, the numerical tools are given which we use to solve the mathematical model. The mathem-atical model and the numerical algorithm are based on the ones given by Prins (1995).
Ignoring the viscous forces in ship hydrodynamics leads to mathematical equa-tions which can be solved easier in an analytical way. Fortunately this
simpli-fication does not lead to results which are only of academic interest. More
than a century ago, around 1861, William Froude supposed that the prediction
of ship resistance consists of two parts, one dependent on the viscosity and one not. Froude's hypothesis found justification by the boundary-layer the-ory, thirty years later, which says that the viscous stress is only significant in a very thin layer next to the ship hull. Together with Krylov, Froude derived differential equations of the ship motions under the assumption that the pres-sure field is not affected by the body. Therefore only the inertial and restoring forces were taken into account. The Froude-Krylov exciting force is still used
as the resultant force of an undisturbed incident wave.
in 1887, Lord Kelvin derived a mathematical explanation for the ship-wave
pattern behind the ship, sailing through a calm sea. Kelvin also made it pos-sible to simplify the mathematical equations by Kelvin's circulation theorem. lt says that the circulation is constant in an ideal (i.e. inviscid) fluid. Thus, any
motion started from a state of rest will be irrotational.
Although assuming the fluid to be inviscid, incompressible and irrotational
sim-plifies the equations, they are still not easy manageable. Therefore other kinds of approximations have to be made. The first theory for the hydrodynamic dis-turbance due to a ship hull is studied by Mitchell (1898). He introduced the thin ship theory of wave resistance in steady-state. This analytical approach,
for a ship with a small beam compared to its length and draught, has been
extended by a number of authors.
From the 1950's lots of engineers and mathematicians have tried to solve the problems in the ship hydrodynamics mathematically, t3 replace the expens-ive experimental techniques. A number of authors emphasized the response
and motions of floating bodies in regular waves. Hakind (1946) extended the thin ship theory to include unsteady motions. Mound the same time,
Ursell (1949) derived a theory for predicting and judging the characteristics
of the flow around a circular cylinder. And John (1950) examined very carefully
an integral-equation formulation based on a free-surface Green function. This formulation forms the backbone for a lot of methods nowadays. Wehausen & Laitone (1960) give a survey of potential equations and Green functions of dif-ferent kinds of surface waves.
Also, the slender body theory, which originated in the aerodynamics, was
changed into the hydrodynamics theory with a free surface. Cummins (1962) was the first, who calculated the steady-state wave resistance, under the as-sumption that the ship hull was slender, with the beam and draught small
compared to the length. Several authors extended the slender-body theory to the unsteady motion. For a vessel without forward speed and a wavelength of the order of its length reasonable predictions can be made.
Then, in the 1960's, digital computers began to make it possible to compute the hydrodynamics theory using numerical solutions for general bodies. For a more detailed overview of the theoretical history cf ship hydrodynamics, the reader is referred to Newman (1978).
Strip theory
The first numerical solutions for a general body were developed using two
di-mensional results for the three-didi-mensional strip theory. Korvin-Kroukov-sky (1955) transferred the aerodynamics theory in ship hydrodynamics. The
theory is theoretically not satisfying, but the results agree well with experi-ments. Ogilvie & Tuck (1969) gave a consistent analysis for higher order theory
and nonlinear free surface. Salvesen, Tuck & Faltinsen (1970) computed the motion of a ship using a linearized free-surface condition, their results were confirmed by the experimental results of Vugts (1968). Gerritsma & Beukel-man (1972) were able to compute the second-order forces, called the added
z,
1.1 Overview 9
resistance, using strip theory. The strip theory is valid in the short wavelength
regime.
Three-dimensional approaches
To tackle the three-dimensional problems in the sea-keeping problem, several
solutions have been reported. These numerical solutions are obtained using
the Neumann-Kelvin formulation. This widely known free-surface condition
assumes that the total disturbance due to the presence of an object is small.
To solve these numerical problems two kinds of panel methods are used. With
the first one, using the transient free-surface Green function, only the ship
surface is discretized, Brard (1972) gives the original formulation. With the
second one, the free surface is also discretized by using the Rankine source
distributions, see Dawson (1977).
Due to linearizations, the three-dimensional problems can be separated into a steady and unsteady part.
The steady problem
The steady problem concerns the effects of the ship steady forward motion in calm water. The generation of waves on the bow and stern of the ship leads to
a power loss of the ship, called wave resistance. Wehausen (1973) gives a
com-prehensive survey of calculations in the steady state problem. Research done nowadays in this topic is for example: The SWAN-code of Nakos (1990) con-cerned seakeeping problems in the frequency domain and linear steady prob-lems. In Nakos's thesis an accurate derivation of a stability criterion is given
as well. At the MARIN, Raven (1996) developed the linear code DAWSON and
the nonlinear code RAPID, which uses raised panels in order to obtain
conver-gence. The RAPID results turn out to be very good, but breaking waves are not taken into account. Tulin & Wu (1996) show that the calculations with their 2D+T simulation agree well with RAPID.
The unsteady linear problems in the frequency domain
The unsteady problem, the problem due to the scattering of the incident waves and due to the motions of the ship, is first computed in the frequency domain. Because of the absence of fast computers, the restriction to harmonic waves was the only way to handle the complex unsteady problem. Assuming small unsteady motions, the unsteady part can be split into a radiation and a diffrac-tion problem.
In the early seventies, several methods were developed to compute the
first-order unsteady motions of a ship in waves. These methods are based on the
application of Green's theorem or source distributions. Yeung (1973) applied the Green function of the Rankin type to the complete fluid boundaries, while
Oortrnerssen (1976) used a source distribution along the hull, while the source function obeys the linearized free-surface condition. Oortmerssen's results of the first-order reactions of a vessel due to motions and waves, were confirmed by experimental results. The importance of the second-order forces has been
recognized in the begin of the 1970's by Remery & Hermans (1972) and others.
They showed that the low frequency components of the wave-drift forces in irregular waves could excite large horizontal motions. Maruo (1960) derived a formulation for the wave-drift forces based on conservation of momentum and energy. Pink.ster (1980) however, made a great effort by deriving a pressure integration technique to compute the second-order forces in regular waves. A
review of the developments in the theories concerning the prediction of second-order wave forces is given in his thesis. At this moment some efficient codes are
available. For instance, the WAMIT-code (1988), a radiation-diffraction panel
program for wave-body interactions, also calculates the fLrst- and second-order
forces.
Wichers (1988) showed in experiments that the low frequency motions were influenced by the wave-drift damping, especially in survival conditions. He also gave a relation between the wave-drift damping and the drift forces, with forward speed. Hermans & Huijsmans (1989) presented a fast way to compute the second-order forces on an object, due to waves, in a uniform flow, there-fore the wave-drift damping could be computed. Nossen, Grue &Palm (1991) presented a method based on conservation of momentum to compute the drift forces with small forward speed. Grue & Palm (1993) improved this method and extended it to the drift moment. A recent overview on wave-drift damping
will be published in Herm.ans (1998).
The unsteady linear problems in the time domain
The disadvantage of studies in the frequency domain is their restriction to har-monic waves. Due to the recent development of large computers it becomes possible to study the time-dependent equations itself inì a finite time.
At the MIT, the time-domain wave-analysis computer code, TiM IT, is
de-veloped for bodies with zero forward speed. Bingharn (1994) extended the method with forward speed and obtained good results for the first-order forces, using the Neumann-Kelvin approximation for the steady potential.
Prins (1995) has developed a two- and three-dimens:tonal time-domain
al-gorithm to compute the behaviour of a cylinder, a hemisphere and a comrner-cial tanker in current and waves. The method can handle general time signals
and the steady potential is approximated by the double-body potential and can compute drift forces.
1.1 Overview 11
Nonlinear approaches
A'though lots of the problems can be solved using linear theory, it is very
pop-ular nowadays to develop nonlinear computer codes. Still it takes a lot of
computing time to get accurate and interesting results.
Kring, Nuang & Sciavounos (1996) extended their SWAN-code, for linear steady
and seakeeping problems in the time domain, to a nonlinear formulation, by combining the linear solution of the surface wave disturbance and the nonlin-ear treatment of the hydrostatic and Froude-Krylov effects. They showed that the linear computed motions for a Series 60 Wigley hull agree well with non-linear calculations and measurements. However to predict the motion of the Snowdrift hull, with an overhanging stern above the water line, the nonlinear
effects are significant.
Van Dalen (1993) presented a two-dimensional nonlinear algorithm, which
computes the forces on oscillating bodies with no mean forward speed. Berk-yens (1996) extended this algorithm to three dimensions for a heaving hem.i-sphere. The method is time consuming and still in a developing stage.
Scorpio, Beck & Korsmeyer (1996) presented a nonlinear method using an Euler-Lagrange approach, which requires at each time step the solution of the linear boundary value problem and the time integration of the
nonlin-ear boundary conditions. They use a multipole algorithm to decrease the time
and memory requirements. They mention that this significant improvement
in efficiency will help in advancing nonlinear free-surface computations from a research project to a usable tool. Ferrant (1996) also showed a fully nonlinear method based on the Euler-Lagrange approach.
At the MIT, a higher order panel program, HIPAN, is developed using
B-splines, see for instance Lee, Manjar, Newman & Zhu (1996). This frequency domain approach makes it possible to compute linear and nonlinear wave
in-teractions with floating or submerged bodies. One of the applications is the
prediction of the phenomenon of 'ringing', a hydrodynamic/structural reson-ance which has been observed for large platforms in extreme wave conditions, caused by the third-harmonic wave loads. This effect is first published by
Fait-insen, Newman & Vinje (1995), and Malenica & Malin (1995).
Another nonlinear approach is that of Ohkusu & Wen (1996), who presented an
analytical method to predict the nonlinear fluid pressure on a ship in waves, using information of the waves obtained by measurements.
Our program handles the linear case and can be extended to moderate
non-linear cases. In this thesis boundary conditions are fulfilled on the mean free surface. Applications on a steady free surface, computed with RAPID, is con-sidered by Bunnik & Hermans (1997), while the extension to a moving free surface can be carried out in principle.
1.2
Mathematical model
In this section, we give the mathematical equations which describe the physical
problem. Consider a three-dimensional object in a fluid with a free surface.
The object, for instance a ship, sails through an incident wave field with a
velocity U(t) in the negative x-direction; this is equivalent to an object with
zero speed in a current U(t) in positive x-direction. The incident waves are
travelling in the water surface in a direction at an angLe ¡3 with the positive x-direction; see figure 1.1.
2d 0
In naval architecture the motions in these six directions are called surge, sway, heave, roll,
yaw (3
pitch and yaw, see figure 1.2. The coordinatepitch system is chosen such that the undisturbed
heave
ç>
free surface coincides with the plane z = Osway and the centre of gravity of the object is on
surge roll the z-axis, with z pointing upwards, thus the
< coordinate system is fixed to the mean posi-tion of the object and the object in the fluid
domain has negative z-values. The
coordin-Figure 1.2: The stx degrees of ate system moves relatively to the earth, thus
freedom.
waves with a frequencyw in the space-fixed frame have a frequencyw relatively to the object, called the frequency of
en-counter. Furthermore, the fluid depth is constant, h.
The fluid, which is water in this problem, is assumed to be ideal, incompress-ible and homogeneous, and the flow irrotational. Assuming these restrictions,
the fluid velocity may be described by the gradient of a scalar velocity potential
. Then, the physical principle of conservation of mass is represented as the
Laplace equation
a. In waves and current. b. Sailing in waves. Figure 11: The geometry of an object.
Furthermore, the object is free to move in all directions, both to translate in three directions and to rotate around the three axes. Therefore there are six
1.2 Mathematicaf modes 13
The pressure p in the fluid is given by Bernoulli's equation,
P=
_P(t+
.+gz+C)
+Po,
(1.2)where p is the atmospheric pressure, which is assumed to be constant. By
combining the dynamic boundary condition on the water surface, which says that the pressure is equal to the atmospheric pressure, and kinematic bound-ary condition on the water surface, which says that the fluid cannot leave the surface, the surface elevation (can be written as
And the free-surface boundary condition can be written as
+ 2 t +
) + g
= Oon z = (.
(1.3)This free-surface condition is a second-order differential equation, therefore two initial conditions are required,
for t < O,
where is the steady potential, which is the potential due to the object's
uni-form translation at forward speed in calm water, see equation (1.6). These conditions assume that the object is sailing through an undisturbed fluid at
t = O, thus that we start with a steady wave field.
The boundary condition on the hull of the object, called the body-boundary
condition, is as follows
on9i,
(1.4)where Ç is the object's velocity in the coordinate system fixed to the average position of the object. This condition is explained physically by assuming that the ship velocity must be equal to the fluid velocity. The boundary condition on the bottom of the fluid domain also represents that the boundary is a rigid wall, therefore
=O
onz=h.
(1.5)Finally, the potential has to satisfy the radiation condition, which states that the waves generated by the object travel away from the object and will go to zero at spatial infinity.
Li flea rizations
To solve the model, given in the previous equations, we linearize the bound-ary conditions and Bernoulli's equation. In the overview 1.1 a few nonlinear approaches are also given, and the most promising results are given using an extension of a linear model. Unfortunately we are just developing the linear model and still have no extension to nonlinearity.
The boundary conditions are linearized in two stages. Firstly, we assume the nonlinear potential to be a linear combination of the steady (time-independ-ent) arid an unsteady (time-depend(time-independ-ent) part
4(1,t) =()+(,t)
. (1.6)The steady potential is the potential due to the object's uniform translation at forward speed in calm water. For small values of U we assume that the
steady potential can be approximated by the double-bc'dy potential. The un-steady potential is the potential due to the prescribed, forced motions of the
object (radiation problem) and due to the incident waves (diffraction problem).
Furthermore, we assume the steady and unsteady potential to be small. There-fore, combining equations (1.3) and (1.6), we get for the free-surface condition for the unsteady potential
onz=(,
(1.7)where we assume for the steady potential quadratic terms and for the unsteady potential linear terms to be sufficient. Linearizing Bernoulli's equation (1.2), we get
p=
On the free surface, when z = (, the fluid pressure
r
and the atmospheric pressure Po are the same, therefore we can write for the surfaceelevationwhere the constant C is replaced by - U2, because there is no elevation at infinity.
Secondly, we assume the surface elevation ( small, therefore the free-surface
1.2 Mathematical model 15
z = O. The Taylor series expansion a round z = O of condition (1.7) is
U2)
(_
-
_zz
(1.8)
on z = O.
And the body-boundary condition (1.4) is linearized around the mean position of the hull, , as in Timman & Newman (1962),
=O and
onL0,
(1.9)
where ä is the total displacement vector, .
+
x-
), with X thetrans-lational, and the rotational motion of the object relative to the centre of gravity 2
Equation of motion
The dynamics of a ship unsteady motions are directed by a balance between the inertia of the ship and the external forces acting on the ship. Newton's law can be written as
with=().
(1.10)where M is the mass matrix, and A, .8 and C respectively added mass, damp-ing and the restordamp-ing force matrix. The motions Y are assumed to be decoupled into a first- and second-order part
=eY)+2Y(2)
withj=1...6,
where is related to the wave height of the incoming wave, which is assumed to be small. We use the following incoming potential,
- A
I L.\Cosh(k(z+h))
.ÇCOS wt
-h k-h (1.11)
with w0 = w - kU. A = ga/wo, and ktanh(kh) = w/g. The first-order
motions are related to the wave frequency, w. The second-order motions are large amplitude low frequency, w motions, if one considers, for instance,
The total wave exciting force in regular head waves consist of three parts, a stationary, a first-order and a mean second-order part,
Fk(t) = Pk +EF'(t)
+E2F2)The way to compute these wave forces by pressure integration is given in the next section.
As long as the unsteady perturbations are small, it is reasonable to consider the system to be linear. Ogilvie (1964) showed that the equation of motion can be written as
j6
s)ds =F',
(1.12)
where a dot means time derivative and where M' is the ship's inertia matrix,
K is the step-response matrix, and , b and c are the asymptotic values of
the added mass, damping and the restoring-force coefficients, and they are
only dependent on the geometry of the hull. In section 3.3, we show how we compute the step-response matrix.
We can also write the equation of motion for pure harmonic motions. Then
the equation of first-order motions can be written as follows
[(îvrj
+
aki((.))1?i(1)+ bkj(w)J'1 + ckjYJJ
F'
, (1.13)3=1,6
where a, b and care matrices with respectively the added mass, added damping and restoring-force coefficients, The indices kj indicae the direction of the force in the k-th mode due to a motion in the j-direction. The added mass
and damping coefficients are frequency dependent, therefore the equation of motion (1.13) can only be used for regular waves.
Before we are able to compute the first-order motions, we have to know the
hydrodynamic coefficients added mass a and added damping b. We do this
by forcing the object to oscillate pure sinusoidal in a given direction, thus we know the motion. t'low calculating the force gives us the coefficients by fitting the force to the acceleration and body's velocity
akjY + bk." = Fk,
The restoring-force coefficients c are given in Prins (1995).
First- and second-order forces and moments
The forces and moments acting on the hull of a ship are obtained by integrating
the pressure p, given in Bernoulli's equation over the wetted surface. Unfortu-nately the actual position of the hull, 9L, is not known due to the linearization.
1.2 Mathematical model 17
Pinkster (1980) approximates this position, knowing the mean position of the
hull and using perturbation series. Prins (1995) extended Pinkster's method for the forward speed case. Grue & Palm (1993) include additional terms in
the formulations of the forces and boundary conditions, which can be found in
Pinkster (1980) in the slow-drift forces, forz.w, calculations. In this section we pay attention to the difference between the derivation of Pinkster and Prins, and that of Grue & Palm. In the results of the section 3.3, we show the
differ-ent results.
The unsteady potential can be split into a first- and second-order part as
= E' +E22 + O(E3),
where the small parameter is proportional the wave height. According to
Grue & Palm (1993), the second-order part can be split into an unsteady and a
steady part
(2) (2)
+
(1.14)Prins (1995) neglected the steady second-order potential &(2), therefore he
neglected an E2-term in the formulation of the pressure
U2+E{1)+.(1)}
+ e2
{(2)
+ (2) + (1)+ O(e3)
=
+
+
e2p2 +O(e3) on owhile this term influences the drift forces and moments according to Grue. To compute the forces, we need the actual position of the hull, while the
pres-sure in equation (1.2) is on the mean position of the hull. Applying a Taylor series expansion to the pressure in the mean position, we get the pressure on the actual position of the hull
=
+
p(°) +p(1)}+ 2 {
(i
2 (°)+
(2) . (o)+ &' .
+
p)
} +
0fr3)(0) (1) 2 (2)
= p
+ ep+
E p+
O(e3)on li.
The first-order forces are found by integrating the products of the pressure and normal vector which give a first-order contribution over the wetted surface
F1
=
/
- J
t\P(1)
This integral can be split into an integral over the mean surface of the hull
and over an oscillatory disturbance of this surface, for more details see Pink-ster (1980) or Prins (1995). The first-order moments are given by
M1
=
_J (p
(( &g) X il(1))+p
((i- g) x il(o))) dS.
The second-order forces are found by integrating the products of the pressure
and normal vector which give a second-order contribution over the wetted surface
F2
=
f
()
il2 +p
+p
il(o))
In this thesis we do not pay attention to the second-order moments. The difference between Prins (1995) and Grue & Palm (1993) in the formulation for
the drift forces is one term,
To compute the contribution of this term Grue rewrites it in a suitable form
without actually solving (2) This derivation is given in section 3.2. By using the derived equations, we are able to compute the extra 'term of the drift forces and study the difference for the floating hemisphere, see section 3.3. lt seems to be a contradiction that the tI'(2)-term gives no contribution when the drift forces are computed using the far-field method instead of pressure integration.
Grue & Palm (1 993) notice that at infinity the contribution of the steady
second-order potential is completely determined by the first-second-order quantities.
Wave-drift damping
To solve the low frequency motions of a moored tanker exposed to irregular waves, the hydrodynamic input for the equation of motion, being the low fre-quency reaction and excitation forces, has to be known. Wichers (1988) shows
by means of extinction tests that for low frequencies damping exists. This
damping is assumed to be of viscous origin. However, the predicted surge mo-tions, computed with the equation of motion and the wave-drift forces, over-estimated the measurements. Wichers shows, on the basis of model experi-ments and computations, that a large part of the damping could be attributed
to the velocity dependency of the wave-drift forces. He also shows that the
mean wave-drift force changes approximately linearly with the low values of the vessel speed U, therefore the wave-drift damping can be computed as the derivative of the mean wave-drift force to the speed. Although this assump-tion is widely used it has never been shown that this assumpassump-tion is correct.
1.2 Mathematical model 19
Since the added damping for low frequency motions is negligibly small and
the fluid is assumed to be inviscid the second-order part of the equation of
motion (1.10) is in surge
.(2) -(2) (2)
-r all W2)) X1 +
c11X1-
F1Wichers assumed that the second-order damping consists of a viscous part and a part attributed to the velocity dependency of the second-order forces. Here, we only look at the inviscid problem, therefore only to the velocity
depend-ency.
Slowly oscillating the object's velocity gives a velocity dependency of the ad-ded mass, damping and exciting forces in the equation of motion. Because a regular wave can be described by
((t) =
cos(wot - k2)
(a cos(wt - kx2)where Ç is the wave amplitude, k the wave number, w is the wave frequency, w the frequency of encounter and the coordinate in the earth-fixed system, the first-order wave force in surge is modulated by the frequency of encounter and a phase shift. Therefore the first-order force will vary but in the wave fre-quency range, and the wave will not contribute to the low frefre-quency damping.
The origin of the wave-drift damping has to be found in the second-order force. Applying Taylor series of the second-order force to the low velocity 2) we get for equation (1.16)
(2) (2)
(M11 + a11 (w2)) (2)
+
=
F2 (w, 0) +(,X1 )
(2)+
p,
where P is the slowly oscillating force acting on the object. Withers derived
from this equation the assumption that the low frequency damping will be
equal to the derivative of the second-order force to the velocity. Two model tests were carried out to analyse this assumption, extinction and towing tests,
and they justify the assumption.
In section 3.2, we carry out two kinds of computer simulations to verify this
as-sumption. Firstly, we compute the drift forces on an object with zero and small uniform speed. Then we compute the wave-drift damping using the derivative of the drift forces to the speed,
(1.16)
B1(w)=
i (w,U)au
U=U (1.17)This formulation is used by Prins & Hermans (1996) to compute the wave-drift damping of a 200 kDWT tanker and the results agree well with the
Secondly, we compute the drift forces on an object with a slowly oscillating
velocity
u(t) = Ucos('2t)
The wave-drift damping is the part of the slowly oscillating wave-drift force that is phased with the velocity. We use filter theory to eliminate the slowly oscillating part of the second-order force. ln the latter case, computations for the wave-drift damping are only done in the two-dimensional test problem, because of the time consuming fact that the matrices have to be updated every time step.
Computing the wave-drift damping using equation (1.17) includes that the
method is able to compute the wave-drift forces on an object with small for-ward speed. Aranha (1994) derived the so called engineering view, i.e. a way to compute the wave-drift damping without knowing the wave-drift forces on an object with small forward speed. In the sections 3.3 and 5.2, we compare this engineering view with equation (1.17).
Steady potential
We approximate the steady potential by the double-body potential, i.e. the
potential of a flow past the sailing body and its reflection about z = O. This
approximation is valid because we assume the steady waves small compared to the incoming waves and it is more accurate than the Neumann-Kelvin ap-proximation Ux, which is only a good apap-proximation for slender bodies. More accurate approximations are discussed in the overview 1 .1. We use the
condi-tions as given in Hess & Smith (1964).
1.3
Numerical model
In this section, we give the numerical model to solve the equations given in the
previous chapter. We use a lower-order panel-method, where we divide the boundaries into panels on which the velocity potential is assumed constant. In the two-dimensional (2D) problem, in section 3.2, we use an equidistantial grid, while we use increasing panels in the three-dimensional (3D) problems
in section 3.3 and chapter 5. We discretize notonly the equations in space,
but also in time. Therefore we introduce a time step t. Through numerical time-integration we will know the potential on every time step. In chapter 4, we give a stability analysis of this numerical model.
Boundary-integral method
We use a Green function and a boundary-integral method to solve the Laplace equation (1.1). The Green functions representing a source and satisfying the
1.3 Numerical model 21
bottom-boundary condition (1.5) as well read
(1
1I logr + logr2 in2D
G(,e)
2ir1 2ir47rr 4'rr2
where r =
-
and r2 = - &j, with ( the mirror image of (with respect to the flat bottom. The boundary-integral method we use is Green's secondtheorem, an integral over all the boundaries of the domain
in3D
5.t)
=
f
(((t)aG(
G(8t))
On dF, &S\bwith ÛS all the boundaries of the do-main S, and
(1
thES
(S O elsewhere=
çbj J" clS -
Lf G, dS,
j=1 A j=1 A-b (1.18) 5withi=1,...,N,
(1.19) where N is the number of panels in the discretization and A the area of apanel. The integrals over the Green functions and their derivatives are com-puted analytically, for which we use the theory given by Yeung (1973) for the 2D problem and for which we use a corrected version of the theory given by Fang (1985). After integrating the Green function we can write equation (1.19)
where the matrices D arid E contain the integrals over the Green functions
and their derivatives.
The boundaries used in the integral
equation are given in figure 1.3. The Figure 1.3: Theboundaries. physical fluid domain is an infinite or
large domain, whereas the computational domain cannot be infinite. There-fore we need to introduce an artificial boundary, ß. In chapter 2 we discuss
the conditions on this boundary.
To solve the integral equation (1.18), we use the discretized form of this equa-tion. Assuming the potential and its normal derivative constant on a panel we
can write
On
D14 = D2
+
+
jn+1,
n+1
Time integration
Prins (1995) showed that several algorithms suggested for time integration, based on integrating the Laplace equation separately from the boundary
con-ditions, can have an unstable solution. Therefore we use an integration
al-gorithm combining all equations.
Firstly, we write the boundary conditions on the hull (1.9), on the free sur-face (1.8) and on the bottom (1.5) explicit for ç. For in5tance, the free-sursur-face condition with Ux for the steady potential becomes
n+1
+1
1(92±
O2-
- g \Ot2
OxOt Ox2where superscripts denote the time steps. Secondly, we discretize the first- and second-order time derivative by a backwards second-order difference scheme.
Thirdly, we approximate the spatial derivatives by second-order difference
schemes. For more details see section 4.3. We deal with the other boundary conditions the same way, thus now we have discretized boundary conditions. Filling in the of these discretized boundary conditions in the discretized Green's theorem (1.20) leads to the following matrix equation
(1.21)
where Jis a time-dependent vector. This time-integration method is, accord-ing to Prins (1995), stable for all time steps and mesh s:izes.
In contrast with the model of Prins (1995), we do not have a condition on the outer boundary yet and therefore no relation between and ct, on the outer
boundary 5. Our matrix equation is
D1?1 =
+
+ f'
(1.22)where is a vector containing the potential on the free
sur-face, on the hull and on the artificial boundary and the normal derivative of
the potential. The dimensions of this matrix equation are not correct yet, but at the end of section 2.4 the correct version will be given. In that section we derive the missing relation between and ç on the outer boundary 5, using
23
2
Absorbing boundary condition
This chapter gives, after an introduction to absorbing boundary conditions, an overview of different methods which are used to absorb tree-surface waves. Subsequently the semi-discrete DtN method is described as given in Givoli (1992). On this method we based our semi-discrete DtN method, which is, both in 20 and in 3D, described in the final sections of this chapter. Parts of this chapter have been published in Sierevogel & Hermans (1 996a).
2.1
Introduction
Modelling the open sea, the physical fluid domain is an infinite (or large) do-main, which is horizontally unbounded. Due to the boundary-integral method,
which takes the integral over a closed boundary, the numerical solution re-quires a truncation of the domain at finite distance. Because the
computa-tional domain cannot be infinite, we have to introduce artificial boundaries.
The use of a wall as artificial boundary will cause reflections and therefore rep-resents not the real sea state. Thus we need an artificial boundary and proper boundary conditions, which absorb the outgoing waves. We want the artificial
boundary to be as close as possible to the body, because then we need less
panels and therefore smaller matrices.
In the literature several methods have been proposed to absorb free-surface
waves. On the basis of a literature search, Prins (1995) decided to use an ex-tension of the Sommerfeld radiation condition for two families of waves. The disadvantage of this Sommerfeld condition isthatit is dependent on the wave frequency, so it cannot handle non-harmonic waves, and on the forward ve-locity. Keller & Givoli (1989) introduced a semi-discrete DtN method1, using an artificial boundary, dividing the original domain into a computational and a residual domain (the interior and exterior). In our method, we use a boundary
condition independent of the wave frequency, using the idea of Givoli's method
with Prins's algorithm. In the interior domain, we use the same mathematical mod el as Prins (1995) used, which is described in sections 1.2 and 1.3, but we do not implement a Sommerfeld radiation condition on the artificial boundary.
'A DtN relation or Dirichlet-to-Neumann relation is the relation between the Dirithiet datum u and the Neumann datum u'.
2.2
Overview
In the literature several methods have been proposed to absorb free-surface waves. These are reviewed in, for instance, Romate (1989). A short description
of the most frequently used methods and some references are given in this section.
Artificial damping I sponge layer / numerical beach
A possibility to absorb the outgoing waves is the use of artificial damping. This
method is also referred to as sponge layer or numerica! beach method and is
used by, for instance, [sraeli & Orszag (1981).
In this method, an artificial dissipative term is added to the free-surface con-ditions near the artificial boundary of
the truncated domain, so that outgo-ing waves are absorbed with as little wave reflection as possible. The
dis-z sipative term u can be added in the
kinematic or in the dynamic
free-sur-face condition or in both, and can be depending on the distance to the
ar-Uficial boundary, see figure 2.1. The dissipative term u can be acting on a function which can be chosen freely, for instance a function of the potential. Our free-surface condition (1.8) for zero-speed could be written as
t+v(x)t+g&=O
onz=O,
but instead of Ø other functions can also be chosen.
The advantage of this method isthat it is easy to implement and that it has
good reflection properties for a wide range of frequencies, but the disadvantage
is that the matching between damping zone and the common free surface has to be smooth, thus a large domain is needed for the damping zone, especially
for low frequency waves.
Partial differential equations
Sommerfeld's radiation condition, Sommerfeld (1949), is required to make the problem well posed at infinity and the condition only corresponds to the
out-going waves. In the frequency domain, it provides a simple linear relation
between the partial time-derivative and the normal derivative. For harmonic solutions with wave number k it reads
asrcc.
Figure 2.1: The linear dissipative term u in the free surface.2.2 Overview 25
A transposition of this condition to the time domain gives
Cr
cft + Oas r *
,with e the local phase velocity of the wave to be absorbed. Applying this
con-dition on the artificial boundary gives a set of partial differential equations. Using partial differential equations as an absorbing boundary condition can
be done in somewhat different approaches. The technique which is probably the most widely used was published by Orlanski (1976). He used a first-order
equation. The novelty in his approach was that the phase velocity needed in
this condition was evaluated numerically in the vicinity of the boundary.
For the acoustic wave problem Engquist & Majda (1979) derived a higher-order differential operator on the outer boundary (see also Jones & Kriegs-mann (1990)). Prins (1995) used an extension of the Sommerfeld radiation
condition for the two wave families, a second-order partial differential equa-tion, which can be written, after substituting the phase velocity, as
02A
7i
\
+u(+2)
r
j Ont9t Onwith r = (Uw/g) <0.25. This condition is easy to implement, but it only
ab-sorbs the waves of which the phase velocity is included. Other waves are partly reflected. Therefore this condition is dependent on the local wave frequency.
Use of exterior solutions
Another way of modelling the absorption of waves on an artificial boundary is the use of simple exterior solutions. In the exterior, boundaries and boundary
conditions are simplified such that an analytical solution can be found. This technique of matching analytical far-field solutions and numerical solutions was, among others, applied by Bai & Yeung (1974) to the linearized sea-keeping
problem in the frequency domain.
Yeung (1985) gave a time-domain solution of a swaying axisymmetric structure
with no mean forward speed. This formulation uses a 'simple-source' repres-entation for the inner domain, while the time-dependent Green function (used by Newman (1985b), for instance) is used in the exterior. A disadvantage of
this 'shell method' is that the Green function has to be computed every time
step, but the free surface of the exterior does not need to be covered with
pan-els. To date, there have been no results available from the shell method that
include forward speed. However, by including forward speed and especially taking into account the double-body potential, we also need to define panels on the free surface.
There are also no results available yet of the combination of a nonlinear interior
Piston-like absorbing boundary condition and numerical beach
Clément (1996) coupled two absorbing boundary methods featuring comple-mentary bandwidths on the right side of the 2D domain, see figure 2.2. On the
left side, B, a wave maker is implemented.
The first one is a numerical beach, fd
f
which absorbs particularly the highfre-quency waves. The second one is a Neu-mann condition, modelling a vertical pis-ton, t) = v(t) on 5,. The function
y is omputed with a convolution integ-ral over the hydrodynamïc force on the
piston B. Unfortunately, this force is not known in the future, using a
time-iteration method. Therefore it has to be approximated, which can only be done
for w + O, thus for low frequency waves.
The method is efficient and straightforward to implement for both linear and nonlinear cases, but the results are only presented in two-dimensions with no current.
2.3 Semi-discrete DtN method (orthogonal functions)
In this section, the DtN method for time-dependent problems, as given in Givoli (1992), is applied to our problem. The semi-discrete DtN method is
chosen, which can be applied to a problem discretized in time. Figure 2.2: A piston-iike
absorb-¡ng boundary condition and nurner-ica beach.
DtN relation
For the acoustic wave equation, Givoli (1992) derived a DtN relation on the outer boundary used in combination with a finite-element approach in the in-terior. This method is very efficient. In the method, a DtN relation is derived, in principle with the help of an explicit solution, by means of the expansion of orthogonal functions. In the frequency domain, the DtN relation becomes homogeneous, while in the time domain an inhomogeneous term originates from the fact that the complete propagating exterior solution has to be taken into account. in the shallow-water case, an orthogonal set of eigenfuncúons is available in the exterior. A direct application, however, is much less efficient than in the acoustic case. Too many terms have to be taken into account due
to slow convergence. The main message of Givoli is that. the complete exterior
field has to be taken into account.
Because we modified Givoli's DtN method in this thesis, this method is
2.3 Semi-discrete DtN method (orthogonal functions) 27
The idea of the DtN method is based on the division of the original domain into a computational domain and a residual domain, and on the matching of
the solutions of the two domains on the artificial boundary, B. The solution in the computational domain is an accurate numerical solution, while the solution in the residual domain can be an analytical approximation of the time-discrete problem.
Given a boundary value problem 2 in an infinite domain S,, the DtN method can be described in four steps:
Step 1. Introduce an artificial boundary 5, which divides S into a computa-tional domain, the interior S, and a residual domain, the exterior D. Step 2. Solve the problem 22 (analytically) in D to derive a relation between
the unknown function and its derivative on 5.
Step 3. Use the relation as a boundary condition on 5. This condition
com-pletes the boundary value problem Ps in S. Step k. Solve in S numerically.
We now use our 2D mathematical model given in section 1.2 to describe the DtN method in the four steps. For simplicity we only look at the problem with
no speed and we neglect the object. Thus the boundary value problem P is given by
I
2=
ç5tt+gç=O
onf(z=O)
[x=(x,z)J
= O
on b (z = h)
Step i The artificial boundary
The introduction of the artificial boundary, Step 1, is shown in figure 2.3.
ID
x=X5 f
z=X
IDz
-z = h
Figure 2.3: The division of the infinite domain into an interior S and exterior D by an artificial boundary B.
Step 2 The exterior solution
In Step 2, we have to solve the problem in D. Although the problem is also
find a relation between the potential and its normal-derivative on the artificial boundary. This relation can be written in general as
q=-M5+H
onß,
(2.1)where M and H have to be defined in this section. Firstly, we mention the predictor-correction problem P as described in Givoli (1992). Secondly, we describe the problem g using the same time díscretization as in the interior.
Let t' be the time after n time steps, n.T, and let u, v' and a' be the
ap-proximation of , and at time t'i. Then the problem becomes
=
{2n+1
=
a' + 9U
= Ou+1
= Ou1 = if + ¿tv + L1 ((i
- 2ß)a' +
2,a'')
u+ = -Mu' + H'
n= n + zt ((1 - y)an + ya1)
LEV
Lonb (z=-h)
Lonjv (z=O)
L onwhere ß and -y are two parameters which determine the stability and accur-acy of the scheme. Givoli (1992) wrote the problem as a predictor-correction problem by matching the equations of P. Then every time step the 'solution'
has to be solved after the updating in the 'correction' and 'prediction'.
Instead of the predictor-correction method we use the same time
discretiza-tion as we use in the interior, see secdiscretiza-tion 1.3. Then the problem becomes
X
with ,a = 2/(g(L\t)2). We solve this problem by first solving the homogeneous
counterpart of the free-surface condition, r(H) = O. Searching a solution
dependent on z and z, we find the homogeneous solution by the method of
variation of parameters. The general part of the homogeneous counterpart can be found by the Fourier transformation and the use of a convolution integral.
Combining the homogeneous and particular solution o 2g, we can derive a
condition on the artificial boundary like equation (2.1),
000
n(XB, z) = (X5, z) =
f
nn(z)n(z')(X5, z') dz'
00 (2.3)
