4TECBNISCHE UNIVERSITEI%
ScheepshydromechanicaArchief
mbiekelweg 2, 2628 CD Delft
Te1:015-2786873/Fax:2781836_1
Elm Marita Haugen
Hydro elastic analysis of slamming on
stiffened plates with application to
HYDROELASTIC ANALYSIS OF
SLAMMING ON STIFFENED PLATES
WITH APPLICATION TO
CATAMARAN WETDECKS
Ming. Thesis
by
Elin Manta Haugen
Department of Marine Hydrodynamics
To Ole Andreas
Without his patience and encouragement this work would never have been accomplished
Abstract
A two-dimensional method for local hydroelastic analysis of wetdeck slamming on high speed catamarans is presented. The stiffened plates in the wetdeck are modeled by Euler beams, and the beam deflection is expressed by a linear combination of the eigenmodes. The eigenmodes are obtained from a free vibration analysis of the structure in air. The hydrodynamic forces are found by solving a hydrodynamic boundary value problem
by introducing a vortex distribution on the wetted part of the model. The vibration
velocity of the wetdeck model is approximated by a Fourier series, and semi-analytical
expressions for the hydrodynamic pressure are derived. The wetted length of the wetdeck is found by a generalized Wagner approach. The effects of air pockets are also investigated. Theoretical predictions are compared with experimental results for an elastic two-dimensio-nal plate model which is dropped towards a water surface. Different drop velocities, forward speeds, wave conditions and relative impact angles are studied. Stresses in several locations of the model are investigated. The comparisons indicate that the presented method is well suited for the prediction of impact loads and corresponding response of a beam model. The local response, in terms of deflections, stresses and pressure time histories, are generally
satisfactory predicted by the theory. Creation of air pockets during the drop tests gives
some difference between the theoretical and experimental results.
The theoretical and experimental results from the beam model analysis are applied to
fullscale catamarans. Estimated maximum stress levels as well as operational limits based
on slamming probability are found. The results are also compared to fullscale measure-ments of stresses obtained from a test with a fullscale sea-going catamaran. The results
indicate that wetdeck slamming may be a problem, although the presented theory seem to overpredict the maximum stress levels.
The important parameters governing the slamming induced response are found to be the relative normal impact velocity, the relative angle between the impacting body and the free surface, as well as the structural bending stiffness. The global accelerations are also important, as they act to reduce the relative normal impact velocity during the impact.
Acknowledgments
This work has been carried out under the supervision of Professor Odd M. Faltinsen at the Department of Marine Hydrodynamics, Norwegian University of Science and Technology, whose guidance and support are appreciated.
I am also grateful for the encouragement from and the many valuable discussions with
my colleagues and good friends at the Departments of Marine Hydrodynamics and Marine Structures and at MARINTEK.
This work was made possible by a research grant from the Research Council of Norway
'Contents
Abstract
iv ,Acknowledgments likNomenclature
ix
General . No) P7 ix Roman Symbols . . ixGreek Symbols 4.: 4 ,, xii
Mathematical Operators and other Symbols 0 VA0. xiii
Introduction
'11.1 Background and Objective
1.2 Scope and Limitations .
, .. ...
21.3 Previous Work ,3
1.4 Present Study . ,
2 Simplified Structural Model
112.1 The Single Euler Beam Model .
... .
132.2 The Wetdeck Modeled by Three Euler Beams 14
2.3 Modal Analysis 15
2.3.1 The Eigenvalue Problem
2.3.2 The Governing Modal Beam Equation of Motion 17
3 'The Hydrodynamic Formulation
193.1 The Hydrodynamic Boundary Value Problem 19
3.2 The Solution Procedure 1 VI 22
.3.3 The Simplified Impulse Method 27
3.3.1 The Initial Impulse Phase .
3.3.2 The Forced Vibration Phase '31
3.4 The Wetted Length 32
3.4.1 Vertical Velocity of the Free Surface outside the Wetted Length . 33
3.4.2 Initial Conditions for the Wetted Length 35
3.4.3 Difficulties with the Wetted Length Estimates 35
1
1
6
15
vi Contents
3.5 Cavitation and Ventilation 36
3.6 Air Pockets in the Initial Phase of the Impact 37
3.6.1 Creation of the Air Pocket 37
3.6.2 Collapse of the Air Pocket 38
3.6.3 Initial Conditions for the Collapse of the Air Pocket 45
3.6.4 The Forced Vibration Phase 47
3.7 Rigid Quasi-static Approximation of the Hydrodynamic Boundary Value
Problem 47
4 Experiments
494.1 Description of Test Model 50
4.1.1 Test Rig 50 4.1.2 Test Sections 51 4.2 Instrumentation 53 4.3 Test Program 53 4.3.1 Tests in Air 53 4.3.2 Tests in Water 54
5
Results and Discussions
575.1 The Numerical Time Simulation Procedure 57
5.1.1 Three-Beam Model and Jet Propagation 59
5.1.2 One-Beam Model and Air Pockets 59
5.2 Verification and Validation 60
5.2.1 Validation of the Eigenvalue Analysis 61
5.2.2 Convergence Tests 62
5.3 Sensitivity Studies 70
5.3.1 Initial Impulse Phase and Initial Conditions 70
5.3.2 Rotational Spring Stiffness 72
5.3.3 Effects of Different Model Scales 75
5.4 Comparison between Numerical and Experimental Results 75
5.4.1 Description of the Numerical Calculations 75
5.4.2 Eigenvalue Analysis 76
5.4.3 Experimental Results 78
5.4.4 Comparisons and Discussions 80
5.5 Importance of the Eigenmodes 84
5.6 Parameter Studies 87
5.6.1 Position of Initial Impact 87
5.6.2 Radius of Curvature of the Incident Wave 88
5.6.3 Structural Mass 89
5.6.4 Total Relative Normal Impact Velocity 90
5.6.5 Forward Speed and Pitch Angle 92
5.6.6 Relative Angle and Total Relative Normal Velocity 94
5.7 Non-Dimensional Maximum Stresses 97
. .
.
..
Contents vii
5.8 Stress at Different Locations 105
5.9 Comparison of Different Structural Models 108
5.10 Importance of Hydroelasticity 110
5.11 Results of the Air Pocket Analysis 115
5.11.1 Sensitivity Study 116
5.11.2 The Initial Conditions 119
5.11.3 Comparison between Experimental and Numerical Results 122
5.11.4 Summary of the Findings in the Air Pocket Model 124
6
Application of Theoretical Model to Fullscale Wetdeck Slamming
1276.1 Operational Criteria due to Slamming Probability 127
6.2 Slamming Measurements on Fullscale Catamaran 131
6.2.1 Numerical Estimate of Stress Level 132
6.3 Error Sources in the Fullscale Application 133
6.4 Lessons Learned from the Fill'scale Experiments and the Investigation of
Operational Criteria 136
7 Conclusions and Recommendations for Further Work
139References 143
A The Hydrodynamic Pressure
149Generalized Hydrodynamic Forces
151C Equation System yielding the Initial Conditions
155Derivation of the Vertical Velocity outside the Wetted Length
157Analytical Evaluation of the Singular Integral
159F Derivation of Surface Deformation due to Air Pocket
163. .
.
Nomenclature
General
Symbols are generally defined where they appear in the text for the first time. Matrices and vectors are represented by bold face characters.
Overdots signify differentiation with respect to time.
Primes signify differentiation with respect to the x-coordinate. Subindices in and n refer to mode numbers.
Subindex k refers to the Fourier cosine coefficients.
Roman Symbols
A(t), Amn(t) Generalized added mass matrix
Ai Derived coefficients in parts of the integrand in expression for
the vertical velocity outside the wetted length
Ak(t) Fourier cosine coefficient
Coefficient in the eigenfunction of the one-beam model
AW) Coefficient in the eigenfunction of the three-beam model
a(t) a( t)
coordinatea(t) Position of the right jet root
ao Initial position of the right jet root
B(t), B,(t)
Generalized hydrodynamic damping matrixDerived coefficients in parts of the integrand in expression for the vertical velocity outside the wetted length
Coefficient in the eigenfunction of the one-beam model Coefficient in the eigenfunction of the three-beam model
Bstr Bgt; Generalized structural damping matrix
Position of the left jet root bo Initial position of the left jet root
Generalized structural stiffness matrix
ix
Principal
Nomenclature'
Derived coefficients in parts of the integrand in expression for the vertical velocity outside the wetted length
Coefficient in the eigenfunction of the one-beam model
CY) Coefficient in the eigenfunction of the three-beam model
c(t) Half the wetted length
ce Speed of sound in water
Coefficient in the eigenfunction of the one-beam model Coefficient in the eigenfunction of the three-beam model
D,,,,t(x) Vertical distance between mean water level and the wetdeck
clwet(x) Vertical distance from x-axis to the wetdeck
Young's modulus
F(t), Fexe,m(t) Generalized excitation force vector
F(x, z. t) Function describing the instantaneous position of the wetdeck
F(t)
Local hydrodynamic added mass forceFda,a(t) Local hydrodynamic clamping force
Fexc(t) Local excitation force
GA Part of integrand in the expression for the vertical velocity
outside the wetted length
GB Part of integrand in the expression for the vertical velocity
outside the wetted length Acceleration of gravity
h(x, t) Air pocket thickness
Height of the outflow throuth of the air pocket
Hs Significant wave height
Maximum operational significant wave height
Area moment of inertia of cross section of beam model Imaginary unit
Counter variable
Subindex for Fourier cosine coefficient
ko Rotational spring stiffness at the beam ends
Overall length of vessel
Vessel's length between perpendiculars
LB Length of one beam
Bending moment at the beam ends
1(t) Position of the ends of the air pocket (±1)
M(t), Amn + mnMmn Total generalized mass matrix
MB Mass per unit area of beam
11:laut Mass outflow per unit time
Mmn Generalized structural mass coefficient
Subindex for mode number
Number of Fourier cosine coefficients
Roman Symbols xi
Nee, Number of time steps in time integration
Subindex for mode number
Probability that the stress is larger than a given value
p(x, w, t) Hydrodynamic pressure
Po Atmospheric pressure
Pac Acoustic pressure
Pap(t) Air pocket pressure
Plow Nap po. Lower limit of the hydrodynamic pressure,
yielding cavitation
pn Real wave number of the beams
Total pressure inside air pocket
Ptot(t)
Nap Vapor pressure
Radius of curvature of the incident waves in the impact region Oscillation period
TN Time scale of the second time phase
Time scale of the initial time phase Time parameter
Time parameter in the numerical time integration Time duration of creation of air pocket
ti Time at time step j
to Time duration of initial time phase
At Time step in numerical time integration
Forward speed of the vessel
fUnk (t) Generalized velocity components
u(t) Velocity of the air flow at the outflow throuth of the air pocket
V(t) Total vertical velocity of vessel
Ve(x,t) Effective normal velocity of the beams
1 (t) Mean effective normal velocity along cut of the single beam
Vem(x, t) Modified effective normal velocity of the single beam
Vem(t) Mean modified effective normal velocity along cut of the single beam
Vind(x, t) Induced normal velocity along the beams
VT Total relative normal impact velocity at moment of initial impact
VT(t) Total relative normal impact velocity
Wn Complex wave number of the beams
w(x,t) Elastic beam deflection
ro(t) Mean elastic beam deflection in space
Space coordinate Transformed coordinate Transformed coordinate
re
x-coordinate (local) of center of gravity of vessel178ta x-coordinate (local) of position of initial impact
X0 x-coordinate (local) of the incident wave crest
xii Nomenclature
The complex coordinate (Z = 2L-F Space coordinate
Transformed coordinate
Distance from neutral axis to position where stress iS calculated
Greek Symbols
Coefficient in an approximation of parts of the integrand in the expression for the vertical velocity outside the wetted length
are
Relative angle between the beams and the free surface in impact areaa wet(X) Trim angle of the wetdeck
Oi Coefficient in an approximation of parts of the integrand in the expression for the vertical velocity outside the wetted length
Ratio between specific heat at constant pressure and at constant volume
-yrx,t)1 Vortex density
Kronecker delta function Indication of a small distance ((x4)1 Surface deflection due to air pocket
Amplitude of incident waves
779(x,t)! Air gap between the undisturbed free surface and the wetdeck model]
713 (t) Heave motion of vessel
lls Pitch motion of vessel
0' Transformed coordinate
Rotational angle of beams relative to the x-axis at the beam ends
Transformed spatial integration variable
ii Wave number of incident waves
Spatial integration variable
fl Density of water
Pair (t) Density of air
a(x,t)
Bending stressa'
Non-dimensional maximum bending stress amplitudeC Maximum bending stress amplitude
Cr Standard deviation of the relative vertical motion in impact area av Standard deviation of the total relative normal impact velocity
Time integration variable
it
Total velocity potentialVelocity potential of the incident waves Velocity potential of the diffraction waves
Transformed coordinate outside the wetted length Transformed coordinate outside the wetted length:
0a(4
Eigenfunction of the beam vibration mode n (t)Mathematical Operators and other Symbols xiii
(x) Eigenfunction of beam j's vibration mode n
52(t) Volume of the air pocket
Angular frequency of the diffraction wave con dry eigenfrequency of vibration mode n
wet eigenfrequency of the governing vibration mode
Mathematical Operators and other Symbols
V Gradient (Hamilton) operator
N:72 Laplace operator
Summation
Absolute value (modulus)
Chapter' IL
Introduction
'Li
Background and Objective
High speed marine vessels, both monohulls and catamarans, have become increasingly pop-ular during the last decades due to increasing requirements of faster transportation at sea.
Until recently, marine vehicles with speeds above 25-30 knots have been used primarily for passenger transport and naval applications, and a considerable number of high speed vessels with lengths between 20 and 40 meters have been put into service worldwide_ During the last few years, there has also been an extended use of high speed ships forcargo
transportation as well as in ferry transportation. As a result, the size of the vessels has
grown significantly. This development can be connected to increased safety requirements' due to severe accidents during the last 'decade.
High forward speeds, 'complex structural outlines and the use of novel materials are factors which make the analysis of these vessels more challenging than for conventional ships. An 'important parameter in the design of high speed craft is the weight. The weight influ-ences e.g. the number of passengers that a vessel can carry, i.e. it influinflu-ences the economics of transportation. Weight optimization is also sought because low weight reduces the re-sistance and improves the performance of the vessel. It also reduces the material costs. The weight optimization must be carried out so that the structural strength of thevessel is,
satisfactory. On catamarans, the part of the vessel referred to as the wetdeck, is important
for the weight optimization. The wetdeck is the lowest part of the cross structure, which
connects the two side hulls of the catamaran. The wetdeck consists of plate fields that are supported by longitudinal stiffeners and transverse frames, as shown in Figure 2.1. The design loads on this part of the vessel are hydrodynamic loads due to wave impacts, which are also referred to as 'slamming.
The wave impacts on the wetdeck may cause severe local! loadings. These local
2 Chapter I . Introduction
namic loadings and the corresponding response depend on the elastic deformations of the
wetdeck. The slamming loads will also give rise to transient global elastic deformations of the ship hull, often referred to as whipping. Slamming occurs most frequently in sea
states where large relative vertical motions due to resonant heave and pitch motions of the vessel occur. The load level due to these impacts depends on the time history of the rela-tive normal velocity between the wetdeck and the water surface, and an accurate estimate of the load and corresponding response becomes more important as the forward speed of the vessel increases. The hydrodynamic loads due to the wave impacts have usually been obtained from empirical formulas describing the slamming pressure, where the elasticity
of the wetdeck is not taken into account. Lack of adequate experience and knowledge
have therefore brought forward the need of practical and reliable estimates of the load and response on the wetdecks due to wave impacts.
Different physical effects may influence the slamming load. Compressibility of water will
have a small effect, but unless maximum pressure levels are investigated, it maybe ne-glected. When the local angle between the water surface and the body is small, an air pocket may be trapped between the body and the water surface. The compressibility of
air influences the creation and subsequent collapse of the air pocket. When the air pocket
collapses. air bubbles will form and mix with the water underneath the body. The large
hydrodynamic loads that occur during wave impacts may yield local time dependent
de-formations of the structure, which again will influence the load. This effect is referred to as hydroelastic. Hydroelasticity is particularly important for wave impacts on flexible structures with small angles between the structure and the water surface, which is often the case for wetdecks. The local dynamic behavior ofthe elastic structure may subsequently lead to cavitation and ventilation, but this normally occurs after maximum response is reached. The shape of the structure is also important for the load and corresponding
re-sponse. The shape of wetdecks may vary between the completely flat (horizontal) wetdeck
to the wave-piercers where the wetdeck is more to be considered as a
third hull. A flat
wetdeck will experience a different load and response than a wedge-shaped wetdeck. An
important parameter for the wedge-shaped structures is the deadrise angle.
1.2
Scope and Limitations
This study presents a method to estimate the local hydrodynamic load and corresponding local response due to wave impacts on stiffened plate sections. The present study is gen-erally valid for all stiffened plate sections with and without forward speed and trim angle, but the study will particularly focus on catamaran wetdecks. Both the structural as well as the hydrodynamic model are two-dimensional, considering the fluid flowand structure in
the longitudinal ship direction. This means that only flat wetdecks areconsidered, and it
includes the assumption that only the longitudinal stiffeners of the wetdeck are deformed.
The incident waves are assumed to propagate in a direction parallel to the longitudinal stiffeners. The structural model does not include shear and tangential deformations.
Ef-I ii
Previous Work
fects of curvature of the wetdecks can thus not be accounted for. The effects of air pockets are not included in the numerical model of the impact problem, but is. treated separately in separate sections. Compressibility of the water is neglected.
In order to investigate slamming on wedge-shaped wetdecks, an approximative
three-dimensional model of both the structure as well as the hydrodynamic fluid flow is needed.
This is not treated in the presented work, but the most general influence of a deadrise angle in the cross-sectional direction can be derived from the results of varying relative
angles between the two-dimensional wetdeck model and the wave profile.
Emphasis is put on the description of the hydrodynamic load, but since this load and the
corresponding structural response is strongly dependent on the elasticity of the wetdeck, focus is also directed towards the description of the structural flexibility of the wetdeck. The
investigation will concentrate on local hydrodynamic loads due to wave impacts and the
corresponding structural responses. Typical structural responses are the stresses induced
in the wetdeck plating and longitudinal stiffeners. These stresses are important for the design of the wetdeck structure locally. The investigation will lead to better insight in
how the various physical effects will influence the slamming load and the response of the
wetdeck.
1.3
Previous Work
Slamming has been Widely studied during the last decades, both numerically and ex-perimentally. However, the majority of the investigations are on slamming against rigid
two-dimensional bodies. The earliest investigations are those by von Karman (1929) and
Wagner (1932), who introduced the local jet flow analysis which is valid for horizontal
bodies only (a first order approximation). Self-similar solutions (based on analytical for-mulations) for slamming against wedges with large deadrise angles was introduced by Do-brovol'skaya (1969). In more recent studies by e.g. Cointe and Armand (1987) and Cointe (1987,1991), Wagner's theoretical basis is further developed by the use of matched asymp-totic expansions, making the analyzes valid for wedges with small deadrise angles and for
circular cylinders. Zhao and Faltinsen (1992,1993) present a fully non-linear numerical method based On a boundary element technique to study slamming on two-dimensional bodies of arbitrary shape. They also present a simplified method. which is a generaliza-tion of Wagner's theory. The simplified method is extended to include separageneraliza-tion from
knuckles in Zhao et al. (1996). In Faltinsen and Zhao (1997), both methods are gener-alized to three-dimensional axisymmetric bodies. Muzaferija and Peric (1998) presenta three-dimensional, non-linear method based on a Navier-Stokes solver and a finite volume method in order to be used on slamming problems. However, viscous effects are generally
not important for slamming processes, and the method may therefore seem unnecessary
cumbersome and computationally expensive. Other works on slamming on two-dimensional sections are those by e.g. Arai and Tasald (1987), Arai and Matsunaga (1989) and Howison
Chapter 1, Introduction
et al. (1991).
Compressibility effects of the water at an initial stage of the impact is studied by Korobkin
and Pukhnachov (1988) and Korobkin (1994a,1994b). These studies show that the fluid
flow is initially supersonic, and compressibility should be accounted for when determining
the maximum pressure. However, except for a very small time duration after initial im-pact, the compressibility has little effect on the pressure level. Since the maximum local
response levels of the wetdeck occur in a much larger time scale than when the compress-ibility effects matter, water compresscompress-ibility is neglected in the present study.
The creation of air pockets is studied theoretically by Verhagen (1967), Koehler and
Ket-tleborough (1977) and Falch (1986). They study the compressible air layer between a falling body and a horizontal free surface, assuming incompressible fluid. More detailed
analyzes of the creation of air pockets are performed by Iwanowski et al. (1992,1993). All investigations show that the pressure gradient of the air near the edges of the falling body
causes an elevation of the free surface. When the deadrise angle between the body and
the water surface is small, air will be trapped underneath the body, forming an air pocket. The theoretical studies of air pockets are confined to the creation of the air pockets. What happens after the air is trapped is generally not treated, except in Verhagen (1967). Addi-tionally, Faltinsen (1997) introduces a simplified method to investigate the situation when an air pocket between a water surface and a flexible body collapses. Experimental studies of air pockets are performed by e.g. Verhagen (1967), Yamamoto et al. (1983). Miyamoto and Ta.nizawa (1985) and Watanabe et al. (1988). The present investigation of air pockets is motivated by observations of air pockets, such as in the experiments reported by Aarsnes
(1994,1994). The theoretical method derived herein is primarily based on the studies by
Verhagen (1967) and Faltinsen (1997).
When the angle between the impacting body and the free surface is small, the
elastic-ity of the structure will influence the slamming load. However, theoretical investigations
of slamming against elastic structures are limited. Except Meyerhoff (1965) who
stud-ied slamming on elastic two-dimensional wedges penetrating an initially calm water
sur-face, extensive studies of slamming against elastic structures have only been carried out during the last few years. Vasin (1993) studies the hydroelastic effects of an elastic
two-dimensional wedge entering a horizontal free surface. Kvalsvold (1994), Kvalsvold and Faltinsen (1993,1994.1995) and Faltinsen (1997) present theoretical investigations of wave
impacts between elastic beams and a wave surface. Additionally, Korobkin(1995.1996a,
1996b) and Khabakhpasheva and Korobkin (1997,1998) have also investigated different
aspects of wave impacts on elastic wetdecks using a two-dimensional model of the wetdeck. The studies by Meyerhoff (1965) and Vasin (1993) may be considered as a two-dimensional cross-sectional model of a wetdeck, while the studies by Kvalsvold and Faltinsen represent a model of a wetdeck in the longitudinal direction, between two transverse frames. Kvalsvold
and Faltinsen model the wetdeck as a two-dimensional beam and express the slamming
load as a function of the beam deflection. Focus is set on the maximum stress levels in the
1.3. Previous Work
structure due to the slamming load, as opposed to slamming studies in general, which often focus on the pressure peaks. The same problem is additionally investigated experimentally by Aarsnes (1994,1996), Kvalsvold et al. (1995) and Faltinsen et. al. (1997). Kvalsvold
(1994) and Kvalsvold and Faltinsen (1995) suggest that the impact can be divided into
three time phases, where different forces dominate in the different phases. This simplifies
the solution method and is utilized in later studies of elastic wetdeck slamming by e.g. Faltinsen et al. (1997) as well as in the present study. The present investigation of
slam-ming against elastic wetdecks is a further development of the investigations by Kvalsvold and Faltinsen.
An approximative three-dimensional theoretical investigation of hydroelastic wetdeck
slam-ming is performed by Faltinsen (1998). This study focuses on a wedge-shaped wetdeck, where a more three-dimensional analysis is needed to obtain a correct description of the
hydrodynamic loading. Orthotrope plate theory is used to describe the wetdeck between two transverse frames. The deflection of the plate in the longitudinal direction gives a
lon-gitudinal variation of the fluid flow, and strip theory is used to describe the hydrodynamic
load in the cross-sectional direction. The results from this investigation are compared to
the results obtained from a fullscale experiment of wetdeck slamming reported by Aarsnes and Hoff (1998).
Experimental studies of wave impacts between a horizontal three-dimensional wetdeck panel and a wave surface have been done by Samuelides and Katsaounis (1997), while
Havsga.'rd et al. (1998) have performed experiments of wave impacts between a
three-dimensional sandwich composite plate model of a wetdeck and a wave surface. Hayman et al. (1991) have carried out experimental studies of slamming against wedge-shaped elas-tic ship sections of both aluminium and sandwich material, which impact a horizontal free
surface. The deadrise angle of the sections used in these studies was 300, but the actual angle between the section and the free surface was varied from 50 to 30° by giving the
section a heel angle. It was found that the slamming load was not significantly influenced
by the elasticity of the structure. A fullscale experiment of wetdeck slamming has also been performed and is reported by Aarsnes and Hoff (1998). This experiment revealed considerable stress levels in the wetdeck due to slamming. which according to the DNV
Rules for High Speed and Light craft should not occur.
Global loads of multihull vessels, such as shear forces and bending moments have recently become increasingly interesting. These loads may be a result of wetdeck slamming. Slam-ming against rigid wetdecks has in this context been studied by e.g. Kaplan and Malakhoff (1988) and Kaplan (1987,1991,1992). These studies apply a simplified momentum consid-eration to find the slamming load and resulting vertical rigid accelconsid-eration as well as shear forces and bending moments in a global analysis of the vessel. A local slamming analysis is not focused on. Zhao and Faltinsen (1992) have also investigated the effect of wetdeck slamming on the ship motion and acceleration. In this study a more elaborate model for the slamming load is used. All these studies find out that the slamming loads are important for
6 Chapter I. Introduction
the global accelerations of the vessel, but less important for the ship velocities and motions.
Global loads due to the hull girder flexibility have also become an increasingly studied
topic. These loads are also typically bending moments and shear forces. The global
elas-tic vibrations can be separated into two kinds; the steady state elaselas-tic vibrations, called
springing, and the transient whipping vibrations. Slamming on monohulls and catamaran side hulls will primarily influence the global loads, whereas wetdeck slamming may also influence the ship motions. Wetdeck slamming induced whipping vibrations and the corre-sponding global loads on multihulls have been studied by e.g. Kaplan and Dalzell (1993),
Okland et al. (1998) and Okland and Moan (1998). These studies find that the global
structural vibrations can influence the slamming load. The same is seen in the results from the fullscale slamming experiments reported by Aarsnes and Hoff (1998). These theoreti-cal studies apply simplified approaches to determine the slamming loads on the wetdeck, such as momentum considerations. and the local elasticity of the wetdeck is not accounted
for. These investigations have found that there is a mutual interaction between the ship
motions, whipping vibrations and the slamming load on the wetdeck. This indicates that the ship motion and velocity as well as the global hull girder vibrations must be known in order to accurately estimate the slamming load on a multihull wetdeck. Additionally,
the effects of the local elasticity of the wetdeck should be included when estimating the
slamming load, which yields the whipping vibrations and affects the ship motion.
1.4
Present Study
The fluid-structure interaction problem is approached by first introducing a structural model and a method to solve the structural problem. Then a method for the formulation
and solution of the fluid flow is selected, and these two problems are coupled together by the governing beam equation of motion.
The wetdeck between four transverse frames is modeled by two-dimensional Euler beams,
which represent the longitudinal stiffeners and the surrounding plating. This model will in the following be referred to as the three-beam model. The transverse frames are in-troduced as beam end supports, where also rotatory springs are located. A larger partof
the wetdeck is modeled in this study, compared to what is done in the previous studies of elastic wetdeck slamming by Kvalsvold and Faltinsen. In those studies, the beam model includes the wetdeck between two transverse frames, and is hence referred to as the one-beam model. The governing one-beam equation of motion is solved by a modal analysis, which means that the beam deflection is expressed by a linear combination of complete functions, in this case chosen as the eigenfunctions of the beam model. The details of the structural formulation are given in Chapter 2.
Chapter 3 describes the formulation and solution method of the hydrodynamic part of
1.4. Present Study
theory is assumed, water compressibility and gravity are neglected. The problem is lin-earized and is hence limited to impacts where the angle between the wetdeck model and the free surface is small. The theory includes effects of forward speed and a pitch angle. A time dependent drop velocity is accounted for. No symmetry assumptions in the fluid flow or beam response are made. Hence, the wave may initially hit the beam model at an arbitrary position along the beams. A non-zero pitch angle causes the beam end to
penetrate the free surface first, or causes the beam model to hit the wave outside the wave crest, depending on the wave profile relative to the beam model.
A time-domain solution of the problem is found by a numerical time integration of the governing modal beam equation of motion. Based on the earlier findings by Kvalsvold and Faltinsen, the problem is divided into two time phases; an initial impulse phase and
a forced vibration phase. In the initial phase, a large hydrodynamic impulse force causes
a large acceleration of the structural mass of the wetdeck model. The time duration of
this phase is short relative to the following time phase. and the details of the load and re-sponse in this phase are not needed to describe the behavior in the second time phase. The maximum response and stress levels occur in the second time phase. Hence, the simplified impulse method is introduced: Only the beam response in the second time phase is solved in the numerical time integration, with initial conditions determined from the impulse load
in the initial time phase. When the simplified impulse method is used, a smaller number
of eigenfunctions is needed to describe the beam deflections, compared to the studies by Kvalsvold (1994), Kvalsvold and Faltinsen (1993,1994,1995). In the one-beam model of the wetdeck, the lowest eigenmode dominates the response. When the simplified impulse method is applied on the one-beam model, the beam deflection is described well enough by one eigenmode (Faltinsen 1997, Faltinsen et al. 1997). In the more complex three-beam
model, at least the three lowest eigenmodes influence the response and must be included in the modal description of the beam deflection.
The hydrodynamic pressure in the second time phase is found by solving a hydrodynamic boundary value problem where the beam flexibility is accounted for. The wetted length at the beginning of the second time phase is given by the time duration of the initial impulse phase. The subsequent wetted length propagation is found by a generalization of Wagner's method (1932), which includes the effect of the beam vibration and forward speed. Details
at the spray root of the jets are not incorporated, which implies predictions of infinite pressure at the spray roots. The beam vibration velocity is described by a Fourier cosine
series, which allows an arbitrary variation in the vibration velocity. When the wetdeck is
modeled by three beams as in this study, it is important that the beam vibration velocity is accurately described. A simplified analysis of a three-beam model where the vertical beam vibration is assumed constant in space, given by Kvalsvold (1994), will not give a
correct slamming load due to the vibrations of the beam model.
The hydrodynamic force is expressed in terms of added mass, damping and excitation forces. The damping and excitation forces are proportional to the velocity of the wetted
8 Chapter 1. Introduction
length propagation, and are hence zero when the beam model is fully wetted. From this moment on, the beam model is subjected to a free vibrationwhere the added mass force
and the structural stiffness force govern the vibration. In the previous studies where the
simplified impulse method is used on a one-beam model, as in e.g. Faltinsen(1997), the
initial phase is defined so that the beam is fully wetted at the beginning of the second
time phase. The entire second time phase is thus a free vibration of the beam, as opposed
to the second time phase in the study of the three-beam model. The three-beam model is first subjected to a forced vibration due to the hydrodynamic damping and excitation
forces, until the model is fully wetted.
The presented theoretical model describing the initial conditions due to theimpulse load
in the initial phase of the impact is derived for impacts without air pockets. Air pockets
may cause a different loading in the initial phase, and the model will then give incorrect
initial conditions for impacts where air pockets are present. Hence, a simplified analysis
of impacts with air pockets is carried out. For simplicity, this investigation focuses only
on a one-beam model of the wetdeck. The investigation is based on several assumptions and strong simplifications, and the results can thus be used only in a qualitative discussion about the general physics of the air pocket collapse and subsequent beam response.
Chapter 4 gives a brief description of the slamming experiments with an elastic three-beam wetdeck model, that have been carried out at MARINTEK. Theseexperiments are
reported by Aarsnes (1996).
The presented theoretical model is validated by comparing the numerical results by the
results reported from the experiments. The numerical results are in satisfactory agreement
with the experiments. It is seen that the influence of several eigenmodes on the response
makes it important to determine the eigenmodes accurately. Some differences between the numerical and experimental results can be observed for impacts where the anglebetween
the model and the free surface is zero, and this is explained by the creation of air pockets
in the experiments. Various parameter studies are carried out, wherethe influence of im-pact velocity, forward speed, structural mass, wave radius of curvature, position of initial
impact and angle between the beam model and the free surface, on the stress levels in
the wetdeck model is investigated. Systematic studies of the importance of hydroelasticity
as function of the angle between the beam model and the free surface and the impact velocity are presented. The stress level in the wetdeck model is expressed in terms of a non-dimensionalized parameter, which is introduced by Kvalsvold (1994). The present
study suggests that the non-dimensionalized stresses should be presented as a function of a non-dimensionalized parameter describingthe wetting time of the beam model relative
to the highest eigenperiod of the beam model. This isalso suggested by Faltinsen (1998). The results from these studies and the following discussions are given in Chapter 5. Results from the simplified air pocket analysis are also presented.
1.4. Present Study
on the dynamic amplification factor given by the impulse loading. The investigation shows
that the dynamic hydroelastic effects are important when the angle between the beam
model and the free surface is small. The maximum stress levels decrease when the angle increases, and the response will finally become quasi-static. Both the maximum stress lev-els and the influence of the hydroelastic effects depend on a combination of impact velocity and angle between the beam model and the free surface. For impacts with zero angle, the maximum stress levels are approximately proportional to the impact velocity and almost independent of the radius of curvature of the incident waves, unless for very small values of the radius of curvature. The global acceleration of the model/vessel, which reduces the impact velocity during the impact, may reduce the stress levels significantly.
In Chapter 6, the results from the investigation are applied in an analysis of a fullscale
catamaran. A method is presented where the operational criteria based on the probability of wetdeck slamming are investigated. However, when the stress level in a fullscale wetdeck is estimated from the non-dimensionalized stresses obtained by the presented theoretical
model, very large stress levels are found. Differences between the idealized theoretical method and the real world, that may explain the unrealistic large stress levels, are pointed out. The influence of e.g. the vessel acceleration which acts to reduce the impact velocity, and an unsmooth wave surface with air bubbles and ripples, are believed to be important factors that will reduce the slamming load and corresponding response significantly. The influence of the side hulls on the free surface elevation in the tunnel of the catamaran, as discussed by Arai et al. (1995), is also believed to be important. On the other hand, Fahilt-sen and Zhao (1998) show that the side hull shape will tend to increases the impact velocity. Finally, conclusions and recommendations for future work are given in Chapter 7.
Parts of the presented theory and results have been published during the course of the
Chapter 2
Simplified Structural Model
The wetdeck of a catamaran will generally consist of a flat plate which is supported by stiffeners in the longitudinal direction. This stiffened plate is supported by transverse
frames at given intervals, see Figure 2.1. The spacing between the stiffeners in a wetdeck is typically 15-30 cm, while the frames are located about 1 m apart. The transverse frames are much stiffer than the longitudinal stiffeners, and by considering them as rigid, they are not subjected to any deformations.
Three-dimensional structural effects will occur, but when the width of the plate section (B) is large compared to the length of the plate (LB), the loading in the center will be transferred directly to the supports at the transverse frames (see Figure 2.1). By neglect-ing the forces that act in the transverse direction, the estimated stresses in the plate are
slightly too large. The error in neglecting the forces in the transverse direction decreases as the ratio BILB increases. The closely spaced longitudinal stiffeners increase the stiffness in the longitudinal direction, and this causes even more of the loading to be transferred to the transverse frames. With a uniform load in the transverse direction, the response of the plate section may be estimated by considering a longitudinal strip of the section with unit width.
The wetdeck is hence approximated by a beam model; where axial and shear deforma-tions and rotary inertia are neglected. This beam model is referred to as the Euler beam model. Kvalsvold and Faltinsen (1995) have shown that shear deformations and rotary
inertia effects are not essential. The wetdeck may be curved in the longitudinal direction, particularly in the front part of the vessel, and axial forces will occur. However, realistic curvatures of the wetdeck will not significantly influence the response. Large deformations in the wetdeck due to large loading will also introduce axial forces, which is often referred to as the membrane-effect. The predicted load-carrying capacity of the wetdeck increases if membrane-effects are included, but these effects are not considered in the present study. Standard beam theory is applied, assuming that plane cross sections remain plane during deformation.
As a very first attempt of describing the wetdeck, one beam is used to describe the part of
12 Chapter 2. Simplified Structural Model
the wetdeck between two transverse frames. The beam is then a two-dimensional model of one longitudinal stiffener and the adjacent plate field, which acts like a flange. Half
the width of the flange is assumed to equal half the distance between two longitudinal
stiffeners, The effective flange may be somewhat smaller. This model can be seen as the
shaded part of Figure 2.1 between two of the transverse frames. The beam is supported by the transverse frames at the beam ends. Boundary conditions must be imposed at the
beam ends, and the stiffness of the parts of the wetdeck structure outside the modeled part must be included in these boundary conditions. These effects are included by introducing
rotational springs at the ends. The rotational springs are defined by the spring stiffness ko = M/O, where M is the moment and 0 is the angle of rotation at the beam ends. No inertia effects from the parts of the wetdeck outside the modeled part are included. This
structural model of the wetdeck is basically the same as the one used by Kvilsvold (1994).
side hull
longitudinal stiffener
wave profile
Figure 2.1: Part of wetdeck. The shaded part is included in the structural beam model.
A more accurate way of describing the wetdeck is to include a larger part of the wetdeck,
MI
2.1. The Single Euler Beam Model 13
such as increasing the number of transverse frames in the model. The part of the wetdeck
that is now included in the structural beam model is shown as the shaded part in
Fig-ure 2.1. The effects from adjacent structural parts of the wetdeck are then more correctly
included compared to when only one beam with rotational springs at the ends, is used.
Including a larger part of the wetdeck in the model proves to be important for the response, as will be pointed out later.
Figure 2.2: Wetdeck modeled as 3 beams.
In the following modeling of the structure, as well as in the hydrodynamic description of the impact, a local, right-handed coordinate system that follows the speed U of the vessel
will be used. The x-axis is in the longitudinal direction, pointing aft, while the z-axis points upwards and the y-axis to starboard. The origin is located at the center of the
beam model. The coordinate system can he seen in Figure 2.2.
2.1
The Single Euler Beam Model
The wetdeck is modeled by one beam with length LB. which can be seen as the middle
beam shown in Figure 2.2 or as half of the shaded part in Figure 2.1. The governing beam equation of motion is given as:
MB (711 (X , t) + (i))
EI54w(I't)
art
=p(x,w,t)
where MB is the mass of the beam per unit length and unit width. The beam consists of one longitudinal stiffener and the plate flange with width equal to the distance between
two transverse frames. w(x,t) is the local deformation of the beam, while V (t) is the global vertical velocity of the wetdeck due to the global ship motions. El is the bending stiffness of the beam and p is the hydrodynamic pressure. Dot means differentiation with respect to time. It is here particularly noticed that the hydrodynamic pressure is not only a function
of the time and space coordinates, it is also a function of the deflection of the wetdeck
model.
(2.1)
The equation also needs a set of boundary conditions. These are:
w(x,t)= 0
for x = ±LBI202w(x,t) ke Ow(x,t)
ax2
El
axThe first equation satisfies zero deflection at the beam end supports. while the last
equa-tion yields continuity of the moment at the supports. The rotaequa-tional spring is introduced to account for any rotational stiffness due to the transverse frames and the parts of the wetdeck outside the modeled part. The spring constant may not necessarily be the same at the two supports.
2.2
The Wetdeck Modeled by Three Euler Beams
The more accurate model of the wetdeck is obtained by including the part of the wetdeck
between four transverse frames. This three-beam model is shown in Figure 2.2. The governing beam equation of motion is still given by Equation 2.1, while the set of boundary conditions becomes:
Chapter 2. Simplified Structural Model
for x = ±LB/2 (2.2) w(j)(x,0 o aw(i1(x,0
au0+1)(x,0
Ox Ox kp0)(x.0
8,0+1)(x,0
ax2 ax2El
Ox kg aw(1)(i.,t) Ox2El
Ox kg aw(3)(x.t) ax2El
axThe superscripts j refer to the beams numbered by 1 to 3 as shown in Figure 2.2. xj
de-notes the x-coordinate of beam end supports, which are numbered from 1 to 4 from the left hand side. The first equation gives zero deflection at the supports, while the second
equa-tion requires continuity in the rotaequa-tional angle at the supports. The three last equaequa-tions
yield continuity of the moment at the supports. As in the one-beam model, the rotational springs are introduced to account for any rotational stiffness due to the transverse frames
and the parts of the wetdeck outside the modeled part. The spring constant may not
necessarily be the same at all four supports.
for x = xi,xj+1 j = 1, 2, 3 for x = j = 1, 2 for x = xj+i
j = 1,2
(2.3) for x = for x = x4=
2. 3 . Modal Analysis 15
2.3
Modal Analysis
One possible way of solving Equation 2.1 with corresponding boundary conditions is by
introducing an approximate solution of the problem by separating the variables and
ex-pressing the unknown function W(X, t) by a sum of known functions *ii(x):
W(X,t)
= E
an(t)0.(x)Ng
E (t)On(x)
(2.4)n=I
where an(t) are generalized coordinates and /1),(x) are coordinate functions. It is required
that each coordinate function satisfies the boundary conditions given by Equation 2.2 or
Equation 2.3 separately. The result of this is that the total solution w(x, t) also satisfies the boundary conditions for all given values of the generalized coordinates. Additionally, the coordinate functions have to satisfy the requirement that they are linearly independent and that they form a complete set, of functions, i.e. they must form a basis, on the domain where
w(x, t) is defined. This method of approximating the unknown function implies that the
differential equation (Equation 2.1) is assumed linear. It will in Section 3.2 be shown that
this is strictly not the case, as weak non-linearities will occur, but the described method
is still applicable for this case. The deflection w(x, t) can now be described as accurate as
wanted by choosing a large enough number of terms, AT eig, in the sum.
Korobkin (1996a) shows that the solution of the beam response during the very early stage of the impact problem has a self-similar behavior, i.e. the deflection w(x, t) does not only
depend on x and t separately, but also on the combination x/Ifi. However, this occurs only as t --+ 0, and it will later be discussed more in detail that the exact solution of the initial impact problem is not important for the further time development of the impact and the beam response. The deflection is thus described by Equation 2,4 with no more
attention payed to the self-similar solution that arises initially.
2.3.1
The Eigenvalue Problem
As long as the requirements mentioned above are satisfied and the coordinate functions are sufficiently continuous and differentiable, they can be chosen relatively arbitrarily. A good choice is to use the eigenfunctions of the problem described by Equations 2.1 and 2.2 for the single Euler beam case, and Equations 2.1 and 2.3 for the three Euler beam model. These functions automatically satisfy the boundary conditions, and they form a complete
orthog-onal set of functions on the two domains LB/2 < x < LB/2 and 3LB/2 < x < 3LB/2, respectively. Orthogonality implies that the functions are linearly independent. Orthog-onal functions also have the advantage that they will make all off-diagOrthog-onal elements in
the structural mass-, damping- and stiffness- matrices to be zero (cf. Clough and Penzien
Therefore, ib(x) in Equation 2.4 is chosen as the dry eigenfunctions of the structural
model. This is an approach widely used in the literature. In addition to Kvalsvold (1994) who uses it in a local structural analysis, it is also used by e.g. Bishop and Price (1979) in
a global structural analysis. In a hydrodynamic problem, the coordinate functions could
also have been chosen as the wet eigenfunctions. By wet is meant that the added mass
of the structure is included. However, the wet eigenfunctions are not orthogonal and will hence give a more complex equation system to solve. It is additionally more cumbersome to find the wet eigenfunctions because the added mass associated with each eigenfunction is not known a priori.
The eigenvalue problem is given by Equation 2.1 with the right hand side and1:7(t) set
equal to zero. The solution is assumed to be of the form:
w(x,t) = Cnew^xe'^` (2.5)
Cn is a constant, TV, is the wave number and is the eigenfrequency. These are not yet known. The assumed solution of w(x,t) is substituted into the governing equation of mo-tion, and an equation giving the relation between the wave number and the eigenfrequency is obtained, (Cn 0 since we want a non-trivial solution):
114
wra
= p
E I "
This means there are four wave numbers, both complex and real:
p,
1477, = ipn (2.7) where i is the imaginary unit. Since there are four different wave numbers, there are four
different solutions. These solutions are linearly independent and they therefore form a basis. The eigenfunctions are thus given as a linear combination of the four solutions.
The Single Euler Beam Model
The eigenfunctions can be written as:
/i(x) = An sin
pa
+ 13,, cos pnx + Cn &UM px + .131,, cosh px (2.8)The eigenfrequency Lan and all except one of the coefficients A,...,D are determined by the boundary conditions. In order to obtain non-trivial solutions, the determinant of the
coefficient matrix describing the boundary conditions must be zero, and this requirement is used to determine the eigenfrequencies of the problem. There exists an infinite number
of eigenfrequencies causing the determinant to be zero, and this gives the solutions for
n 1, no. When the eigenfrequency for mode n is found, the corresponding coefficients
are found by satisfying the boundary conditions. There are four equations for the boundary
(2.6)
16 Chapter 2. Simplified Structural Model
,t4.
Modal Analysis 11conditions, and one is already used to find the eigenfrequency. Three equations are there-fore left to determine four coefficients. The fourth coefficient is hence chosen arbitrarily; here the coefficient& are normalized so that the sum of the four coefficients equals one The slamming analysis using the one-beam model of the wetdeck will in the further study be simplified by assuming the fluid flow and the beam response to be symmetric about the
center of the beam model. The number of boundary conditions is then reduced to two
and the eigenfunctions are reduced to:
= B! Cos Pflx + D,, cosh pnx
The 'Three Euler Beam Model
Due to, the different boundary conditions for the three beams, each 'beam has its- own
deformation pattern. The eigenfunctions can be written as:,
4) sin px + B$,I) cospx + C1,1)'sinh px + Df;') cosh px
42) sin px + X2) cos
px + Cr sinh px + D?) coshp,x,
A?) sin px + .13,3) cos pnx + 0,3) sinh pnx + Ii2) coshpx.
{2.10
The superscripts 1 to 3 again refer to the three different beams. The structural mass MB
and the bending stiffness El may vary for each beam. The wave number will then also vary, while the eigenfrequency is always the same for all three beams. The eigenfrequency con and; all except one of the coefficients AV , j = 1, 2, 3 are determined by the boundary
conditions. In order to get non-trivial solutions, the determinant of the coefficient matrix describing the boundary conditions must be zero, and this requirement is used to determine the eigenfrequencies of the problem. There exists an infinite number of eigenfrequencies causing the determinant to be zero, and this gives the solutions for 7i = 1, ..., oo. When the eigenfrequency for mode 72 is found, the corresponding coefficients are found by satisfying the boundary conditions. There are twelve equations for the boundary conditions, andone
is already used to find the eigenfrequency. Eleven equations are therefore left to determine twelve coefficients. The twelfth coefficient is hence chosen arbitrarily, here the coefficients are normalized so that the sum of the twelve coefficients equals three.
2.3:2
The Governing Modal Beam Equation of Motion
After solving the eigenvalue problem, the original problem 'described by Equations 2.1 and 2.2 or 2.3 is considered again. The modal decomposition of w(x,t) given by Equation 2.4
is substituted into Equation 2.1. Since each one of the eigenfunctions 0(x) satisfies the boundary conditions, so will also the linear combination of the eigenfunctions do. The
boundary conditions for the problem are therefore satisfied. The beam equation of motion
is multiplied by Wm(x), m = 1, ..., A re,g and then integrated over the domain. A set of
(2.9)
18 Chapter 2. Simplified Structural Model
New equations needed to determine the New unknown variables an(t) is then obtained. The governing modal beam equations of motion can now be written in the same way as
Kvalsvold (1994) did. For the three Euler beam model:
31.1 3 /2 b(t)
E Almnez,i(t) + E Cmnan(t) + A/113V (t) 0,7,(x) dx =
f
p(x,w,t)71.),(x) dx (2.11)n=1 n=1 B /2 a(t)
where Illmn and Cm, are entries in the generalized structural mass and generalized stiffness
matrices, respectively. The hydrodynamic pressure acts from a(t) to b(t). This wetted length will generally increase rapidly in time. How to determine the wetted length is
focused on in Section 3.4. The generalized coefficients M, and Cmn are given as:
3L B /2 Mm. = MB f Om(X)011(X)C1X (2.12) B /2 31,8 /2 Cm
El f 0,(x)0" (x)dx
(2.13) 3L3/2where primes signify differentiation with respect to the spatial coordinate x. The single
Euler beam model yields a similar system of modal beam equations of motion, except for
the integration limits which are defined by the domain L8I2< x < LB/2.
Equation 2.11 can be simplified. Due to the orthogonality properties of the eigenfunctions, Mm,. = 0 for m rt. It can also be shown that Cm = 4/1/./m, (see e.g. Kvalsvold (1994)). The solution procedure described above means in other words that the governing equation is satisfied on a weighted and averaged form. When, as here, the chosen weight functions
are the same functions as those used to approximate the unknown solution w(x,t). the
method is commonly known as the Galerkin method, which is described in e.g.Zienkiewicz
and Taylor (1989).
Chapter 3
The Hydrodynamic Formulation
The hydrodynamic pressure p(x,w,t) in the governing modal equations of motion,
Equa-tion 2.11, remains to be determined. The wave impact is described as a hydrodynamic
boundary value problem, and the pressure is obtained as the solution of this. The bound-ary value problem used to describe the impact is similar to the one Kvalsvold (1994) used,
i.e. it is a generalization of the boundary value problem introduced by Wagner (1932).
The hydrodynamic boundary value problem will be solved only for the three Euler beam
model, since the impact problem with a single Euler beam model is already thoroughly
described by Kvalsvold (1994). The one-beam model will be used in an investigation where the influence of air pockets are studied.
3.1
The Hydrodynamic Boundary Value Problem
Two-dimensional incompressible potential theory is assumed, implying that viscous effects are not accounted for. The two-dimensional approximation requires that the incident waves
are long crested and propagate in a direction parallel to the longitudinal stiffeners in the catamaran wetdeck. The disturbance of the free surface due to the waves generated by
the side hulls are neglected. Assuming two-dimensional fluid flow and hence loading is a
requirement for the approximation of the plate section with a beam model, as was dis-cussed in Chapter 2. The problem is linearized at each time step, and forward speed is
included. A velocity potential (I) is introduced to describe the fluid flow around the body. (13 = 6+ Ux +61, where the forward speed U is included in the total velocity potential as a constant flow in the positive x-direction (see Figure 3.1). 0 is the velocity potential of the disturbed fluid flow and is the velocity potential of the incident waves. The
hydrody-namic boundary value problem is then described by the two-dimensional Laplace equation
V20 = 0 in the fluid domain, a free surface condition stating that a fluid particle on the free surface remains on the free surface and the pressure on the free surface equals the
atmospheric pressure, as well as a body boundary condition requiring that a fluid particle on the body boundary and the body boundary have the same velocity normal to the body boundary.
20 Chapter 3. The Hydrodynamic Formulation
Half the wetted length of the beam model, which varies in time, is given by:
C()=
a(t) b(t)2
where a(t) and b(t) are the locations of the jet roots in the local coordinate system. The jet roots are here defined as the intersection points between the beams and the disturbedfree
surface, describing the part of the beams that is wetted, i.e. subjected to hydrodynamic loading. The wetted length of the beam model depends on the hydrodynamic boundary value problem and must be found as a part of the solution of the problem. The
develop-ment in time of the hydrodynamic boundary value problem is hence non-linear.
zi (19- = 0 ax X I
/
b(t) a(t) Uta0/az
V20 =0
Figure 3.1: Linearized hydrodynamic boundary value problem.
The linearized free surface condition is derived from14 = 0 on the free surface and
trans-ferred to z = 0.
Acceleration of gravity is neglected because in impact problems, it isgenerally much smaller than the accelerations of the fluid flow. The fluid velocities due to
the potentials 0 and 0] are additionally assumed small, and products of thesefluid veloc-ities are hence neglected. The free surface condition is further divided into two separate
cases, depending on the magnitude of the forward speed relative to the vertical velocity of the beam model. When the forward speed is large, U/V 1, the linearized hydrodynamic
boundary value problem is as shown in Figure 3.1. The linearized free surfacecondition is given by + U = 0 on z = 0, which simplified can be written as
= 0.
Theinitial condition for the free surface is 0 = 0 on z = 0. By following a fluidparticle on
the free surface upstream of the body,= 0 causes 0 = 0 on z
0 to remain the freeDDt
surface condition. This also leads to 2, = 0 on z =- 0 upstream of the body.
Behindthe body, the free surface must be split in two parts: The wake and behind the wake. A
free surface particle downstream of the body starts from the trailing edge and is convected
further downstream as the time increases. Hence, in the wake, the free surface condition
0 on z = 0 applies. Behind the wake, extending to infinity, the freesurface
Ot ex
is not affected by the body and 0 0 on z = 0 due to the initial condition. The wake
extends to the distance Ut downstream of the body, where t is the time from initial impact. This boundary value problem has been studied by Ulstein and Faltinsen (1994) for impacts
= 0 (3.1) 0 = 0 >> +
=
3.1. The Hydrodynamic Boundary Value Problem 91
on an air bag and by Faltinsen (1997) for impacts on the one-beam model. These studies
conclude that the effects of the forward speed U from the free surface condition are not
important for wetdeck slamming under realistic conditions.
The linearized free surface condition for smaller values of U, including U = 0. is described
by = 0 or (17 = 0 on z = 0 outside the wetted length. Due to the conclusions from the
studies of large forward speed, it is believed that the hydrodynamic boundary value prob-lem described by this free surface condition also gives an approximately correct solution for large forward speeds. In the present study, this free surface condition will be applied. The kinematic body boundary condition can be expressed as:
aF
at
= 0 onF(x, z,t) = 0
F(x, z, t) = z - 773(t) + (x - x0)775(t) - (d,t(x) + w(x,t))
F(x, z, t) refers to the local coordinate system and is defined so that F(x,z,t) = 0 on
the wetdeck structure. xG is the longitudinal center of gravity. 713(t) and 715M are the
global heave and pitch motions, referring to a global coordinate system. &et is the vertical distance from the undeformed wetdeck to the x-axis. Evaluating Equation 3.2 gives:
00(x, t) . = 713(t) - (x - xG)7i5(t) (E:
aos,t)
06(x,
t)) (g5(t)
a(x)
aw(x.t)
as
as
as
)
(3.3)au.(s,t)
0 01(x , t) +at
az
onF(x,z,t) = 0
where the trim angle of the wetdeck, when assuming that the wetdeck is close to horizontal, is given as awe = . The body boundary condition can be simplified by assuming that
and 'lit are an order of magnitude smaller than U for realistic values of U. This is
OX OX
not satisfied for 2 close to the jet roots, where an inner solution must be applied. If an
impact without forward speed is to be analyzed, 2 and 12,jL can not be neglected compared
to U. However, it is also assumed that the pitch angle as well as the trim angle of the
wetdeck are small so that + 6490:1t. multiplied by 7/5 (t) - awe (x) can still be neglected. The
body boundary condition can be further simplified by investigating the magnitude of the
term U--L-1:'' relative to°1/'t . The ratio between the two terms can be approximated as QI, where T is a characteristic time scale, and LB is used as the length scale. This ratioLB is much smaller than one for realistic values of U. T and LB, yielding >> U kJ' . The
where (3.2)
-22 Chapter 3. The Hydrodynamic Formulation
0 (1)(x, t)
=VT(t)+ti;(x,t)
az on z = 0 (3.6)
The total relative normal velocity can be explained as the actual impact velocity, i.e. the
normal velocity that is felt by the wetdeck when impact occurs.
The study of the one-beam model with large forward speed by Faltinsen (1997) concludes
that forward speed effects are not important for 0.5LBwIU > 2.5, whereLow is the
gov-erning wet eigenfrequency of the beam model. The only influence from the forward speed is a modification of the impact velocity due to the term U(7)5(t) awn).
3.2
The Solution Procedure
To simplify the solution procedure of the hydrodynamic boundary value problem, a trans-formed coordinate system is introduced, which is defined so that the origin is in the center of the wetted area. The transformed coordinates are given as:
= x (a(t) c(t))
(3.7)
= z
The described hydrodynamic boundary value problem is shown in Figure 3.2. Thisproblem
is similar to the steady lifting problem in infinite fluid. To utilize these similarities, a
term U --Liatuo:'' can hence be neglected. The global ship velocity can further be written as 7)3(0 (x sc,)7)5(t) V (t) when the spatial variation of the global ship velocity in the
impact region is neglected. The radius of curvature of the incident wave is large compared
to the length of the impact region, so that the vertical particle velocity of the incident
waves,19etl,,is also assumed constant with respect to x in the impact region. The value of
is estimated at x = 0. Additionally, the impact is of very short time duration compared
to the oscillation period of the incident wave. The time dependency of the incident wave
is thus neglected. A further simplification can be made by assuming that the curvature of the wetdeck is negligible in the impact region, i.e. °wet constant. The body boundary
condition is additionally transferred to z = 0 in the local coordinate system. The simplified kinematic body boundary condition thus becomes:
00(x,t) \ Or
Or = V (t) + (775(0 aw)e az
on = 0 (3.4)
By defining the total relative normal velocity VT(t) the body boundary condition can be
expressed in the same way as in livalsvold (1994):
vr(t),v(t)
u(775(t) awe) az (3.5)3.2. The Solution Procedure 23
0 = 0
*I X
Figure 3.3: Definition of the transformed coordinate
By expressing the beam vibration velocity with modal components as in Equation 2.4 and
2-0 =
-c(t)
eft)
,00/Or
720 =
Figure .3.2r Transformed and linearized hydrodynamic boundary value problem.
vortex distribution on the wetted length is used to solve the hydrodynamic boundary value problem. The normal velocity on the wetted length induced by this vortex distribution is:
e(t)
tt) =1
I 7(e, t)
(3.8),e
c(t)
where ,y(C t) is the vortex distribution which is integrated along the wetted length, and
is the integration variable. No Kutta condition is applied to the trailing edge. A Kutta
condition will lead to a flowpattern leaving the trailing edge tangentially, which happens when U>> V. In the study of wave impacts at large forward speed by Faltinsen (1997), a
Kutta condition is introduced. As the further description of the boundary value problem
will show, theflow pattern is in the present case split into two parts, one that is symmet-ric and one that is antisymmetsymmet-ric about the -axis. The hydrodynamic pressure is thus
singular at both the leading and trailing edge.
The effective normal beam velocity is defined as V, (x t) = VT (0 + iv (x,0. The transformed
coordinate 01 is introduced to simplify the derivations. Its definition follows from Figure 3.3.
2 = c(t) cos fe .(3.9)
