Nonlinear effects of a flexible stern seal bag on cobblestone oscillations of an SES

TORE ULSTEIN

NONLINEAR EFFECTS

OF A FLEXIBLE STERN SEAL BAG

ON COBBLESTONE OSCILLATIONS

OF AN SES

TEcHNIScEE tThIIVERS ITEIT

Scheepshydromechanìca

Archief

Mekeiweg 2, 2628

D Deif t

Tel: 015-786873/Fax:781836

DOKTOR INGENIØRAVHANDLING 1995:60

INSTITUYF FOR MARIN HYDRODYNAMIKK TRONDHEIM

Nonlinear effects

of a flexible stern seal bag

on cobblestone oscillations

of an SES

Dr.ing. Thesis

by

Tore Ulstein

Department of Marine Hydrodynamics

A numerical and theoretical study of wave induced vertical accelerations

(cobble-stone oscillations) of an SES in small sea states is presented. Resonant spatially

uniform and nonuniform dynamic cushion pressure variations are then important.

In the present study a nonlinear time domain solution is used. The

nonlineari-ties concentrated on in this study are mainly caused by the flexible stern seal bag

behavior. In this connection the local elastic vibrations of the stern seal bag and its interactions with the free water surface and the spatially varying pressure due to air leakage underneath the seals are accounted for. "Dry" modes are used to describe the flexible response of the bag. Hydroelasticity is important during the impact between the bag and the water. The analysis of the hydrodynamic part of

the impact problem has similarities with the high speed two-dimensional planing problem. The effect of spatially varying pressure underneath the flexible stern seal

bag due to air leakage is also studied. It is demonstrated that both the spatially

varying pressure underneath the flexible stern seal bag due to air leakage and the impact between the bag and the water are important for the vertical accelerations

of the vessel. The influence of main parameters characterizing the stern seal bag

are discussed. TECHNISCHE UNIVERSIIFJy Laboratorium voor

Scheepshydromecij

kchlef

Mekeiweg 2,2628 _{CD De!ft}

Abstract

1 _{015.788873.. Fax: 015 781833}

Acknowledgement

This study has been carried out with guidance from Professor Odd M. Faltinsen

whose engagement and encouragement during this work are appreciated.

I am also grateful for the many valuable discussions with my colleagues and good

friends at the Department of Marine Hydrodynamics, the Department of Marine

Structures and at MARINTEK.

This work has received financial support by The Research Council of Norway

(NFR). The computer time is supported by the Norwegian Supercomputing

Contents

Abstract

i

Acknowledgement

2

Contents

3

Nomenclature

6

i

Introduction

13

2 The global formulation of the SES motions

24

2.1 The global equations of motions of the SES 25 2.2 The global hvdrodvnamic approximations 26

3 The air cushion boundary value problem

30

3.1 The boundary value problem 30 3.2 Discussion of the air cushion model 36

3.2.1 The bow skirt wave generation 37

3.3 The numerical solution procedure 42

3.3.1 Verification 48

4 The local air flow underneath the stern seal bag

58

4.1 The 2-D stead air flow underneath the stern seal bag 59

4.1.1 The nonlinear steady panel method 60

4.1.2 Verification of the nonlinear steady panel method 67 4.2 The 1-D local air flow underneath the stern seal 71

5

The structural modelling of the bag

80

.5.1 The static solutions for a i-ioop and 2-loop bag 81

5.2 The linearized dynamics of the bag structure 84

5.2.2 The eigenvalue problem 90

5.2.3 The linear response 97

5.2.4 Discussion of the structural model 99

5.2.5 Verification of the structural model 101

6 The bag as an unsteady planing surface

103

6.1 The theoretical approach 103 6.2 The hydrodynamic boundary value problem 104

6.3 The simplified HBVP 109

6.4 Approximation of the wetted length 109 6.5 The modal hvdrodvnamic force 111

6.6 The numerical implementation 112

6.7 Verification of the hvdrodynamic model 114

7 Scaling of the local bag impact

116

8

Coupling between the motions of the SES and the flexible

bag 120

8.1 Coupling from the flexible bag to the motions of the SES 120 8.2 Coupling from the motions of the SES to the flexible bag 124 8.3 Coupling between the air cushion and the stern seal bag pressures 127

9

Results and discussion

131

9.1 The numerical time simulation procedure 131

9.2 The local hydroelastic analysis of the flexible bag 134

9.3 Results for the global SES response 151

10 Conclusions and recommendations of further

work 166

References

169

A The analytical solution for 2-D steady air flow

174

B The 2-D steady surface wave generation

180

C The principal coordinate

185

E The modal hydrodynamic force

195

E.1 The simplified modal force calculation 197

E.2 The complete modal force calculation 199

E2.1 The simplified hydrodynamic model 208

Nomenclature

a The half element length

a (t) The principal coordinate

A Hvdrodynamic added mass coefficients of the SES side-hulls

A(t)

\lodal added mass matrix (see equation (6.27))

A,f (X) Hydrodynamic added mass coefficient in heave of a side-hull section

A,A

Hydrodvnamic added mass coefficient in heave of a side-hull at the aftermost section

A 2-D louver area of the bag

Af 2-D inlet area of fan

b(X) Sectional beam of the SES side-hulls

B Beam of the air cushion

Hydrodynamic damping coefficients of the SES side-hulls

B(t)

Modal damping matrix (see equation (6.27))

c(t) Half wetted length of bag

c Coefficients in the j'th bag eigenfunction (i 1.8)

Ca(t) Restoring matrix (see equation (3.37))

C,(t)

Restoring matrix for one element (see equation (3.40)) Hydrodynamic restoring coefficients of the SES side-hulls C3'(t) Modal restoring matrix (see equation (5.47))

Modal restoring matrix (see equation (5.47))

c1(t)

Modal restoring matrix (see equation (6.27)) Constant defined in equation (3.61)

Cern Factor accounting for the effective structural mass of the bow skirt

C Coefficient in the eigenvalue analysis of the cable C Coefficient in the eigenvalue analysis of the cable

1/if = length of time interval used in the FFT

Static difference pressure across cable segment i (see Figures 5.1 and 5.2)

see Figure 5.3)

Time step in numerical time integration Complex wave number of the cable e(s, t) Strain (= e0 + ¿(s, t))

EA Axial stiffness of cable

EI

Bending stiffness of cable

f

Frequency in [Hz]

f(t)

Tapering function defined in equation (8.16)

F'(t)

Hydrodvnamic excitation force in direction j due to the SES side-hulls

F(t)

Force in direction j caused by the unsteady excess air cushion pressure

F'(t)

Force in direction j due to the bag

F

Froude number (U//L)

F1

Froude number (U//)

Fa(t) Forcing vector (see equation (3.37))

Fae(t) Forcing vector for one element

F(x: y, t) Function that describe the wetted surface

F&cc,j(t) Modal excitation force vector (see equation (6.27))

FwD. FwD Vectors defined in equation (3.41)

F[. Vector defined in equation (3.42)

FQ'S _{Vector defined in equation (3.45)} g Acceleration of gravity

h(x. t) Height distribution underneath the stern seal bag (see Figure 4.6,

In chapter 6. h Vt

Leakage height underneath seals, = hLo + hL(t) (h'L = kahL) (see Figure 4.6)

Hb Vertical height of bag (see Figure 5.1 and 5.2)

H

Height of air cushion (see Figure 4.2)

H, Significant wave height

i Complex unit (also used as counter variable)

15 Structural mass moment of inertia around the Y-axis of the SES

Function defined in equation (3.52)

k Incident wave number (wave component n. k) ka Air jet contraction coefficient

k Real wave number of the cable

k Imaginary wave number of the cable

k0

=

Characteristic wetted length (see Figures 3.2 and 5.10)

L Length of SES air cushion and side-hulls

L(t) Unsteady hydrodynamic lift

L5 Horizontal length of stern seal (bag/panel) (see Figures 5.1 and 5.2)

L5 Length of all cable segments

L Length of cable segment i (see Figure 5.10)

L5 Characteristic length scale

m Counter variable

M Structural mass of the SES Ma(t) Mass matrix (see equation (3.37))

Ma,e(t) Mass matrix for one element (see equation (3.39))

Mm Structural mass per unit length of the bag and bow skirt

M(t)

Modal mass matrix (see equation (5.47))

M(t)

Modal mass matrix (see equation (5.47)) M Number of vortex elements in the wake

n Counter variable

n Normal vector (nj,n2) (see Figure 4.2)

N()

Trial function matrix

Number of mode shapes used in modal analysis of the bag Ne(e) Trial function matrix of one element (see equation (3.38))

Nm Number of modes

N5 Characteristic deformation scale of the bag (see equation (7.11))

N Number of wave components

p(s,'r,t)

Hydrodynamic pressure acting on the bag during impact

(see Figure 5.10. In the chapter 4 p is the aerodynamic pressure)

Pa Atmospheric air pressure

p5(t) Total bag pressure ( Pso + p5(t), see Figure 8.2)

Complex air cushion pressure amplitude (= j _{± Psp)}

3(x. t)

Unsteady spatial varying pressure underneath bag (see Figure 5.10)

Q(±;i)

=log1-Q(t)

Volume flux of air into air cushion (through a fan)

Qo Steady volume flux of air into air cushion (In section 8.3 it is the steady volume flax into the bag)

(t) Unsteady volume flux of air into air cushion (In section 8.3 it is the unsteady volume flux into the bag)

Q(t)

Volume flux of air out of air cushion underneath the seals

Q0o Static volume flux out of bag

0(t) Unsteady volume flux out of the bag

()

Static 2-D air cushion fan slope Static 2-D bag fan slope

r Wave number of acoustic wave (r =

- ) in chapter 4)

R Radius of curvature

Steady radius of curvature of cable segment i (see Figures 5.1 and 5.2)

RB Length between point O and B (see Figure 5.2)

Rc Length between point O and C (see Figure 5.2)

s Arc length coordinate (see Figures 5.6 and 5.7)

S Length of element i (see equation (4.9) and Figure 4.3)

S(w0)

Wave spectrum

S(f)

Energy spectrum defined in equation (9.3)

S()

Surface area (only used in chapter 4) t Time variable

t Time instant of initial impact

t

Tangential vector (see Figure 5.3)

T(s, t) Total tension in cable ( = To + t(s. t). see Figure 5.3

Peak period of the spectrum Characteristic time scale

Static tension in cable segment i (see Figures 5.1 and 5.2)

U Forward velocity of the SES

(x, t) Part of vertical velocity (see equation (630))

Velocity of sound in air

Vbag(t) Equivalent horizontal velocity due to unsteady bag motion

(t) Equivalent horizontal velocity due to air leakage. bag motion and

air flow between air cushion and bag

v0(x, t) Vertical velocity over the wetted length

(= Re{5oe'}, in subsection 3.2.1)

Generalized vertical velocity (see equation (6.17))

1(t) Generalized vertical velocity (see equation (6.17))

vol(t) Integrated transverse deformations along cable segment bounding the air cushion (see Figure 5.6)

V Local air flow velocity underneath the bag

V

Velocity vector (u,v)

V(t)

Outflow velocity underneath the seals

V0 + l(t) = Vo +

Re{et})

Static volume of bag

(t) Unsteady volume change of bag

V 2-D air cushion volume (=LH)

Vertical global velocity at the bag in the hydroelastic impact problem

Vga Vertical global velocity amplitude at the bag in the local

hydroelastic impact problem

s, y s and y-coordinate of local coordinate system

Global coordinate transformation (=

- X)

Coordinate vector to a point on the surface used inchapter 4 (x,y) Field point vector used in chapter 4 (z',y')

X, Y, Z X, Y and Z-coordinate of the global coordinate system

(see Figure 2.1)

X1 X position of fan

lIw

Vertical motion of free water surface (= Re{et})

Greek symbols:

ii

Iii Pa pb(t)

p(X, t)

PcO Pw 1

(X,t)

{}

Wave length

Complex wave number of mode j Real wave number of mode j Local coordinate

Relative critical damping ratio for the bag structure

Mass density of atmospheric air

Mass density of air in the bag (= P ± Pb, see Figure 8.2)

Mass density of air in air cushion Static mass density of air in air cushion Mass density of water

Local time variable

Velocity potential of air flow in air cushion

Vector containing node point values of p (see Figure 3.4) (angle of attack in subsection 6.7)

-(z,t)

Vortex density

Ratio of specific heat of air (-ya = 1.4) Phase angle of wave component n

((X, t) Incident wave deflection (In chapter 4 ((z) is free air

surface coordinate)

(a Incident wave amplitude (wave component n. (an)

1?3(t) Heave motion of SES

Pitch motion of SES

17bs (t) Vertical motion of the wetted part of the bow skirt

(= Re{et})

i(s, t)

Transverse motion of bag (cable)

T/t(S, t) Longitudinal motion of bag (cable)

The unsteady free water surface deflection caused by the bag

ag(X, t) Function describing the bag geometry in the impact area

( ag(') íag(X: t))

9(s, t) Angle defined in Figure 5.3 (= 90(s) + Ô(s, t))

9, i = 1,2,3 Angles defined in the Figures 5.1 and 5.2 9, j = 2,3,4 Angles defined in Figure 5.9

(x, y, t) Velocity potential of water flow caused by the bag (bow skirt)

(In the chapter 4 it is the steady velocity potential

of the 2-D air flow underneath the bag)

thj(x, y, t) Velocity potential due to incident waves

cf(s) Transverse mode shape function of mode j of the bag

(s) Longitudinal mode shape function of mode j of the bag

1'(X, t) 1-D velocity potential (see equation (3.49))

Complex spatial varying air cushion pressure amplitude

(=

3cosx')

bsp(X) Complex spatial varying air cushion pressure amplitude

due to the bag motion (= cos rx')

We Encounter frequency (wave component n, Wen) Wo Circular wave frequency (wave component n, Won)

Wj Eigenfrequencv of mode j of the bag

Abbreviations:

ACV Air Cushion Vehicle

AP, FP Aft and fore perpendicular (X =

BVP Boundary Value Problem

CG Center of gravity (X = O) FEM Finite Element Method

FD Finite Difference method

FFT Fast Fourier Transform

HBVP Hydrodvnamic Boundary Value Problem 1. 2, 3-D One-, two- and three-dimensional

RCS Ride Control System

i

Introduction

During the last decade an increasing new interest for high speed marine vessels is observed. This interest has also occurred earlier and can be represented by for instance the Surface Effect Ship (SES) development by the US Navy (see Butler

1985). The SES concept is one out of many high speed craft designs (catamaran,

foil-catamaran, wave-piercer and hydrofoil to mention some). The principal idea

with an SES is to partly lift the vessel out of the water by trapping an air cushion

between two catamaran hulls, a bow skirt and a stern seal (see Figure 1.1). A

consequence is that the resistance of a typical SES is lower than the resistance of a similar sized catamaran in most sea states of practical interest. The excess pressure

in the air cushion is provided by the fan system that is blowing air into the air

cushion and the SES is in this way lifted partly out of the water. The bow skirt is usually a finger seal, consisting of a row of vertical loops of flexible material. The most common type of stern seal is the flexible bag, consisting of loops of flexible material open against the side-hulls with one or two internal webs restraining the

aft face of the loops. The flexible bag is most often equipped with two or three

loops.

A problem with the SES is high vertical accelerations in very small sea states.

This phenomenon is often referred to as the cobblestone effect and is a resonance

effect due to the compressibility of air in the air cushion. One important source

of excitation of the cobblestone effect is the dynamic change in air cushion volume caused by the incident waves. This resonance phenomenon occurs at high frequen-cies relative to the resonance frequenfrequen-cies for the rigid body motions of displacement

ships of similar length. The two lowest resonance frequencies in the air cushion

of a 30-35 m long SES are approximately 2 Hz and 5-6 Hz. Due to the frequency

of encounter effect, there are waves with sufficient energy in small sea states to

excite these resonance oscillations. The eigenfunction for the dynamic air cushion pressure is constant in space for the lowest eigenfrequency and represents acoustic wave resonances for the higher eigenfrequencies.

Figure 1.1: Sketch of an SES air cushion with a bow finger seal and a 3-loop flexible stern seal bag. Toyama et al 1992.

Originally, only the lowest. spatially constant, pressure resonance was considered,

see for instance Kaplan et al 1981. However, full scale trials of SES reported by Steen 1993 show that the second resonance corresponding to the first spatial pressure resonance of the air cushion can be of the saine importance as the first

uniform pressure resonance when it comes to vertical accelerations. An important

contribution to this first spatial pressure resonance is the one-dimensional (l-D)

standing pressure wave. This wave has a node approximately midships and

max-imum pressure at the bow and at the stern. This implies that the first spatial

pressure resonance mainly excites the pitch motion of the SES. Hence, the

max-imum accelerations occur at the bow and at the stern of the SES, while smaller

effects are seen at the center of gravity. This was observed in the full scale trials.

The rigid ship motions in the frequency range of the air cushion resonances are

small, but the vertical acceleration level is high. The hydrodynamic damping due to the rigid ship motions is negligible in thisfrequency range. Important damping mechanisms are due to the air flow into the air cushion through the fans and the air leakage underneath the seals and through louversthat are part of a Ride Control System (RCS).

In practical designs a ROS is designed to damp the cobblestone oscillations. This

is usually done by measuring the unsteady air cushion pressure. Based on this

air cushion pressure is minimized. To do this properly, one needs a simplified but

rational mathematical model that accounts for the unsteady air cushion pressure coupled to the motions of the SES. Possible models of RCS can be found in for instance Sørensen 1993 and Sørensen et al 1993. In the design of RCS it will be

important to account for the first spatial pressure resonance in the air cushion. This

can be done by placing the ride control louvers to the ends of the air cushion in

order to obtain maximum damping. This is explained by the fact that the pressure

component corresponding to the first spatial pressure resonance has maximum pressure at the ends and a node point at midships.

The seakeeping of Air Cushion Vehicles (ACV) and SES has been studied by a

number of authors in the literature. Recent progress in numerical and experimental

investigations can be found in for instance the proceedings of ITTC 1993. The

main difference between an ACV and an SES is that the SES has rigid side-hulls. The ACV has flexible seals instead of these side-hulls. The most common way to model the hydrodynamics of an SES has been to model the air cushion as a uniform pressure distribution travelling on the free water surface and approximate the effect of the side-hulls. In this connection, Doctors 1992 presented an extensive survey of research done in the field of hydrodynamics of travelling pressure distributions. This, and other studies consider spatially uniform pressure oniy.

Nakos et al 1991 studied theoretically the three-dimensional (3-D) free surface

deformation due to an oscillating pressure patch with constant velocity. They

assumed that the pressure distribution was uniform in the transverse direction and constant in most of the longitudinal direction. At the ends of the air cushion they used a tapering function that accounts for the pressure fall off. The hydrodvnarnic analysis was based on the linear in.homogeneous unsteady free surface condition

due to the air cushion pressure. This hyd.rodynamic problem was solved by a

higher order panel method. The authors also presented results for the free surface deflections where the effect of two slender side-hulls (b/L 0.05) were accounted for. Those results indicate at least qualitatively, that the wave pattern underneath

the air cushion is little influenced by the presence of the two slender side-hulls. In the panel method used by Nakos et al panels are distributed on a part of the free water surface, since the free water surface in this case extends to infinity.

One should be aware of potential numerical problems at high Froude numbers and high frequencies. These problems are mainly related to the truncation of the computational domain and the fact that the difference between the wave lengths corresponding to the divergent and transverse wave systems are large. For high

Froude numbers the divergent wave length is small while the transverse wave length

is large.

McHenry et al 1991 presented a study where numerical results as well as

experi-mental results were compared for two different vessels. These were the XR-5 SES

(LIB, = 6.5, thin side-hulls) testcraft presented in Magnuson & Wolff 1975 and the AGNES SES (NES200, L/B, = 3.5. relatively thick side-hulls) presented in

Bertrand 1989. The numerical results are based on two different computer codes,

namely \VASES and FASTSEA. WASES is based on Kaplan 1988 and the method accounts in a simplified way for the hulls and neglectsthe water velocity potential caused by the air cushion pressure. FASTSEA accounts for the hulls as described in Faltinsen & Zhao (1991a.1991b) and the effect of the air cushion pressure is in-cluded according to Nestegârd 1990. McHenry et al point outthat the skirts may have an important effect on pitch damping for vessels with thin side-hulls. This is observed in the comparison with experimental results for the XR-5 SES testcraft.

The comparison between theoretical and experimental results for the XR-5 SES

show acceptable agreement. In the comparison between theoretical and

experi-mental results for the AGNES SES the agreement is improved. The two different computer codes do not show any significant difference in the prediction of heave and pitch motions. It is also pointed out by McHenry et al that the water velocity

potential due to the air cushion pressure is not of significant importance. This is

also consistent with Kaplan 1989. He studied the effects of motion induced surface wave generation for a range of speeds, sea states and vessel configurations. His ana-lytical treatment of the surface wave generation due to a translating and oscillating

pressure distribution was based on a two-dimensional (2-D) linear hydrodynamic

model. In this analysis Kaplan neglects the hydrostatic and hydrodvnamic forces due to the side-hulls and the seals. Kaplan states that the overall influence of

the generated waves due to the translating and oscillating pressure distribution, is found to have only a small effect on the motion response and spectral rms values.

He concludes that the neglection of such waves in models predicting the motion

response is vaiid.

The local air flow underneath the seals has also been recognized to be an important phenomenon when predicting the motion response of an SES. This air flow depends on the air cushion pressure at the seals together with the local geometry depending

both on the SES motions and the local seal motion. In a wave condition, the

escape area (leakage area) is varying and will for some time instants be zero. This implies that the air leakage underneath the seals is nonlinear. This nonlinear effect

is easiest to account for in a time domain approach and is usually included in nonlinear computer codes. When it comes to local air flow effects, such as for

instance air flow separation from the lowest region of the stern seal, the problem is more difficujt to treat. To determine the air flow separation line, a viscous analysis

should be applied. No theoretical analysis that takes this effect into account is known to the author. The choice of air flow separation line is influencing the pressure distribution underneath the flexible bag. Earlier in the text it has been

stated that the overall effect of the far-field radiated waves caused by the air cushion pressure and the side-hulls can be neglected. This is probably a fair approximation when the integrated effect of these waves over the air cushion area is considered. In this connection it may be appropriate to separate between the far-field radiated

waves and the local wave effect. The reason for doing so is that the far-field

radiated waves disappear when the frequency of encounter goes to infinity, but the local wave effect will still exist locally near the structure. Locally at the seals both these wave effects could be important when calculating the air leakage. One reason for this is that the air leakage height underneath the seals is small. A typical value

of the air leakage height underneath the stern seal bag is 2 - 3 cm. This means

that even small wave effects may be important locally in the calculation of the air

leakage.

In computer codes predicting the SES response it is customary to use a quasi-static

fan characteristic. This is not completely correct since the air cushion pressure is oscillating in time. Dynamic effects of the lift fan and the corresponding ducting system may therefore become important. These dynamic effects are related to unsteady lift on the rotor blades, difference pressure across the fan, change in

rotational speed and air inertia of the ducting system. Experiments show that

the dynamics are dependent on fan geometry. rotational speed, difference pressure

and length of the ducting system. The experimental analysis of Sullivan et al

1992 found that dynamic effects of the fan system affect the overall air cushion dynamics. Sullivan et al conclude that experiments should be carried out in each

case to obtain the dynamic effects of a fan system. According to Durkin & Leuhr

1978 the major source to fan dynamics was due to the air inertia in the ducting

system.

Witt 1993 presented experiments for a centrifugal fan. Based on his experiments

the pressure difference across the fan was presented as a function of the volume flow rate. A hysteresis effect was observed in these pressure discharge curves. He clearly demonstrates that the hysteresis effectbecomes more important as the

frequency of oscillation is increased. At a frequency of oscillation equal to 2

-3 Hz a pronounced hysteresis was observed. In order to relate this frequency

to the resonance frequencies of the air cushion, the uniform pressure resonance and the first spatial pressure resonance frequency of the air cushion are equal

to approximately 2 and 5 - 6

Hz respectively for a 30 - 35 m long SES. This

indicates that the hysteresis may become a problem for SES. Witt also finds that

the hysteresis effect is significantly affected by the length of the ducting system.

He concludes that the hysteresis effect is most important when a number of fans are operated in parallel. Further. he states that when only one fan with a fiat fan

characteristic is used, the hysteresis effect should have small or no direct impact. A practical consequence of the hysteresis effect may be that the working point of

the fan is not uniquely defined, this means that different working points exists. A

result may be that the lifetime of the fan is drastically reduced and that severe

vibrations may occur.

The acoustic resonance phenomenon addressed above was pointed out by Nakos et al 1991. In this reference they show how the acoustic resonance frequencies can

be found and concludes that the first spatial pressureresonance of the air cushion can be triggered in small sea states for actual velocities of the SES. Nakos et al

1991 did not relate this effect to the global motions of the SES.

al (1992. 1993) and Steen 1993 (see also Steen & Faltinsen 1995). In these works the coupling between the acoustic pressure waves in the air cushion and the global motions of the SES was included. In this connection Sorensen 1993 concentrated on the consequences of this effect for the RCS. while Steen 1993 concentrated on

the passive response of the SES (no RCS). In Steen's study of the cobblestone effect, he found that the dynamics of the stern seal bag was important for the global acceleration level in low sea states. He analyzed the cobblestone effect by using a quasi-linear frequency domain solution. The effects of a dynamically varying leakage area underneath the seal and the deformation of the bag due to a change in the air cushion pressure at the stern were considered. The deformation of the bag was analyzed quasi-statically. The dynamically varying leakage area and the deformation of the bag will have a similar effect on the air cushion as a moving piston at the end of a long finite tube. Acoustic waves in the air cushion were shown to be significantly affected by this mechanism. It was for instance

shown that the effect of the flexible bag reduces the first spatial pressure resonance

frequency from approximately 6 Hz down to approximately 5 Hz, relative to the air cushion supported with a rigid stern seal for a 30-35 m long SES. This is consistent with the full scale trials reported by Steen 1993. Steen 1993 does not account for the hydroelastic interaction between the stern seal bag and the free

water surface. Hvdroelastic interaction means that the hydrodynamic loading is a function of the structural deformations resulting from the hvdrodvnamic loading. Lee & Rhee 1991 have studied theoretically the effect of skirt deformation on the

response characteristics of an AC\. They do not account for this fluid structure

interaction either.

The flexible stern seal bag behaves hydrodynamically as an unsteady planing

sur-face at high Froude numbers. The author is not aware of any theoretical analysis of the unsteady interaction between a flexible bag structure and the free water

surface. However, Doctors 1977 studied the 2-D steady planing of a flexible skirt

modelled as a flexible beam with bending stiffness. He used linearized potential theory to solve the hydrodynamic boundary value problem. The effect of gravity was included. Bessho & Komatsu 1984 studied the 2-D unsteady planing

the effect of the wetted length change, but does not consider deformations of the

planing surface.

Many different flexible seal concepts exist. A major concern is the tear and wear

of the flexible seals mainly caused by the water surface. In this connection

exper-imental investigations have been carried out by for instance Malakhoff & Davis 1982. They studied various designs within two main categories, namely Flexible

Seals and Semi-Flexible Seals.

The background for the present work is that the cobblestone effect is a major

problem connected to the SES design. The aim of the present study is to improve

the understanding of the cobblestone effect of an SES. Based on this increased

knowledge an improved SES design can be made. When starting

this work it

was believed that the flexible stern seal bag was of significant importance for the cobblestone effect. Due to this choice of working hypothesis, the work presented in this thesis is mainly concentrating on how the flexible stern seal bag is influencing the cobblestone effect.

Initially when this work was started, the analysis of theflexible bag and its in-teraction with the free water surface was focused on. This approach was mainly

motivated by the work of Steen 1993, where it was pointed out that the bag dynam-ics was important for the cobblestone effect and that the effect of fluid structure

interaction between the bag and the water surface could be of significant impor-tance. In this connection two physical quantities were of special interest. These

were the volume change of the bag affecting the air cushion volume and the vertical motion of the lowest point of the bag which has relevance to the air leakage height

underneath the flexible bag. The motion of the bag was at an early stage of this

work only caused by the hydrodynamic interaction with the water surface (impact

loads). Most of the effort has been concentrated on this partof the work.

In order to demonstrate the importance of the local bag motion on the cobblestone effect, a simplified global model of the SES has been established. This simplified model of the SES is described in the chapters 2 and 3. Here the motion of the bag is not only caused by the hydrodynamic interaction with the water surface, but in addition the effect of both the unsteady pressure variationsin the air cushion and the bag are considered.

Since the purpose of the global model of the SES is to demonstrate the importance

of nonlinear effects mainly caused by the bag, with regards to the vertical accel-erations of the SES. the hydrodynamics of the side-hulls are approximated in a

simplified way by using the high frequency asymptote of the hvdrodynamic

coeffi-cients. In this connection it is important to have in mind that the effect of the air cushion pressure acting on the wetdeck of the SES is dominant relative to the

ef-fect of the hvdrod namic pressure acting on the side-hulls in the frequency domain of interest. The simplified hydrodynamic model of the side-hulls are described in chapter 2.

In chapter 3 a 1-D analysis of the air cushion pressure is presented. The volume change in the air cushion due to the diffraction of incident waves are neglected.

The solution of the air cushion pressure is obtained by solving the 1-D wave

equa-tion where the boundary condiequa-tions are linearized about the mean posiequa-tion of the

boundaries. The wave equation is solved numerically by the Finite Element Method (FEM) approach. A quasi-static fan characteristic is assumed and a RCS is not ac-counted for in the present work for the sake of simplicity. The analysis is based on a time domain solution that accounts for the nonlinear air leakage underneath the bow skirt and the stern seal bag which is an improvement of the analysis presented

in Steen 1993. The local air leakage underneath the stem seal bag is introducing

a spatially varying pressure underneath the flexible bag that is accounted for here. Steen 1993 did also account for the spatial pressure variation underneath the stern

seal bag, but his analysis do not account for the nonlinearity in the air leakage underneath the bag that causes the spatially varying pressure. The local air flow underneath the stern seal bag and the bow skirt is discussed in chapter 4.

The coupling between the flexible bag motion, the air cushion pressure and the

vertical plane motion of the SES are accounted for. So is the air flow between the

air cushion volume and the bag volume. This coupling is described in detail in

chapter 8.

In chapter 5 the structural model of the bag is described. The bag is pressurized with air. It is deformed due to the hvdrodvnamic pressure distribution on the wetted surface of the bag and the air pressure in the bag and the air cushion. The unsteady deformation of the bag is found by a numerical time integration.

High numerical accuracy is needed. This has been achieved by using the "dry" mode superposition approach for the flexible behavior of the bag in combination

with extensive use of analytical expressions for the excitation and reaction forces.

"Dry" mode superposition implies that the eigenvalue problem is solved without

accounting for the hydrodynamic reaction forces and the pressure forces due to the compressibility of air in the bag and the air cushion. The finite radius of curvature of the bag and the coupling between the elastic longitudinal and transverse

oscil-lations are found to be important. The flexibility is mainly due to axial stiffness

in the bag structure. However, the effect of bending stiffness is also included.

The foundation for a numerical and theoretical study of the unsteady interaction

between a flexible bag structure and the free water surface (local impact) is given

in chapters 5 and 6. The hvdrodynamic part of the problem presented in chapter 6. has similarities with the linearized unsteady foil problem. An importantdifference

is that the wetted length of the structure changes rapidly with time. The wetted length is found from a nonlinear integral equation, by generalizingwhat Wagner

1932 did in the case of slamming. A difference is that the forward speed of the bag is included and it is assumed large relative to the relative vertical velocity between the bag and the water. The scaling of the local bag impact is described in chapter

7.

The computational results presented in chapter 9 show that the flexible Stern seal

bag has a significant influence on the cobblestone effect. It is demonstrated that

the flexible bag reduces the first spatial pressure resonance frequency from approx-imately 6 Hz down to approxapprox-imately 5 Hz for a 30-35 m longSES relative to the air cushion supported with a rigid planing seal. This was also pointed Out by Steen

1993.

The results show that the effect of the spatially varying pressure caused by the air leakage underneath the bag is important. It is demonstrated that this effect cause the bag to minimize the air leakage. Hence, the bag tries to "follow" the local water surface. This behavior is a characteristic feature of the flexible stern

seal bag.

The results indicate that the effect of the fluid structure interaction between the bag and the water surface is significant in terms of vertical accelerations around

the first spatial pressure resonance frequency. It is found that the effect of fluid structure interaction between the flexible bag and the water surface reduces the

vertical accelerations here.

The results also clearly demonstrate that the height to length ratio of the bag effects the cobblestone oscillations and that an increase in the height to length

ratio of the stern seal bag reduces the cobblestone effect.

It is also noted that the vertical accelerations are nonlinear with respect to the

significant wave height. Based on the present study, this implies that the response in vertical accelerations of the SES is overpredicted if a linear transfer function is

2

The global formulation of the SES motions

The aim of the present work is to improve the understanding of the cobblestone effect of the SES. Earlier work by Steen 1993 found that the flexible Stern seal bag is important in this connection. Hence, the present study are focusing on the

influence of the flexible bag. To do this, the local bag motion must be related to the

global motions of the SES and the air cushion pressure. Since the purpose of the global model of the SES is to demonstrate the the importance ofnonlinear effects

mainly caused by the bag, the hydrodynamics of the side-hulls areapproximated

in a simplified way. In this connection one should also have in mind that the

effect of the air cushion pressure acting on the wetdeck of the SES is dominant

relative to the effect of the hydrodynamic pressure acting on the side-hulls in the

frequency domain of interest. The frequency region in focus is around the two lowest resonance frequencies of the air cushion. For a 30 - 35 m long SES these

frequencies correspond approximately to 2 and 5 - 6 Hz. The first corresponds

to a uniform pressure resonance, while the second corresponds to the first spatial pressure resonance of the air cushion.

In this chapter the equations for coupled heave and pitch motions are formulated for an SES and the global hvdrodynamic approximations are presented. Consequently, only head sea waves are considered.

In Figure 2.1 the coordinate system used in the analysis of the heave and pitch motions of the SES is defined. This is a right-handed XYZ-coordinate system that is moving with the forward velocity U of the SES and is fixed to the mean

oscillatory position of the vessel. Here the X-axis is pointing upstream in the

direction of the forward velocity of the SES. and is going through the center of gravity (CG). The Y-axis is pointing to the port side of the vessel and the Z-axis

CG

X

7

Figure 21: This figure shows the definition of the global coordinate system used in the present

analysis.

2.1

The global equations of motions of the SES

The equations of coupled heave and pitch motions can now be formulated in the

time domain as follows,

(M + .4)i3 + B3h,)3 +

3r3 + .45i5 ± B5 ±

= F31(t) + F30(t) (2.1)

and

(15 + .45)i5 + Bi)5 + C5is + A3i3 + Bi)3 ± C'3rì3 = F5h(t) ± F5(t). (2.2)

Here 773 and 775 are both functions of time and dot stands for the time derivative.

M is the structural mass of the SES. A, B and are the hydrodynamic added

mass, damping and restoring coefficients respectively of the two side-hulls of the

SES in direction i due to motion in direction j It is here important to note that the left hand side of the equations defined in (2.1) and (2.2) are correct only if the high frequency limit of the hydrodvnamic coefficients can be applied in the frequency domain of interest. This is true in the present analysis. If this was not

in order to account for the memory effect in the wave radiation. I is the structural

mass moment of inertia around the Y-axis. i is the motion in direction j. Heave motion is positive in the direction of the Z-axis and the pitch motion is positive

in the case of bow-down. The left hand sides of equation (2.1) and (2.2) are

linear due to the small heave and pitch motions. F is the linear hydrodvnainic

excitation force on the side-hulls and F is the nonlinear force acting on the SES

due to the integrated unsteady excess pressure in the air cushion in direction i. The nonlinearity in F is mainly caused by nonlinear effects related to the flexible

stern seal bag. This implies that the bag is coupled to the motions of the SES

through FC

The bag and air cushion pressure represents additional unknowns. Equations

that

describe their behavior will be discussed in chapter 3 and 8.

2.2

The global hydrodynamic approximations

In the following text some hydrodynamic simplifications are presented. One reason for making these simplifications is that the main focus in the present analysis is put on the nonlinear effects caused by the flexible stern seal bag. These simplifications will be appropriate for this purpose.

In the present analysis hvdrodynamic interaction effects between the two side-hulls are neglected. The wave generation caused by the flexible bow skirt, the two side-hulls and the unsteady air cushion pressure is also neglected. The wave generation caused by the flexible bow skirt is discussed in the second section of the following chapter.

The hydrodvnamic coefficients A,. B and C are found, based onthe Salvesen

et al's 1970 strip theory. This theory will only serve as a rough estimate for the hvdrodvnamic coefficients in the present analysis. This strip theory isbasically a high frequency theory. When wave generation from the side-hulls is important one

must limit this strip theory to moderate

Froude numbers (F = U//L < 0.4).

Since it is here focused on the high frequency range where the wave generation

from the side-hulls is negligible, there is no restriction on the Froude number. In

the present analysis the high frequency asymptote is used as an approximation.

ap-proximated by zero. Based on these approximations the hydrodvnamic coefficients can be found by integration along the side-hulls as.

4h

_{2f A(X)dX}

33 L B

= 2UA,A

4h

_{_2fXA(X)dX}

35 L

- UA3 ±

ULAA

--

_2fXAdX

1153

-L B

= UA3 +

ULA,A

A5

2f X2A(X)dX

L B =

UL2AA.

(2.3)

Here L is the length of the SES, A(X) is the 2-D added mass coefficient along

one of the side-hulls. means the added mass coefficient for the aftermost

section. The center of gravity (CG) is assumed to be at the midship. Examples on 2-D frequency independent added mass coefficients in heave corresponding to the

infinite frequency asymptote. can be found in Faltin.sen 1990 p. 52. An example of a more refined hvdrodynamic analysis that accounts for the wave generation

caused by both the side-hulls and the air cushion pressure can be found in Nakos

et al 1991. One should note that their analysis is based on a frequency domain

approach.

The hvdrodynamic restoring coefficients can be written as follows based on inte-gration along the side-hulls as,

= 2Pw9 f b(X)dX = 2pgAw

L

=

2Pw9 f Xb(X)dX + 2USA,A

L

=

_2pgfXb(X)dX

= 2pgMw

C

= 2Pw9 J

X2b(x)dX - U2A3 + U2LAA

L

= 2pgI, -

U2A3 + U2L4A

(2.4)

Here b(X) is the sectional beam of a side-hull and A, M and 1w are the area, moment and moment of inertia of the water plane of a side hull, respectively. Pw

is the mass density of water and g is the acceleration of gravity. The additional terms in C and C are due to forward speed effects.

In the case of a typical SES. around 80% of the displacement is carried by the excess

pressure in the air cushion. This indicates that the excitation force and moment

are dominated by the contribution from the air cushion pressure variation, caused by the incident waves. The contribution from the hydrodynamic excitation due to the side-hulls can be approximated by using the Salvesen et al's 1970 strip theory.

Based on this theory the excitation force in heave and the excitation moment in

pitch can be written as follows.

=

2Im{(

anJ e

-+

e_ko4A )

_} (2.5) and

F =2>21m{(

-

¡ e

e

-

wonWenA1 + iUw0A)de

+

ianUe_ie_

LA2DA)

k,.

(2.6)

Here N,, is the number of regular wave components used in the simulation, d is the draft of the side-hulls and s is the sectional area coefficient. It is assumed

that the longitudinal coordinate of the aftermost section is - and that the 2-D

excitation force and moment is based on the following definition of the incident

wave potential,

1(X, Z. ) kZ

cos(k(

- X) -

_{t + ).}

(2.7)

n=i On 2

Here w

= kg and Wen = w + kU. For further explanations, see equation

(3.19).

3

The air cushion boundary value problem

In the previous chapter the equations describing the global heave and pitch motions of the SES were described. Here it was pointed out that the effect of the air cushion pressure acting on the wet deck of the SES is dominant relative to the hydro dynamic pressure acting on the side-hulls in the frequency domain of interest. In order to solve the equations of heave and pitch motions defined in the equations (2.1) and

(2.2). a boundary value problem (BVP) that describes the unsteady behavior of the air cushion pressure must be formulated. A solution of this BVP. makes it possible to calculate FC and FC.

In this chapter it is focused on the air cushion BVP. A 1-D model of the air cushion pressure is presented. Here the air cushion BVP is described in the first section. In the following section some aspects of the air cushion model is discussed- The last

section describes the numerical solution procedure used to solve the air cushion

BVP. An important part of this section is the verification of the numerical solution procedure.

The effects on the air cushion pressure due to the flexible stern seal bag is included

and discussed on a general basis in this chapter. Further, more details related to the coupling between the flexible bag and the aircushion pressure are presented

in chapter 8.

3.1

The boundary value problem

In Figure 3.1 the idealized model of the air cushion is shown. The unsteady air

cushion pressure is assumed to be constant in the transverse direction (Y-direction)

of the SES. It is also assumed that there is no pressure gradient in the vertical direction (Z-direction). These assumptions are discussed in the next section. The

governing equation in the air cushion is the so-called wave equation (see Landau & Lifshitz 1959 p. 245). The derivation is based on an inviscid compressible fluid.

The fluid motions are assumed to be small so that products of small quantities can be neglected. An irrotational flow is assumed so that there exist a velocity

/

Free water surface

Figure 3.1: This figure shows the idealized model of the air cushion of an SES, used in the present

analysis.

potential. The equation of continuity and Euler's equation are combined with the

following result,

- vV = 0.

(3.1)

Here is the velocity potential of the air flow in the air cushion and y5 is the

velocity of sound in air. V2 is the Laplacian operator. v is defined as follows,

=

(3.2)

where p is the aerodynamic pressure and Pc is the mass density of air in the air cushion. The subscript s under the square root sign stands for the isentropic

motion of the fluid. Hence. the compression of the fluid is assumed adiabatic which means that energy dissipation in the fluid is not accounted for and as a result, the entropy of the fluid is constant. Such fluids are said to be ideal. The pressure and density in the air cushion are decomposed as follows,

Pc = JQ + 7

where p and p are the mean aerodynamic pressure and mass density of air respectively. j5 and 6 are small perturbations. is written as,

= PcO (3.4)

It should also be noted that the wave equation in equation (3.1) is reduced to Laplace equation if the first term on the left hand side of the wave equation is

neglected. This term represents the effect of compressibility of the fluid.

To solve the differential equation defined in equation (3.1), boundary conditions are needed at the boundary enclosing the air cushion volume, as shown in Figure 3.1.

On this boundary the normal velocity is prescribed in the present analysis. Here

n is the normal vector

pointing out of the enclosed volume. Small oscillations

are assumed around the mean position. Based on linear theory these boundary conditions can therefore be satisfied at the mean position of the boundaries. The

boundary conditions are defined as follows:

Rigid part of wetdeck:

= 13 - Xr15 ôn

At the fan:

(3.5)

The volume flux of air through the fan (per unit length in the beam direction of

the SES) into the air cushion is modelled as follows,

Q = Q + ()g

(3.6)

Here Q0 is the steady volume flux entering the air cushion It should be noted

that the static fan characteristic is used to approximate the 2-D fan slope at the mean pressure in the air cushion, denoted by Pco One should also note

that

the fan slope (%), is negative. The unsteady part of this volume flux defined in

the last term on the right hand side of equation (3.6), is represented by a uniformly distributed normal velocity over the area ofthe fan as

c»p \ôPìO

--

Pc

8n Af

Here Af is the 2-D inlet area of the fan.

Velocity at the stern seal bag:

=

(3.8)

Here Vbag is an equivalent horizontal velocity due to the unsteady bag motion. The velocity Vbag is constant over the height of the bag. and is found by dividing the

rate of change of the air cushion volume due to the bag motion by the height of the air cushion. This can be done due to the assumption that there is no spatial variation of the unsteady air cushion pressure over the height of the air cushion. A more detailed description of this quantity will be given in chapter 8.

In that

connection the coupling between the flexible stern seal bag and the air cushion and the coupling between the air cushion and the bag volume (see Figure 8.2) will be focused on. The coupling between the air cushion and the bag volume also results in an equivalent velocity contribution. This contribution is written as.

¿c

l8Qb-

- L

=

_ap

(Pb(t) - Pc(: t))

(3.9)

Here () is the 2-D static fan slope at the steady equilibrium difference pressure

between the bag and the air cushion and is the unsteady bag pressure. H is

the height of the air cushion.

Leakage underneath seals:

The volume flux of air out of the air cushion underneath the seals (per unit length in the beam direction of the SES) is modelled as follows,

Q = Vakahj.. (3.10)

Here Va is the outflow velocity, ka is the air jet contraction coefficient and hL is the

leakage height underneath the lowest point of the seal. The air flow underneath the bag is analyzed as a jet flow. Based on this model the air flow is contracted

when it separates from the lowest point of the bag. This effect is accounted for by

the air jet contraction coefficient ka. The local air flow is discussed in chapter 4.

underneath the seals is incompressible. This approximation can be used due to

the moderate air flow velocities relative to the velocity of sound in air denoted by y5. Another reason is that the characteristic wave lengths of the standing pressure

waves in the air cushion are long relative to the analyzed domain. If unsteady

terms in Bernoulli's equation are neglected (this approximation is discussed in the following chapter), V can be written as follows.

=

+

/2cü +c - Pa)

(3.11)

Here Pa is the atmospheric pressure and Pa S the atmospheric mass density of air. The leakage height denoted by hL is written as follows.

= hi. + hL (3.12)

Here hLo and h1. are the steady and unsteady leakage height underneath the seals.

It is now assumed that both j3, and fr1. are small quantities compared to Pco and

hLo. Based on this assumption the following approximation can be made,

Q V0khO ± VaOkaL +

VOkahLO

(3.13)

2(pc - Pa)

Here products of small quantities are neglected. The equivalent normal velocity at

the seal is set equal to the unsteady part of Q divided by the height H of the air

cushion. Hence.

IlL Vaokahi.o

= l/ooka

+

Pc (3.14)

On

fI

2H(po - Pa)

One should here be aware of the limitations of the local air flow model described in equation (3.14). This air flow model is limited to small values of the ratio /Va0.

Hence 15c/(pco - Pa) must be small. This air flow model is also limited to moderate

time variations, since unsteady terms in Bernoulli's equation is neglected. This model is mainly used for the air flow underneath the bow skirt. For the air flow

underneath the stern seal another model is used. This model is not limited to small values of the ratio pc/(Pco Pa) and small time variations. The air flow underneath

the stern seal bag is further discussed in thefollowing chapter.

In the case of the stern seal bag the

equivalent velocity due to the air leakage

and the air flow between the air cushion and the bag volume as shown in chapter

8.

Free water surface:

The normal velocity at the free water surface can be written as follows,

V,, = VVF.

ax

az

(3.15)

when products of small quantities are neglected. Here V (U +

, ) and

Ff = Z - Ç(X, t) where is the velocity potential due to the incident waves and Ç is describing the free water surface elevation of the incident waves. The kinematic free surface condition is written as follows.

- u

at_ ax

az

Hence,

= Ç(X,t)

where the subscript t denotes the time derivative. Ç is defined as.

Ç(X, t) = Çasin(k(

- X) - wet).

(3.18)

Here Ç is the wave amplitude, k the incident wave number and w the encounter

frequency of the incident wave relative to the SES. It is written as, w = w0 ± kU, where kg = w. w0 denotes the circular wave frequency.

Since the present analysis is a time domain analysis, the single incident wave defined in equation (3.18) can be generalized to irregular waves (see Faltinsen 1990 p. 23).

Since the present analysis is linear with respect to the incident wave amplitude, this is done by summing a finite number of wave components. The free water

surface elevation of the irregular incident waves is now written as,

Ç(X. t) = - ÇanSjTh(kn( - X) - went ± 6,,). (3.19)

(3.16)

S(w0) =

0.31wH -1.25()

e . (3.21)

Here w is the circular peak frequency of the wave spectrum and H. is the significant wave height.

3.2

Discussion of the air cushion model

In this section some aspects of the air cushion model described in the previous section are discussed. In the previous section it is assumed that the air cushion

pressure is constant in the transverse and the vertical directions of the air cushion. To justify these assumptions. it is worthwhile focusing on the eigenvalue problem

of a rectangular box with the length L, beam B and height H. The governing

equation in this box-volume is the so-called Helmholz equation defined as,

(wnmP)2

+

= o. (3.22)

where = Re{çe.m.9t}. i is here the complexunit. The eigenfunctions denoted by , satisfies the boundary conditions of the box. The eigenlrequencies for this

problem can now be written in non-dimensional form as.

Here is the n'th random phase angle distributed between O and 2ir and N is

the number of regular wave components used. The incident wave amplitude of the n'th component is found by the following relation.

12

_{= S(w0)A0.}

_(3.20)

Here WOn is the difference between successive frequencies. S,.. (won) denotes the

wave spectrum. A modified Pierson-Moskowitch wave spectrum is chosen. It is

defined as follows,

wn,m,pL ₂

()2 + (.)2

_(3.23)

7rv

For a typical SES, the length to beam and length to height ratios of the air cushion

can be L/B = 3.5 and L/HC =

14 respectively. Based on these ratios, one notes

direction of the air cushion (X-direction) corresponding to the non-dimensional

frequencies 1, 2 and 3. For this geometry the first non-dimensional eigenfrequency

including the transverse standing wave will be equal to 3.5 and the first eigen-frequency including the vertical standing wave will be equal to 14. Because the frequency range of interest in this analysis is far below 3.5, the 1-D approxima-tion is found to be reasonable. One should also have in mind that the problem in the present analysis is symmetric about the center plane of the SES. The first

transverse mode will therefore not be excited.

In the previous section the influence of the stern seal bag on the air cushion is modelled as a vertical plate that is allowed to move in the X-direction. The fact

that the bag has a finite length in the X-direction can be neglected since the

characteristic wave length of the standing pressure waves in the air cushion is

much larger than the length of the bag. The motion of the bag will therefore have

the same effect on the air cushion as a moving piston at the end of a long tube.

3.2.1

The bow skirt wave generation

The wave generation caused by the bow skirt has not been studied by anyone as

far as the author is aware of. This effect is usually neglected in theories predicting

the motions of SES. However, the wave generation caused by the bow skirt may

cause some wave generation and in this way modify the volume pumping of the air cushion volume caused by the incident waves.

In this subsection it will be shown that the wave generation is mainly depending on the unsteady air cushion pressure at the bow and the effective accelerated structural mass of the bow skirt. This analysis is only meant to be a preliminary study of parameters of importance for the bow skirt wave generation. A greatly

simplified engineering approach is therefore applied.

The planing bow skirt may as a first approximation be modelled as a "very" flexible

beam, hanging down from the bow as shown in Figure 3.2. The skirt is assumed to extend along a straight line in t.he transverse direction. In head sea this bow skirt will behave like a high aspect ratio planing surface. The aspect ratio is here defined as the ratio between the wetted length and the beam of the bow skirt. Since the aspect ratio is high a 2-D analysis of the water flow and the bow skirt

Figure 3.2: This figure shows the idealized 2-D model of the bow skirt of an SES. p, isthe air cushion pressure and lis the wetted length of the bow skirt.

structural deformation can be used.

The relative vertical velocity at the wetted part of the bow skirt can be directly related to the wave generation. In the following text this coupling will not be

focused on. but a qualitative approximation of the relative vertical velocity at the wetted part of the bow skirt will be presented and discussed in relation to the wave generation. It is assumed that if the relative vertical velocity at the wetted part of the bow skirt is small relative to the vertical velocity due to the incident wave, the effect of the wave generation caused by the bow skirt can also be assumed small.

It is here focused on the unsteady problem, and it is assumed that the static

equilibrium is found based on the balance between the steady air cushion pressure

and the steady lift obtained by the wetted length of the bow skirt. The vertical

velocity y0 on the wetted part of the bow skirt is assumed to be constant in space

and is set equal to,

vo(t) = ts(t) -

L(t) = Re{ioe_t}.

(324)

Here lbs is the local vertical motion of the wetted partof the bow skirt, c5 is the velocity potential of the incident wave, y is the vertical coordinate and i is the

complex unit. Harmonic time dependence (e_2t) is assumed. Since the bow skirt

PC

structure is not completely flexible, a larger part than just the wetted length of the

bow skirt structure is accelerated. This effect is accounted for by the coefficient

Cern, that will be larger than or equal to one. It is also assumed that the unsteady

pressure is acting over the accelerated length. This in not necessarily true, but is

used as a simplifying assumption in this analysis. Based on these approximations the following simplified equation of vertical motion of the wetted part of the bow

skirt can be written as,

MrnCernlbs(t) + Ab5o = L(t) -

=

t)Ceml. (3.25)

Here JvIrn is the 2-D structural mass per unit length of the bow skirt, i is an

approximation of the wetted length of the bow skirt and is the unsteady air

cushion pressure. A, is the hydrodynamic added mass of the bow skirt and L is

the unsteady lift that can be written as follows in this simplified model,

Ab5 = ir

L(t)

- pUivo(t).

(3.26)

This formula of the lift is based on steady foil theory. Since the problem is un-steady this is not completely correct, but it will serve the purpose of the present

approximation. The steady lift is overpredicting the unsteady lift as can be seen by Theodorsen's function shown on p. 228 in Newman 1977. Theodorsen's function

represents the reduction and shift in phase of the lift force due to a harmonically

oscillating foil relative to the steady lift. Equation (3.25) is now rewritten as.

2 - . A5 ir L

- WeMmCemIì&s - iWey'Vo = "Pw(JVO - P(X )Cem. (3.2i')

Here is the vertical motion amplitude of the wetted part of the bag and is

the pressure amplitude in the air cushion at the bow. î'c is written as,

So = zw. -

(3.28)

where c5 is the incident wave amplitude. Based on equation (3.27) 7bs can be

i

pco(iwl + U) -

Cem

- W(MmC'em+pwl)iWepwU

From a hvdrodvnamic point of view the relative vertical velocity at the wetted part

of the bow skirt is an important quantity. If this quantity is small the generated waves caused by the bow seal is also small. The relative vertical velocity on the

wetted part of the bow skirt is made non-dimensional with respect to the vertical

velocity due to the incident wave. This yields the following result,

=

+ 1 (3.30)

WO(a

'o

Ç

The expression for the relative motion defined in equation (3.29) is substituted into equation (3.30). The different quantities are made non-dimensional and the

following result is obtained.

0Pcof Cern - KíacCern

e(.t<íacCem + ¡-vr) - Z5We

Here the following definitions are used,

U

= w

g PC gMm pwU2 U Pcof Rfac F1 (3.29)

One should here note that Pcof generally is frequency dependent, but is in this connection assumed constant. This assumption is used here since the aim of the

present analysis is to focus on important quantities for the planing bow skirt wave generation. Typical values of the non-dimensional parameters defined in equation

(3.32) can be as follows; = 0.5,Kfac 7. iO and F1 = 25.

The relative vertical velocity at the wetted part of the bow skirt is defined in

equation (3.31) and is presented graphically in Figure 3.3 for different values of the non-dimensional parameters. vo vr = WOÇa (3.31) (3.32)

0.25 0.200.15 -> 0.10- 0.05-0.00 0.0 2 Cern 2 PC0f 0.5 O ri g i n al 2 _ac F1-50 100

_{1e U/g}

200

Figure 3.3: This figure shows the relative vertical velocity at the wetted part of the bow skirt for different values of the non-dimensional parameters defined in equation (3.32). The relative vertical velocity vr is defined in equation (3.31). Crm is the ratio between the effective accelerated

structural mass and the structural mass of the wetted length of the bow skirt ( 1). The "Original" curve refers to the following set of non-dimensional values; pj = 0.5, K1 = 7 iO.

F1 = 25 and Ce,,. = 2.

If orte focuses on the results presented in Figure 3.3 one notes that there is only a

small effect of reducing the Froude number (F1). If it is for one moment focused on the right hand side of equation (3.31). it is seen that the term proportional to

t in the denominator is dominated b the second term (term involving F1). This

term represents the hydrodvnamic added mass effect of the wetted part of the bow skirt. Based on typical quantities for the different non-dimensional parameters one observes that the added mass is approximately four times larger than the structural mass, represented by the first term (KfacCern). In this connection one should also

have in mind that the damping term (caused by the unsteady lift force) denoted by -i7rJe/2 is dominating the denominator. Based on this observation one may conclude that the unsteady lift force is most important.

It is now focused on the numerator of the right hand side of equation (3.31), where three non-dimensional parameters are involved. That is p, cernand Kfac'

The importance of these non-dimensional parameters are illustrated in Figure 3.3.

one might expect an increase in the effective structural mass increases the relative

vertical velocity at the wetted part of the bow skirt. When it comes to Kf ac this parameter is found to have only a small influence on the results, at least in the

region of primary interest around the first spatial pressure resonance frequency in the air cushion (WeU/g 90). The results also show that Pcof has a significant

importance on the relative vertical velocity at the wetted part of the bow skirt. When setting this parameter equal to zero one observes that the relative vertical

velocity is significantly reduced. These calculations indicate that p and Cern are practically speaking of the same importance when it comes to wave generation.

Based on this simplified model one may conclude that the disturbance due to the

bow skirt is small compared to the incident waves when the bow skirt is "very"

flexible, that is when Ce.,,. is below approximately 2 and when Pcof is below ap-proximately 0.5. This implies qualitatively that the wave generation caused by the flexible bow skirt is small.

3.3

The numerical solution procedure

In this section the numerical procedure that is used tosolve the air cushion BVP

defined in the first section of this chapter, is described and verified. The governing equation of the air cushion volume is the wave equation and it is defined in

equa-tion (3.1). The boundary condiequa-tions are also described in the same secequa-tion. The numerical method that will be used to discretize the wave equation is referred to

as the Finite Element Method (FEM).

The boundary value problem can be formulated mathematically as follows,

2

la2

V

an

= 0in1'

= V(f,t)n on F.

(3.33)

Here V(. t) n is the normal velocity on thesurface F that encloses the air cushion volume denoted by V. n is the normal vector pointing out of the fluid domain. The Galerkin method is used to formulate the equations used for the spatial dis-cretization. In words this method can be derived b summing the integral of the