Ship motions due to sudden water ingress. Theory

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SHIP MOTIONS DUE TO SUDDEN

WATER INGRESS.

THEORY.

A.W. Vredeveldt - TNO

J.M.J.. Journée TUD

Report No. 981-P

TNO-Report B-92-1040

January 1993

Doift University Of Technology Ship Hydromechanics Laboratory Mekelweg.2

2628 CD Deift The Netherlands Phone015 - 78 6882

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Centre for Mechanical Engineerfrig

TNO Building and:

Construction Research

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inctudingmarine structures. Sponsor Project no. Authors Approved Visa Number of pages Number of tables : NUmbers of figures Leeghwaterstraat5 P.O. 'Box 29 2600 AA DeIft The Netherlands Telex 3 8192 Fax +31 155641 02 Phone +31 15608608 Netherlands Directorate General of Shipping and

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A.W. Vredeveldt - TNO J.MJ. Journée - TL.JD G. Th. M. Janssen

37, Al, Bi-3

2 4

TNO-re port _B-92-1040 _{SHIP MOTIONS DUETO SUDDEN WATER INGRESS,}

THEORY

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TNO-report Date Page B-92IO4O November 1992 2 CONTENTS NOMENCLATURE 3 SUMMARY 4 INTRODUCTION 5 EQUATIONS OF MOTION i i 6

2.1 Motions in the frequency domain 6

2.2 Motions in the time domain 8

DETERMINATION OF HYDROMECHANIC COEFFICIENTS 11

3.1 Hydrodynamic potential coefficients in the frequency domain 11

3.2 Hydrodynamic potential coefficients in the time domain

...12

3.3 Viscous roll damping 14,

3.4 Restoring moment 23

4 DETERIATION OF INCLINING MOMENT 24

4.1 Inclining moment 24

4.2 Flow caiculätion cross openings 24

4.3 Flow calculation airvents 26

CURVES OF RESIDUAL STABILITY i i 27

VALIDATION 28

CONCLUSION 34

ACKNOWLEDGEMENT. 35

REFERENCES 36

APPENDÏX A REVIEW OF COMPUTER PROGRAMS

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NOMENCLATURE

angular roll acceleiation angular roll velocity pressure loss density of water

sectional area coefficient radial frequency

flow area

hydrodynamic moment of inertia beam

hydrodynamic damping coefficientj hydrodynamic damping factor damping coefficient for roll motion damping coefficient for roll motion damping coefficient for roll motion damping coefficient for roll motion breadth of 'dùct

sectional breadth breadth of wing tank centre of gravity spring coefficient' spring coefficient depth

diameter of damage orifice sectional draught

interval for 'integration along ship length aceleration of gravity (9.81)

metacentric height (varying with angle of heel) height of duct

half breadth to' draught ration height of wing tank

rigid moment of inertia of ship pressure loss coefficient

height of centre of gravity above base retardation function

length of ship length of duct length..of tank

sectional added mass in direction k due heeling moment

hydrodynamic mass coefficient solid mass. of inertia coefficient

sectional damping in direction k due 'to motiOn number of compartments

distance of c.o.g.. above the waterline water 'flow

gas constant draught

external load in K-direction roll angle 'p p

a,

O) A a44 B b44

b'

h (2) u44, b (2) 44f h (2) '-'44k bd B, b, c.o.g. C44

ci

D D,, D, dab g GN, hd H0 h K KG kki L mk Mb Mk n_kj nc

0G

Q R T Xk (p frJ/s21t Erad/sl [N/mi [kg/rn3] i-] [rad/s]' [rn2] [kgm2]: [m'] [Nms/rad] [Nms2/rad2]

viscotis part, eddy maken part viscous part,, friction part viscous part, bilge keel part viscous part Em]' [m] '[rn] i-] [Nm/rad] [N/m],[Nm/rad] [m] [m/s21 [m] [m] E-] Em] [kgm2] i-1 [m'] [kg/s2],[kg mis2] [rn] [m] [m]

to motion in. directionj

[Nm]' [kg],[kg rn2] .[kg],[kg m2j in direction j [-i [m3/s] [Ji(kg k)i [m] [Nh[NmI [rad]

TNO-report Date Page

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TNOrepon _Date _Page

B-92-1040 _{November 1992}

SUMMARY

Roll on Roll off vessels appear to. be sensitive to rapidcapsizing due to sudden ingress of water; Rapid capsizing is caused by a drastic decrease of static stability properties dueto free surfaces, as well as by inertia effects with regard to the roll motiOn.

!Ifl this report it is shown that the dynamic behaviour of the ship dùe to sudden ingress of water cannOt be neglected To this end a calculation model was developed which was venfied by ingress tests For these tests a wing, tank cross duct configuration was chosen.

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B-92-1040 November 1992 5

1. LNTRODUCFION

In many cases ship collisions have caused a rapid capsize of the struck ships. Most of these ships had a qualification of positive residual stability when damaged. A clear insight into the reasons of suchan unexpected capsize is not available Several matters were suggested by various experts and regulatory bodies for further investigaLiQn One of these matters refers tothe dynamic behaviour with regard to ship motions of damaged ships [1], [2] and [3].

The stability of ships is presently determined by applying quasi static methods. Inertia effects with regard to the (roll) motions of the shipare neglected. Usually flow calculations are not carried Out either, although there is one exception which refers to the design of cross ducts. When a cross duct is applied a flow calculation is carned out 'in order to establish the minimum required cross sectional area.

The results of the research effort reported in [12], showed that it is not apriori, justifiable to neglect dynamics while assessing the damage. stability of ships. However, some assumptions which were made with regard to inertia effects and damping of the fluid are considered questionable Moreover the coupling of the roll motions to sway and yaw was neglected as well In order to verify the assumptions and simphfications the Netherlands Directorate of Shipping and: Maritime Affairs commissioned TNO-CMC to develop the calculation method further.

The present calculation method determinessway,roll and yaw motions simultaneously. Moreoverthe calculation of the hydrodynamic coefficients is now incorporated. Convolution integrals are applied in order 'to use hydrodynamic 'data which is available in the frequency domain only, in transient calculations The 2nd order differential equations are now solved by a Runge Kutta tune integration

Curves of residual stability duringeachstage' of flooding can 'now be generated.

A direct interfacing is established to withdraw hydrostatic data and tank data from an existing hydrostatics program.

Thisreport presentsthetheory of theapplied calculation method. There also existsa user manual which explains details about the usage of the computer program [13] A third manual [14] is available for programmers which explains details on the progràm code.

The acronym DYNTNG stands for DYNamic iNGress which 'refers to the dynamic response of a ship sudden ingress of water.

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B-92-1040 November 1992 6

2. EQUATIONS OF MOTION

2.1 Motions in the frequency domain

For a simple harmonic excitation of a linear system, the equations of motions for sway, roll and yaw motions can be written as:

E

Xk (1) J=Z4.6 for

k= 2,4,6

2 refers to sway 4 refers to roll 6 refers to yaw, inwhich: xi Xk

M,

ak Ckj

acceleration of harmonic' oscillation in direction j velocity of harmonic oscillation in direction j displacement of harmonic oscillation m direction j :harmonic-exciting force' or moment in direction k

solid mass or inertia coefficient hydrodynamic mass. coefficient hydrodynamic damping coetficient spring coefficient

In these equationsof motions.the hydrodynamiç mass and':dampingcoefficients and the external loads depend on the frequency of oscillation. So these equations describethe motions of a vessel in the 'frequency domain. The restoring spring terms do not vary with the frequency.

To calculate the hythodynamic mass anddamping coefficients for the sway, roll and yaw motions of a ship,. two or three dimensional potential flow theories can be used. Here the program SEAWAY-C uses the two dimensional or strip theory method.

For the determination, of the two dimensional coefficients of ship like cross sections, these sections are conformally mapped to the unit circle by the so called two parameter Lewis transformation.

The advantage of confòrmal mapping is that the velocity potential of the fluid around an arbitrary shape of a cross section in a complex plane can be derived from the more convenient circular section m another complex plane. In this manner hydrodynainic pmblems can be solved directly with the coefficients of the mapping

function.

The advantage of making use of the two parameter Lewis conformal mapping is that the on diuiensionalised frequency dependent hydrodynamic potential coefficients depend on two parameters only. These parameters are represented by the sectional breadth to draught ratio and the secûònal area coefficieni Then, according tot he strip theory, the hydroclynainic coefficients for the ship can be found easily by integrating the sectional values along the ships length.

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B92 1040 November 1992 7

Studies, carried out in the past on this subject, have showed that this approach yields a fairly good prediction of the hydrodynamic potential coefficients.

Restrictions to Hull Forms of Ships When defining:

B,, : sectional breadth on the waterline

D, sectional draught

a,

sectional area coefficient

H sectional half breadth to draught ratio

the half breadth to draught ratio is defined by:

H,,

and sectional area coefficient by:

Bj2

D,

A simple:transformation' of the crosssectionaihullform to theunit circle will be obtained for B,, H0 and a, with the well known Lewis transformation A descnption of the representation of ship huilforms by Lewis two parameter conformable mapiing is given in chapter 3.

Frame shapes of ships not covered by this approach are the re-entrant ones and the non-symmetric shapes. Conventional hull shapes, bulbous bows and tunnelled stems are covered.

The limits for the sectional area coefficient a, are:

for H0 < 1.0:

[2 -

H01 < a, < [10 + H,, +

for H > 1.

₀ [2

< a < -- [10 + H +

Il

32 H0

'

32 0 H0

If a value of a, is outside of this range it has to be set to the value of the nearest border of this range,to

calculate the Lewis coefficients.

Numerical problems, for instance with bulbous or aft cross sections of a ship, are avoided when the following requirements are fulfilled:

Bj2> al),>

B,/2

with for instance a = 0.01. A,

0g

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B-92-l040 November 1992 8

2.2 Motions in the time domain

As a result o the formulation in the frequency domain, any system influencing the behaviour of the vessel should have a linear'relation to the displacements, the velocities and the accelerations of the body. However, in many cases there are several complications which pensh this requuement such as the non linear viscous damping and non linear external fornes and moments To include these non-linear effects in the prediction of the vessels behaviour, it is necessary to formulate the equations of motions in the time domain, which relate

instantaneous values of forces, moments and motions. Memory functions have to be used to represent the motions dependent hydrodynamic inertia and damping terms by the known frequency dependent values of these terms. The principle of this method has been given by' Cummins [19] and Ogilvie [20].

ComminsEquations

The floating object is consideredio be a linear system with the translational velocitiesas input and ,the reaction 'forces andLmoments of the surrounding water as output. The object is supposed to be at rest at time t = to. 'Then

during a short time At an impulse displacement Ax, with»a constant velocity V, is given to theobject. So: VAt

During this impulsive displacement, the water particles will start to move. 'When 'assuming that thefluid is:free of rotation, a velocity potential, linear proportional to V, can be defined:

'b=Vjr

for

t0<t<t0+At

in which is the normalised velocity potential.

After this impulsive displacement Ax, the waterparticles are stiff moving Becaùse the system isassumedto be linear, the motions of the' fluid, described 'by the velocity potential p, are proportional 'to the impulsive displacement Ax. So:

I=XAx

for t>t0At

Here

_x

is a normalised velocity potential.

The impulsive displacement Ax during the period (t0, t0+At') does not influence the motions of the fluid during' this period only but also further on in time This holds that the motions during the penod (t0 t0 + At) are

influenced also by the motions before this period.

When the object performs an arbitrary in time varying motion this' motion can be considered as asuccession

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TNOreport Date Page

B-92-1040 November 1992

Then the resulting total velocity potential 1'(t) dwing the period (t,., t,. + ist) becomes:

«'(t)

=

fv

iP + ETxj(t t

+ At) .

AtJ]

niiniber of itimesteps

to + nAt

t,. + (n-i).At

j th velocity component during period (t,. t,. + At)

j-th velocity component during period (t,., L + At)

normalised velocity potential caused by a displacement in direction j during period (t,., t,.

+ At)

normahsed velocity potential caused by a displacement in direcuon j dunng period (t,., t,,,

with:

xi

The pressure in the tlúid follows from the linearised equation of Bernouilli;

Integration of these pressures over the wetted surface S of the floating object yields an expression for the hydrodynamic reaction forces and: moments Fk. With k the generalised directionalcosine. Fk becomes:

Fk =

_S1Pk

= _. _dS] . + J-2.4.6 When defining:.

= p

fJ

Jk1S

ô (tt)

= p Is1 k

[pi3 f

ôXjt)

ndS] .

+ At)

Letting ¿t go to zero, yields:

«'(t)

2 i(r)

'

+ Xj (t-r)

dt]

J-Z4.6

in which:

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TNO-report Date _Page

B-92-1040 _{November 1992} ₁₀

The hydmdynamic forces and moments become:

=

i6m . i (t)

JC

(t-t)

. 11(t) .

d}

for k 2,4,6.

Together with the linear restoring spring terms. "Cx" and the linear external loads "X,1(t)',.Newton'S Second law of dynamics. yields the linear equations of motionsin the time dOmain:

translational or rotationalacceleration in direction j at time t

i(t)

translational or rotational veiocity in. directiOrìjat time t.

x1(t) = translational or rotational displacement in directionj at time t

= solid mass or 'inertia coefficient

m = hydronamic mass coefficient

= retardation function

C = spring coefficient

Xk(L): = external load in direction k at time t.

Whenreplacing in thethmpingpart"t" by "t-t" andchangingthe integration._{boundaries thispart canbe written} in a more convenient form:

(M1 +m1)i1(t) +fK (t)1j(t-t)dt +CxJ=Xk

i-Z4,6.

for j = 2,4,6 for k = 2,4,6.

Referring to the basic work on this: subject by Cummins [19], these 'equations of motions are caìled, the "Cummins Equations".

(M

+ n) ;(t). + f

for k = 2,4,6 in which:

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TNO-report Date Pàge

B-92-1 040 November 1992 11

3. DETERMINATION OF HYDROMECHAMC COEFFICIENTS

3.1 Hydrodynamic potential coefficients in the fiquency domain

H.ydrodynamic Potential Coefficients

Thetheory on the calculation of the two-dimensional hydtody!lamic potential coefficients is given by Ursll [16] andTasai [17] All algorithms are descnbed m detailby Journée [18]

These calculations will be canied Out here by the preprocessing program SEAWAY-C. This prOgram calculates the potential mass coefficients for an infinite frequency = °° and the potential damping coefficients for a range of frequencies, O 1.

These two dimensiònal coefficients, defined'with respect to an axes system with the origin in the waterline, are

given, by:

rn22 and n2 rn44 and n44..

rn24 and '12.4

rn42. and n42.

According to the. strip theory, the three-dimensional hycirodynamic potential mass coefficients in the_{G(xb, 'ib'} potential mass and damping coefficients for sway

potential mass and damping coefficients. for roll

potential mass and damping coupling coefficients for roll to sway potential mass and damping coupling coefficients for sway to roll

zb)coordinate system are defined by:

a2,2 82.4

=fm'.dx,

= dx1, +

0G .

L 8z6

= fm2,2"xbdb

-

a24 84.4 ₌ _ffl!4,4' ch1, +

th1, +0G '82,4

84.4 = 'Xb

0G 'a26

86 _8Z6 = a46 84,4 f1112,2 .

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B-92-1040 November 1992 12

In the same manner the total hydrodynamic potential damping coefficients can be found:

=

tn22'

dx

b4

b6

= 21 = b4,2

= b

=

fn44' 'dxb +OG'.f 42' dx +0G b2A

b46 fn4,2' Xb -

+ 0G

L

b=

ba6,4 = b4 b6,6

Jlxb2clxl,

'Pl ease ñote that in the frequency domain external loads have to be of an harmonic type.

3.2 Hydrodynamic potential coefficients in the time domain

The hydrodynamic coefficients in the time domain are not readily available. This section shows how use can be made of the frequency dependent coefficients which are available from strip theory.

To this end we 'have to proceed along the outlines presented in section 2.2, Le the Cumniins equations To' determine mk and K, the velocity potentials j and j have to be fôund, which is very complex. A much. more easy method to' determine ni and Kk. can. be obtaned by making use of the hydrodynamic mass and damping data found by existing two or three dimensional potential theory based computer programs in the 'frequency domain. Relativàly simple relations can be found between m and Kk and: the calculated data of the

hydrodynamic mass and damping in the frequency domain.

'The floating objçct is supposed to carry out an' harmonic oscillation with an amplitude' 1 in the ditection j:

= I

. sin' (mt)

Xb 'dAb

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ThO-repôrt Date Page

B-924040 November 1992 13

A substitution m the Cummins equation yields;

_2

_{(n) . siñ($) +}

.

f K(r)

cos(wt-wr) .4v + c

sm(còt)

= )Ç(t)

for k =2,4,6. This results into:

(r) sin(wr) dv sin(wO +

.coswt) +

sinwt

X(i)

for k = 2,4,6.

lii the;classic frequency domain descriptiòn these equations. of motions are presented by:

w [M

+a(cò)] sin(wt)

[b(w)] cos(t) + c

Sin(t) = Xk(t)

for k = 2,4,6. With:

ak,(w) = frequency dependent hydrodynamic mass coefficient

b,(o)

= frequency-dependent hydrodynamic damping coefficient restoring spring term coefficient

When comparing the time dömain and the frequency domain, equations, both with linear terms, it is found:

A FOurier re-transformation, the damping term yield the retardation function:

fb1(c.) cOs(ør) d

= m--- f)ÇCv)sin

r) d t

=

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ThO-report Date Page

B-92-1040 November 1992 14

Then the mass temi fo1lows from;

in =

+

.-!-f

_{K1(r) sin() d}

_t

This expression is valid for any value of o, so for co 00,_{one finds:} mk = a,(w = oo)

Addition of NonLinearities

So far, ;these equations of motions are linear. But non-linear contributions can be added now to )Ç easily. Fòr instance, non-linear viscous roll damnng contributions can beaddçd to X4:

AX4 = -N«

i I

Nonlineàr spring ierms can now be included, by considering them as an external load and. shifting, their contributions to the right hand side of the equations of motions, for instance:

C4,4 -0

AX4 = -pgVGNp(q» sin ip

in whichGÑp((p)isthe.transverse metacentric height at arbitrary, heeling angels.

3.3 Viscous rH damping

Generally, the roll damping, term in the left hand side of the equation of motions, for roll is given by:

b« .

+ +

with:

b = linear potential' roll damping coefficient

b"

= linear or linearised additiOnal roll damping coefficient non-linear additional roll damping coefficient

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B-92-1040 _{November 1992} 15

The additional roll.damping coefficients bandb2 are caused by viscouseffects. Until now it is flot possible todetermine these coefficients in a theoretical way. They haveto be estimated by free rolling model experiments or with an empirical method, based on a large number of model experiments with systematic. varied ship forms. In time domain simulations a linear as well as anon-linear roll damping coefficient can be used. However, in frequency domain calculations an equivalent linear all damping coefficient has to he estimated.

Here, the algorithms of the empirical method Of Ikeda, Himeno and Tanaka [22] will be given,

Then the linear and non-linear equations of pure róll motions, used to analyse free rolling model experiments, are given. Relations between the dampingcoefficients in these linear and non-linear equations will be discussed.

Empirical met hod ofIkeda

Becausethe viscous'paitof-the roll damping issignificantlyinfluehced by the viscosityof the -flùid,lt is.not possible td. calculate. the total roll damping in apure theoretical way.. Besides this, experiments showed.íilso a nonlinear behaviour of the viscous partsoftheroll damping;

The roll:damping term in the left hand. side of the equation of motions for.roll is given. by:

4) I

For theestimation of the additional non-potential parts of the roll damping,.use has been made of work published by Ikeda, Himeno and Tanaka [22].

Their empiric method is calledhere the "Iketh method'.

The Ikeda methodestimates:the following components of the additional roll damping coefficient ola ship at zero forward speed:

= non-linear friction damping non-linear eddy damping = non-linear bilge keel damping

Then the non-linear additional roll damping. coefficient b«,2 isgiven by:

k (2) .- h (2) _k (2) _h (2)

44v

- 44f

+ 44c + 44t

An equivalent linear additional roll damping coefficient can 'be found by requiring that the equivalent linear damping dissipates an equal amount of energy as the non linear damping so

b«'»

.

7. . $ . dt

=

7is

i . .4)

dt

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TNO-report Date _Page

B-92-1040 November 1992 16

Then the equivalent linear additional roll damping coefficient b«" becomes:

b (»

44v Ca) b44

Ikeda, Himeno and Tanaka claim fàirly good agreements between their prediction method and experimental results. They conclude that the method'can be used safely for ordinary ship forms.

But for unusual ship forms very full ship forms and ships with a large breadth to draught ratio the method is not always sufficiently accurate.

For numerical reasons two restrictions, have to be made during the sectional calculations: if a> 0.999 then a1 = 0.999

if 0G <-D,.cr, then 0G. =

Nomenclature of Ikeda

In the main description of the Ikeda method,ihe nomenclature of-Ikeda is maintainedas far as possible p = density of water

V = kinematic viscosity of'water

acceleration of gravity

O) = circular roll frequency Pa = roll amplitude

R,, = Reynolds number L

= length of theship

B = breadth of the ship

D = average draught of the ship

CB = block coefficient Sf = hull surfâce area

0G

= distance of centre of gravity above still water level

B, = sectiOnal breadth on the waterline

D1 = sectional draught

= sectional area coefficient

H,, = sectional half 'breadth to draught ratio

a1 = sectiónal' Lewis coefficient

a3 = sectional Lewis coefficient

M,, = sectional Lewis scale fáctor

r, = average distance between roll axis and hull' surface

bk = height of the bilge keels

= length of the bilge keels

= distance between roll axis and 'bilge 'keel

= correction for increase of flow velocity at the bilge = pressure coefficient

L

= lever of the' moment

Rb = local' radius' of the bilge circle

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B-92-1040 November 1992 17

Friction damping, b4P

Kato deduced semi-empirical formulas for this component from experimental results of circular cylinders, fully iirimersed in the fluid.

An effective Reynolds number for the roll motion was denotedby: 0.512 .

For ship forms the average distance between the roll axis and the hull surface can be approximated by:

(0.887 + 0.145 _CB)

S,fL +' 2.0. 00

'In here, for ship forms the wetted hull surface area S, can be approximated by: S, = L. (1.7.D + CB.B)

When eliminating the temperature of water, the kinematic viscosity can be expressed into the density of water by the following relation' in the kg-ms system:

fresh water:

v.106 = 1.442 + 0.3924: .

(p - 1000) + 07424

. (p - 100O)

sea water:

v.106 = 1.063 + 0.1039 .(p - 1025) + 0.02602 .(p - 1025)2

Kato expressed the skin friction coefficient as: C, = 1.328 .. R° + 0.014

The first part in this expression represents the laminar flow case. The second' part has been ignored by llceda but has been included here.

When using this, a non-linear roll damping coefficient due to skin friction at zero forward speed can be expressed as:

Eddy damping, b44;2

At zero forward speed the eddy componentof'the roll damping for the naked hull is mainly caused by vortices, generated by a two-dimensional separation. From a number of experiments with two-dimensional cylinders it was found'that for a naked hull this component of the roll moment is proportional to the roll velocity squared. This means that the non-linear roll damping coefficient does mit depend on the period parameter but on the hull form only.

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TNO-report _Date _Page

B-924040 _{November 1992} ₁₈

Whenusing a simple form for thepressure distributiònon the hull surface it appeal-s that the pressurecefficient C, is a function of the ratio y of the maximum relative velocity U to the mean velocity U on the hull

surface:

The C-y relation was obtained from experimental roll damping data of two-dimensional models. These experimental results axe fitted by:

C= 0.435 ;exp(-y - 2.0

. exp(-0;.187.y);+ 1.5Ò

The value of y arourida cross section is approximated by the potential flow theory for a 'rotatingLewis-form cylinder than infmite fluid

An estimation of the sectional maximum distance betweenihe mIl axisandthe huIlsurface, r,,, hasto be ixiade Values of r,,,((p) have to be 'calculated 'for:

(p(p0.0

and:.

(1'+a)

4a3

The values of r(p) follOw from:

r(p)

=:_{MsJ((J+a1);sth(p)-a3.sin(:3(p))2+((l-a1).cos(p)}

a3.cos(3p))2]

With these two results, a valuer and' a valúe p follow from the conditions: for

r(cp1,> r(p):

r

r(1)

and p=

for r(p <r(p):

r =

çp2) andç= P2

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TNO-report _Date

B92- 1040 _{November 1992}

The relative velocity ratiO y on a cross section. is obtained by:

WZ.f3 _2.M

y =

r

2D0(c,1+OG/D)]hf2

D2Z H

with:

H I + a12 -- 9a2 + 2a (.1-3a) . cos(2p) - 6a3.,cos'4p) a = -2a3 . cos (Sp)' + a1 (1-a). cos (3p) + [(6-3a1)a32 +

(a12-3a1).a3 + .a19 cos((p)

b -2a3,. sin (Sp)' + a1. (i-ai) .sin (3p)' + [(6-3a1).a32.+ (a1-3a1).a3 ± a12]

siip)

f3

=' i + 4'exp[-l.65.. iø

(1o]

With this a non-linear sectional eddy making damping coefficjent for zero forward speed' follOws fro

with: 1/2 p 'H

(a' -1)'

r1, = =D3

= B/2

4 .

,

fj:R D1j 'D1 H for: rb < D3 and rb < Bj2 for: 'H0> i and rb> D3 for: H0 < i and r> H0.D3 Page 19

For 'a three-dimensional' ship form the zero fOrward speed eddy 'making damping coefficient is found by'an

integration over the 'ship length:

=

Bilge Keel Damping.

b2

The bilge keel component of the non-linear roUi damping coefficient is divided into two components: 'a' component BN, due to the normal force of the 'bilge keels

'a component B due 'to the 'pressure on the hull surface created by the bilge keels.

'f1

= 05 .[1 +tanh(2a2 - 14)]

f2 _{= 0.5 . [1 - cos(E.a)J, - 1.5} .

[1 - exp(55a3)]si(ir.a)

Theapproximation bfthe localh.radiusof:the bilgeciztle R is: -"

0G

_H

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The normal' force component BN, of the bilge keel daffiping can be deduced from experimental results of oscillating flact plates The drag coefficient CD depends on the penod parameter of the Keulegan Carpenter number. Ikeda measured this non-linear drag also bycarrying out free rolling experiments with an ellipsoid with and without bilge keels.

'This results in a non-linear sectional damping coefficient:, BN' = i3. h . . CD

h

= 22.5

k

+24O

= 1.0 + 0.3.exp.[-l60.(1.0-aj]

The local' distance between theroll axis and the bilge keeFr will be determined further on. Assuming apressure distnbution on the hull caused by the bilge keels a non hnear sectional roll damping coefficient can be defined

B' =

1/2

r2

C

Ikeda carried out experiments to measure the pressure on the hull swface created by bilge keels. He found that the coefficient C of the pressure on the front face of the bilge keel does not depend on the penod parameter whilethecoefficient Cof thepressureontheback4aceof the bilgekecland:,the length 'of the negative pressure region depend on the period parameter.

TNOreport Date Page

B-92-1040 _{November 1992} 20

with:

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m1 m6 Rb, D, 3 . (H0-O.215

-0G

D, = 1.0 m1 -= H0- rn1

O.414.H0 + 00651.m12 - (O.382H0 + 0106),iii1

-

(H0 - O.215.m1) . (I - 215.m1) O.414.H0 .'- O.0651.m12 - (0.38240.0106 . 110)m - 0215rn1) (1 - O.215.rn1) +

6(1-215

m1) + .m1(in3,i; + ni4 +

s

m7 = - O.25.n.in.1 D, = .0.0 rn8 = rn7 + 0.414.m1 = rn7 + O.4l4.ml.[I-cos(SJRb)] for: S0> 0.25 .it.Rb for: S0 < O.2S.lt.Rb for: S0 > O.25 .,t.Rb for: S0 < O.25.lt.Rb

TNO-repott _Date Page

B-92-1040 _{November 1992} ₂₁

Ikeda defines an equivalent length of a constant negative pressure region S0 over the height of the bilge keels,

which is. fitted to the following empirical forinula S0 = 030 . t.fk.rp, + l.95hk

The pressure coefficient on the front-face of the bilge. keel is given by:

C 1.20

The pressure coefficient on the backface of the bilge keel isgiven by: h

= -22.5 k

-

1.20

7t.rk.fkç$

The sectiònal pressure moment is given by:

f

C1,1dh = D?

(-A.c

+ B)

o with; (m3 + m4).m3 - m72 B m3 m4

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B-92-1040 November !992 22

The approximation of the lÒcal radius of the bilge circle Rb is given before.

The approximation of the local distance between the roll axis and the bilge keel r is given as

The total bilge keel damping coefficient can be obtained now by integrating the sum of the sectional roll damping coefficients BN' and B' over the length of the bilge keels

1.0 + - 0.293 Dg (4) tk

= Dg

J

[..293

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B-92-1040 November 1992 23

3A Restoring moment

The restoring moment is detennined by the fóllowingexpression:

M = C

ç

with:, C44 = nonlinear spring ( = angle of roll M = restoring. moment Cki is determinedby: Cq = pgV (Kìskp sirup KG sirup). with: p density of water g

=9.81

V displaced volume

KNp = Metacentric height above keel

ç

= angle of heel

KNç vanes with draught, angle of heel and angle of trim KNç is determined by interpolanon within the KNçi sinq) curves as available from the hydrostatics program.

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B-92-1040 November 1992 24

4. DETERMINATION OF INCLINING MOMENT

4.1 inclining moment

The heeling moment Mmay be determined by asummation of the heeling moments caused by the weight of the fluid in each compartment.

Mk = pgv1

= heeling moment

= density of water

= acceleratiOn due to gravity

= volume of water in each damaged compartment i due to water ingress

= number of compartments

= transverse position of centre of gravity of water volume

distance of entre of gravity of water volume to keel (measured parallel withcentre plane)

= distance of centre of gravity of dry ship to keel

= angle of heel

It should be noted that the centre of gravity of each volume varies with angle of heel, trini angle and the volume of water..

4.2 Flow calculation cross openings

The flOw calculation through .the daiiage orifice and the crossflooding openings are carried oût by applying Bemouilli's law. In order to cater for the hydrostatic pressure difference along the height of: the opening .a subdivision into horizontal strips.

The flow of water is determined by:

2K 2dA with:

Q = volume of waterflow

¿W = pressure through each strip

K pressure loss coefficient

p density of water

dA = sectiona1 area of flow' strip.

-ijcosç

zsmç - 'KG Sin,]

(4) with: Mk, p vi, nc

ii-Zg7c,i KG (

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TNO-report _Date

B92-1040 _{November 1992}

The pressure difference through a strip dA equals: AP = [Pl1 +hw1pg] - [Pl2 - hw2pg]

with:

Pl = air pressure in compartment

hw _{= height of water level}

p = density of water

g

= 9.81

Refer to Figure 4.1 'for additional clarification

water level

incomp I

Cetre of gravity dry ship

ygrcs'

Centre of gravity of watervolume in

comp. ¡ zgrcs

Figure 4.1 Definition

of

centreofgravity and water level parameters

The flow of air through each strip is carried out in a similar way. However, density p is not kept constant.

Page 25

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Applying Boyle-Gay Lassac's law the following formula' can be derived: P1RT

Thedfrection of flow is from.compartment with' high pressure tocompartment' With, lowpressure.

'Note:

The pressure loss coefficient is defmed as given belòw:

2

K-2APA

The explànatory notes issued by 1MO (Annex 6 of STAB XXXII/14) ['10] uses as different definition The relation between both coefficients is given by

1+K

with:

F _{dimensionless factor of speed through flow openings' as used in [1O]}

K = dimensionless factor.of pressure loss through flow openings as usedLin this report..

4.3 Flow calculation äirvents

No subdivision is madè into flow strips. Further it is assumed that an airvent is either dry of wet. Thus only air flow is calculatedor water floW The. same formula's are applied as in section 4.2 however dA is now replaced by the full area of the vent opening.

with:

=

N (11+P2)K

Q1, = volume of air flow

s

API = air pressure loss (P - P2) N

R = pecific gas constant of air 287.17

HJ

kg °K'

T = air temperature _[OK.]

= air pressures at front and rear of strip N.

K

dA

= pressure loss coefficiént = sectional area of flow strip

[1

B-92-1040 _{November 1992} 26

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B-92-1040 _{November 1992} 27

CURVES: OF RESIDUAL. STABILITY

During the roll motion the residual stability caii be determined. To this end at several instants the amount.of flood water in each compartment is fixed to the amount present at that instant Subsequently the residual stability is determined by determanmg the inchnmg moment caused by the water in each compartment and the restoring moment. These moments are determined as function of the angle of heel. Please note that there will usually be an inclining moment even if the ship symmetrically flooded due to free surfaces

The residùal stability is now defined as the diffèrence between the inclining moment due to water in the compartment and the restoiing moment.

Mr = pgV (KNp - KG) smp - Mk((p)

with:

p

density of water

g

= 9.81

rn/s2

V displaced volume m3

KNp = metacentric height above heel as function of p rn KG height of centre of gravity above heel rn

Mk(p) = inclining moment as function of p Nm

p

= angle of heel rad

The tennpgVKNp :isdeterniinedbyinterpolationwithinthe crossturves.data generated by the hydiostatics program.

Mk(p) is determined as pointed out in a previous chapter section 3.3.

GZ(p)

Mr

pgV

Finally, the moment of residual stability isdivided by pgV in order to obtain levers (GZ(p) which is commonly used in naval architecture.

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B-92-1040 November 1992 28

6. VALIDATION

The calculation program was validated against test data available fòr a simple 3m pontoon with a wingtank crossduct configuration as reported m [12] Moreover a partial validation was camed out against calculations in the frequency domain. Tothis endflowoffluid was assumed non existent. Theparticulars of the testpontoon are listed below.

Figure 6.1shows some detäils of the testpontoon.

Length L 3.000m

Breadth B = 2.100 m

Draught T

= 625 rn

Verticali centre of gravity KG (1800 ni

Transverse metacentric height GM = 0.101 m

Gyradius for roll

kjB

0.337

Natural rofl period

Tp

= 5.139 s

Linear roll damping coefficients k = (1050.

Non-linear roll damping coefficients k2 = 0.000

Length of wingtanks L = 1.000m

Breadth of wingtanks

B.

= 0.400 rn

Height of winglanks H = 1.250 m

Width ofcrossduct WC = 0.200m

(30)

L ng tanks

Figure 6.1.. General. arrangement :estpontoon

:...

daxnge orifice

cxoss flow

doss d.uct

L

3.Oi0

rn

B

2.10 in

D

.1.2.5

rn

T

0.6.25 in

Cross duct

:

rn

in

i

1.30 b

'0.2.0

h

0.40

Page 29 TNO-report B-92-1040 Date November 1992

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The checks against the test results are shown in Table 6;2.

Figure 6.2 shows the roll motion. for one test case (run 53) both calculated and measured. Figure 6.3 shows cross curves if residual stability.

Freq. Sway Amplitude Roll AmplitUde Phase Lag

Seaway Dyning Seaway Dyning Seaway Dyning

(p C(p

(radis) (rn/rn) (rn/rn) (rn/rn) (%) (deg/rn) (deg/m) (deg/m) (%) (deg) (deg) (deg)

0.50 0.990 1.009 +0.019 +1.9 1.61 1.60 -thOl -0.6 267.2 67.3 +0.1 1.00

954

0.958 H +0.004 +0.4 11.54 11.67

0.13

1.1

256.4 255.8

-6

L50 0.923 . 0.925 H

0.002

+0.2 8.50 8.47

-03

-4

104.1' 103.4 -0.7 2O0 0.845

0.847 L 0.002 +0.2,

1.28. 1.21

-07

-5.4 275.5 275.6 +0.1

230

0.75

0.75.1

+001 0.1

5.91 5.84 -1.2 274A 274.2 -0.2 300 0638 0638' 0000

00

910

901

009

10 2740 2739 01 350 0507 0S08 +00011+01 1085 1073 008 07 2736 27'3 03

400

0.363 0360 1 -0.003 -0.8 1'86 10.65 -0.21 -1.9 2710 272.7 -03 450 0215 0215 0000 00 912 902

006

07

2722 2717

05

5.00 0.083 0.082 -0.001 -1.2 6.04 5.93 -0.11 -1.8 271.1

279

-0.2 =0.003 =05 =0.10 =1.5 =03

B-92-1040 November 1992 30

The check against results of the frequency domain calculation procedure was carried out by temporarily programmmg wave loads into the source code of the time domain program These loads were calculated by general purpose ship motionsprogram,the wave amplitude was taken equal 1.00 m.

Table 6.1 shows the results.

Table 6.1 Comparison of transfer functions generated by time domain program Dyning and frequency domain program.

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Max angle of heel Time required to reach zero heel

Area below curve

Variatión (degs) (sec) (degs*s) Run

of M C M C M C Nr. Duct Orifice 0.018 18.6 20.6 22.0

290

184 207 ' 43 (m2) 0.027 18.2 20.4 16.5 22.0 133 149 (53) 0.036 183 * 14.0 * 115 * 49 Airhole Orifice O 0.000078 17.0 16.2 17.8 18.5 .-16.8 -22.0 H ₁₂₈ -, 146 58' 55 (m2)

000157

18.2

24

16.5 22.0 133 149 (53)' 0000235 193

208

170 195 132 150 51 Damage Orifice

'029864

10.9 146 21.5 26O 127 168 59 (m2 0.059828 13.6 16.4 17.S 24.0 130 156 63 0.125663 18.2 H 20.4 16.5 22.0 133 149 (53) Kg H 0.747 18.2 20.4 16.5

220

Ì33 149 (53)

;m2)

8

_23.0 24.1 20.5 24.0 224 186 120

B-92-1040 November 1992 31

Table 6.2 Comparison of calculated and measured characteristics of the roll versus time curves. (M refers io measured and C refers to calculated.)

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TNO-report 'Date Page

B-92-1040 November 1992 32

VER IFICATON CALCULATION TESTPONTOON

L x B x T = 3.00 x 2.10 x 0.625 m 24 22 20 18 16 r-' 14' Q, Q 12 o' Q 10 LJ Q C o o L RUN 53 KG = 0.75 rn time (s)

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0.2

0.18

0.16

0.14

0.12

-'0.1

0.08

0.06

-0.04 -'0.02

0.02 -0.04

0.06

0.08 -0.1

O

,nct

+ mc ane (2.4

A after 16 ec

GZ curves of

(residual) stability

Figure ó.3 Cross curvesofresidualstability

I -I I '1 I I I

12 16

angle of Pel (degre]

X after 23 ,ec

O after 9

24

TNO-report Date Päge

'B-92-1040 November 1992' 33

4 0

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B92-i 040 November 1992 34

7. CONCLUSION

From Table 6.1 itcan beconcluded thatafairly good agreement is found between timedomaìcalculatións and frequency domain calculations. Thisshouldnothetooswprising since thehythodynamiccoefficients come from the same origin.

Table 6.2 shOws thata reasonable agreement is foundlbetweenmeasuredandcalculatedroll motions dtito water ingress.

Since the calculated roll motion show a pronounced periodic character it must be concluded that dampmg is underestimated in the present methodL

Comparison of the roll motions as calculated with the three degrees of freedom program Dyning and the one degree of freedom program ROLLINGRESS shows no spectacular differences.However, it shOuld be ñoted that this obsesvation is only Valid for thepontoon.

Nothing definite can be said about actualLships.

Thenlain conclúsionis that-amoderatelyaccurate predictionofroll motion4ue to sudden wateringressis quite

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B-921O4O November 1992 35

ACKNOWLEDGEMENT

The authors v'ish to acknowledgethe valuable contribution of Mr. WQ Westgeest' and Mr. JJ. Uwland who have carried out a great deal of the coding of the computer program.

Finally the valùable discussions and encouragement from Ir H. Verrneer2 should not be left unmentioned

TNO Centre of Mechanical Stnictures

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B-92-1040 November 1992 36

REFERENCES

[1]. SpoUge, JR

The technical investigation of the sinking of the RO/Rl ferry European Gateway. RINA MaE No. 3, 1986.

Boltwood D.T.

Ro/Ro ship survivability; Comments on damage stability modelling. Ro/Ro 88, Gothenburg, 7-9 June 1988.

Braund, N.A.

Damage stability; research forïhe.future. Sale ship/Safe cargo conference, London 1978. Dand, I.W..

Hydrodynamic aspects of the sinking of the.feny Herald of Free Enterprise. The Naval Architect; May:1989

[51 Peach, et al

The radii of gyration of merehant ships.

North East Coast of Engineers and ShipbUilding Transactiòns. June 1987, pp. 155-117. Journée, J.MJ.

Seaway-Delft, User manual and theoretical background of release 3.0.

of Technology..Repoa No. '849;.January .199 Ikeda. Y. et al

Predictionmethod for ship rolling.

Department of Naval Architecture,University of OsakaPrefecture Japan, Report No.. 00405, 1.978.

Blevins, R.D.

Applied Fluid Dynamics Handbook.

Van Nostrand Reinhold Conipany New York, 1984. [9j Ireland, N.

Damage stability model tests. Project No. 34620.. British Maritime Technology, May 1988.

[101 IMCO

Explanatory notes to the regulatiòns on subdivision and damage stability of passenger ships as equivalent to part Bof chapter II of the International Convention for Safety ofLife at Sea, 1960.. ANNEX II STGAB Xvii 1.

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B-92-1040 November 1992 37

[:1 1] Technical University Deift Numerical analysis Cl. Lecture Notes (in Dutch).

[12] Vredeveldt, A.W. and Journée, J.M.J.

Roll motions of ships due to sudden water ingress; Calculations and experiments.

International Conference on Ro/Ro Safety and Vulnerability the Way Ahead, London, U.K., April

1991.

[131 Uwland, JJ.,Vredeveld, A.W.

Ship motions due to sudden water ingress, computer program, user manual.

Report Nr 93 CMC R0205 TNO Bouw Centre for Mechanical Constructions March 1993 Deift, The Nétherlands.

Uwland J.J., Vredeveldt A.W:,

Ship motions dueto sudden wateringress, computer program, programmers manual. Report Nr.93-CMC-R0204, TNO-Bouw, Centre for Mechanical Constructions; March 1993. Dëlft, The Netherlands.

Vredeveldt, A.W. and Journée, J.MJ.

Dynamisch Hellingsgedrag van een Lek Schip. Schip en Werf, Jaargang 57, Nr. 7, 1990. Ursell,F.

On the Rolling Motion of Cylinders in the Surface of a Fluid.

Quarterlyiournai oLMechanics and Applied Mathematics, Vol. II, 1949:

TasaiF.

Hydrodynamic Force and Moment Produces by Swaying and Rolling Oscillation of Cylinders.on the Free Surface.

Research Institute for Applied Mechanics, Kyushu University, Japan, Vol. IX, No. 35, 1961. Journée, J.MJ.

Theory and Algorithms of Two Dimensional Hydrodynamic Potential Coefficients

Deift University of Technology, Ship Hydromechanics Laboratory, Deift, The Netherlands,Report Nr. 884, November 1990.

Cummins, WE.

The Impulse Response Function and Ship Motions.

Symposium on Ship Theory, Institüt für Schiffbau der Universität Hamburg, Germany, 25-27 January 1962.

Ogilvie, T.F.

Recent Progress Towards the Understanding and Prediction of Ship Motions. Fifth Symposium on Naval Hydrodynamics Bergen, Norway, 1964

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TNO-repoij _Date _Page

8-92-1040 _{November 1992} ₃₈

[21;] Journée, J.M..J.

'SEAWAY-DELFT"

User Manual and Theoretical' Background of Release 3.00.

Deift University of Technology 'Deift Ship Hydromechanics Laboratory Delft The Netherlands Report Nr. 849, January 1990.

[22]: Ikeda, Y., Himeno,, Y. and Tanaka, N. A Prediction Method for Ship Rolling.

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APPENDIX A REVIEW OF COMPUTER PROGRAMS

(41)

APPENDIX.B ADDITIONAL CONSIDERATIONS WITH REGARD TO FLOWCALCULATIONS For each compartment, theintegration formula's forair volurneV and'airmass M are (thesubscriptP refersto the previous value, i.e. one time step back):

E= V- Vi, -

4-

(J'J)

₌₍₎

(84)

=

M - M

(J+)

O (85)

Based on the mass balance (equation of continuity) the derivatives of the air volúme and the air mass with respect to time are (see Fig. 1):

V -Q(i)

The incrementalformofthe integration formula!s as theyareused for the Newton-Raphsonsolûtionmethod are: