Dynamics of a ship with partially flooded compartiment

Deift University of Technology

Ship HydromechaflicS Laboratory

Library

Mekeiweg 2, 2628 CD Deift. The Netherlands

Phone: +31 15 2786873 - Fax: +31, 15 2781836

DYNAMICS OF A SHIP WUIH PARTIALLY FLOODED

cOMPARTMENT

jan O. de Kat

Maritime Research Institute Netherlands (MARIN)

Preseüted

at

The Second Workshop on Stability aiid Operatiönal Safety of Ships

18-l9Nov. 1996

DYNAMIOS OF A SHIP WITH PARTIALLY FLOODED

COMPARTMENT

Jan 0. de Kat

Maritime Research Institute Netherlands (MARIN)

Current address: David Taylor Model Basin, NSWC, Carderock Division Sealceeping Dept. Code 5500, Bethesda, MD 20084-5000, USA

Tel. +1 - 301 -2273945 Fax: +1 - 301 - 227 5442

SUMMARY

This paper presents a brief outline of a six degree-of-freedom, nonlinear time domain model that is capable of simulating the large amplitude motion response of an intact or damaged ship in waves and wind. The part of the mathematical model related to the damage fluid effects is discussed in some detail; the present model neglects sloshing. Experiments were conducted with a tanker model in low steepness beam waves and with different amounts of fluid inside the ship Roll decay tests in calm water provide useful material for validation of the simulation model. Fúrthermore, predicted and measured heave and roll responses in waves are compared and discrepancies in roll due to sloshing are highlighted.

INTRODUCTION.

o describe the complete behavior of an intact ship in waves and wind is not a trivial undertaking; yet the presence of fluid inside a ship may add another dimension of

complexity. The dynamics of a ship with internal fluid lias been investigated

extensively in the past in the context of roll-stabilizing tanks, water trapped on deck, dock

ships, LNG carriers and oil tankers, among others. A ship in a damaged condition presents

some sinulanties within this context,but the degree of complexity increases as weconsider water ingress through a damage opening, possibly with progressive flooding through and between compartments. The purposeof this paper is to revisit the elementary case of a ship with a rectangular compartment, which is not connected to the ocean and which can be

partially flooded with a fixed amount of water.

Recent research on this topic has been reported by Armenio et al. (1996) and Francescutto and Contento (1994), who describe a time domain simulatiOn model that couples the roll motion of a ship to the fluid dynamics inside a tank, where the internal fluid problem is dealt with by solving the Navier-Stokes equations numerically. The ship roll motion is

described by a single degree of freedom model; the approach has been a applied to a fishing

vesseL Petey (1986) also describes a siftuilation model that can solve the coupled roll and fluid motions in the time domain. .Rakhmanin and Zhivitsa (1994) derive equations for the

coupled sway and roll motions for a hip with a partially filled compartment, and they

illustrate that the fluid being a non-solid should result in the adjustment of, for example, the

moment of inertia of the vessel-liquid combination. Rakhmanin (1962) provides detailed expressions that determine the extent to which the moiñent of inertia of a "solidified" fluid

is reduced by considering it as a liquid.

This paper starts with a brief description of the mathematical model for a ship with one or more damage openings. Rather than embarking with the more general damage scenarip

involving water ingress and progressive flooding, we first consider the case of a ship with a

fixed amount of fluid inside an isolated compartmenL To this end experinients have been cathed out with a tanker model in beam waves. A number of the test conditions have been simulated and comparisons between simulations and model tests are discussed The paper

concludes with óbservations and recommendations.

THEORY

The

mathematical model described here has its roots in intact stability:

It was

developed to simulate the large amplitude motions of a steered ship in severe seas and

wind. The model consists of a nonlinear strip theory approach, where linear and nonlinear pOtential flow forces are added to maneuvering and viscous drag forces. It is capable of predicting seakeeping and maneuvering behávior in moderate to severe seas, as well as extreme motions related to surfriding, broaching and capsizing (see De Kat, 1994, and De Kat et al., 1994). The mathematical model has been extended to account for the effects of damage fluid inside the ship, with the possibility of modeling water ingress and

oufflow through openings between compartments.

The derivation of the equations of motions is based on the conservatioñ of linear and angular momentuín. These are given in principle in the inertial (earth-fixed) reference system, defined by the system of axes (f,yC,z). As the exciting forces are more easily described in the ship-fixed coordinate system, the Euler method is applied for deriving the equations of motion in terms of a rotating, ship-fixed coordinate system. Here the fluid Inside the ship is considered in a dynamics sense as a free particle withconcentrated mass;

using this approach classical rigid body dynamics can be used to derive the equations of

motion. A similar methodólogy has been applied by Letizia andVassalos (1995.

For a damaged ship the equations of motion are expressed with respect to the center of

gravity of the intact ship, G. The Origin of the local, ship-fixed coordinate system with axes

(x,y,z) is located at G. Although in terms of locäl coordinates G isindependent of time, the

center of gravity of the overall damaged ship will be time dependent which introduces a

number of cross-coupling terms in the equations of motion.

ThecenterofgrvityOftheflOOdwaris1bYbet0tthmsP«tt00E The

location vector r = (Xf, yf, z) and its derivatives (velocities and accelerations) are defined in the local ship-fixed coordinate system Giyz with respect to G.

([Mo] -i- (a()] + [M]).0 =

The combined set of equations for the conservation of linear and angular momentum gives

the equations of motion for the ship system with six degrees of freedom.

Here it is

convenient to consider the full 6 x 6 mass matrices, as isdone for the intact case All terms that contain linear or rotational ship accelerations are transferred to the left hand side (LHS) of the equations of motion. Using the generalized mass matrices, the equations of motion m the ship-fixed coordinate system are written as follows:

'Z F'

+ ni1 ).( WG q - va ri' Y.F,

(nzo+mj).(uar-wap)

(ino+rnj).(vap-uaq)

(IQ-I,.0)qr

(I,0-I0)pr

(,,O-JEO)pq)

+ additional ternis

The matrix [M0] is the generalized mass màtrix of the intact ship, [a] is the added mass matrix that is part of the linear radiation forces (the convolution integrals are part of the terms in the RHS) mo is thetune-independent mass of the intact (dry) ship, mf is the mass of the flood water inside the compartment [Mf] is the matrix containing all

ship-acceleration related inertia terms associated with the flood water in a damaged compartment and it is given by:

- [Mr].xc +

The summation signs in the RuS represent the sUmof all external force contributions,

which resUlt from:

Froude-Krylov force

Wave radiation (convolution integrals) Diffraction in1 O O O m1 . zj - .y1 O m1 O - rn1 . O in, .x1 O O in1 flj1.y1 -m1.x1 O

[Mf]=

O iflj.Zj .m1.Y1 -m1.z1 O m1.x1 m1.)'1. -rn,.x1 O

Ij.+m1.r,

m1.x1Y, -rn1.x1z1 -mj.x1Yy

I,.,,+rn1.r,2

-m1.Y1z1 -m1.X1Z1 -m,.Y1Z1

If.zz+mj.r/

Viscous and maneuvering forces

-Propeller thrust and hull resistance Rudder and appendages

Wind

Internal fluid

The "additional terms" in the RHS of the equations of motion stem from cross products, which appear when expressing the conservation of momentum in a ship-fixed coordinate

system, and from the motion of the fluid relative to the ship.

For example, the

conservation of imear momentum in the ship-fixed system is given by the following

eq,uatiom

F = (1m,+1nf).('G+O.®vG) + ,nj.(thør+$j + û®(Dør)+ 2.w®v1)

+ ,iif.(irG+vf+o®r)

where the vectors VG = (UG, VG, w0) and w = (p, q, r) represent the linear and angular

velocity vectOr in the ship-fixed coordinate- system, respectively; Vf iS the velocity vector

of the center of gravity of the flood water expressed in the ship-fixed reference system All

terms resulting from the above expression are retained and, except for the ship

acceleration terms, they are put in the RHS of the equations of motion The conservation of angular momentum can be derived in a similar fashion.

Forces âssociated with internal fluid

Taking a quasi-static approach, the damage fluid causes an additional vertical force to act on the ship. lEi the inertial (earth-flxed) reference frame, the force vector is the following

o o

rn1 . g

The above vector can be expressed jn terms of the ship-fixed coordinate system by the-following ttansfotmation:

where the matrix [Tfl] is the tranformatioñ matrix containing the Euler angles. The moment vector about G (in the ship-fixed system) caused by the flood water is given by:

-= r®F

An important assumption made here is that the water level remains horizontal at all times,

which implies that slohiñg effects -are neglected.

Possible consequences of this

assumption are investigated below Another critical assumption is the homogeneous

distribution of permeability inside, any compartment.

EXPERIMENTS

Model

tests have been carried out at MARIN with a 200 kDWT tanker at scale 1:82.5

inabasinmeasuring200mx,l5mxlm.

ThepathcUlaSOfthetankrgiVefl

in table i and in figure

1 The tanker was equipped with a compartment

symmetrically positioned amidships, the length of the compartment is 82.50 m and the width is 31 76 m The tanker was tested with the compartmentpositioned at two levels with respect to the basehne (1) at a height of 520 m above the keel and (2) at 1650 m above the keel Different fill levels were achieved by adding color-dyed water to the compartment, which was closed off with a perspex lid to prevent fluid from spilling out in the event of soshing. This paper discusses results pertaining only to the case with the

compartment located closest to the keel.

I 7r.w,

T2i

,r.h

g. tanh()

Length, Lpp (m) 3i0.20

Beam,B m)

47.20 Draft,T(m) 16.00 Depth,D(m) 2607

Displaced weight (ti) 202,600

Center of gravity above baseline, KG (rn) 10.0

Metacentric height, GM (m) 9.50

Natural roll period intact (s) 10 Table 1. Particulars of tanker

Tests comprised roll decay tests and tests in regular beam waves with periods ranging from 8 to 12 s The wave height (crest-trough) was between 2 mand 3 m The model was kept m position by means of a soft-spring mooring arrangement as shown in Fig

i, at zero

forward speed Fluid levels in the compartment ranged from zero ("intact't ship) to 16 m fill depth, wich corresponds to a maximum fill ratio h/w = 0.5. Fig. 2 shows the natural period of the fluid in transverse direction as a function of fill ratio; the period for the first

For shallow fill depths (h/w <0.15) the period can be based on the critical shallow water

wave speed:

RESULTS AND DISCUSSION

We

Results for heave are presentedand roll motion responses are the primary items of interest in thisinterms of Response Amplitude Operators (RAOs),study. given by the heave amplitude per unit wave amplitude Roll decay tests provide

'useful information on the nalurài roll period and damping in calm water. Roll RAOs are obtained by the ratio of the roll amplitude per unit wave amplitude These results are

presented for the following casés: 1. Intact ship

'2. Fluid levél: h = 16 m (h/w = 0.5; volume = 41927 m3)- no sloshing

Fluid level: h = 1m (h/w = 0.03; volume =2620 rn3) subresonant cOndition Fluid level: h = 4m (h/w = 0.13; volume = 10482m3) resonant sloshing

In case 2, with h = 16 in, the natural sloshing period lies well below the range of wave

frequencies tested Hence, for the conditions tested the fluid level remains horizontal and the simulatiòns should be able to represent the behavior quite well.

Case 3 presents a vry shallOw fill depth; here the sloshing period, lies well above thç wave ..tS _{frequenetos However, the fluid level does not stay horizontal in those test conditions the}

fluid motion is characterized by hydraulic bores - at any one time three bores could be running across the tank simultaneously Onemight be tempted to believe thatinspite of the

fluid dynamics the motion of the ship may not bé affected significantly, as the mass of fluid is small compared with the mass of the ship.

Case 4 is a true sloshing conditiqn: the sloshing period lies within the range of wave periods and is approximately equal to the natural roll period of the ship In the model tests the water

would splash forcefully against the compartment's lid for all wave periods tested.. Here we

would expect large deviations between simulations and measurements.

An overview of the figures showing comparisons between tests and simulations is given in

table 2.

Table2 Ovérview of figures with eìperimental and simulation results Heave RAO Roll decay Roll RAO

Intact ship (no fluid) Fig.3 F*g.7 Fig. 11

imfiliheight

Fig. 4 Fig. 8 Fig. 12

4mflllheight

Fig 5 Fig. 9 Fig. 13

Heave in beam waves

Figures 3, 4, 5 and 6 show that the heave response in beam waves is, predicted

consistently well for all fluid fill levels considered.' The characteristics of the' heave response change graduálly with the amount of fluid in the tank; heave tends to decrease with increasing fluid levels, but not at all exciting frequencies

Roll decay

For the intact case, the predicted roll decay follows the measured 'roll decay very closely (see figure 7) - both the roll 'period and decay are very close. This suggests that viscous

roll damping is modeled adeqùately in these conditions, which then

leads to the

assumption that hull roll damping is modeled properly also in the cases with internal fluid present. For the lärgest fill depth considered here, with fill ratio h/w = 0.5 and fluid weight being around 20% of the displacement çf the intact ship, the predicted roll decay is quite similar to the measured results (see figure 10) Here the natural penod of the tank is much shorter than the roll period of the ship - 5 s versus 10 s, and no sloshing occurs (the fluid surface stays honzontal) It is mteresting to note the slightly higher damped behäviôr in the simulation model than in the case of the model test. The natural period of the ship with fluid is fractionally longer than the intact ship - the increase in natural period is about i second.

Figures 8 shows that even the presence of a relatively small amount of fluid (h/w = 0.03 with fluid Weight being' approximately equal to 1% of the displacementof the ship) inside the tank can have a significant influence on the roll decay The measured roll decay shows a monotonically, albeit not continuous, decrease in successive roll amplitudes, whereas the simulated roll decay underpredicts the roll damping. During the first roll period the simulated and measured periods are approximately equal; subsequently, however, the predicted roll period stays constant, while the measured period does not remain the.Same (it decreases from around lito 10 secOnds).

The discrepancies in roll become even more significant for the 4 m fill level case. The measured röll motion decays rapidly during the first two cycles, but then shows a beat-type of behavior. The sloshing of the fluid inside the ship keeps it from attaining static equilibrium for some time. The natural period is the same as the predicted périod during the first cycle (around 11 s), but then increases arid increases again. Obviously a different mathematical model would be requfred to simulate properly the observed dynamics.

Roll response in beam waves

For the 'ship with no internál fluid, the RAO for roll obtained by simulation iS reasonably close to the RAO derived fim the model tests (see figure ii). The correspondence in the resonant peak is good, while the roll is overpredicted at the highest wave frequencies; th same applies to the case with the largestfill depth (see figure 14). In the latter case the fluid level remains horizontal at all times and the roll moment caused by the fluid is in phase with the roll motion.

For the case of shallowest fill depth tested (hlw (103) the measured and predicted roll response agree quite well for frequencies higher than the roll resonant frequency, see figure 12. At the resonant roll frequency the measufed roll response is lower than the simulation jesuits.

The largest discrepancies. between simulation and model test occur for the true sloshing case, where the fill height is 4 m, see figure 13. Here the fluid acts as an active damper over the whole range of wave frequencies tested, rather like a well tuned anti-roll tank. The roll moment exérted by the fluid on the ship is out of phase with the roll motion and therefore provides an effective source of roll damping.

CONCLUSIONS

This

paper presents a six degree-of-freedom mathematical model that is capable of simulating the dynamics of a damaged (and intact) ship. To validate the simulated dyñarnic response of a damaged ship in a stepwise fashion, model tests have concentrated initially on a vessel with a floodable compartment without any openings to the sea. For this purpose a tanker model with different fluid fill levels was tested in low steepness beam waves Predicted heave motions in beam seas compare very well with measurements. Roll motions, however, show less consistent agreement: in non-resonant conditions the roll motion amplitudes agree fairly well with measured values, but m

(fluId) reSonant conditions the simulation mOdel overpredicts the roll response

significantly.

REFERENCES

Armenio, V., La Rocca, M. and Francescutto, A., "On the Roll Motion of a Ship with Partiàlly Filled Uiibaified Tanks," Part i & 2, Paper no. IJOPE-JC-154 to be published in the International Journal of Offshore and Polar Engineering, 1996

De. Kat, 1.0., "Irregular Waves and Their Influençe on Extreme Ship

Motions," Proceedings oftlze SympOsium on.Naval Hydrodynamics, Santa Barbara, Aug. 1994, pp.

39-58

De Kat, 1.0., et al. (1994), "Intact Ship Survivability: New Criteria from a Research and

Navy Perspective," Proceedings of the STAB '94 Synzposiwn, Melbourne (FL)1 Nov. 1994

Francescutto, A. and Cozitento, G., "An Experimental Study of thé Cotipling Between

Roll Motion and Sloshing in a Compartment," Proceedings of the Fourth (1994)

International Offshore and Polar Engineering Conference, Vol III, Osaka, April 1994, pp.283-291

Letizia, L. and. Vassalos, ])., "Formulation of a Non-Linear Mathematical Model for a Damaged Ship Subjected to Flooding," Proceedings of the Sevastzanov Symposium,

Kaliningrad, May 1995

Petey, F., "Numerical Calculation of Forces and Moments Due to Fluid Motions in Tanks and Damaged Compartments," Proceedings of the Third Internationtil Conference on

Stability of Ships and Ocean Vehicles STAB '86, Gdansk, Sept. 1986, pp 77-82

Rakhmanin, N.N., "The E,perimental Study of Dynamic Properties of the Ship with Partially Flooded Compartments," Troodi of CNII of A N Krylov, Vyp 191, Sudpromgiz,

1962 Çm Russian)

Rakhmanin, N.N. and Zhivitsa,

S., 'Prediction of Motion óf Ships with Flooded

Compartments m a Seaway," Proceedings of the STAB '94 Symposzwn, Melbourne (FL),

List of figure captions

Figure 12.

Figure 13.

Figure 14.

Particulars of tanker and test setup

Sloshing period fundamental mode

of fluidin ¡de rectangular tank

as a function of fill ratio

Rponse Amplitude Operator for heave in

beam waves

hip without internal fluid)

Response Amplitude Operator for heave in beam waves (fluid level inside compartment: i m)

Response Amplitude Operator for heave in beam waves (flüid level inside compartment: 4 m)

Response Amplitude Operator for heave in beam waves (fluid level inside compartment: 16 m):

Roll decay for ship without interflal fluid Roll decay for ship with 1 m level

Roll decay for ship with 4 In fluid level

Roll decay for ship with 16 mfiuidlevel

Response Arnplitudè Operator for roll in beam waves (ship without internal fluid)

Response Amplitude Operator for roll in beam waves (fluid level inside compartment: i rn)

Response Amplitude Operator for roll in béam waves (fluid level inside compartrnent:4 m)

Response Amplitude Operator for roll in beam waves (fluid level inside compartment: 16 'n)

Figure 1. Figure 2. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7. Figure 8. Figure 9.

Figure lo.

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