hydro structure interactions in seakeeping sime … · hydro-structure interactions in seakeeping...

International Workshop on Coupled Methods in Numerical Dynamics

IUC, Dubrovnik, Croatia, September 19th-21st 2007

Hydro structure interactions in seakeeping

Sime Malenica

Bureau Veritas - Researche Department, Paris La Defense, FRANCE

Abstract. Even without considering the ship’s structural responses, thenumerical modeling of the ship seakeeping behavior remains an openproblem for a general case of the ship advancing in waves with arbitraryforward speed. This is true both for, the most commonly used, potentialflow models and for the general CFD codes based on solving the Eulerand/or Navier Stokes equations. The main problems of the modelingconcern the correct representation of the waves generated by the inter-action of the ship with the sea waves, and the presence of the free surfacewhich is not only unknown in advance but, at the same time, supportsa highly non-linear boundary condition. The impossibility to solve thecomplete non-linear seakeeping problem at once, led to the different lev-els of simplification of the original non-linear boundary value problem.The common practice consists in identifying the most dominant physicalaspects, of the particular problem, and in application of the dedicatedsimplified numerical models free of the ”non-important” parts.

Only the potential flow models for fluid flow are considered in this paperand the main aspect is put on hydro-structural coupling issues and cor-responding structural responses. In that respect and for some parts ofthe analyses, the hydrodynamic part is supposed to be known i.e. it isassumed that the boundary value problem for the velocity potential wasefficiently solved. Both local and global hydro-structural issues are con-sidered and that in the context of the ultra large ships (LNG carriers andcontainer vessels) for which the common rules of classification societiesreach their limits and direct calculation procedures are necessary.

Key words: hydro-structure interactions, potential flow, quasi static

loads, hydro-elastic interactions, impact loads

1. Introduction

It is enough to take a look on Figure 1., to understand how difficult thenumerical modelling should be for general seakeeping problem. Indeed, lot ofdifferent physical phenomena are involved (waves, ship speed, large ship oscil-lations, slamming, sprays, wind, ...) and it is impossible to take all them intoaccount at once. It is fair to say that, up to now, there is no efficient numerical

∗Correspondence to: [email protected].

1

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Figure 1. Ship sailing in rough waves

tool for global modeling of these kind of situations with sufficient accuracy. Thatis why the usual procedure consists in introducing the additional assumptionsand subdividing the complex general problem into relatively small problems forwhich the dedicated tools are then developed.

The present paper deals with the following hydrodynamic problems relatedto the hydro structure interactions in seakeeping:

• Global hydro-structural issues

– quasi static linear rigid body structural response

– quasi static weakly non linear rigid body structural response

– linear hydro-elastic ship response

– weakly non-linear hydroelastic ship response

• Local hydro-structural issues

– slamming loads

– fatigue loading of the side shell longitudinals

General computational scheme which was put in practice in Bureau Veritas isshown in Figure 2.. Within this scheme, it is possible to perform all the abovementioned types of calculations under the assumptions which are discussed inthis paper.

2. Linear quasi static global loadings and responses

As mentioned before, even the linear general seakeeping problem of shipadvancing with forward speed in waves is still an open problem from the mod-elling point of view. Only the case without forward speed can be handled withsufficient confidence and the inclusion of the forward speed usually requires ad-ditional simplifying assumptions. Regardless of these fundamental difficulties insolving the boundary value problem for velocity potential, the hydro-structuralcoupling issues remain the same, and that is why we will illustrate them on thecase without forward speed. Before considering the hydro-structure interaction

Hydro-structure interactions in seakeeping 3

)

Scatter Diagram

)

Scatter Diagram

)

Scatter Diagram

Figure 2. Overall computational scheme

problem in more details, and for the sake of clarity, first we briefly describe thebasics of the seakeeping model which is used in most of the seakeeping toolsbased on Boundary Integral Equation techniques.

2.1. Basics of the linear seakeeping analysis

Thanks to the linearity, the problem is formulated in frequency domain. Thetotal velocity potential is decomposed into the incident, diffracted and 6 radiatedcomponents:

ϕ = ϕI

+ ϕD− iω

6∑

j=1

ξjϕRj(1)

where :

ϕI

- incident potential

ϕD

- diffraction potential

ϕRj

- radiation potential

ξj - rigid body motions

At the same time, the corresponding dynamic pressure is found from the linearBernoulli equation, and the similar decomposition is adopted:

p = iωϕ = pI

+ pD

+

6∑

j=1

ξjpRj(2)

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In order to obtain the the total hydrodynamic pressure, the dynamic variationof the hydrostatic pressure should also be added to the above expression:

phs = −g[ξ3 + ξ4(Y − YG) − ξ5(X − XG)] (3)

where the subscript ”G” denotes the position of the center of gravity, with respectto which the motion equation is written.

It is important to note that the motion equation is written in the so calledearth fixed reference system, or in the system parallel to it, if the body is ani-mated with forward speed. For that reason the restoring matrix is not obtaineddirectly by integration of the hydrostatic pressure (3), but also the change of thenormal vector should be taken into account.

F hs = [ C ]ξ =

∫ ∫

SH

B

[phsn − gZΩ ∧ ndS] (4)

where Ω denotes the rotational component of the motion vector Ω = (ξ4, ξ5, ξ6),and SH

B denotes the hydrodynamic mesh of the wetted body surface. Note thatthe compact notation is used throughout whole the paper, so that the normalvector n denotes (nx, ny, nz) for i = 1, 3 , and (R − RG) ∧ n for i = 4, 6.After integrating the pressure over the wetted body surface, the correspondingforces are obtained and the rigid body motion equation, in frequency domain, isusually written in the following form:

(

−ω2([ M ] + [ A ]) − iω[ B ] + [ C ])

ξ = F DI (5)

where:

[ M ] - genuine mass matrix

[ A ] - added mass matrix

[ B ] - damping matrix

[ C ] - hydrostatic restoring matrix

F DI - excitation force vector

The final expressions for the excitation, added mass and damping coefficientsare:

FDIi = iω

∫ ∫

SH

B

(ϕI + ϕD)nidS (6)

ω2Aij + iωBij = ω2

∫ ∫

SH

B

ϕRjnidS (7)

Note also that, in the general case, the total restoring matrix is a sum of thepressure part (4) and the gravity part which is zero in the present case becausethe motion equation is written with respect to the center of gravity.


2.2. Solution of the boundary value problems

Within the Bureau Veritas’s numerical code HYDROSTAR, the BoundaryIntegral Equation (BIE) method based on source formulation, is used to solvethe Boundary Value Problems (BVP) for different potentials.In the case of zero forward speed, the general form of the BVP is:

∆ϕ = 0 in the fluid

−νϕ +∂ϕ

∂z= 0 z = 0

∂ϕ

∂n= Vn on Sb

lim[√

νR(∂ϕ

∂R− iνϕ)

]

= 0 R → ∞

(8)

where Vn denotes the body boundary condition which depends on the consideredpotential:

∂ϕD

∂n= −∂ϕ

I

∂n,

∂ϕRj

∂n= nj (9)

Within the source formulation, the potential at any point in the fluid isexpressed in the following form:

ϕ =

∫ ∫

SH

B

σGdS (10)

where G stands for the Green function, and σ is the unknown source strengthwhich is found after solving the following integral equation:

1

2σ +

∫ ∫

SH

B

σ∂G

∂ndS = Vn , on S

H

B (11)

This equation is solved numerically, after discretizing the wetted part of thebody into a number of flat panels over which the constant source distribution isassumed.

2.3. Loading of the structural model

The loading of the structural model is composed of two parts:

1. Inertia loads

2. External pressure loads

Inertia loads can be included straightforwardly by associating the accelerationvector to each finite element. Concerning the pressure loading, most of themethods nowadays use the different interpolation schemes in order to transferthe total hydrodynamic pressure (2,3) from hydro model (centroids of the hy-dro panels) to the structural model (centroids or nodes of the finite elements).

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Besides the problems of interpolation, it is important to note that the motionamplitudes, which are present in the definition of the total pressure, were cal-culated after integration over the hydrodynamic mesh. For that reason it isimpossible to obtain the completely equilibrated structural model. Indeed, theFEM model has its own integration procedure which is usually different.In order to obtain the perfect equilibrium of the structural model here we intro-duce two main ideas:

• Recalculation of pressure in structural points, instead of interpolation

• Separate transfer of pressure components, and calculation of hydrodynamiccoefficients (added mass, damping, hydrostatics & excitation) by integra-tion over the structural mesh

Here below we discuss these two points in more details.

2.4. Hydrodynamic pressure

What we propose here is the recalculation of the pressure at the requiredlocations instead of interpolation from the hydrodynamic model. This becomespossible thanks to the particularities of the BIE method which gives the con-tinuous representation of the potential through the whole fluid domain Z < 0.In this way the communication between the hydrodynamic and structural codesis extremely simplified. Indeed, it is enough for the structural code to give thecoordinates of the points where the potential is required and the hydrodynamiccode just evaluates the corresponding potential by:

ϕ(xs) =

∫ ∫

SH

B

σ(xh)G(xh;xs)dS (12)

where xs = (xs, ys, zs) denotes the structural point and xh = (xh, yh, zh) thehydrodynamic point.

In the case of linear seakeeping without forward speed, this operation issufficient because the pressure is directly proportional to the velocity potentialand, within the source formulation, the potential is continuous across the bodywetted surface. This is very important point because, due to the differences inthe hydrodynamic and structural mesh, the structural points might fall insidethe hydrodynamic meshes.Once each pressure component has been transfered onto the structural mesh, the”new” hydrodynamic coefficients are calculated by integration over the structuralmesh:

FDIS

i = iω

∫ ∫

SS

B

(ϕS

I + ϕS

D)nidS (13)

ω2AS

ij + iωBS

ij = ω2

∫ ∫

SS

B

ϕS

RjnidS (14)

where the superscript ” S ” indicates that the quantities are taken on the struc-tural mesh.


2.5. Hydrostatic pressure variations

Here we concentrate on the calculation of the hydrostatic restoring matrixwhich is obtained after the integration of the hydrostatic pressure variations dueto the body motions (3). The procedure is rather similar to the hydrodynamicpressure, and we just need to integrate the expression (3) over the structuralmesh. For the sake of clarity, let us first rewrite the hydrostatic pressure varia-tions in the following compact form:

phs =

6∑

j=1

ξjphsj (15)

where phsj = 0 except for:

phs3

= −g , phs4

= −g(Y − YG) , phs5

= g(X − XG) (16)

With these notations, the first part of the hydrostatic restoring matrix becomes:

Cpij =

∫ ∫

SS

B

phsj nidS (17)

In order to obtain the complete hydrostatic restoring matrix, one additional termaccounting for the change of coordinate system should be added to the aboveexpression as shown in equation (4). In the earth fixed coordinate system, thisadditional term is accounted for by the change of the normal vector (4). However,the structural response is calculated in the body fixed coordinate system inwhich the normal vector do not change. It can be shown that, in the body fixedcoordinate system, the change of normal vector is equivalent to the change ofthe gravity action, so that we can write:

F g = −mgΩ ∧ k = [ C ]gξ (18)

where the only non zero elements of the matrix [ C ]g

are Cg

24and C

g

15which

will be canceled by the contributions implicitly present in [ C ]p

.The total restoring matrix becomes:

[ C ]S

= [ C ]p

+ [ C ]g

(19)

where the superscript ” S ” indicates that the pressure related part was calculatedby integration over the structural mesh.

2.6. Motion equation and final loading of the structural model

The final motion equation can now be rewritten in the form:

(

−ω2([ M ] + [ A ]S

) − iω[ B ]S

+ [ C ]S)

ξS

= F DIS

(20)

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Solution of this equation gives the body motions ξS

so that the total linearpressure can be written in the form

pS

= pS

I+ p

S

D+

6∑

j=1

ξS

j (pS

Rj+ p

hs

j) (21)

In summary the final loading of the structural model will be composed of thefollowing 3 parts:

−ω2miξS

i - Inertial loading (to apply on each finite element)

pS

i - Pressure loading (to apply on wetted finite elements only)

−migΩS ∧ k - Gravity term (to apply on each finite element)

It is clear that the above structural loading will be in perfect equilibrium becausethis equilibrium is implicitly imposed by the solution of the motion equation (20)in which all different coefficients were calculated by using directly the informa-tion from the structural FEM model.

One example of calculations is shown in Figure 3..

Figure 3. Example of the linear quasi static ship structural response for 7800 TEU

container vessel.

3. Linear hydro-elastic ship structural responses

Recent trends in increasing the ship size, especially for container vessels andLNG carriers, raise the new hydro-structural issues in ship design both fromextreme loading point of view and fatigue point of view. Indeed, due to theirextreme size (almost 400 meters in legnth) these ships become much ”softer”which means that their hull natural frequencies will be significantly reduced. Atthe same time ship speed remains relatively high (around 25 knots) so that therisk of hydroelastic resonance between the waves and the structure (springing)is present. In addition, due to the large bow flare of these ships, the importance


of slamming induced vibrations (whipping) is also increased. Unlike the globalquasi static hydro structural model, where it is possible to perform the hydroand structural calculations separately, the hydroelastic model requires the fullcoupling between the hydro and structural calculations i.e. they need to besolved at the same time. Regardless of the structural model (3D or 1D), themost common models for global hydroelastic simulations are based on the socalled modal approach. Within this approach the total ship displacement isrepresented as a sum of different modal displacements:

H(x, y, z, t) =

N∑

i=1

ξi(t)hi(x, y, z) (22)

=

N∑

i=1

ξi(t)[hix(x, y, z)i + hi

y(x, y, z)j + hiz(x, y, z)k]

where the vector functions hi denote the modal functions (usually dry structuralmodes) and ξi their amplitudes. This decomposition implies the definition ofsupplementary radiation potentials with the following body boundary condition:

∂ϕRj

∂n= hjn (23)

After solving the different boundary value problems for the potentials, the corre-sponding forces are calculated and the following modal motion equation written:

−ω2([ m ] + [ A ]) − iω([ B ] + [ b ]) + ([ k ] + [ C ])

ξ = F DI (24)

where [ m ] is the structural mass, [ b ] is the structural damping, [ k ] isthe structural stiffness, [ A ] is the hydrodynamic added mass, [ B ] is thehydrodynamic damping, [ C ] is the hydrostatic restoring, ξ are the modal

amplitudes and F DI is the modal hydrodynamic excitation. The solution ofthis equation gives the modal amplitudes and the (linear) springing problem isformally solved.

The equation (23) implies the transfer of the structural modal displacementsfrom structural mesh onto hydrodynamic mesh. In the case of simplified beammodel this transfer is rather straightforward but in the case of 3D FEM modelit involves non-trivial interpolation procedure from one mesh to another. Oneexample of the modal transfer is shown in Figure 4.. Few validation results areshown in Figure 5. and 6. where the comparisons with experiments on highlyelastic barge model are shown. We can see that the agreement is fairly goodboth for vertical and torsional type of responses.

4. Time domain simulations

In order to simulate the transient ship response and/or nonlinear type ofresponses, it is necessary to work in time domain. This is usually performedusing the well known procedure proposed by Cummins [1].

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Figure 4. Structural mode shape and its transfer onto hydrodynamic mesh.

Figure 5. Experiments on the flexible barge and comparisons with numerical results

for vertical bending.

0

2

4

6

8

10

12

14

0.5 1 1.5 2 2.5

RA

O_T

[deg

/m]

T [s]

ExpeNumerical

Figure 6. Experiments on the flexible barge and comparisons with numerical results

for torsion.


Let us first write the time domain equivalent of the motion equation (24):

([ m ]+[ A∞ ])ξ(t)+([ k ]+[ C ])ξ(t)+∫ t

0

[ K(t−τ) ]ξ(τ)dτ = F (t)+Q(t)(25)

where overdots denote the time derivatives and:

[ A∞ ] - infinite frequency added mass matrix

[ K(t) ] - matrix of impulse response functions

It was shown in [1], that the impulse response functions can be calculated fromthe frequency dependent damping coefficients:

Kij(t) =2

π

∫

∞

0

Bij(ω) cos ωt dω (26)

The advantage of the time domain method lies mainly in the fact that we canintroduce the non-linear components in the excitation forces F (t) and Q(t).Once the impulse response functions Kij have been calculated (by numericalintegration), the motion equation (25) is integrated in time using the RungeKutta 4th order scheme.

Figure 7. Initial deformation of the elastic barge and time history of the vertical

displacement of the first pontoon.

It is important to note that this approach is valid both for quasi static shipbehavior as well as for the hydroelastic one. Indeed, the only difference is thenumber of degrees of freedom which is increased in the hydroelastic model.

One example of calculations using this approach is shown in Figure 7.. Itconcerns a kind of decay test for elastic barge from Figure 5.. The decay testconsisted in pulling out of the water the first pontoon and then releasing it. Aswe can see the agreement between numerical and experimental results is verygood.

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5. Weakly non linear Froude Krylov model

As mentioned above, once in the time domain, it is possible to add any typeof non-linear loadings into the motion equation. The simpliest weakly non linearwave loading concerns the so called Froude Krylov model which we briefly explainhere below. This model is relevant in practice, both for the local fatigue loadingof the side shell longitudinals close to the waterline, and for the evaluation ofthe non-linear internal loads in sagging and hogging conditions.

According to the linear theory, the hydrodynamic model ”stops” at the wa-terline (z = 0) so that locally (close to the waterline), negative hydrodynamicpressure may occur. The Froude Krylov model is rather intuitive and consists inadding the hydrostatic part of pressure (−gz) below the wave crest (in linearsense) and by putting zero total pressure above the wave trough. The problemreduces to the evaluation of the (linear) wetted part of the ship.

5.1. Linear pressure and relative wave elevation

According to the linear theory, the linear hydrodynamic pressure, at anypoint below z = 0, is composed of two parts:

• pure hydrodynamic part associated with the time derivative of the velocitypotential ϕ

• hydrostatic variation due to the ship motions

We write for the wave of unit amplitude:

p(x, y, z, t) = ℜiωϕ(x, y, z)e−iωt + ℜ−gζv(x, y, z)e−iωt (27)

where ϕ is the velocity potential and ζv is the vertical displacement due to theship motions. Both quantities are complex and represent the standard output ofthe, above mentioned, linear frequency domain seakeeping codes. Furthermorethe potential can be decomposed as shown by (1) and, on the other side, thevertical displacement is equal to:

ζv(x, y, z) = ξ3 + ξ4(y − YG) − ξ5(x − XG) (28)

In order to simplify the notations, it is useful to express the pressure in metersof water height. We write:

p(x, y, z, t) =Ap(x, y, z, t)

g= ℜ iωA

gϕ(x, y, z)e−iωt + ℜ−Aζv(x, y, z)e−iωt

(29)where A is the incident wave amplitude A = H/2.In this way the relative wave elevation (usually denoted by ηR) is equal to thepressure p at z = 0:

ηR(x, y, t) = p(x, y, 0, t) (30)


Let us finally adopt the following notations:

p(x, y, z, t) = pz (31)

so that we can write:

ηR(x, y, t) = p0 (32)

5.2. Modified pressure distribution near the waterline

As mentioned before, the Froude-Krylov correction means that we will addthe hydrostatic pressure −gz for the points below the wave crest, and we willput zero for the points above the wave surface. To put the things into a moreclear perspective we consider separately two cases p0 > 0 and p0 < 0. Note thatthe discussion below concerns arbitrarily time instants, not necessarily the oneswhere the p0 reach its maximum or minimum value.

5.2.1. Positive relative wave elevation p0 > 0

Wave profile

Total pressure (>0)

z=0 z=0

Hydrostatic pressure (>0)

Hydrodynamic pressure (>0)

Ξ (>0)

Figure 8. Pressure distribution for p0 > 0.

In this case we can identify two wetted regions with corresponding total pressuredistributions:

P (z, t) =

p0 − z for 0 < z < Ξ

pz − z for z < 0(33)

The choice of the maximal wetted point Ξ is quite natural:

Ξ = p0 (34)

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5.2.2. Negative relative wave elevation p0 < 0

In this case the situation is more complicated and we have to be careful.As far as the pressure distribution is concerned, the situation is similar to theprevious case, and the choice is quite natural (even if, strictly speaking, not fullyconsistent):

P (z, t) =

0 for 0 > z > Ξ

pz − z for z < Ξ(35)

The main problem appears to be the choice of the minimal wetted point Ξ. The

z=0

Wave profile


Hydrodynamic pressure (<0)

Ξ (<0)

Total pressure (>0)

z=0

Figure 9. Pressure distribution for p0 < 0.

usual choice for Ξ is the same as in the previous case:

Ξ = p0 (36)

The problem is that this choice introduce the jump in the pressure distributionbecause:

p0 6= pz , for z = Ξ (37)

which means that the pressure at the point imediatly below the wave profile willnot be zero but:

P (Ξ, t) = pΞ − Ξ = pΞ − p0 (38)

The reason for this is the inconsistency of the adopted approach which do notinclude all non-linear terms, so that all necessary conditions can not be satisfiedat the same time. We have to ”sacrify” something.Let us now choose the value of Ξ as a solution of the following equation:

Ξ = pΞ (39)


Note that an iterative procedure should be used to obtain the solution of thisequation.

It is easy to see that this choice eliminates the jump in the pressure dis-tribution. The disadvantage is that we have to solve the equation (39). The”physical” consequence of the above choice is that the wave profile is changedfrom the assumed one (ηR). However, this slight move of the free surface is withinthe order of the approximations we adopted at the begining. Strictly speakingboth choices are not fully consistent but the second one has the advantage toallow for more physical pressure distribution.

5.3. Pressure range

In the spectral fatigue analysis we need to know the pressure range. In orderto obtain this range, we have to consider the time instants where the relativewave elevation reachs its maximum and minimum values. The figure 10. isparticularily useful. The range of pressure variation is defined by the differencebetween the maximum an minimum pressure value. In both cases we can identify3 different wetted zones with the associated pressure ranges:

∆P (z, t) =

|p0 | − z for 0 < z < Ξmax

|pz | − z for Ξmin < z < 0

2|pz | for z < Ξmin

(40)

Ξ min

z=0

Ξ max

Minimum total pressure (>0)

Maximum total pressure (>0)

Hydrostatic pressure (>0)RANGE

z=0

Ξ max

Ξ min

Minimum total pressure (>0)

Maximum total pressure (>0)


RANGE

Ξmax = |p0 | , Ξmin = −|p0 | Ξmax = |p0 | , Ξmin = −|pΞ |

Figure 10. Maximum and minimum pressure envelope.

In the first case (Ξmin = −|p0 |) we can clearly see how the jump in the pressure

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distribution affects the pressure envelope and subsequently the range.Let us finally mention another possibility for evaluation of the pressure range.It is in fact the slightly modified first method, because the value of Ξmin is thesame Ξmin = −|p0 |, but the exponential decay of the pressure with depth isassumed. The final expression for the pressure range is:

∆P (z, t) =

|p0 | − z for 0 < z < Ξmax

|p0 |eνz − z for Ξmin < z < 0

2|p0 |eνz for z < Ξmin

(41)

where ν is the wave number ν = ω2/g.

z=0

Ξ max

Ξ max

Ξ min Ξ min2

Figure 11. Pressure range when Ξmin = −|pΞ |.

It is clear that the choice of the decay function eνz is arbitrary and does notcorrespond to the physical situation. Indeed, the total hydrodynamic pressure iscomposed, not only of the incident wave pressure, but also contains the diffractedand radiated parts as well as the hydrostatic variation. This means that thepressure jump will still remain.

With respect to all the discussions, the use of the second method i.e. Ξmin =−|pΞ |, is recommended. Indeed it appears to be most consistent because itdoesn’t introduce the pressure jump, and hydrodynamic pressure, below thewaterline, decays according to the results of the liner seakeeping calculations.On figure 11. we show the pressure range as a function of the vertical coordinatefor this case.


5.4. Numerical implementation

Once the pressure distribution determined, we have possibilities to work ei-ther in frequency or in time domain.

5.4.1. Time domain

In this case the things are quite clear because we know exactly the pressuredistribution at each time instant. We can ”charge” the structural model quasi-statically at each time and obtain the time signal of the stresses which canbe postprocessed for fatigue analysis. One example of the modified pressuredistribution is shown in Figure 12..

Figure 12. Pressure distribution on the wetted part of the ship at an particular time

instant.

As mentioned before, it is important to note that the Froude-Krylov modelis also relevant for the evaluation of the internal loads in hogging and saggingconditions. Indeed, according to the linear theory these loads remains the samewhich of course is not tru in reality. These informations about the interal loads insagging and hogging conditions, are essential for determination of the ship designloads. One example of the results is shown in Figure 13. where the sagging andhogging vertical shear forces and bending moments are presented.

5.4.2. Frequency domain

If we want to perform spectral fatigue analysis in frequency domain, weshould first linearize the local nonlinear pressure model. We note that thislinearization is implicitely done when the range was defined. Indeed, the halfof the pressure range defines the amplitude of pressure variations in frequency

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Figure 13. Internal loads in sagging and hogging conditions.

domain. A mean value can also be defined:

Pmean(z, t) =

1

2|p0 | − z for 0 < z < Ξmax

1

2|pz | − z for Ξmin < z < 0

−z for z < Ξmin

(42)

All the expressions above are valid for the incident wave of finite amplitude A.For some purposes, it is sometimes useful to define a kind of non-linear RAO.In that case we should divide the above values by the wave amplitude i.e. halfof the wave height.

6. Slamming & whipping

Slamming is a very important source of ship structural loading both fromlocal and global points of view. Indeed very high localized pressures appearduring the slamming, but also the corresponding overall forces are very high.This means that not only the local ship structure will be affected by slamming,but whole ship will feel the slamming loading through the so called whippingphenomena. Whipping is defined as the transitory global ship vibrations due toslamming. One exampe of the whipping response is shown in Figure 14., whereit can be clearly seen that the quasi static internal wave loads are significantlyincreased by the high frequency whipping vibrations. The hydrodynamic mod-eling of slamming is extremely complex and still no fully satisfactory slammingmodel exists. However, the 2D modeling of slamming is well mastered todayand 2D models are usually employed to assess the slamming loads on ships.Within the potential flow approach, which is of concern here, several more orless complicated 2D slamming models exist, starting from simple von-Karmanmodel and ending by the fully nonlinear model. In between these two modelsthere are several intermediate ones such as Generalized Wagner Model (GWM)and Modified Logvinovich Model (MLM) which will be discussed here.


Figure 14. Whipping.

6.1. Determinaton of the critical impact conditions

The slamming calculations are usually decoupled from seakeeping calcula-tions. Simply speaking, it is assumed that slamming does not influence the shipmotions. This seems to be a reasonable assumption for most of the typical ships.In order to clarify the determination of the impact conditions, we show in Fig-

Figure 15. Determination of impact conditions.

ure 15. the typical impact situation which is likely to occur for a ship advancingin waves with forward speed U . As far as the linear theory is considered, thevelocity of the point P attached to the ship, might be written as:

vB = ξ + Ω ∧ (RP − RG) − U(Ω ∧ i) (43)

where ξ denotes the translational velocity of the ship’s center of gravity RG,the Ω is the corresponding rotational velocity, RP is the position vector of thepoint P and i denotes the unit vector in x direction. On the other hand, thefluid velocity at the same point P can be written in the form:

vf = ∇ϕ + U∇(φ − x) (44)

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where ϕ denotes the unsteady velocity potential and φ the steady velocity po-tential. It is important to note that, within the linear approach, the velocityfield can not be calculated above the mean free surface z = 0, so that the fluidvelocity is calculated at the point (XP , YP , 0).The relative velocity at point P is the difference between the body velocity andfluid velocity:

vR = vB − vf (45)

Within the 2D approach the ship structure is subdivided into several strips inthe direction defined by the unit vector s (see Fig. 15.). This means that theabove defined relative velocity should be projected onto the strip direction:

vR = (vB − vf ) · s (46)

Moreover, we also need to calculate the relative motion of the point P , in order toidentify the time instant when slamming occurs. Similar to the relative velocity,the relative motion between the point P and the free surface, can be evaluatedusing the following formula:

ηR = ZP + ζv − η (47)

where ZP is the initial position of point P with respect to the initial free surface(z = 0), the ζv denotes the vertical displacement of the point P and η denotesthe wave elevation at (XP , YP , 0):

ζv = [ξ + Ω ∧ (RP − RG)] · k (48)

η = −1

g[∂ϕ

∂t− U(∇φ − x) · ∇ϕ] (49)

With these notations, the impact will occur when the relative motion ηR changethe sign from positive to negative. Usually one additional condition is alsointroduced in order to eliminate the cases with very low relative velocity. Thiscondition, usually known as Ochi condition, states:

vR ≤ −0.093√

gL (50)

where g denotes the gravity acceleration and L is the characteristic ship length(usually length between perpendiculars).

The main goal of slamming analysis is the determination of the design loadsfor a given ship and given operational profile (loading conditions, sea state, speedand heading). The sea state is usually defined by the scatter diagrams whichgive the probability of occurrence for different combinations of wave heights andwave periods. In order to identify the design (worst) sea state with respectto slamming loads, first we need to perform the spectral analysis of relativevelocities and relative motions for all sea states contained in the (design) scatterdiagram. One typical result of these calculations is shown in Figure 16.. Fromsuch a figure, we can deduce the critical operational conditions (combination ofthe sea state, loading condition, speed and heading) when the maximum relative


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Figure 16. Maximum relative motions and velocities for different sea states and time

history of relative motion and velocity for a given sea state.

velocities occur. Once the critical operating conditions has been identified, thetime domain simulations are performed for this particular sea state and exactimpact conditions are obtained by using the expressions described in the previoussection. One typical time signal of the relative motion and relative velocity isshown in Figure 16.. From this diagram we can easily deduce the slammingoccurrences and corresponding slamming velocities.

Within the simplified models of slamming the original problem is reduced tothat of a body entering the calm water surface at a prescribed velocity. Thesimplest model is the so called von-Karman model which is followed by the well-known Wagner model. The main difference between these two models is thedetermination of the wetted part c(t) of the body, as shown in Figure 17.. Thevon-Karman model assumes the wetted part as the intersection of the enteringbody with the initial free surface, while in the Wagner model the wetted partof the body is unknown in advance and is determined as a part of the solution.Within the original formulations of these models the boundary conditions onboth the body surface and the free surface are linearized and imposed on theinitial position of the liquid boundary before the impact. The experience showsthat the Wagner model well predicts the wetted part of the entering body andthe liquid flow induced by the impact, however, it overpredicts the hydrodynamicpressures acting on the body and the total hydrodynamic force. The model byvon Karman underpredicts the loads in 2D case but provides rather reasonableresults in 3D calculations, as it will be shown later.

The Generalized Wagner Model, represents an improvement of the Wagnermodel in the sense that the body boundary condition is satisfied on the exactbody surface (Figure 18.), the condition on the free surface is the same as inthe Wagner model but it is imposed on the horizontal lines at the splash-upheight. The splash-up height is unknown in advance and is determined with thehelp of the linearized kinematic condition and the ”Wagner condition”, whichstates that the elevation of the free surface is equal to the vertical coordinateof the body surface at the contact point. Within the GWM the pressures arecalculated by using the nonlinear Bernoulli equation.

The main idea of the MLM is that the velocity potential, which is numeri-cally calculated within the GWM, can be approximated by the Wagner solution

22 S. Malenica

Figure 17. Von-Karman and Wagner model.

with account for the body shape up to the second order with respect to the pen-etration depth. MLM is an approximate second order model, within of whichthe body boundary condition is satisfied at z = 0 (Figure 18.) but in calcula-tions of the hydrodynamic loads the body shape is taken into account. In MLM,the quadratic term in the Bernoulli equation is retained but negative pressuresare disregarded in evaluation of the total force acting on the body. The latterfeature of the MLM is the same as in the GWM. The main advantage of theMLM, when compared to GWM, lies in its simplicity and incomparably lowerCPU time requirements. The disadvantage lies in the fact that the pressure isnot provided with sufficient accuracy within MLM model. However, predictionsof the total hydrodynamic force by the MLM is surprisingly good compared withboth more accurate numerical results and experimental data.

Figure 18. Generalized Wagner model (left) and MLM model (right).


Figure 19. Geometrical definition of different sections for 2D slamming calculations.

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Figure 20. Slamming forces for different sections.

As far as the 3D slamming methods are concerned, the Wagner type methodsare less developed, the main reason being the difficulty in determination of thewetted part of the body. That is why the so called 3D Generalized von-KarmanModel (GvKM) becames interesting. Within this model the body boundary con-ditions are satisfied on the actual body surface but the free surface boundaryconditions are linearized and imposed on the initial position of the liquid bound-ary. The wetted part of the body is known at each time step and efficient 3Dpanel methods can be applied to solve the associated boundary value problemswith respect to the velocity potential. A main problem now is the determina-tion of the mesh at each time step (Figure 21.) but this can be done with ratherrobust algorithms.

Now we present some results of comparisons between the GWM and MLMmethods for 2D slamming. For that we chose one typical example from practice,

24 S. Malenica

Figure 21. Remeshing of the ship hull at different draughts.

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Figure 22. Vertical force on a ship bow part using the 3D Generalized von-Karman

method.

where the slamming of 7800 TEU container vessel is considered. In the generalcase the ship hull is subdivided into 20 to 30 sectons, but here we chose the 4 mosttypical ones (Figure 19.). Sea state is defined by Pierson Moscowitz spectrumwith Hs = 12m and Tp = 17s. In Figure 20. the results for the sectional forcesare presented and, as we can see the two methods agree fairly well with eachother in spite of quite strong assumptions of the MLM method. This is veryimportant point because this means that the MLM method can be used quitesafely for calculations of the 2D slamming forces and due to its very low CPUtime requirements, as compared to GWM, the long time slamming/whippingsimulations can be performed within reasonable CPU time. In addition theMLM method is much more stable and robust than the GWM method.

Finally we present few validation results for 3D slamming calculations usingthe GvKM method. The passanger ship from Figure 21. is considered. The


experiments were carried out for forced water entry of the fore ship part and theexperimental results for the vertical slamming force are presented in Figure 22.together with the numerical results obtained with the GvKM. As we can see theagreement is surprisingly good.

7. Conclusions

We have presented here, what we beleive to be the most important aspectsof the hydro-structure interactions for the problem of ships sailing in waves.The difficulties associated with this problem and the recent progress in theirmodelling were discussed. It is clear from the discussions that we are still farfrom the final solution of the overall problem, but the significant progress inunderstanding and modelling of the different aspects was made in the last decade.The most critical points which require further investigations concern:

• seakeeping at forward speed• non linear hydroelastic interactions• 3D slamming• sloshing• water exit• local hydroelastic impacts• ...

It is important to mention that only the potential flow models were discussed inthis paper. This is because these models allow for relatively fast calculations interms of modelling and CPU time, and they are most commonly used in practice.However, these models includes several more or less realistic simplifications whichshould not be forgotten. The other class of methods (RANSE, VOF, SPH,...) based on the solution of the Navier Stokes or Euler equations were notcommented in details. This, of course, do not means that these methods shouldbe excluded in the future. Indeed, these methods have several advantages namelyin terms of flexibility for inclusion of several important physical aspects at thesame time (viscosity, flow separation, waves, wind, ...). However, their limiationsin terms of CPU time requirements, user friendliness, and some purely numericalproblems (meshing, choice of time step, convergence, ...) seems to be the reasonfor their relative limited use in seakeeping, except for some particular problemslike wave resistance, sloshing etc. In spite of these limitations, it is however likelythat these methods will gain more importance in the future especially because ofthe computers improvements which lead to extremely fast calculations. Anyway,and at least in the near future, the most reasonable way of proceeding wouldprobably be in using the hybrid methods which will combine the advantages oftwo approaches.

References

[1] W.E. Cummins. The impulse response function and ship motions. Schiffstecknik,1962.

26 S. Malenica

[2] R.E.D. Bishop and W.G. Price. Hydroelasticity of ships. Cambridge UniversityPress, 1979.

[3] O.M. Faltinsen. Sea loads on ships and off shore structures. Cambridge UniversityPress, 1990.

[4] A.A. Korobkin. Analytical models of water impact. Euro Journal of Applied Math-

emaics, 2005.

[5] B. Molin. Hydrodynamique des structures off-shore. Technip, 2002.

[6] S. Malenica B. Molin, F. Remy and I. Senjanovic. Hydroelastic response of abarge to impulsive and non impulsive wave loads. 3rd International Conference on

Hydroelasticity, 2003.

[7] J.N. Newman. Ship hydrodynamics. MIT Press, 1977.

[8] J.N. Newman. Wave effects on deformable bodies. Applied Ocean Research, 1994.

[9] R. Zhao O.M. Faltinsen and J.V. Aarsnes. Water entry of arbitrary two dimensionalsection with and without flow separation. 21st Symposium on Naval Hydrodynam-

ics, 1996.

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