Three-Dimensionnal Hydroelastic Water Entry: Preliminary Results

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Three-Dimensionnal Hydroelastic Water Entry:

Preliminary Results

Bundy Donguy, Bernard Peseux, Laurent Gornet, Emmanuel Fontaine

To cite this version:

Three-Dimensional Hydroelastic Water Entry: Preliminary Results

B.

Donguy,

B.

Peseux and

L.

Gornet

Ecole Centrale de Nantes

Nantes, France

E.

Fontaine

Institut Fran�ais du Petrole Rueii-Malmaison, France

ABSTRACT

Slamming loads during the hydrodynamic impact of a three· dimensional body are studied numerically. First, the finite element method is used to solved the fluid part. For the water entry problem of a rigid body, the numerical results are successfully compared with analytical solutions for wedge, cone and the three­ dimensional case of an elliptic paraboloid shaped bodies. In the second part of the paper, the coupled formulation and the coupling matrix between fluid and structure part which will be used to treat the hydroelasticity impact problem is presented and validated by solving for sloshing in a tank with an elastic wall problem.

KEY WORDS: slamming, hydroelasticity, asymptotic, three­ dimensional.

INTRODUCTION

In severe sea conditions, impact loads with high pressure occur when the hull of a ship strikes the water surface. These impulsive loads may generate plastic deformations of the local hull structure. Fractures have also been observed as the result of severe slamming events. In extreme cases, the integrity of the overall structure may be threatened due to a large increase of the global bending stresses. The ability to better predict the structural response of the ship hull to slamming loads, both locally and globally, appears therefore necessary.

Reviews on the subject of slamming have been proposed by Korobkin & Pukhnachov

(1988),

Mizoguchi & Tanizawa

(1996)

and more recently by Faltinsen (2000) who focus attention on the influence of hydroelasticity effects. From a theoretical point of view, slamming loads have been mostly studied within the framework of potential flow theory, assuming blunt and rigid body, and planar flow. Under these assumptions, first order asymptotic solutions were found for the case of a wedge with small deadrise angles (Wagner,

_1932),

a cylinder (Cointe,

_1987),

and more generally, for arbitrary two-dimensional blunt body shapes (Cointe,

_1989,

Howison, Ockendon & Wilson,

1991).

The latter also consider arbitrary axi-symmetric bodies for which the solution is very similar. These classical two-dimensional asymptotic results were successfully validated against experiments in the cases of wedges (Chuang,

_1967,

Fontaine & Cointe,

1997),

cylinders (Cointe & Armand,

1987),

and realistic ship cross-sections (Zhao et al,

_1996,

Magee & Fontaine,

1998).

Good agreement is generally reported although the asymptotic solution tends to over predict the maximun of the peak pressure. Comparisons require sometimes to take into account three-dimensional side effects. The first order asymptotic solution was also compared with fully non-linear simulations using Boundary Element Method together with a cutting algorithm for the jet (Zhao & Faltinsen,

1992,

Fontaine & Cointe,

1997,

Iafrati & Korobkin,

2000).

For a wedge, the comparisons show that the pressure distribution is correctly predicted as the deadrise angle goes to zero. For larger values of the deadrise angle, the agreement remains very good if second order effects are accounted for in the asymptotic pressure distribution. For constant vertical impact velocity, most of the second order effects are accounted for when the matching technique proposed by Cointe (

₁₉₈₇₎

is applied.

From a practical point of view, the two-dimensional asymptotic solutions for the flow around rigid bodies are classically used to represent the three-dimensional flow within the framework of a strip theory. Nevertheless, the use of a strip theory assumes the ship hull to

be

slender near the impact point, which is not always the case, thus the need for three-dimensional solutions. Some problems also arise when strip theory is applied to study locally the hydro-elastic effects associated with severe impact events. Indeed, the structural deformations of the hull near the impact point exhibit a strong three­ dimensional character (see e.g. Donguy, Peseux, Fontaine,

2000).

Unlike the two-dimensional problem for which analytical solutions can

In the present paper, the asymptotic equations describing the fluid

flow during the first time of impact are first presented together with the

physical assumptions sustaining the modelling. Within the framework

of this analysis, the problem is solved once the velocity potential on the

body and the contact line between the body and the free surface

are

known. The numerical procedure used to solve the problem is then

described. The velocity and displacement potentials

are

computed

based on a variational formulation of the problem together with the

Finite Element Method. The displacement potential is introduced for

iterative resolution of the contact line. Once the wetted area is known,

pressure distribution is then computed using velocity potential. The

accuracy of the numerical scheme is tested through systematic

comparisons between numerical and analytical results. Classical rigid

body solutions are recovered for the two-dimensional case of a wedge,

the axi-symetric case of a cone, and the three-dimensional case of an

elliptic paraboloid, therefore validating the approach for the rigid body

case. The second part of the paper is devoted to hydroelasticity. The

equations describing the structural deformations are included in the

analysis and a coupled approach is proposed to solve the resulting fluid

- structure interaction problem. Numerical simulations for the impact

of deformable bodies impact will be presented �in subsequent

publications. In the present study, attention is focussed on the

validation of the coupling terms and of the resolution procedure. More

specifically, the numerical scheme is successfully applied to determine

the high frequency resonant frequencies of a sl()shing tank with an

elastic wall. For this test case problem, the numerical results by

Peseux, Cartraud and Argouar'ch

(1999) are successfully recovered.

THREE DIMENSIONAL RIGIG BODY WATER ENTRY

Fluid flow modelling

The water entry problem is classically formulated within the

framework of potential flow theory. The fluid is assumed to be ideal

and incompressible, and the flow to be irrotational. Under these

assumption, the velocity field can be deduced from a velocity potential

<{!=1/l(x,y,z,t).

according to

v = grlid<{! •

z

free surface

y

(ilt)

Figure

I.

Geometrical definitions

The velocity potential must satisfies the following boundary value

problem:

il<{!=O

in the fluid

_(I)

iJ<{! .:.

-on the body

(2)

-=u n iJn ,.

iJ<{! +2.(grad<f!)2+gz=O _dt

₂

on the free surface

(3)

dh d<{!

on the free surface

(4)

-=-dt -=-dt

where the displacement of the body is given by its

iijx,y,t)

and the

free surface elevation is written as

z = h(x,y,t).

The body boundary

condition, Eq.

2,

expresses the continuity of the normal velocity on the

body surface. The kinematic and dynamic free surface conditions, Eq.

3 and Eq. 4 respectively, state that the free surface is a material surface

along which the pressure is constant. Moreover, the fluid is assumed to

be initially at rest:

h(x,y,O) =

0

1/l(x,y.z.O) =

0

and the flow to be unperturbed far from the body

:

lgradtfll-+

0

when

(K+l+iJm � oo

(5)

(6)

Once the velocity potential is known, the pressure on the body is

calculated by Bernoulli's equation:

p _{iJ<{! I} - ₂

-= ----(grad<{!) -gz ₍₇₎ p dt

2

Asymptotic formulation

The above mentioned boundary value problem is not easily

tractable, even numerically, due to the violent deformations of the free

surface. The latters are experienced as jets developing along the side of

the body (see e.g. Greenhow,

1987). Classically, the hydrodynamic

impact problem

(1)-(4) is simplified into:

il<{!=O d<{! .:. -iJn = u,.n <{!=0 iJh = iJ<{! dt iJz

inD1

(8)

on rs

(9)

onrL

(10)

onrL

(11)

Where

rs

is the projection on

Z =

0 Of the Wetted body SUrface and rL

the position of the unperturbed free surface. q. represent the linearized

fluid domain (see fig.

1). The simplified equations can be fomally

derived using perturbation technique (see Cointe,

1989, Wilson, 1989).

The perturbation procedure relies strongly on the blunt body

assumption since the small parameter used in the asymptotic expansion

is the ratio between the immersion and characteristic length scale of the

body wetted width. Within a far field point of view, the body boundary

condition, Eq.

2,

can be written on

z =

0 without introducing

significant error, therefore justifying Eq.

9. The exact dynamic

condition, Eq.

3, has been replaced by a Dirichlet condition for the

potential on the undisturbed position of the free surface, Eq.

I

0.

Physically the acceleration in the fluid is assumed to be large compared

to gravity which can be neglected during the first instants. Finally, the

simplified kinematical free surface condition, Eq.

!I,

states that the

vertical displacement of the free surface is equal to the fluid vertical

motion, evaluated on the undisturbed position of the free surface. The

quadratic terms have to remain small compared to the linear ones for

this approximation to be valid. The resulting problem, Eqs.

8-11 is

often referred as the generalised Wagner

(1932) problem. In the

present formulation, three-dimensional effects are retained through the

Laplace's equation, and structural deformations of the body surface are

taken into account in the body boundary condition. In this section the

body is assumed to be rigid. Structural equations driving the

displacement ii, and hydroelastic effects will be presented in the

second part of the paper.

far-field solution is matched near the intersection to a local solution describing the formation of a jet following Wagner (1932), leading to a composite solution which is regular by construction. Since the

jet

solution is completely defined by the far-field solution, attention is focused on the latter.

Determination of the wetted surface

Due to the deformations of the free surface, the wetted surface r8 is part of the unknown. An additional equation is therefore needed to close the problem. Physically, the contact line is determined by imposing the existence of an intersection point between the outer expansion of the free surface elevation and the body:

h(x.y,t) =

u,.z

o2>

This condition, intuitively introduced by Wagner (1932), also states volume conservation to the leading order (Wilson, 1989, Fontaine & Cointe, 1997). In order to satisfy this condition, the asymptotic problem is formulated for the displacement potential:

'l'(x,y,z;t)= .(1/>(x,y,z;s)ds (13)

as suggested by Korobkin (1982). The main advantage of introducing this transformation is that solving for

'I'

does not require the knowledge of the temporal evolution of the free surface elevation. The boundary value problem satisfied by 'I' is similar to the one satisfied by 'except for the body boundary condition which reads:

d'l' -

-a;=u,.n

on the body (14)

The problem for 'I' is solved through an iterative procedure until Eq.

12

is satisfied. A precise description of the method is given in Donguy et al.

(2000).

Numerical resolution

The resolution procedure for the velocity and displacement potentials is based on the Finite Element Method. The weighted residue method is applied considering the weighting functions q> which verify the boundary condition q> =

0

on rL . The problem comes to

minimize the integral quantity:

W(l/>) =

_Jn,

C

q>R(tfJ)dD

₍₁₅₎

where the residue R(l/>) is set equal to t1,. Then, applying Green's identity and taking into account of boundary conditions, Eq.

16,

leads to the weak formulation:

r gradrp.griidlf>.

dD =

r

rp(ii,.n).

dS

�I

�

(16)

The potential 1/> and weight Galerkin's method leading to:

functions q> are approximated using

q> _{= <Nt> {}

_rp}'

_{1/J= <N1>{1/>J'} (17)

where the shape functions N1 depend on the types of elements, and { rp}' denotes the nodal potential vector of the finite element (e). Finally, the following linear system is deduced from the discretization of the bilinear forms, Eq.

16:

[Hkt/Jl=

{G•}

Hij

=

Jn•Ni,j•Nj,;dD

ro,f

=

Ir�

{N

f}(iis.n.klS

(19) (20) (21)

and for the displacement potential:

Validation

[H Yl/1} =fq,}

(G"

}e ₌

C

• { N 1

}(ii,.ii.klS

Jra

(14)

(15)

The numerical method previously described is applied to solve several test case problems of rigid bodies penetrating with constant velocity a free surface initially at rest.

The two-dimensional test case of a wedge is presented in Fig.

2

and

3 where the free surface elevation and the pressure distribution are plotted, respectively. Despite a relatively coarse mesh based on triangular and quadrilateral elements is used, the agreement between the numerical and analytical solution remains excellent, even close to the intersection point were the solution is singular since the pressure and the free surface slope tends to infinity. The singularity of the pressure distribution is nevertheless correctly captured as can be seen in Fig.

3

where only the far-field solution has been represented. One should keep in mind that the singularity is to be removed by matching the far-field solution to a jet solution near the intersection.

The axisymetric case of a cone is presented in Fig.

4

and 5.

Although the body shape is axi-symmetric, the computation is performed on a three-dimensional mesh based on hexahedron types of elements. This allows to check independently the three-dimensional resolution of the Laplace's equation and the iterative procedure for solving the wetted area. Good agreement between numerical and analytical results is obtained again. All parameters being similar to the previously described two-dimensional computation, the relative error remains of the same order, despite the mesh is this time really three­ dimensional.

0.1

0.05

0

0.5

y(m} analytical -­ numerical • • •

1.5

Figure

2:

Wedge shaped body. Comparison between numerical and analytical solutions for the outer free surface elevation.

0.04

0.02

·0.02

0.2

0.3

0.4

analytical -­ numerical • • •

" ...

... .._

..

_

....______

0.5

x(m)

0.6

0.7

Figure

4:

Cone shaped body. Comparison between numerical and analytical solutions for the outer free surface elevation.

0.008

0.004

·0.004

·0.008

0

A

!*

I

analytical

--I

I

numerical • • •

;\_

i

..

---�-��

I

I

I

I

j/

0.1

0.2

_x(m)

0.3

0.4

0.5

Figure

6:

Elliptic paraboloid shaped body. Comparison between numerical and analytical solutions for the outer free surface elevation.

18

16

35

25

l

20 .. _j:l £::

15 ...

Figure

3:

Wedge shaped body. Comparison between numerical and analytical for the outer pressure distribution. The error distribution is plotted on the right.

20r---

--

---

---·

40

14

6

4

analytical numerical • • •

2�-.����--�

o����wa������ru0

Figure

5:

Cone shaped body. Comparison between numerical and analytical for the outer pressure distribution. The error distribution is plotted on the right.

1200

�

analytical

1100 Lj

numerical

1000

900

800

"!:...

700

�600

""

;g

500

l:i..400

.I

I

300

200

100t---�.-!�,._.l

...

-�

0

_./

"100o

_{0.2 0.4 06}

y(m)

0.8

...

50

45

40

35

30�

--25 ...

_e

20

�

15

10

5

1.�

3.2

2.4 · 1.6

0.6

Figure

8:

Elliptic paraboloid shaped body. analytical --­

numerical • • •

0.02 0.04 0.06

t(ms) 0.08 0.1

Figure

9:

Elliptic paraboloid shaped body. Comparison between numerical and analytical solutions for the wetted length along the two main axis. The blue line represents the penetration depth.

HYDROELASTICITY

In this section, the preyious analysis is extended to treat the case of a deformable body. The equations governing the structural deformations are first recalled for the sake of completeness, together with the boundary conditions. A coupled method to solve the fluid­ structure interaction problem is then proposed and validated on an example.

Governing structural equations

The usual assumptions of small perturbations is made. The equilibrium equation reads:

:: - -

-pu,

=divdi.+

f

inn,

(22)

where

I

denotes the stress tensor,

ii,

corresponds to the structural displacement, and f corresponds to volume force associated with the gravity field. If gravity effects do not contribute to the local deformations here, they nevertheless influence the global motion of the body before impact. Boundary condition on the structure is:

onrs

(23)

where the subscript f and s refer to the fluid and the structure,

respectively.

The structural modeling is performed within the framework of linear elasticity. Hooke's law is used to express the relation between the

stresses and the strains which are given by the linearised Green-

₌

Lagrange tensor e :

e. '· 1

= .!.Ju.

2 � '· 1

+u1

·' .

)

(24)

Coupled problem

The structural problem is also solved using the Finite Element Method. The principle of virtual works is applied to Eq.

22

together with boundary condition, Eq.

23,

leading classically to the discretised form:

(M, JP}+ (K ]{u }= {F}

{F

_{f =}

_{fr• (N.J {n}

}pdS

8

(25)

(26)

where the mass and stiffness matrices, respectively [JW:,] and [K], together with the generalized load vector {F), are obtained by assembly of the elementary terms calculated on each finite element (e). The nodal displacements {U} is linked to the interpolation of the displacement field through:

�,

}=

_[N,

]{u}

(27)

By definition, hydroelastic effects arise as the result of a strong interaction between the fluid flow and the structural response. From Eq.

26,

it can be seen that the pressure imposed by the fluid will drive the structural deformations. On the other hand, the structural deformations influence the pressure field through the body boundary condition as shown by Eq.

2.

In order to obtain a robust method to solve the coupled fluid - structure interaction problem, all the evolution equations are integrated simultaneously. The structural deformations are included in the fluid flow discrete formulation, while the pressure is expressed as the time derivative of the velocity potential in the discrete structural analysis. The following problem is then obtained:

[H ){if>}= (FS

_JP}

[MJp}+(K){U}=-p1(FSf �}

[FS]' =

_{fr,{N1J{nY[N,}ls}

8

(28)

(29)

(30) where the coupling terms

[FS]

representing the fluid - structure interaction appear explicitly. Eq.

30

shows that the coupling term involve only the values of the structural displacement

ii,

and the velocity potential � on the wetted surface of the body, r8• For the numerical resolution, Eqs.

28

and

29

are rewritten using with a new nodal vector

{ W}

r = { � ,

U}

r , leading to:

where:

[M]=

[

[o] [o]

_{[o] [M ,]j}

1

(M� }+(B� }+[K){W}=O

(31)

[

[ol -[Fsn

[B)= Pr[Fsf

[o)

j (K)=

[

(H)

[o] [K.]j

(o)l

The fluid - structure interaction problem, Eq.

31,

is then integrated starting from the initial conditions:

W(t=0)=W0

Sloshing in a tank with an elastic wall

The numerical resolution of Eq . 31 is performed using commercial Finite Element Method numerical code Castem. An external procedure is implemented for the evaluation of the coupling matrix [FS]. The correct implementation of the approach is checked for the test case problem of sloshing in a rectangular tank including an elastic beam in its center, as shown in Fig. 10. The mechanical excitation consists in a harmonic load applied to the beam extremity. A frequency scanning is performed, see Fig. 12, to evaluate the eigenfrequencies which correspond to the peaks of the beam displacement. The free surface elevation for the first four modes of the coupled problem is shown in Fig. 11, together with the modal shape of the elastic beam. The numerical value of the eigenfrequencies are presented in table I, column C l. They are compared to the results by Peseux et at (1999), column entitled "Ref. A toolbox that allows the direct computation of the resonant frequencies was also applied, see column C2. All the results are relatively closed to each other, especially, the two last columns. The computation by Peseux et al (1999) were performed using another f.e.m. code. which may explains the small differences in the results. The agreement between the three method is very satisfactory, which give confidence into the external numerical procedure that has been developed in the present study. Developing this procedure, and controlling precisely the coupling terms resulting from the fluid - structure interaction is necessary to treat the water impact problem of an elastic body.

CONCLUSIONS

In the present paper, a numerical method is proposed to solve the three-dimensional water entry problem of a blunt and rigid body. A variational formulation together with a finite element method are used to solve the so called Wagner problem. The wetting correction is obtained through an iterative procedure. The numerical resolution is validated by simulating simple problems, such as the water impact problem of a wedge with small deadrise in 2D, or a cone for the axi­ symmetric case, and, an elliptic parabolord in the three-dimensional case. Good agreement is obtained between numerical and analytical solutions. The approach is then extended to treat the case of a deformable body, for which a strong fluid-structure interaction is expected. Structural elastic deformations are computed simultaneously in the f.e.m. analysis. The fluid - structure interaction arise through a coupling matrix. The correct evaluation of the coupling term has checked for the sloshing problem of a tank with an elastic wall. Earlier numerical results are recovered, therefore validating the methodology of the coupled approach.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support of the French Navy, Delegation Generale de l'Armement and Bassin d'Essais des Carenes.

REFERENCES

Castem2000. Departement de Mecanique et Technologie du Commissariat a I'Energie Atomique.

Cointe, R (1987). "L'impact hydrodynarnique en deux dimensions,"

I'" Journees de l'Hydrodynamique, Nantes, France.

Cointe, R, Armand, JL (1987). "Hydrodynamic impact analysis of a cylinder," J. Offshore Mechanics and artic Engineering, Vol 9, pp 237-243.

Cointe, R (1989). "Two dimensional water solid impact," Journal of Offshore Mechanics and Arctic Engineering, Vol Il l , pp 109-114. Chuang, SL (1967). "Experiments on slamming of wedge - shaped

bodies," Journal of Ship Research, pp 190-198.

Hertz. r, fl fl f4 fs

.-Elastic beam

fluid

1�

L Mechanical characteristic E = 2,1.1011 Pa p, = 7800 kg.m·� e=6.IO"'m H=0,266 m Fluid characteristic p1= 1000 kg.m·� L=0.231 m ht= 0,210 m

Figure 10: elastic beam in a heavy fluid problem

Ref. Cl C2 Hertz. Ref. Cl C2 1.49 1.60 1.61 f6 3.33 3.29 3.21 1.86 1.84 1.83 f, 3.46 3.51 3.42 2.39 2.46 2.44 fs 3.95 3.91 3.73 2.66 2.63 2.61 f9 4.0 3.99 3.81 2.97 3.06 3.01 Table 1. Eigenfrequencies in Hz f= !,6Hz f= !,84Hz

-

. .

-< f=2,46 Hz f= 2,63 Hz

Figure 11: Eigenmodes shapes.

./

\

1.5 2 2.5 3

frequerrck� fHl) 3.5 4

Donguy, B, Peseux, B, Fontaine, E (2000). "On the ship structural response due to slamming loads," Proc. of European Congress on Computational Methods in Applied Sciences and Engineering,

Barcelona.

Faltinsen, OM (2000). "Water impact in ship and ocean engineering,"

Keynote lecture,Proc.4th lnt.Conf.on Hydrodynamics,Yokohama.

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Hagiwara, K, Yuhara, T (1974). "Fundamental study of wave impact loads on ship bow," Journal of the Society of Naval Architect, Vol

135.

Howison, SD, Ockendon, JR, Wilson SK (1991). "Incompressible water entry problems at small deadrise angles," Journal of Fluid Mechanics Vol222, pp 215-230.

lafrati, Korobkin, AA (2000). In Proc. 15th Workshop on Water Waves and Floating Bodies, Dan Caesarea, Israel.

Korobkin, AA (1982). "Formulation of penetration problem as a variational inequality," Din. Sploshnoi Sredy, Vol58, pp 73-79.

Korobkin, AA, Pukhnachov, VV (1988). "Initial stage of water impact," A.R.F.M., Vol 20, pp 159-185.

Magee, A, Fontaine, E (1998). "A coupled approach for the evaluation of slamming loads on ships," Proc.

'J'h

lnt. Symp. on Practical Design of Ships and Mobil Units, The Hague Netherlands.

Mizoguchi, Tanizawa (1996). "Impact Wave Loads due to slamming -A review," Ship Technology Research, Vol. 43, pp. 139-154 .

Peseux, 8, Cartraud, P, Argouar'ch, Y (1999). "Dynamic response of partially filled tanks - Explicite time integration method," Proc. of ASME Pressure Vessels and Pipe Conference, Vol394, pp 191-199.

Scolan, YM, Korobkin, A.A (2001). '"Three-dimensional theory of water impact. Part I. Inverse Wagner problem".

Wagner, H (1932). "0ber Stoss und Gleitvorglinge an der Oberfliiche von Fliissigkeiten, "Z. Ang. Math. Mech., Voll2, pp 193-215.

Wilson, SK (1989). '"The mathematics of ship slamming,'' Ph.D. thesis, University of Oxford.

Zhao, R, Faltinsen, OM (1992). "Water entry of two-dimensional body," Journal of Fluid Mechanics, Vol 246, pp 593-612.

Zhao, R, Faltinsen, OM, Aarsnes, JV (1996). "Water entry of arbitrary two-dimensional sections with and without flow separation,'' Proc.