two coordinate systems are parallel to each other. Let denote the position of the global reference point by the vector R and that of the local referen

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1 Dynamic coupling of seakeeping and sloshing Malenica» S., Zalar M. and Chen X.B. BUREAU VERIAS -DR, 1bis Place des Reflets, Courbevoie, France ABSRAC he methodology for dynamic coupling between liquid motions in ship's tanks (sloshing) and rigid body motions of the ship (seakeeping) in frequency domain, is considered. he problem is formulated under the classical assumptions of linear potential theory and Boundary Integral Equations method is used to solve both sloshing and seakeeping hydrodynamic part. BASIC MEHODOLOGY We consider the sloshing and seakeeping parts separately and, for the sake of clarity, only one partially filled tank is considered, the extension to arbitrary number of tanks with arbitrary fillings being straightforward. Seakeeping Z Figure 1: Basic notations and definitions. In the classical linear rigid body seakeeping analysis we end up with the motion equation in the form: where :! ([M ]+[A ]) i![b ]+[C ] fο g = ff DI g (1) X Zi Xi fο g - rigid body ship motions [M ] - genuine mass matrix of the ship [A ] - hydrodynamic added mass matrix [B ] - hydrodynamic damping matrix [C ] - hydrostatic restoring matrix ff DI g - hydrodynamic excitation force where subscript " " indicates that the quantity is written with respect to the global reference point. Sloshing Only the linear case is considered. Similar to the seakeeping part, an interior boundary value problem can be formulated for the potentials associated with six generic motions of the tank. he details are explained in the sections to follow, and here we just note that the final result gives the added mass matrix [A ] associated with each tank motion [in the local coordinate system (X i;y i;z i)]. Note that, since the linear potential theory is assumed, no damping can be generated by the liquid motions in the tank (we will see after, that an artificial damping term can be introduced). Also, as expected, no hydrodynamic excitation vector is present. On the other hand a (negative) hydrostatic restoring matrix [C ]isalsodefined. COUPLING If the two problems were defined in the same coordinate system, their coupling would be straightforward, once the different matrix calculated. It is clear that this will not be the case in general, and the transformations from the local coordinate system (tank) to the global (ship) will be required. In the section below we concentrate on this aspect of the problem. Coordinate transformations What we have to do is to transform the action (forces and moments) of the liquid motions from the local (tank) coordinate system to the global (ship) coordinate system. We assume that Paper No. -JSC- Malenica 1

2 two coordinate systems are parallel to each other. Let denote the position of the global reference point by the vector R and that of the local reference point by R : R = X i + Y j + Z k ; R = X i + Y j + Z k () Let also define the vector joining two points ( and )by R : R = fx ;Y ;Z g = R R = fx X ;Y Y ;Z Z g () In the case of the added mass matrix (similar for any other matrix), the general situation in the local coordinate system of the tank is described by the following equation: [A ]fο g = ff g () he goal is to express the force vector ff g in the global coordinate system with respect to the point. For instance we denote this vector as ff g. As noted before, the above equation () is written in compact form which means that it is six dimensional. he generalized motion and force vectors describes both translational and rotational components: fο g = ρ fο g fω g ff ρ ff ff ; ff g = g fm g It is also convenient to write the added mass matrix in the following form: " 11 # [A [A ]= ] [A1] () [A 1 ] [A] where [A ij ] denotes ( ) parts of the total matrix. We know that the motions and forces in each coordinate system are related to each other by the following relations: () ο = ο + Ω ^ R ; Ω = Ω = Ω () F = F = F ; M = M + F ^ R (8) so that we can write: where the matrix [V] is: fο g = fο g [V]fΩg () fm g = fm g [V]fF g (1) [V] = Z Y Z X (11) Y X With this in mind we can now decompose the equation (): [A 11 ]fο g+ [A 1 ] [A11][V] fωg = ff g (1) [A 1 ]fο g+ [A ] [A1][V] fωg = fm g [V]fF g which can be rewritten in a compact( ) form: [A ]+[A ] fο g = ff g (1) with: [A ]= (1) [A 11 ][V] [V][A 11 ] [V][A1] [A1 ][V] [V][A11 ][V] his is the required final result for the transfer of the added mass matrix from the point to the point. We can now write the motion equation of the coupled system:! ([M ]+[A ]+[A ]+[A ]) (1) i![b ]+([C ]+[C ]+[C ]) fο g = ff DI g HYDRO Here we consider in details the evaluation of different quantities in the motion equation (1), which are associated with the liquid (water, oil, LNG,...) pressure actions. As in the previous sections, the frequency domain approach is assumed, and we consider seakeeping and sloshing parts separately. Seakeeping he pressure is calculated from linear Bernoulli equation: p = %gz + i!%' (1) where we recognize the so called hydrostatic part associated with the hydrostatic pressure %gz and the pure dynamic part associated with the potential '. Hydrostatics Let us first note that, for seakeeping part, the liquid in tanks is considered as a rigid part of the body, which means that the draught of the ship is the real draught and static position of the center of gravity takes into account "rigid" mass of the liquid in the tanks. Even if the procedure for evaluation of the restoring matrix is rather well known, we found useful to detail the derivation of the hydrostatic coefficients in order to make clear analogy between seakeeping and sloshing. he general expressions for the hydrostatic force and moment are: F hs = %g M hs = %g z~nds z(r r ) ^ ~nds (1) hese expressions are written for the instantaneous position of the ship and, in order to transfer it to the ship position at rest we have to perform the linearization. Basically, this means that we will assume small motions and use the aylor series expansion to express the quantities at the instantaneous position as a function of their position at rest. By doing that, we obtain the following expressions: z = Z + v ; v = ο +Ω 1(Y Y ) Ω (X X ) (18) ~n = n + Ω ^ n ; r r = R R + Ω ^ (R R ) (1) where capital letters and n denotes the value of different quantities at rest. Note that the normal vector is pointing out of the Paper No. -JSC- Malenica

3 fluid domain. After introducing the above expressions into (1) we obtain: F hs = %g M hs = %g f vn + ZΩ^ngdS () f v(r R )^n+ω^[z(r R )^n]gds hese are the original expressions for the hydrostatic restoring forces, which are usually transformed into more common expressions involving the integrations over the waterplane area. he final expression is written in the compact form: with: ff hs g = [Chs ]fο g (1) [C hs ]=%g () A w A wy A A wx A A wy A I A 11 +VZ B I A A wx A I A 1 I A 1 VX B +VZ B VY B where subscripts " B" and " A" indicate the position of the center of bouyancy and the center of the waterplane area respectively. A w and I A ij stand for the waterplane area and corresponding area moments while V denotes the bouyant volume. In the case where the reference point and center of gravity G do not coincide, the restoring part associated with the mass displacement should be added to the above matrix: [C m ]= mg Z G X G Z G Y G so that we can finally write for the restoring part: () ff C g = [C ]fο g ; [C ]=[Chs ]+[Cm ] () Let finally note that, for the purpose of the numerical calculations, it is much wiser to perform the integration directly over the wetted body surface () instead over the waterplane area (). Hydrodynamics As usual, the calculation of purely hydrodynamic part of the forces is much simpler than hydrostatic part, provided the potential is efficiently calculated. Let just recall the basic principles. Body moves under the action of an incident wave field. In the context of the linear theory, we consider a generic case of an monochromatic sinusoidal wave whose velocity potential can be found: ' I = iga! eνz e iν(x cos fi+y sin fi) () where g is gravity, A is the amplitude of the wave,! is its frequency and fi is the incident angle i.e. the angle between the direction of wave propagation and the positive x axis (fi = ß for head waves). he fluid flow induced by this incident wave will be modified by the presence of the body, so that the total flow potential ' will be the sum of the incident potential ' I and the perturbation potential ' P : ' = ' I + ' P () Under the assumptions of linearity, it can be shown that potential ' P should satisfy the following boundary value problem (BVP): ' P = ν' P + P = z = P = I i![ο +Ω^ (R R )]n on S B pνr limh P iν' P = R!1 () where p ν is the infinite depth wave number ν =! =g and R = X + Y. In order to write the motion equation of the body in the convenient form, we make use of linearity andwe further subdivide the potential ' P into elementary potentials: ' P = ' D i! X j=1 ο j' Rj (8) he boundary value problem for each of these potentials will have the above form () and only the boundary condition on the body will be different. Adopting the compact notations the boundary condition on the body for each potential can be written in the form: D Rj = I = n j () where we recallthatn j = n for j =1; ; and n j =(R R )^n for j =; ;. he potential ' D is called diffraction potential and it represents the potential due to the interaction of the incident wave with the fixed body, while the six potentials ' Rj are called radiation potentials and they represent the potentials due to the six elementary body motions (ο 1 - surge, ο -sway, ο - heave, ο =Ω 1 - roll, ο =Ω - pitch, ο =Ω -yaw) in otherwise calm water. It is easy to check that the present decomposition allows the fullfilment of the complete body boundary condition (). he forces are obtained by integration of the pressure over the mean wetted surface of the body: ff hd g = i!% = i!% 'fngds () [' I + ' D i! X j=1 which can be rewritten in a compact form as: ο j' Rj ]fngds ff hd DI g = ff g +(! [A ]+i![b ])fοg (1) Paper No. -JSC- Malenica

4 where ff DI g is the excitation force and [A ]and[b ] (both real) are the so called added mass and damping matrix respectively: F DI i = i!%! A ij + i!b ij = %! (' I + ' D )n ids () ' Rj n ids () his completes the evaluation of the seakeeping hydrodynamic coefficients in (1). Sloshing Hydrostatics Similar procedure is repeated for sloshing problem and the expressions (1) are directly used but now with respect to the local reference point. It is essential to note that, due to the fact that the free surface is always horizontal, the expression for the vertical coordinate z which have to be used for hydrostatic pressure is: z = Z Z A +Ω 1(Y Y A ) Ω (X X A ) () where we recall that A denotes the center of the waterplane area, which in the case of sloshing means the center of the free surface. With this in mind, the expressions for the hydrostatic forces and moments become: F hs M hs = %g = %g f[ω 1(Y Y A ) Ω (X X A )]n+zω ^ ngds f[ω 1(Y Y A ) Ω (X X A )][(R R ) ^ n] +Ω ^ Z(R R ) ^ ngds () In order to derive the compact form of the tank restoring component ff hs g = [ Chs ]fο g () the same kind of transformations, as for the seakeeping part, is used. After some manipulations, the following expression is obtained: [ C hs ]= %g () I AA 11 + VZ B I AA 1 VX B I AA 1 I AA + VZ B VY B Note that the area moments of the waterplane have to be calculated with respect to the center of the free surface A and not with respect to the calculation point as it was the case for seakeeping. his represent the total static action of the water in tank. However, since we treated the water in tank as a rigid mass, for the seakeeping problem, we should substract this contribution here. hus we define the fictive mass restoring matrix: [ C m ]=mg Z G X G Z G Y G (8) Furthermore, for the liquid in the tank, the positions of the center of gravity and the center of "bouyancy" are the same and the bouyancy is equal to the liquid mass so that the final hydrostatic matrix can be significantly reduced in this case: [C ]=[C hs ]+[Cm ]= %g I AA 11 I AA I AA 1 I AA 1 () Note that the above restoring matrix is negative, so that the global restoring of the coupled system is reduced, as expected. Hydrodynamics Due to the fact that the linear theory is assumed, the situation is quite similar to the seakeeping part. he main differences concern the absence of diffracted potential and the slight changes of the free surface condition for the perturbation potential. Indeed, contrary to the exterior (seakeeping) problem, in the sloshing problem we have to take into account the fact that the free surface is moving also due to the overall motions of the ship. In order to properly derive this condition we consider the situation presented on figure. Dynamic and kinematic free surface conditions may be written in the form: %g(ξ Z A v () where Zv A is the vertical displacement of the center of the waterplane area. After taking time derivative of the dynamic condition, two conditions can be combined in + = (1) In frequency domain (Φ = <f'e i!t g; Zv A = <f v A e i!t g) we obtain: ν' + = i!a v () where the vertical displacementofthewaterplane area center may be expressed as: A v =(ο +Ω^R A )k () On the other hand the body boundary condition remains the same: = i![ο +Ω^ (R R )]n () Paper No. -JSC- Malenica

5 Z A Ξ Z A v Figure : ank motion. so that the linear perturbation potental may be decomposed into radiation potentals as before: ' = i! X j=1 X A ο j' Rj () Each radiation potental satisfy the following BVP: ' Rj = ν' Rj + Rj = f j z = Rj = nj on () In the above equation n j has similar definition as for the seakeeping problem ( n j = n for j =1; ; andn j =(R R ) ^ n for j =; ; ) while the forcing function f j comes from the modified free surface condition () with: f j = 8 >< >: j =1 j = 1 j = Y A j = X A j = j = () Once again, the normal vector is pointing out of the fluid domain, which, in this case, means opposite to the seakeeping problem. Note that the present BVP has some similarities with the BVP for seakeeping of air-cushion supported floating bodies (the same is true for hydrostatic part) (see Malenica et al. ). As far as the forces are concerned, and since there is no damping in the closed problems within the potential theory (the potential ' Rj is real valued in this case), the total hydrodynamic force is written in the form: where : ff hd g =! [A ]fοg (8)! A ij = %! ' Rj n ids () his basically completes the motion equation (1) and problem formally can be solved. However there are few questions about the damping of the sloshing motion. Indeed, since the linear potential theory assumptions were used, we can expect some problems around the sloshing resonant frequencies, because there is no damping and the results may be unrealistic around these frequencies. An simplified method will be adressed in more details in the last section. It is important to note here that the dynamic inertia component of the liquid action in the tank is completely described by the above defined added mass matrix [A ] and the genuine mass of the liquid should be excluded from the total mass matrix [M ] in the equation of ship motions (1). Solution of the hydrodynamic BVP's Seakeeping As explained in the secton, the generic BVP for seakeeping part has the following form: ' = ν' + = z = = vn lim h pνr iν' on = R!1 () In order to solve this problem we usethewell known Boundary Integral Equation method based on the so called source formulation. For that purpose a Green function is defined satisfying the following set of equations: G = νg = z = lim h iν' = R!1 (1) It is then possible to show, using the Green's theorem, that the solution for the potential ' can be represented in the following form: ' = ffgds () Fulfillement of the body boundary condition gives the Boundary Integral Equation for source strength ff: 1 ff + ds = vn () which is solved by usual numerical routines after discretizing the body surface in a finite number of flat panels. Sloshing Since the sloshing problem is governed by the similar kind of BVP (), the same method can be applied. In the case of horizontal motions (surge, sway, yaw) the above presented BIE method can be used directly, and following integral equation is obtained: 1 ff + ds = nj ; j =1; ; () In the case of vertical motions (heave, pitch, roll), due to the non-homogeneous free surface condition () some clarifications Paper No. -JSC- Malenica

6 are necessary. Let us consider the following generic BVP: ' = ν' + = fj z = = nj on he solution is divided in particular and homogeneous part: () ' = ' P + ' H () First we seek for the particular solution satisfying the Laplace equation and the following free surface condition: ν' P P It is easy to see that constant is a solution: ' P = fj ν = fj () (8) Since the constant does not introduce the normal velocity on the tank boundaries the BVP for homogeneous part becomes: ' H = ν' H + H = z = H = nj on () his is the same type of BVP as those for the horizontal motions, and same method can be used. In the case of heave motion the situation may be further simplified because the solution (total) may be found by inspection: ' R = z () his means that the heave motion will not generate the free surface deformations, so that the corresponding added mass in heave will be equal to the mass of the liquid inside the tank: A = % zn ds = %V = m (1) Let us finally note that the most logical choice for the local reference point R is the center of the free surface R A so that f = f =which means that the free surface condition for roll and pitch becomes homogeneous too. Damping of the sloshing motion As indicated before, the damping of sloshing motion should be taken into account in order to avoid the violent resonant motions in the tank which can appear due to the inconsistency of the linear potental sloshing model which we use. Note that, our intention here, is not to correctly model the sloshing phenomena in the tank, which is a quite complicated business, but just to take, what we beleive to be, the most important part of sloshing dynamics which influences the ship global behaviour. he detailed analysis of real sloshing problem is beyond the scope of the present work and can not be done using the linear (probably even non-linear) potential theory, so that more sophisticated models (c.f. Euler or Navier Stokes) are usually required. We are aware that the damping can be treated only approximately in the context of linear potential theory, and we propose here one simple method which can easily be put in practice and subsequently calibrated using the experimental results. he basic idea is to modify the body boundary condition on the tank boundaries: = ifflk' + vn () his condition imply that the main part of dissipation happens only in the boundary layer. his is probably not true, but the final result in terms of ship motions may be sufficiently accurate especially after calibration of the parameter ffl. In fact, regardless where the energy is dissipated, the final effect on ship motions will be the same, provided the added mass is accurately calculated. Note that we can also choose to modify the free surface condition in a similar manner. Anyway, the consequence of the above trick () is that the resulting potential becomes complex with the imaginary part giving the damping matrix:! A ij + i!b ij = %! ' Rj n ids () From numerical point of view, the expression for the potential remains the same () and only the Boundary Integral Equation () change to: 1 ff + ifflg)ds = nj () he last problem concerns the evaluation of the parameter ffl. What we propose is to calibrate it through the comparisons with experimental results. Indeed, the correct evaluation of damping using numerical tools is almost hopeless today, even if the more sophisticated tools were used. On the other hand, we expect that the global ship behaviour will not be very sensitive to the internal damping in the tank. his means that even rough estimation of ffl may be good enough. NUMERICAL RESULS In order to demonstrate the capabilities of the method, we choose the example from Molin et al (). It is a rectangular barge (length L =m, widthb =1m, draught =:18m) withtwo rectangular tanks (length L = :8m, width B = :m, draught =:1m) located on the barge deck (bottom of the tanks at :m). Only the case of beam waves is considered. On figure we show first the results for added masses coefficients for sway and roll. Different curves are for different values of "damping" coefficient " (). Dotted line, equal to 1, represents the results for the case when the liquid in the tanks is treated as rigid. We can clearly identify the sloshing resonant frequency, and we can observe the effects of damping. Finally, on figure we show the roll RAO of the barge for different cases. We can observe very strong influence of sloshing. Instead of one peak (dotted line for "rigid" liquid) we havenowtwo peaks as expected. Furthermore if we compare these results to the figure in Molin et al. (), we can see that the results are almost the same. hus we conclude that our model is now operational and can be used in common studies of the LNG carriers with partially filled tanks. Paper No. -JSC- Malenica

7 a /(ρv) ω ε=. ε=. ε=.1 Roll RAOs (rad/m) ω ε=. ε=. ε=.1 a /[ρv(l + )/1] ω ε=. ε=. ε=.1 Bodies, Japan Figure : Roll RAO in beam waves. Kim Y. (1) "Coupled analysis of ship motions and sloshing flows.", 1th Int. Workshop on Water Waves and Floating Bodies, Japan Figure : Added mass coefficient for sway (top) and roll (bottom). CONCLUSIONS An efficient method for dynamic coupling between sloshing and seakeeping is presented. In contrast to the usual time domain methods (see e.g. Rognebakke and Faltinsen, 1; Kim, 1), the present method uses frequency domain approach. he basic assumptions (linearity, frequency domain,...) are the same as in Molin et al () but the method is completely different. he method is validated against the results from the literature. he advantage of the present method lies in the possibility to use the existing diffraction-radation codes with very minor changes. REFERENCES Chen X.B., Malenica»S. & Petitjean F. (1) "Off-shore hydrodynamics.", Bulletin echnique BV, pp. -. Malenica»S, Espanet A. & Zalar M. () "enue a la mer des objets flottats supportés par des coussins d'air.", 8- emes Journées de l'hydrodynamique, Nantes Molin B., Remy F., Rigaud S. Jouette (de) Ch. () "LNG- FPSO's: frequency domain, coupled analysis of support and liquid cargo motions.", IMAM, Greece Rognebakke O.F. & Faltinsen O. (1) "Effect of sloshing on ship motions.", 1thInt. Workshop on Water Waves and Floating Paper No. -JSC- Malenica