Advanced hydrodynamic analysis of LNG terminals

Transcription

1 Advanced hydrodynamic analysis of LNG terminals X.B. Chen ), F. Rezende ), S. Malenica ) & J.R. Fournier ) ) Research Department, Bureau Veritas Paris La Défense, France ) Single Buoy Moorings, INC Monaco, Principality of Monaco Abstract Advanced hydrodynamic analyses of floating LNG terminals are presented. They consist of the application of the middle-field formulation newly-obtained by Chen (6) to evaluate not only the drift loads but also the low-frequency wave loads, and the adoption of the notation of fairly perfect fluid to introduce the dissipation in resonant wave kinematics due to complex interaction between floating bodies in offloading operations. The numerical results validated with measurements of model tests show that the usual approximation of QTF is not appropriate in most applications, and that the common near-field formulation gives results of lowfrequency loads with poor convergence. The innovations presented here solve these issues and provide accurate and efficient computation of full QTF. The dissipation model of multibody interaction is shown to be effective by the very good agreement between numerical results and those of recent model tests. Furthermore, the dynamic effect of liquid motion in tanks is analyzed and taken into account in the computation of global responses. Keywords LNG Terminals; Multi-body interaction; Full QTF; ydrodynamic resonance and Seakeeping with liquid motion in tanks. Introduction Recently, there are more and more large floating LNG terminals being developed in remote offshore locations where marine environment can be hostile, in order to distance themselves from neighbors and minimize permitting issues. As the important part of a LNG system, the terminal can be a barge or a converted vessel, including a regasification or a liquefaction plant, a number of storage tanks and offloading facilities. It serves also as a support to moor a LNG carrier during offloading operations. In the design of the mooring systems of LNG import/export terminal and LNG carriers, the side-by-side configuration is being considered as the preferred option. In the design of such mooring system of LNG/FSRU terminals and LNG carriers in deep water or in a zone of shallow water, one key issue is the accurate simulation of low-frequency motions of the system to which the second-order wave loading is well known as the main source of excitation. Associated with this issue, the multibody interaction and the dynamic effect of liquid motion in tanks have to be taken into account in a consistent and efficient way. The formulations of second-order wave loads are summarized after this introduction. The low-frequency wave load is expressed by the sum of one part depending on the first-order quantities and another contributed by the second-order wave field. The first part being function of quadratic product of the firstorder wave field and responses can be directly evaluated once the first-order solution is obtained. The second part can be further decomposed into one component depending on the incoming waves and another relating the second-order diffracted waves. The indirect method (Molin, 979) is used to evaluate the secondorder diffraction load which is formulated by the sum of two integrals of askind type, one on the hull and another on the free surface. The analysis by developing the low-frequency load into a Taylor series with respect to the frequency ( ω) provides the approximation of first-order ( ω) as proposed in Chen (994) and Chen & Duan (7), one order better than the zeroth-order Newman approximation largely used in practice. The important analysis on the classical near-field formulation has been recently realized in Chen (4). A new near-field formulation is obtained by direct application of the variants of Stokes s theorems. Applying the Green s theorem to the domain limited by a control surface, a second new formulation is obtained and involves the integrals on the control surface and along its intersection with the mean free surface. Unlike the formulation given in Ferreira & Lee (994) obtained by applying the momentum theorem and applicable only to the drift loads, this new formulation is general as it can apply to the high-frequency loads as well as the low-frequency loads, to horizontal load components as well as vertical load components. An interesting feature of the formulation concerns the lowfrequency wave load for which the formulation is largely simplified. In particular, the horizontal components of drift loads involve only a surface integral on the control surface and a line integral along its intersection with the free surface. This formulation written on the con-

2 trol surface at some distance from the body is called as the middle-field formulation. It is shown that it has the same advantage as the far-field formulation to have rapid numerical convergence for horizontal drift loads. Furthermore, in the case of multiple bodies, the control surface can be one surrounding an individual body and the wave loads applied on the surrounded body are then obtained, while the far-field formulation provides only the sum of wave loads applied on all bodies. An important application of the developed method is the multibody interaction. The side-by-side situation amplifies the interaction and can yield large kinematics of wave field in the confined zone. Within the framework of the classical linear potential theory, there is not any limit in predicting wave elevations at the free surface while the resonant motion in the reality must be largely damped by different mechanisms of dissipation. Unlike the method developed by Buchner et al. () or that by Newman (4), we apply directly the authentic equations, presented in Chen (4), of the fairly perfect fluid involving the energy dissipation via introducing the damping force. The integral equation extended to a limited zone of the free surface is then developed. Numerical examples show that the method is efficient and provide results closer to the experimental measurements. The results realized recently by SBM in OCEANIC are presented with numerical results using this dissipation model. The effect of liquid motion in partially-filled tanks of a LNG carrier is taken into account in the seakeeping analysis. Classically, only the hydrostatic effect is taken into account by subtracting the corresponding stiffness from the global hydrostatic matrix. This is only valid for very low wave frequencies. The dynamic effect is important especially at a wave frequency close to one resonance frequency of tanks. Under the assumption of linear potential flow, the fluid motion in tanks can be evaluated by solving the boundary value problem involving the same Green function which satisfies the free surface condition. To approximate the damping effect to liquid motion in tanks, the condition on tanks wall is modified by introducing a small positive parameter equivalent to a partial reflection of walls. This implies that the main part of dissipation occurs in the boundary layer. Numerical results show that the approximation is good enough to capture the major coupling effect of liquid motion with the global motion of vessels. Finally, some discussions and conclusion on the foregoing analysis are addressed. QTF of low-frequency wave load Numerous studies have been devoted to the analysis of second-order wave loads as summarized in Chen (4). The general formulation of second-order wave loads can be obtained by directly integration of the second-order pressure on the hull surface of body s mean position and the variation of the first-order loads due to the first-order motions. The second-order wave load is then composed of one part dependent on the quadratic product of the first-order quantities and another part contributed by the second-order potential : with (F,M) = (F,M ) + (F,M ) (a) (F,M ) = ρ ds Φ () t (n, r n) (b) where F = (F x,f y,f z ) stands for the forces, M = (M x,m y,m z ) for the moments and Φ () for the second-order potential. The first part (F,M ) is given in Chen (6) by : F =ρ ds [ ( Φ) n+ Φ t (X n) ] ρg dl [ η n+ξk ] Γ (a) M =ρ ds [ ( Φ) (r n)+(r Φ t )(X n) ] ρg dl [ η (r n) + Ξ(r k) ] with Ξ defined by Γ Ξ = (η ηζ 3 )n 3 / cos γ η(x n) (b) which is involved in the waterline integral giving only a contribution to vertical forces and moments around the horizontal axes. The body motion is represented by the vector X = (ζ,ζ,ζ 3 ). The normal vector n in the waterline integral is contained on the mean free surface, n = n for a wall-sided hull. In general, n = (n,n,)/ cos γ with γ the angle between n and the horizontal plane. For a wall-sided hull, n 3 = so that Ξ = η(x n). The formulation () derived from Eqs.7 & 8 in Chen (6) obtained by applying the two variants of Stokes theorem to the classical near-field (pressureintegration) formulation is more compact. We consider bichromatic waves associated with frequencies (ω j,ω k ) in which the first-order velocity potential written by Φ=R { a j φ j e iωjt} + R { a k φ k e iω kt } (3) with (a j,a k ) being amplitudes of first-order incoming waves. Introducing the form (3) into (), we obtain different components of the second-order load associated with different frequencies equal to (ω j ), (ω k ), (ω j + ω k ), and (ω j ω k ), respectively. Since we are interested here only to the low-frequency load, the components associated with the frequencies (ω j ), (ω k ) and (ω j + ω k ) are ignored. Furthermore, the drift load (at zero frequency) can be obtained by the limit of the low-frequency load associated with the frequency (ω j ω k ) when ω k tends to ω j. Without loss of generality, we may write the low-frequency load by : { } (F,M)=R a j a k (f,m)e i(ωj ω k)t (4) with (f,m)=(f,m )+(f,m )+(f D,m D ) (5) where a k stands for the complex conjugate of a k. This rule to denote the complex conjugate by the over line is applied to all first-order quantities in the following.

3 The first part (f,m ) can be directly obtained by () while the second part is composed of the component (f,m ) contributed by the incoming waves and that (f,m D ) by the diffracted waves : (f,m )= i(ω j ω k )ρ dsφ () (n,r n) (f D,m D )= i(ω j ω k )ρ dsφ () D (n,r n) The second-order incoming velocity potential is written as φ () = ia g cosh(k j k k )(z+h)/cosh(k j k k )h g(k j k k )tanh(k j k k )h (ω j ω k ) E (6) with E = e i(kj k k)(x cos β+y sin β) and A defined by A= ω j ω k [ k j k k + tanhkj h tanhk k h ] ω j ω k [ ] + k j /ω j cosh k j h k k /ω k cosh k k h (7) The contribution by the second-order diffraction potential can be evaluated by Molin s method (979) : (f D,m D ) l = i(ω j ω k )ρ [ () φ ds + i(ω j ω k ) ρ g n N F ] ψ l ds N F ψ l (8) where ψ j is the additional radiation potential at (ω j ω k ) due to a unit motion in the lth direction. The non-homogeneous terms N F and N involve the firstorder incident potentials and diffraction/radiation potentials. The second part can be further decomposed into one term of integration of incoming wave pressure (f,m ), one askind integral on the hull (f,m ) and one askind integral over the free surface (f F,m F ) resulting from the second-order forcing on the free surface. The sum of last two represents the integration of diffraction wave pressure (f D,m D ), according to (8). Thus, we may write the lth component of QTF : F l =(f,m ) l +(f,m ) l +(f D,m D ) l (9) with (f D,m D ) l = (f,m ) l + (f F,m F ) l. Furthermore, F l (ω j,ω k ) is assumed to be regular function of (ω j,ω k ) and a Taylor expansion with respect to ω=(ω j ω k ) can be developed : F l (ω j,ω k ) = F l (ω) + ωf l (ω) + O[( ω) ] () with ω = (ω j +ω k )/. The second-order low-frequency wave loads F l (ω j,ω k ) in bichromatic waves of frequencies (ω j,ω k ) are composed of one component F l (ω) depending on ω = (ω j +ω k )/ and another ω F l (ω) linearly proportional to ω = ω j ω k. Both F l and F l are analyzed and given in Chen & Duan (7). The striking fact is that F l (ω) is a pure real function while F l (ω) a pure imaginary function. The usual approximation proposed by Newman (974) largely used in practice is based on the use of F l so that it is O() approximation. The numerical results presented in Chen (994) using the O( ω) approximation are in good agreement with experimental measurements on the N Kossa FPSO while the wave loads based on Newman approximation are largely underestimated. The time simulation of low-frequency motions confirm the Newman approximation gives results too low: RMS being 35% (and extreme values up to 6%) smaller than those of model tests. Very recently, Newman (4) confirms that the zeroth-order approximation is poor as the waterdepth is below m. It is further recognized that the approximation including the effects of the second-order incoming wave potential, and its diffraction by the body, but not the part resulting from the second-order forcing on the free surface (as proposed by Chen, 994) gives much better results. This study confirms that not only the O() approximation can underestimate largely the secondorder wave loads as shown in Chen (994) but also it provides wrong phase differences with respect to incoming waves since the complete QTF is a complex function while that by the O() approximation is purely real. Middle-field formulation The formulation () to compute the first part of lowfrequency load is called as near-field one since the involved terms are evaluated on the hull and along the waterline. In the particular case of ω k = ω j, the lowfrequency wave load becomes constant drift load which is contributed only by the first part. Another formulation base on the momentum theorem for the horizontal drift forces. This formulation involving first-order wave field in the far field is often called far-field formulation and preferable in practice thanks to its better convergence and accuracy. By performing a local momentum analysis, Ferreira & Lee (994) developed a formulation over a control surface surrounding the body to evaluate the constant drift load. Starting with the near-field formulation and applying the Green theorem in a domain D surrounded by S = C F with the body hull at its mean position, a fictitious (control) surface C surrounding the body and the mean free surface F limited by the intersection Γ of with z = and that Γ c of C with z =, we obtain Eqs.35 & 36 in Chen (6) which are general as it can apply to the high-frequency loads as well as the low-frequency loads, to horizontal load components as well as vertical load components. The control surface C can be at a finite distance from the body or one pushed to infinity. In the first case, C may be pushed back to while in the second case, C may be composed of the surface of a vertical cylinder plus the seabed. Furthermore, in the case of multiple bodies, the control surface C can be one surrounding an individual body and gives the wave loads applied on the surrounded body. Concerning the low-frequency wave load, it can be easily checked that the values of the hull integral and of the first term in the free-surface integral are of order O( ω). Furthermore, the waterline integral as well as

4 y Wave heading x -. Middle-field and far-field Control surfaces Terminal s hull Near-field Figure : Terminal s hull & control surfaces (left) and drift loads F y D (right)..4. Middle-field.3 Near-field. -. Near-field Middle-field Figure : Real (left) and imaginary (right) parts of QTF F y ( ω=.4) in oblique sea the second term in the free-surface integral contribute only to the vertical loads including the vertical force F z and moments around the horizontal axis (M x,m y ). Thus, the horizontal components (F x,f y,mz ) of lowfrequency loads can be expressed as : f y = ρ nφ k y/ω j φ k nφ j ] y/ω k ρ ds [ ω j φ j φ k y ω k φ k φ j ] y g F (b) F x = ρ nφ k x + φ k nφ j x φ j φ k ] n C + ρω jω k dlφ j φ k n + ω f x g Γ c F y = ρ nφ k y + φ k nφ j y φ j φ k ] n C + ρω jω k dlφ j φ k n + ω f y g Γ c M z = ρω jω k dlφ j φ k n 6 + ρ g Γ c n(xφ k y yφ k x) C (a) (b) + φ k n(xφ j y yφ j x) φ j φ k n 6 ] + ω m z (c) with the additional terms (f x,f y,mz ) given by : f x = ρ nφ k x/ω j φ k nφ j ] x/ω k ρ ds [ ω j φ j φ k g x ω k φ k φ j ] x F (a) m z = ρ n(xφ k y yφ k x)/ω j φ k n(xφ j y yφ j ] x)/ω k ρ ds [ ω j φ j (xφ k y yφ k g x) ω k φ k (xφ j y yφ j x) ] F (c) The formulations (-) provide an original way to evaluate the horizontal components of low-frequency wave loads. The additional terms given by () are of order ( ω). If the body s motion is small (X ) in waves of small period, the integral over hull surface is negligible since φ n = X t n on. Since the velocity potentials and their derivatives can be evaluated in an accurate way on the free surface, the integral over the part of free surface is ease. In regular waves, the formulation () reduces to the first two integrals on the control surface since ω j = ω k. The low-frequency loads by () become the drift loads. If the control surface C is put to infinity, the expression () is in agreement with the far-field formulation. On the surface C at infinity, asymptotic expressions of the first-order potential can be used to simplify further the formulation to the single integrals involving the Fourier polar variable. This shows for-

5 Gap meshed as the damping zone Num. ǫ= Num. ǫ=.6 Measurement Figure 3: Side-by-side barges & damping zone (left) and wave elevation in the center of gap (right) mally that the usual near-field formulation and far-field formulation are indeed equivalent. The near-field, middle-field and far-field formulations are first compared in the computation of second-order drift loads on a LNG terminal of size (Length Width Draught = 35m 5m 5m) moored in water of finite depth (h =75m). The meshes of the hull composed of 49 panels, and the control surfaces C F including the part of free surface F are illustrated on the left part of Figure. Only the half of the hull (y ) and that of C F for (y ) are presented in the figure. On the right part of Figure, the nondimensional values of drift load F y D /(ρgl/) with L = 35m in waves of heading β = 95 are depicted against the wave frequency (ω). Three meshes composed of 49, 386 and 784 panels on the hull surface are used. The results using the near-field and far-field formulations are represented by the dashed, dot-dashed and solid lines for three meshes (49, 366 and 784 panels), respectively. The results using the middlefield formulation are shown by the symbols of circles (49 panels), crosses (366 panels) and squares (794 panels). The curves associated with the near-field formulation are separated for ω >.45 rad/s. This shows that the results using the near-field formulation are not convergent in most part of wave-frequency range. This is due to the fact that the near-field formulation is very sensitive to the singularities of fluid velocities at sharp corners of structure s hull. On the other side, the results obtained by the far-field formulation (dashed, dot-dashed and solid lines) are indistinguishable on the whole range of wave frequency. The same feature is observed for the results associated with the middlefield formulation (circles, crosses and squares). Furthermore, the results of middle-field formulation are in excellent agreement with those of far-field formulation. Now, we consider the low-frequency load F y /(ρgl/) at a difference frequency (ω j ω k ) =.4 rad/s in waves of the same heading β = 95. The results in complex are presented on Figure against wave frequencies (ω k ). The real part and imaginary part of F y are depicted respectively on the left and right part of the figure. The results obtained by using the near-field formulation are illustrated by the dashed, dot-dashed and solid lines associated with the meshes of 49, 366 and 794 panels, respectively. The results obtained from the middle-field formulation are shown by the symbols of circles, crosses and squares associated with three meshes. Again, we observe that the near-field formulation gives the results with poor precision while the middle-field formulation provides the results of excellent convergence. Multibody interactions The interaction of multiple bodies includes the mechanical and hydrodynamic interactions. The mechanical interaction is defined by the mechanical properties of the connection between bodies which depend only on the design and operation procedure. The hydrodynamic interaction is more complex and requires a complete solution taking account of full interaction between multiple bodies. In some cases such as side-by-side vessels, the hydrodynamic interaction may annul any motion in the confined zone at some wave frequencies, or create violent kinematics of wave field at other wave frequencies. Particular attention to this resonant phenomena is paid and new method based on the notion of fairly perfect fluid is developed to take into account the damping mechanism in fluid. Recently, a few of publications have been realized on the resonant motion of wave field in the confined zone between two floating bodies. Unlike the resonant response of body s motion associated with the balance of inertia and stiffness loads, this resonant kinematics of fluid is due to the hydrodynamic interaction - wave kinematics annulled or amplified by the complex scattering between bodies. Within the framework of the classical linear potential theory, there is not any limit in predicting wave elevations at the free surface while the resonant motion in the reality must be largely damped by different mechanisms of dissipation. This unrealistic fluid motion magnifies the wave loads on the bodies. To hold the wave motion back to a realistic level, Buchner et al. () developed a method consisting to place a lid on the gap in between the two bodies. The unrealistic wave kinematics is then suppressed. In fact, no wavy elevation is possible under the rigid lid and noticeable perturbation around the ends of the lid due to the diffraction effect can be observed. To make wavy motion allowable on the lid, Newman (4) renders the lid flexible using a set of basis functions of Chebychev polynomials. The deformation of

6 6 5 4 Num. ǫ = Num. ǫ =.53 Measurement Figure 4: Wave elevation along the center line of gap for (ω=.7, β=8 ) (left) and (ω=.76, β= ) (right) the flexible mat (equal to the free-surface elevation) is then reduced by introducing a damping coefficient. Unlike the above methods using an artificial lid, we apply directly the authentic equations, presented in Chen (4), of the fairly perfect fluid involving already the energy dissipation via introducing the damping force. The dissipation term appears in the boundary condition on the free surface : φ z k φ iǫk φ = for P F(z = ) (3) with k = ω /g and ǫ the dissipation coefficient. Following the analysis in Chen (4), we have : φ(p) = dsσ(q)g(p,q) S = F F (4) S and the integral equations to determine the source distribution σ are : πσ(p) + dsσ(q)g n (P,Q) = v n P (5a) S 4πσ(P) dsσ(q)g n (P,Q) = P F (5b) S 4πσ(P) + iǫk dsσ(q)g(p,q) = P F (5c) S The integral equation (5b) on the internal waterplane surface F is necessary to eliminate the irregular frequencies. The integral equation (5c) is written over entire F. owever, we know σ = if ǫ= from (5c). As we need to apply a non-zero value of ǫ only in the zone where the fluid kinematics is susceptible to be violent, the discretization of F is limited. A practical way is to mesh the zone between two vessels on which a constant or a distribution of ǫ varying in space can be applied. Two side-by-side barges of the same dimension in meter (L B T = ) with mechanical properties (z G =. and gyration radii=.87/.57/.57) are free floating in head waves. The mesh of barges and the whole gap (.6 as width) meshed as the damping zone are presented on the left of Figure 3. The free-surface elevation is measured at the center of the gap. The results of numerical computation with two values of parameter ǫ= and.6 are drawn on the right together with those of measurements, against the wave frequency (rad/s). Large free-surface elevations are remarkable at three Figure 7: Set-up of model tests by SBM in OCEANIC wave frequencies. The results with ǫ = (no damping) are much larger than those measured while the results with ǫ=.6 agree well with the measurements. By comparison between the curves of numerical results corresponding to ǫ= and.6, we see that the damping affects only the values in the range of frequencies around one where large elevations occur, as expected. The model tests have been recently performed by SBM in OCEANIC (OCC) located in St. John s, Newfoundland, Canada. A FSRU of 3k m 3 storage capacity and a LNGC of 35k m 3 are moored independently in water of 6m in depth. They are placed side by side with a gap of m, as shown in Figure 7 of the model test set-up. Vessels motions, mooring loads and wave elevations in the gap were measured during this campaign. Monochromatic waves of m in height with periods ranging from 7.5 to s in the vicinity of each gap resonance were tested for four incidences, 9 beam on LNGC exposed, 8 head on, 5 LNGC exposed, and LNGC sheltered. In numerical computations, several dissipation coefficients ǫ in (3) were applied on the free surface in the gap to cope with wave resonance. Comparisons of numerical results associated with ǫ =.53 with the experimental results in terms of wave elevation, motions RAO and mean drift forces are presented on Figures 4, 5 and 6, respectively. The wave elevation RAO along the center line of the gap expressed in m/m is plotted on Figure 4 for ω =.7 rad/s in head waves (β = 8 ) in the left part and for ω=.76 rad/s in oblique waves (β = ) in the right part, against the longitudinal position cen-

7 .8 Num. ǫ = Num. ǫ =.53 Measurement Figure 5: LNGC s sway RAO (left) and LNGC s heave RAO (right) Num. ǫ = Num. ǫ =.53 Measurement Figure 6: Second-order sway drift force on FSRU (left) and that on LNGC (right) in head waves tered at the midship section, from aft position (negative abscissa) to fore position (positive). The numerical results agree quite well with the numerical calculation performed with a dissipation coefficient ǫ equal to.53. The shape of the RAO, with larger amplification near the bow (positive abscissa) is well recovered. LNGC s RAOs are plotted on Figure 5 for sway (left) and heave (right) motions, as a function of wave frequencies. Again numerical computation gives a fair agreement with the model tests results. Away from resonance, the free surface dissipation is not affecting the sway/heave motion response. The mean sway drift forces in head waves expressed in kn/m are shown on Figure 6 on FSRU (left) and on LNGC (right). The measured sway QTFs on the FSRU (positive forces at resonance) and the LNGC (negative forces at resonance) are opposite in sign and have almost identical magnitudes. The drift forces computed with dissipation coefficient ǫ=.53 are in good agreement with the experiments. This implies a reduction of the mean transverse drift forces QTF at resonance from 5 kn/m to 6 kn/m. The maximum force derived from the tests is found at a slightly lower frequency than numerical calculation. Experiments have shown that wave elevations in the gap can be more than twice the incident wave elevation in one of the two SBS layout considered. It is also noteworthy that beat patterns exist in the longitudinal direction. The position of the wave elevation envelope nodes and antinodes shifts with the wave frequency and with the direction of the incident wave. It has been demonstrated that resonance in the gap not only affects the local wave field but also the st order motion response for specific degrees of freedom and the mean wave drift force in sway as well. In summary, the resonance in the gap affects the calculation of the relative motions at amidships manifold critical for loading arm design and the prediction of the mooring lines loads that dictate the offloading operations feasibility and the terminal availability. Effect of liquid motion in tanks The full account of liquid motion in tanks such as sloshing effect is difficult due to the nonlinearity of the phenomena and a time-domain solution of coupling is necessary as presented in Malenica et al. (4). It is assumed that the major effect to vessel s global motion is linear so that a solution in frequency domain is possible. This assumption is indeed valid when the liquid motion in tanks is not violent in the range outside of resonant sloshing frequency. At low wave frequencies, only the hydrostatic effect is important and can be taken into account just by introducing negative values for the non-zero terms of stiffness matrix : C 44 = ρgi yy ; C 45 = ρgi xy = C 54; C 55 = ρgi xx (6a) with I xx,xy,yy are the moments of waterplane with respect to its center. In general cases especially at a wave frequency close to one of resonance, the liquid motion induces

8 ..8.6 Num. ǫ= Num. ǫ=. Test n Test n Test n 3 Test n 4 Test n Figure 8: LNG carrier and two tanks (left) and sway RAOs in beam sea (right) Num. ǫ= Num. ǫ=. Test n Test n Test n 3 Test n 4 Test n Num. ǫ= Num. ǫ=. Test n Test n Test n 3 Test n 4 Test n Figure 9: eave RAOs (left) and roll RAOs (right) in beam sea additional inertia loads and damping if energy dissipation is modeled. In fact, the motion equation of the vessel is modified as : 6 [ ω (M kj +A kj +A kj) iω(b kj +B kj)+c kj +C kj ] aj =F k j= (6b) for k =,,,6 and the inertia matrix M kj associated with the mass distribution excluding the liquid in tanks. The additional mass matrix A kj is equal to that to consider the liquid as a solid mass in classical approximation valid for low frequencies and the damping B kj = in this case. The linear velocity potential due to forcing oscillations of the tank can be solved in the same way as the solution of radiation problem for the vessel. The matrices A kj and B kj can then be obtained for each wave frequency. At low wave frequencies, the contribution of liquid in tanks is nearly like solid mass. When wave frequency approaches the resonant frequency, the value of inertia increases rapidly without limit at the resonance. The added-inertia changes the sign when the wave frequency goes across the tank resonant frequency. This variation of inertia modifies the response of the barge. Instead of one peak without dynamic effect of liquid motion, there are two : one on the left of the tank resonance and another on the right. This can be explained by the fact that the inertia is largely amplified when the excitation frequency is close to but smaller than the first tank resonant frequency, the peak of global response is then shifted on the left. At the tank resonance, the response is largely reduced due to the large value (up to infinity if no damping) of addedinertia. At a higher wave frequency, the large negative values of the inertia due to liquid motion yield a second peak resultant from a new balance between the total inertia force and stiffness force of the system. We consider a LNG carrier of 74m in length, 44.m in width and.58 in draught. The tank No. of size (L B = ) and the tank No.4 of size (L B = ) with a filling height of m are placed at the position 44.55m and 64.5m from the after perpendicular of LNG, respectively. The bottom of tanks is at the height of 3m from the baseline. The mesh of LNG together with the two tanks are illustrated on the left of Figure 8. Different sets of model tests in irregular waves varying significant height S from.5m to 6m and peak periods T P from 8s to 6s associated with the spectrum of Jonswap type were made with a length scale of :5. The results of numerical computations with ǫ = and. are compared with those of measurements. The RAO of sway motion is presented on the right of figure 8 while those of heave and roll are on the left and right of Figure 9. There is not significant difference between the results with ǫ = and those with ǫ=. except the peak values of sway and roll are slightly smaller for ǫ =., as expected. It is shown that the numerical results are in very good agreement with model tests. Not only the position of peaks in sway and roll motions of numerical computa-

9 tion coincides with that of measurements, but also the values of peaks in two sets of results are in excellent agreement. Discussion and conclusion The low-frequency wave load is composed of one part depending on the first-order quantities and another part contributed by the second-order wave field. The zeroth-order approximation (Newman, 974) consists of using only the drift load derived only from the firstpart of wave load is shown to not be appropriate for most applications. On the other side, the first-order approximation in Chen (994) and Chen & Duan (7) gives much better results and is considered to be sufficient for most applications in deep water as well as in water of finite depth. To evaluate the quadratic transfer function of lowfrequency load in bichromatic waves, the near-field formulation derived from the pressure integration is largely used and considered to be the only way to go, unlike the constant drift load for which the far-field formulation based on the momentum theorem is available as well. owever, the near-field formulation is reputed by its poor precision and convergence, especially for structure s hull with sharp geometrical variations. The method using higher-order description of hull geometry (B-spline patches, for example) was hoped to give better accurate results than the lower-order method (constant panels). owever, the higher-order method is more sensitive to the singularities which are present in the velocity field at sharp corners. As concluded in Newman & Lee (), this sensitivity is manifested when the tangential fluid velocity is computed as in the evaluation of the mean pressure or the low-frequency pressure. As a result, the low-frequency load converges slowly or in the worst cases, it may be non-convergent. The middle-field formulation newly-obtained in Chen (4) solves this issue. Its application in the computation of second-order low-frequency loads confirms its important advantages. Firstly, it permits to make the connection between the near-field formulation derived from the pressure integration and the far-field formulation based on the momentum theorem for the constant drift load. Secondly, it accumulates the advantages of both near-field and far-field formulations, i.e. the excellent precision of far-field formulation and the access to the low-frequency wave loads as the near-field formulation. Furthermore, in the case of multiple bodies, the middle-field formulation provides the drift load as well as the low-frequency load on each individual body while the far-field formulation can only give the sum of drift loads on all bodies. Based on the notion of fairly perfect fluid, the damping to reduce, to a reasonable level by comparing to model tests, the resonant kinematics of wave elevation is applied via the boundary condition at the free surface. Following the same principle, the boundary condition on body s hull can be modified as well to include a partial reflection equivalent to energy dissipation in boundary layer. The new integral equations are established following these modifications. The applications to the side-by-side multibody interaction and to the liquid motion in tanks show its soundness and efficiency. It is natural to extend the application to the moonpool issue for which the success can be envisioned. In spite of these successful applications, the method remains to be an approximation to the dissipation mechanism - an important and complex aspect of fluid mechanics. The involved parameters need to be determined by comparing to experimental measurements or results of elaborated CFD simulations. The dynamic effect of liquid motion in tanks is represented by the added-mass and damping (if a dissipation coefficient is applied on tank walls) terms. These terms can be obtained in a similar way as the solution of radiation problems. The important coupling effect on global responses of LNG carriers (or floating terminals) is shown and compared with experimental measurements. The second-order low-frequency load on LNG carriers/terminals must be very affected by these effects. References Buchner B., van Dijk A. & de Wilde J. () Numerical multiple-body simulation of side-by-side mooring to an FPSO. Proc. th ISOPE, Stavanger, Chen X.B. (994) Approximation on the quadratic transfer function of low-frequency loads, Proc. 7th Intl Conf. Behaviour Off. Structures, BOSS 94,, Chen X.B. (4) ydrodynamics in offshore and naval applications - Part I. Keynote lecture of 6th Intl. Conf. ydrodynamics, Perth (Australia). Chen X.B. (6) Middle-field formulation for the computation of wave-drift loads, J. Engineering Math. DOI.7/s x. Chen X.B. & Duan W.Y. (7) Formulation of lowfrequency QTF by O( ω) approximation. Proceedings of nd IWWWFB, Plitvice (Croatia). Ferreira M.D. & Lee C.. (994) Computation of second-order mean wave forces and moments in multibody interaction, Proc. 7th Intl Conf. Behaviour Off. Structures, BOSS 94,, Fournier J.R., Mamoun N. & Chen X.B. (6) ydrodynamics of two side-by-side vessels experiments and numerical simulations. Proc. 6th ISOPE, San Francisco (USA), -7. Malenica Š., Zalar M. & Chen X.B. (3) Dynamic coupling of seakeeping and sloshing, Proc. ISOPE 3, onolulu. Molin B. (979) Second-order diffraction loads upon three-dimensional bodies. App. Ocean Res., 97-. Newman J.N. (974) Second-order, slowly-varying forces on vessels in irregular waves, Proc. Intl Symp. Dyn. Marine Vehicle & Struc. in Waves, Mech. Engng. Pub., London, Newman J.N. & Lee C.. () Boundary-element methods in offshore structure analysis, Proc. th Intl Conf. Off. Mech. Arc. Engeng, Rio de Janeiro. Newman J.N. (4) Progress in wave load computations on offshore structures, Oral presentation at OMAE 4, Vancouver.