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2 Ocean Engineering () 77 7 Contents lists available at ScienceDirect Ocean Engineering journal homepage: Analysis of second-order resonance in wave interactions with floating bodies through a finite-element method C.. Wang a, G.X. Wu b, a Division of Hydrodynamics, China Special Vehicle Research Institute, Hangong Road, Jingmen, 44 Hubei, China b Department of Mechanical Engineering, University College London, Torrington Place, London WCE 7JE, UK article info Article history: Received September 7 Accepted 4 February Available online 4 March Keywords: First- and second-order theory Resonance Multi-floating bodies Time-domain analysis Finite-element method abstract A time-domain method is employed to analyse the resonant oscillations of the liquid confined within the two floating bodies. The velocity potentials at each time step are obtained through a finite-element method (FEM) with quadratic shape functions. The matrix equation of the FEM is solved through an iteration. The radiation condition is satisfied through a combination of the damping zone method and the Sommerfeld Orlansi equation. A detailed analysis is made for two rectangular floating cylinders undergoing forced oscillation. The first-order potential reveals the resonant behaviour of the wave motion at certain frequencies o i, which is similar to sloshing in a tan. More interestingly, the secondorder theory further reveals that when the oscillation frequency is at o i / or half of the resonant frequency, no first-order resonance is observed as expected, but the second-order resonant motion becomes evident, which does not seem to have been extensively investigated so far. Detailed results for two rectangular cylinders are provided to show some insights into the resonant effect due to the interaction between the bodies. The first- and second-order resonant phenomena have been observed and the result has shown that the second-order components have significant influence on the wave and force in some cases, especially at the second-order resonance. & Elsevier Ltd. All rights reserved.. Introduction The motion of a floating body in waves is usually frequency dependent. When the stiffness term due to the hydrostatic restoring force cancels the inertial term due to the body acceleration together with the added mass, resonance can occur and the body may experience large motion. Another type of resonance is the motion of a liquid confined in a tan, or sloshing. When the excitation frequency is equal to one of the natural frequencies of the tan, the wave elevation from the linear velocity potential may become infinitely large. An interesting feature related to large wave loading is that observed by Maniar and Newman (997). They considered the linear diffraction problem by an array of identical vertical cylinders and found that when the wave frequency is near the trapped mode (Ursell, 9) very large force can be found on the cylinders near the middle of the array. Evans and Porter (997) showed that the large waves and wave loading could occur even when there were a few cylinders, such as four, especially when they were close to each other. Corresponding author. Tel.: ; fax: address: gx_wu@meng.ucl.ac.u (G.X. Wu). All these discussions are related to the first-order potential theory. Based on the perturbation theory, when the first-order motion or loading corresponds to a frequency o, the second-order result will correspond to o. Intuitively, one naturally speculates that when the excitation frequency is half of the resonant frequency, the second-order motion or loading might become quite large. This hypothesis seems to be confirmed by the wor of Malenica et al. (999) and the wor by Wang and Wu (7) on a group of cylinders in waves near the trapped mode. However, it is then a big surprise to see that when a rectangular tan is in forced sway motion at half of its first natural frequency, no second-order resonance is observed. This has led to a new speculation of whether the above intuitive speculation is too overspeculative. To uncover the reason for the discrepancy in these two cases, Wu (7) did detailed second-order analysis for the sloshing problem of a rectangular tan in swaying motions. He found that because the motion of the tan is antisymmetric, the first-order potential has only components corresponding to the odd natural frequencies and the second potential to the even natural frequencies. As a result, no second-order resonance occurs when the excitation frequency is half of the first natural frequency or indeed half of any other odd natural frequencies. However, second-order resonance does happen when the excitation frequency is half of any even natural frequencies. This then 9-/$ - see front matter & Elsevier Ltd. All rights reserved. doi:./j.oceaneng...4

3 7 C.. Wang, G.X. Wu / Ocean Engineering () 77 7 shows that the original hypothesis is also valid for the sloshing problem. The present paper considers two rectangular floating cylinders undergoing forced oscillation in the time domain. It is expected that the motion of the liquid confined between the bodies resembles sloshing in a rectangular tan. The difference is that because the confined water is connected to the outside fluid, energy will propagate to infinity and there is a radiation damping. As a result, the liquid motion may not be infinite at the resonant frequency. However, it is still expected to be quite large because of the resemblance of the case to the sloshing. Detailed simulations are made in this paper based on the finite-element method (FEM). These include two floating bodies undergoing in-phase vertical or horizontal motion and out-phase horizontal motion. There has been some second-order analysis for the forced motion of a single body (e.g. Wu, 99). The present analysis is evidently relevant to a catamaran and other similar structures. There have been early papers on interactions between waves and two floating bodies in the frequency domain (e.g. Wang and Wahab, 97; Ohusu, 97; Lee et al., 97; Breit and Sclavounos, 9). However, the second-order resonance investigated in this paper seems to have been overlooed so far. The present wor is therefore of significance in this sense.. Mathematical formulation We consider the hydrodynamic problem of two floating bodies in forced oscillations. As shown in Fig., a right-handed Cartesian coordinate system oxy fixed in the space is defined, in which x is measured horizontally and y points vertically upwards form the still water level. For body, a body-fixed Cartesian system coordinate system o X Ȳ is also defined. When the body is at rest in the calm water, Ȳ points upwards and the origin of o X Ȳ is at x ¼ a, y ¼, taen from the coordinates of the middle point between the two intersections of the body with the still water surface. The body surface is denoted by S and its unit inward normal vector by N ~ ¼ðN x ; N y Þ. The seabed is assumed horizontal along the plane y ¼ h. Let t denote time and be the elevation of the free surface S f relative to the still-water level. When the fluid is assumed incompressible and inviscid, and the flow irrotational, the fluid motion can be described by a velocity potential f, which satisfies the Laplace equation within the fluid domain : r f ¼ in () and is subject to the following boundary conditions: q q qx qx ¼ ons f () þ g þ rf ¼ ons f () qn ¼ ~ N ð ~ V þ ~O ~r Þ on S (4) ¼ ony ¼ h () qn where g is the acceleration due to gravity, V ~ is the translational velocity of body at x ¼ a and y ¼, O * is the rotational velocity around z which points out of the paper passing though (a, ) and ~r ¼ðx a ; y b Þ is the position vector. In addition, the potential satisfies the radiation condition, which will be discussed in detail later. We shall use the perturbation method to solve the above problem up to the second order. The mean position of the body surface S is denoted by S ðþ with the inward normal ~n ¼ðn x; n y Þ. The motion of body can be decomposed into a translation defined by a displacement vector ~ X ¼ðX ; Y Þ and a rotation through an angle Y about the z. The movement of a point ( X Ȳ ) on the body after the translation and the rotation can then be written as ~X ¼ðX þ x cos Y ȳ sin Y ; Y þ x sin Y þ ȳ cos Y Þ ð x ; ȳ Þ () We assume that the body motion relative to its dimension and the amplitude of the generated wave is small. It is then possible to expand the boundary condition on the instantaneous surfaces to their mean positions. Thus we write (Isaacson and Ng, 99) q q qx qx þ q q q þ¼ qx qx þ g þ rf þ þ g þ rf þ¼ ons ðþ f (7) ons ðþ f _~X N ~ ¼½rfþ ~X rðrfþþš N ~ on S ðþ (9) where S () f is the still water surface. The overdot in Eq. (9) indicates the time derivative and thus _~X ¼ V ~ þ O ~ ~r Using the Stoes expansion procedure, we can write f ¼ f ðþ þ f ðþ þ () ¼ ðþ þ ðþ þ () ~ X ¼ ~ X ðþ þ ~ X ðþ () () S f S f () S b o x y S b Y ¼ Y ðþ þ Y ðþ þ () where e is a perturbation parameter related to the wave slope, and the superscripts () and () indicate components at the first- and second-order, respectively. We assume that the body motion is of first order, or S c S c h ~ X ¼ ~ X ðþ (4) Fig.. Coordinate system. Y ¼ Y ðþ () Eq. () becomes ~X ¼ X þ x Y ȳ Y ; Y þ x Y þ ȳy þ Oð Þ ()

4 C.. Wang, G.X. Wu / Ocean Engineering () Similarly, the unit normal vector ~ N can be written as N * ¼ðn x cos Y n y sin Y ; n y cos Y þ n x sin Y Þ ¼ n x þ Y n y Y ; n y þ Y þ n x Y þ Oð Þ (7) The superscript () has been dropped from X, Y, Y for convenience, as their expansions have only one term. Substituting Eqs. () () into the Laplace equation and the boundary condition and rearranging the terms in terms of the order of e, wehave r f ðlþ ¼ in ðþ () and We first consider the hydrostatic term. Assume the body is wall sided. When there is no wave elevation. The hydrostatic term will then become rgðr Y A w Þ ~ j where r is the mean volume displacement of the body and A w is the distance between the two intersections of the mean body surface with the mean free surface. Assume now the runup along the rightand left-hand sides of the body as r and l. When considering the problem to second order, we then have an additional contribution as rg r yð cos Y ~ i sin Y ~ jþ ds þ rg l yðcos Y ~ i þ sin Y ~ jþ ds ¼ rg½cos Yð l r Þ~ i þ sin Y cos Yð l r Þ j* Š ðlþ qðlþ ¼ f l ðþ þ g ðþ ¼ f ðlþ qn ¼ f l on S ðþ f (9) on S ðþ f () on S ðþ () ¼ rgð l r Þ~ i For the dynamic terms in the Bernoulli equation, we write! p d ¼ r ðþ þ ðþ r rfðþ " # r ðx Y Y Þ ðþ x þðy þ Y X Þ ðþ y þ Oð Þ on S ðþ () ðlþ ¼ ony ¼ h () where () is the time-independent fluid domain bounded by the seabed, the mean body surface, and the still-water surface S ðþ. The f terms f l, f l, and f l are given, respectively, as >< ðl ¼ Þ f l ¼ ðþ q ðþ qx qx f ðþ >: ðþq ðl ¼ Þ >< ðl ¼ Þ f l ¼ rfðþ ðþq f ðþ >: ðl ¼ ð X _ ȳ _Y Þn x þð Y _ þ x _Y Þn y!! Y n y X _ ðþ þ Y qx n x Y _ ðþ ><! f l ¼ q f ðþ ðx ȳ Y Þ qx n x þ q f ðþ qx n y! q f ðþ ðy þ x Y Þ n y þ q f ðþ qx n x >:. Hydrodynamic forces ðl ¼ Þ ðl ¼ Þ The pressure due to fluid flow can be obtained from the Bernoulli equation p r rf rgy () The hydrodynamic force F and moment M * on body can be obtained by a direct integration of the pressure over the instantaneous wetted body surface S. ~ F ¼ pn * ds (4) S ~M ¼ pð~r N ~ Þ ds () S Notice S consists of S ðþ and the variation due to the body and wave motion. Combined with this variation, the first term in Eq. () then gives a following second-order contribution to the force rg ðþ r ð ðþ r þ Y þ Y A w =Þ ~ i þ rg ðþ l ð ðþ þ Y l Y A w =Þ ~ i in which Eq. () is used when l ¼. All these derivations give * F ¼ ~ F ðþ þ ~ F ðþ (7) where ~ F ðþ ~ F ðþ ¼ r S ðþ ¼ r S ðþ ðþ " * ðþ þðy K þ Y X Þ ðþ y ry S ðþ þ rg½ ðþ l ðx ðþ l n ds rgy A w ~ j () þ rfðþrf ðþ þðx Y Y Þ ðþ x # ~n ds ðþ ð n ~ y i þ nx ~ jþ ds rgð l r Þ~ i þ Y þ Y A w =Þ ðþ r ðx ðþ r þ Y Y A w =ÞŠ ~ i (9) The corresponding moment components M ~ ðþ and M ~ ðþ may be obtained in a similar way. 4. Finite element discretization and numerical procedures We shall adopt the finite-element method to solve the velocity potential problem at each time step. Quadrilateral parametric element (see Fig. ) is used. The shape functions defined in a local coordinate system ~ x ¼ðx; Þ corresponding to element e with eight nodes may be expressed as N ðeþ i ðx; Þ ¼ 4 ð þ x 9 ixþð i Þð þ x i x þ i Þði ¼ ; ; ; 4Þ >= N ðeþ ðx; Þ ¼ i ð x Þð þ i Þði ¼ ; 7Þ N ðeþ ðx; Þ ¼ i ð Þð þ x i xþði ¼ ; Þ >; ()

5 7 C.. Wang, G.X. Wu / Ocean Engineering () 77 7 y η 7 4 (-,) (,) 4 7 (,) o x (-,) o (,) ξ (-,-) (,-) (-,) Fig.. -node quadrilateral isoparametric element. This gives x ¼ X ¼ x N ðeþ ; y ¼ X ¼ y N ðeþ Once the mesh is generated, the potentials f () ( ¼, ) can be expressed as f ðþ ðx; yþ ¼ Xn J¼ f ðþ J N J ðx; Þ () where f () J is the potential at node J and n is the number of nodes. N j in the equation is equal to N (e) if node J in the global system is node of element e, and (x, y)ae. Otherwise N j ¼. Through the Galerin method, we have r f ðþ N i d¼ () ðþ Using Green s identity and the boundary conditions, we can obtain the following matrix equations: ½KŠff ðþ g¼ff ðþ gð ¼ ; Þ () where K IJ ¼ rn i rn j dðies p & JeS p Þ ðþ F ðþ I ¼ Sn N I f ðþ n ds rn Xn I ðþ I¼ ðjspþ ðf ðþ P ÞrN J dðies p Þ S p in the above equations represents the Dirichlet boundary on which the potentials, denoted by f p () ( ¼, ), are nown, and S n represents the Neumann boundary on which the normal derivatives of the potentials, denoted by f n () ( ¼, ), are nown. The integration in the equation is calculated element by element with respect to (x, ) through the Jacobian. Once the coefficients are found, the matrix equations are then solved through an iteration based on the conjugate gradient method with a symmetric successive over relaxation (SSOR) preconditioner. The first- and second-order derivatives of the linear potential on element nodes are required for the boundary conditions of the second-order potential in Eqs. (9) (). The derivatives have been calculated through a difference method or a Galerin method (Wu and Eatco Taylor, 994; Ma et al., a, b; Wang and Wu, 7; Wang et al., 7). Here, we can differentiate the shape function directly, as they are nonlinear. Thus qx ¼ Xn j¼ f j qn j qx ; ¼ Xn j¼ f j qn j (4) where qn j qx qx 4 qn 7 j ¼ qx 4 qx q qx 7 q qn j qx 4 qn 7 j q () within each element. The second-order derivatives q f/qx, q f/qx and q f/ can be obtained in a similar way through the Jacobian. The mesh used in the perturbation theory is fixed as the problem is solved within the mean domain. For this reason, we may use the following fourth-order Adams Bashforth equation f ðt þ DtÞ ¼f ðtþþ Dt 4 ½f ðtþ 9f ðt DtÞþ7f ðt DtÞ 9f ðt DtÞŠ () for time marching of the wave elevation and the potentials on the free surface, as the information corresponding to the same node at previous time steps can be easily stored. For long-time simulations, an appropriate radiation condition should be imposed on the boundary S c to minimise the wave reflection. Here we use a combination of Sommerfeld condition and a damping zone. Cointe et al. (99) adopted a damping zone technique for their D numerical wave tan. In the damping zone, an artificial viscous term is added into the inematic and dynamic free surface boundary conditions. We have (Cointe et al., 99) q ðþ ¼ ðþ qz f vðþ (7) ðþ ¼ g ðþ þ f nf ðþ () with ( nðxþ ¼ ao x x l x pxpx ¼ x þ bl oxox or x4x where o is the wave frequency, l the linear wave length. The damping zone starts from a point at x and extends to x +bl. Parameters a and b control the strength and the length of the damping zone, respectively, and they may be obtained through numerical experiments. In the following simulations a ¼ b ¼. as in Tanizawa and Swada (99).. Numerical results The cases considered below correspond to the situations where cylinders start motion suddenly. In the numerical simulation, however, the body surface boundary condition is not imposed immediately but satisfied gradually. A modulation function M(t) is

6 C.. Wang, G.X. Wu / Ocean Engineering () applied in Eq. () in a manner similar to that in Isaacson and Ng (99), ¼ MðtÞVðÞ n ð ¼ ; Þ (9) where V () n ( ¼, ) is the first-and second-order normal velocity, respectively, and M(t) is given as ( pt cos T tot MðtÞ ¼ txt and T ¼ p/o is the wave period. The use of the modulation function helps the wave to reach the periodic state more smoothly and quicly. A comparison between results with and without the modulation function was made by Wang and Wu (7), which showed that the adoption of the modulation function greatly improved the result, especially for the second-order component. The alternative is of course to start the motion gradually. The breadth and draught of each cylinder is b and d, respectively. The spacing between the centre lines of the two cylinders is denoted as L cy. Cylinders one and two are located at x ¼ L cy and L cy, respectively. If the liquid motion between the cylinders is approximated as that in a sloshing tan of width (L cy b) and large depth, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi np o n ¼ ðl cy bþ g ; n ¼ ; ;... (4) Fig. 4. Waves at (a) the left side and (b) right side of cylinder one at o ¼ o.---- Linear; second order; linear+second order. then should be a good estimate for resonant frequencies (see Wu, 7). Each cylinder is subjected to the following vertical or horizontal motion X ¼ A sin ot (4) where o and A are the motion frequency and amplitude of each cylinder, respectively. In the following simulations, the water depth is chosen as h ¼ d and the motion amplitude A ¼.d, together with d ¼ b and L cy ¼ b. μ/ρdb The present simualtion Lee et al. (97) ω /g.. Vertical motions We consider a case of both cylinders undergoing the vertical motion or heave. Because the flow is symmetric, the resonant motion may occur only at the even natural frequencies, or n ¼, 4, y in Eq. (4), following the analysis in Wu (7). Thus we choose o ¼ o. Both control surfaces S c are at locations about four times the linear wavelength far from the origin point. On the free surface, segments are used on the left side of cylinder one and on the right side of cylinder two and segments between the two cylinders. There are segments on the control surfaces at the far ends. On each cylinder surface, there are segments and segments along the vertical and horizontal faces, respectively. This corresponds to a mesh of 799 nodes and 4 elements, as shown in Fig.. Results given in Fig. 4 are only for cylinder one because of symmetry. It can be seen that the wave runup along the right side of cylinder one, which is in the confined liquid zone, increases significantly with the time. The runup on the other side of the cylinder also increases with time, but the rate is much smaller. This clearly shows the resemblance of the motion of the confined liquid to wave sloshing in a tan (Wu et al., 99). Fig.. A typical mesh. λ/ρdbω To avoid the possibility that the above observation is due to numerical inaccuracy, the linear result for force F(t) is decomposed into added mass m and damping coefficient l based on the following equations: m ¼ tþt FðtÞ sin ot dt (4) pao t tþt ω /g Fig.. Comparison of vertical added mass and daming coefficient for the twin cylinder in heaving motions. l ¼ FðtÞs cos otdt (4) pa t Comparison has been made with the result obtained by Lee et al. (97) in the frequency domain. Agreement is good, as shown in Fig.. It should be noticed that o ¼ Og is near the natural frequency. At this frequency, the result from time domain

7 7 C.. Wang, G.X. Wu / Ocean Engineering () 77 7 simulation would tae a very long time to get the periodic state and Eqs. (4) and (4) become impractical for calculation. The results at this point are therefore not included in the figure. Based on the perturbation theory, the first-order results will be a forcing term for the second-order potential. When the firstorder potential is big, one would expect the second-order potential to be big too. It is interesting to see that the secondorder runup in Fig. 4 does not reflect this. Similar results have been found for a group cylinder near the trapped mode and detailed analysis has been made by Wang and Wu (7). The reason for this is that while the components of the second-order potential may be large, they cancel each other and therefore the overall second-order runup is not very big. The maximum value of wave peas at the right side reach.4 for first order,. for second order and.7 for the linear plus second order. Fig. gives the results for a single isolated cylinder at the same location. It is seen that the wave amplitude is only about. for first order and. for second order. For the waves at the left side, the results also show an increasing trend in peas, but this is not as fast as that at the right side. It is concluded that the waves in the area between the cylinders have significantly been affected by the mutual interference at the resonant frequency o. As in Wu et al. (99), we run simulations at o ¼.9o and o ¼.o. The results in Figs. 7a and b show that the envelop of the wave runup oscillates with much lower frequency. This is in fact controlled by o o as in the sloshing case. The second-order wave runup in both cases is rather small. We then run a η () /A η () /A Fig.. Waves at (a) the left side of cylinder one at o ¼ o in single-cylinder case ω =.99ω ω = ω ω =.ω ω =.ω 4 Fig.. Waves at right side of the cylinder one ω =.9ω at A at C at B 4 ω =.ω ω =.ω Fig. 7. Waves at right side of the cylinder one Linear, second order; linear+second order Fig. 9. (a) Linear; (b) second order; (c) linear+second order at o ¼ o.

8 C.. Wang, G.X. Wu / Ocean Engineering () simulation at half of the estimated resonant frequency, or o ¼.o. The second-order runup becomes quite significant (see Fig. 7c). However, there is no clear resonance discussed by Wu (7). The reason is obviously is that o is an approximation. Assume the real natural frequencies are o. We first undertae extensive numerical search for o. Fig. gives examples at o ¼.99o, o,.o,.o, which shows that o ¼.o is a better approximation. Fig. 9 gives wave elevations when o ¼ o at three points A, B and C. Here, A is on the right-hand side of cylinder one, B is the middle point between two cylinders and C is the middle point of A and B. The linear free-surface profiles are given in Fig. and the corresponding forces on the cylinder one are given in Fig.. As o is a more accurate result for the resonant frequency, we can investigate the second-order resonance at o ¼.o. Fig. gives the wave runups along the right-hand side of cylinder one. In the figure, is the wavenumber obtained from o /g ¼ tanh h. For comparison, the result at o ¼.4o is also given. The resonant behaviour of the former is far more evident. Further results for the total waves and forces at o ¼.o are given in Fig.. The nonlinearities in the wave and the horizontal force are F x /ρgba F y /ρgba = 9. = 9. = 9.4 = 9. = 9. = x/λ Fig.. Linear free surface profiles at o ¼ o. 4 4 Fig.. Forces on cylinder one at o ¼ o Linear, second order; linear+second order. η () /A F x /ρgba F y /ρgba ω =.ω ω =.ω ' 4 Fig.. Second-order waves at the right side of cylinder one. quite strong. The amplitudes of the first-order wave and forces remain almost constant, but the amplitudes of the second-order wave and horizontal force increase with the time and can even be bigger than the first-order results because of the second-order resonance effect. Further simulations have been made at o ¼ o 4 and o ¼.o 4 and it is found that o 4 is a good approximation for o 4. Fig. 4a gives the first-order wave elevation at o ¼ o 4 and Fig. 4b gives the second-order wave elevation at o ¼.o 4. The resonant behaviour in both cases is obvious. This further confirms the observation in the wor of Wu (7) about the second-order resonance... Horizontal motions in opposite directions We consider the next case in which the cylinders undergo the following motions X ¼ A sin ot; X ¼ A sin ot (44) Fig.. Waves and forces for cylinder one at o ¼.o.----Linear, second order; linear+second order.

9 74 C.. Wang, G.X. Wu / Ocean Engineering () 77 7 η () /A η () /A Fig. 4. (a) first-order wave at o ¼ o 4 and (b) second-order wave at o ¼.o. F x /ρgba F y /ρgba Fig.. Forces on cylinder one at o ¼ o Linear, second order; linear+second order Fig.. Waves for cylinder one at (a) o ¼ o and (b) o ¼.o Linear, second order; linear+second order. F x /ρgba F y /ρgba Fig. 7. Forces on cylinder one at o ¼.o Linear, second order; linear+second order. As in the vertical motion, the flow in this case is symmetric. The resonant behaviour is expected to be similar to that analysed in the vertical motion. Thus, first-order resonance is expected at o ¼ o and second-order resonance at o ¼.o. Figs. 7 give results for cylinder one. The resonance of waves and forces similar to the heaving motions can be observed... Horizontal motions in the same direction We further consider a case in which the two cylinders are undergoing the same horizontal motion defined as X ¼ sin ot or the anti-symmetrical swaying motions. The first-order resonance is now expected at o ¼ o n, n ¼,, y and the second-order resonance at o ¼ :o n (see Wu, 7, for details). Numerically, it is found that o ¼.o, o ¼.o, o ¼ o, o 4 ¼ o 4. Fig. gives wave runups along cylinder one at o ¼ o. The result shows clearly the resonant behaviour of the motion of the liquid confined between the bodies. It is interesting to see that the resonant effect is also quite significant on the left-hand side of the cylinder. Fig. 9 gives the second-order runups along the righthand side of cylinder one at o ¼.o n (n ¼, ). The second forces on cylinder one at o ¼.o and.o 4 are given in Figs. and, respectively. The resonance can also be seen from the figures. All these are consistent with what was discussed in Wu (7).

10 C.. Wang, G.X. Wu / Ocean Engineering () Conclusions A time-domain finite-element method has been used to analyse the second-order resonant liquid motion corresponding to two floating cylinders in forced oscillations. The -node element and quadratic shape functions have been used. The firstand second-order velocity potentials at each time step are obtained through solving matrix equations based on an iteration method. The radiation condition is imposed through a combination of the damping zone method and the Sommerfeld Orlansi equation. Extensive simulations have been made for the two floating bodies in vertical and horizontal motions. The significant effects on wave and force due to resonant interactions have been observed. It has been found that very large second-order waves and hydrodynamic forces may be produced when the excitation frequency is half of the resonant frequencies. This can be Fig.. Waves at (a) the left side and (b) the right of cylinder one at o ¼ o Linear, second order; linear+second order. F x () /ρga F y () /ρga Fig.. Second-order forces on cylinder one at o ¼.o. η () /A ω =.ω ' η () /A 4 ω =.ω ' ω =.ω ' ω =.ω 4 ' η () /A - η () /A Fig. 9. Second-order waves at the right side of cylinder one.

11 7 C.. Wang, G.X. Wu / Ocean Engineering () 77 7 F y () /ρga F x () /ρga obviously extended to the case in which the excitation has multifrequency components. If the sum of difference of any two frequency components is equal to half of one of the natural frequencies, second-order resonance may occur (see Wu (7) for details). It ought to be pointed out, however, that the conclusions made here are based on the perturbation theory. They may be valid only within this framewor. References 4 4 Fig.. Second-order forces on cylinder one at o ¼.o 4. Breit, S.R., Sclavounos, P.D., 9. Wave interaction between adjacent slender bodies. Journal of Fluid Mechanics, 7 9. Cointe, R., Geyer, P., King, B., Molin, B., Tramoni, M., 99. Nonlinear and linear motions of a rectangular barge in a perfect fluid. In: Proceedings of the th Symposium on Naval Hydrodynamics Ann Arbor, Michigan, pp. 9. Evans, D.V., Porter, R., 997. Near-trapping of waves by circular arrays of vertical cylinders. Applied Ocean Research 9, 99. Isaacson, M., Ng, J.Y.T., 99. Time-domain second-order wave radiation in twodimension. Journal of Ship Research 7,. Lee, C.M., Jones, H., Bedels, J.W., 97. Added mass and damping coefficient of heaving twin cylinders in a free surface. NSRDC Report 9. Ma, Q.W., Wu, G.X., Eatoc Taylor, R., a. Finite element simulation of fully nonlinear interaction between vertical cylinders and steep waves. Part : methodology and numerical procedure. International Journal for Numerical Methods in Fluids,. Ma, Q.W., Wu, G.X., Eatoc Taylor, R., b. Finite element simulation of fully nonlinear interaction between vertical cylinders and steep waves. Part : numerical results and validation. International Journal for Numerical Methods in Fluids, 7. Malenica, S., Eatoc Taylor, R., Huang, J.B., 999. Second order water wave diffraction by an array of vertical cylinders. Journal of Fluid Mechanics 9, Maniar, H.D., Newman, J.N., 997. Wave diffraction by a long array of cylinders. Journal of Fluid Mechanics 9, 9. Ohusu, M., 97. On the heaving motion of two circular cylinders on the surface of a fluid. Reports of the Research Institute for Applied Mechanics XVII (), 7. Tanizawa, K., Swada, H., 99. A numerical method for nonlinear simulation of -D body motion in waves by means of BEM. Journal of the Society of Naval Architectures of Japan. Ursell, F., 9. Trapping modes in the theory of surface waves. Proceedings of the Cambridge Philosophical Society 47, 47. Wang, S., Wahab, R., 97. Heaving oscillations of twin cylinders in a free surface. Journal of Ship Research, 4. Wang, C.., Wu, G.X.,. An unstructured mesh based finite element simulation of wave interactions with non-wall-sided bodies. Journal of Fluids & Structures, Wang, C.., Wu, G.X., 7. Time domain analysis of second order wave diffraction by an array of vertical cylinders. Journal of Fluids & Structures (4),. Wang, C.., Wu, G.X., Drae, K.R., 7. Interactions between fully nonlinear water wave and non-wall-sided D structures. Ocean Engineering 4, 9. Wu, G.X., 99. Second order wave radiation by a submerged horizontal circular cylinder. Applied Ocean Research, 9. Wu, G.X., 7. Second order resonance of sloshing in a tan. Ocean Engineering 4, Wu, G.X., Eatco Taylor, R., 994. Finite element analysis of two-dimensional nonlinear transient water waves. Applied Ocean Research, 7. Wu, G.X., Ma, Q.W., Eatoc Taylor, R., 99. Numerical simulation of sloshing waves in a D tan based on a finite element method. Applied Ocean Research, 7.