WAVE-BODY INTERACTIONS FOR A SURFACE-PIERCING BODY IN WATER OF FINITE DEPTH *

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1 ,(6): DOI: /S (09) WAVE-BODY INTERACTIONS FOR A SURFACE-PIERCING BODY IN WATER OF FINITE DEPTH * LI Yong, LIN Mian Institute of Mechanics, Chinese Acadey of Sciences, Beiing , China, E-ail: liyong@iech.ac.cn (Received June 17, 010, Revised August 30, 010) Abstract: Nonlinear wave-body interactions for a stationary surface-piercing body in water of finite depth with flat and sloping bottos are siulated in a two-diensional nuerical wave tank, which is constructed ainly based on the spatially averaged Navier-Stokes equations with the k odel for siulating the turbulence. The equations are discretized based on the finite volue ethod and the schee of the pressure iplicit splitting of operators is eployed to solve the Navier-Stokes equations. By using the force tie histories, the ean and higher-haronic force coponents are calculated. The coputational results are shown to be in good agreeent with experiental and nuerical results of other researchers. Then, the horizontal force, the vertical force and the oent on the surface-piercing body under nonlinear regular waves with flat and sloping bottos are obtained. The results indicate that the botto topographies have a significant influence on the wave loads on the surface-piercing body. Key words: wave-body interaction, surface-piercing body, nuerical wave tank, finite water depth 1. Introduction Many types of surface-piercing structures, such as acket platfors and an-ade islands, have been used in coastal engineering in the offshore oil and gas developent. Usually the structures are subected to strong nonlinear water wave ipacts and are precarious under the extree ocean situations. Therefore, it is iportant to investigate the wave forces on surface-piercing structures in water of finite depth. There are any studies of the interactions between the water waves and structures of various ierged types, for exaple, the fully suberged structures [1-3], vertical bodies reposed on seabed [4-6] and the surface-piercing bodies [7-10]. Aong the, the probles of surface-piercing bodies have attracted uch attention recently. Tanizawa and Minai [7] developed a two diensional nuerical wave tank to * Proect supported by the National Natural Science Foundation of China (Grant No ), the Knowledge Innovation Progra of Chinese Acadey of Sciences (Grant Nos. KJCX-YW-L07, KZCX-YW-1-). Biography: LI Yong (1978-), Male, Ph. D. Corresponding author: LIN Mian, E-ail: linian@iech.ac.cn siulate the radiation and diffraction of a single surface-piercing body. Bai et al. [8] investigated the nonlinear hydrodynaic forces on a surface-piercing body of arbitrary shape in three diensions by a tie doain second-order ethod. Koo and Ki [9] considered the single and double bodies based on potential theory and Boundary Eleent Method (BEM). Wang et al. [10] developed a three diensional tie-doain coupled nuerical odel to obtain the nonlinear wave forces acting on a box-shaped body. In these studies, the effects of the water depth are neglected. In the present article, the wave forces and the rotational oents are investigated for a fixed structure under the finite water depth condition. First, a -D nuerical wave tank is built and verified. The siulated results are copared with the experiental results of Noiri and Murayaa [11], the analytical solution of Maruo [1] and the nuerical results of Koo and Ki [9]. Second, the nonlinear wave-body interactions with a flat botto are siulated in this tank. The ean wave force, the haronic force coponents and the rotational oents on the body are calculated and the results are copared with those under the condition of deep water. Finally, the wave loads on a fixed body with sloping topography is

2 746 studied.. Matheatical forulation.1 Governing equations The governing equations are the Reynolds averaged Navier-Stokes equations + u = 0 t x p ui + uui = + t x xi x u ui e + Du i i + gi xi x where x ( =1,,3) represents the coordinate (1) () coponents, u is the fluid velocity, p is the pressure, is the density, g is the acceleration of gravity, and D i is the daping coefficient and the daping ter D i u i is added to the oentu equation directly. = +, where is the fluid e -6 viscosity, = s, f is the turbulent eddy viscosity, f = C k /. k is the turbulent kinetic energy, and is the turbulent energy dissipation rate. The k two-equation odel is adopted to estiate the turbulence. k+ uk = t x x f k + + Pk k x f (3) + u = + f t x x + CP 1 k fc x (4) k k P k where ui u ui = f + x x i x k and (5) are the turbulent Schidt nubers. The constants in Eqs.(3)-(5) take the following values: C =0.09, C 1 =1.44, C =1.9, k =1.0 and =1.33. In order to capture the water-air free surface, an Eulerian ethod naed the Volue Of Fluid (VOF) ethod is adopted. The equation for the volue fraction is + u = 0 t x (6) where is the volue fraction of water and 1 represents the volue fraction of air. The volue fraction of each liquid is used as the weighting factor to get the ixture properties, such as density and viscosity, i.e., = + 1 w (7) a where w and a represent the density of water and air, respectively.. Nuerical schee The equations are discretized based on the Finite Volue Method (FVM). In order to guarantee the coputational precision, the central difference schee with second-order accuracy and the Quadratic Upwind Interpolation of Convective Kineatics (QUICK) schee with third-order accuracy are eployed to represent the diffusion ter and convection ter, respectively. The Preconditioning Conugate Gradient (PCG) ethod is eployed for solving the algebraic equations. For solving the Navier-Stokes equation of incopressible fluid flow, one will encounter a proble of the pressure-velocity coupling. Here, the Pressure Iplicit Splitting of Operators (PISO) schee is eployed to treat the coupling. k and equations are coupled in the schee and solved by a segregated approach. For the sake of coputational stability, the Courant nuber is used to obtain the tie step at every beginning of the calculation cycle ( =0.1for wave cases), defined as C n C n where u t x = i i t is the tie step and (8) xi represents the esh size. When the solution in the fluid doain is obtained, the tie history of the wave force on the bodies can be obtained by integrating the pressure and the viscous force. The flow chart for the siulation process is plotted in Fig.1, where tnow is the current tie in calculation and tend is the total calculation tie.

3 747 For a free surface siulation, generally, the divergence is an issue, because of the sharp interface between water and air. In this work, a Copressive Interface Capturing Schee for Arbitary Meshes (CICSAM) is used to capture the fluid interfaces with eshes of arbitrary topology. Details about the CICSAM can be found in Ubbink and Issa [13]. 3 aa ch az + d cos ( ax t) 8 4 sh ad aga shaz + d u y = sin( ax t)+ ch ad (9) 3 aa sh az + d cos ( ax t) 8 4 sh ad (10) where A, a and d are the wave aplitude, frequency, wave nuber, and water depth, respectively. x is the distance fro the origin to the wave aking point. At the outlet, the pressure and the turbulence quantities are also specified as of zero noral gradients and the velocity is specified by Soerfeld radiation condition, expressed by u 1 u + = 0 x c t (11) Fig.1 Coputation flow chart The entire coputational doain with the botto topography is shown in Fig. and there are basically five types of boundary associated with the governing equation: inlet, outlet, structure wall, bed, and atosphere. where c is the wave propagation velocity at the end point of the daping zone. At the surfaces of the structure, no slip wall boundary condition is used. The velocity vector at the bed is set to be zero and the zero noral gradient condition is chosen for other quantities. For the atosphere boundary, the total pressure is set to zero and k and are set with zero noral gradient. The botto boundary of the doain is the bed. No slip wall boundary condition and the wall function are used. The wall function can be suarized as follows [14] l u = 0Lexp 1 u (1) Fig. Sketch of the fluid doain with the botto topography Boundary conditions associated with regular waves are prescribed along the inlet of the coputational doain. The pressure and the turbulence quantities including k and are set to be of zero noral gradient and the velocity vector is specified by Stokes and Cnoidal regular waves. The theoretical velocity of second-order Stokes regular waves can be expressed by u x aga ch az + d = cos( ax t)+ ch ad k = u u = l C 3 (13) (14) where l represents the ixing length, 0 is a constant, = , is the Karan constant, =0.4, u is the friction velocity, L is the characteristic length, which is the distance of the

4 748 centre of the near-wall grid to the botto wall..3 Nuerical ipleentation The prograing is based on the open source coputational fluid dynaics code naed Open Field Operation And Manipulation (OpenFOAM), of the version 1.5, which can be downloaded freely through the internet (OpenCFD 008). OpenFOAM provides a fundaental platfor to write new solvers for different probles and the tensorial approach and the obect oriented techniques are used. Xu [15] studied the nuerical wave tank with OpenFOAM, but with the daping zone being ignored. In the present work, the solution for the coputational doain is carried out by the adoption of the turbulence solver for incopressible two-phase flow. Based on Eqs.(1)-(14), the default wall function in OpenFOAM is odified. The inflow boundary and two daping zones are added to the original OpenFOAM solver. As for the inlet location, according to Eq.(9) and Eq.(10), a new wave aking boundary naed wavetievaring is prograed with C++ language. Correspondingly, in order to absorb the wave energy reflection fro the end-wall and the re-reflection fro the input boundary, artificial daping zones are allocated at the two ends of the doain. In this paper, for the right end daping zone, the daping ter Diu i is added to the oentu equation of OpenFOAM solver, as shown in Eq.(). D is expressed as i nuber of the air phase in the surface doain. In order to solve this proble, a odified schee is proposed. In the coputational doain, if the region is occupied by the air phase, the velocity and the pressure of this zone are set to zero. This will not alter the nuerical results because w is uch larger than a. 3. Nuerical results 3.1 Model validation Firstly, we investigate the characteristics of the flow field in the nuerical wave tank. In the present case, d =5.0, T =4.5s, A/ d =0.1, where T is the wave period. When the calculation becoes stable, the tie series of the wave profile at x =78.9 indicate that the daping schees presented in the present paper work well, as shown in Fig.3. In addition, the calculated horizontal and vertical velocities at the point x =78.9 and y =0 are copared with the theoretical results, which shows a good agreeent. It is shown that the two-diensional wave nuerical tank perfors well for regular waves. g Di = i n +1 h 0 n x x (15) l where l is the length of the daping zone, x 0 is the distance fro the origin to the starting point of the daping zone. n and are the daping coefficients, n =, i =0.6. Siilar ethods were used by Dong and Zhan [16]. For the left daping zone, it is also a wave aking region and the velocity in this area is odified by ui = ui + Di ui u i at the end of each tie step. i D i is written as l-x D i = 1 l (16) In addition, the interface capturing schee in OpenFOAM for two-phase flow is iproved. In the course of using this schee, it is found that the schee is indeed easy to use and can capture the surface well except that it is very slow. The tie step becoes very sall and the coputation always takes a long tie. The reason for this is the large Courant Fig.3 Coparisons of tie series for wave profile and velocities Secondly, for the verification of the siulated

5 749 wave forces, the calculation results are copared with Noiri and Murayaa [11], Maruo [1], and Koo [9], respectively. The coputational doain is shown in Fig.4, where is the wave length. A surfacepiercing body is fixed in the iddle of the doain and the radius of the round corner is The input wave properties are listed in Table 1. The incident wave height is Figure 5 shows the coputational eshes used for the siulations. In order to accurately capture the wave surface and calculate the wave forces, the eshes in surface zones and zones along body are refined. results and Koo s nuerical results. Figures 7-9 show the coparison of the force coponents and the oent. The first-haronic force coponents and the rotational oent are calculated by Fourier analysis. The coputational results are shown to be in good agreeent with experiental results except in the region of 1.5. It is believed that the deviation ay be attributed to viscous and other nonlinear effects. It is also noted that the present results are ore consistent with experiental results than Koo s, which is based on potential theory. It is shown that the turbulence of fluid has an influence on the wave-body interactions even in deep water. Fig.4 Sketch of the doain for verification Table 1 Incident wave inputs Case T g/ d A/ d () Fig.7 Coparison of first-haronic horizontal noralized force Fig.8 Coparison of first-haronic vertical noralized force Fig.5 Coputational eshes used for the siulation Fig.9 Coparison of first-haronic noralized oent Fig.6 Coparison of drift force Figure 6 shows the coparison of the drift force with Noiri s experiental results, Maruo s analytical 3. Doain with flat botto topography The coputational doain is shown in Fig. with tan = 0. The length of daping zone is, where represents the axiu wavelength of all

6 750 cases. The water depth for the incident region is 5.0. Tables and 3 give, respectively, the values of diension paraeters and the input wave properties, where T g/ d is the diensionless paraeter to represent the nonlinearity of the incident wave. Table Diension paraeters W () 100 d () 5.0 h () 0.5 B () 5.0 Table 3 Incident wave inputs Case T g/ d A/ d () features with the increase of T g/ d. Figure 11 shows the coparison of horizontal force coponents for finite depth and deep water conditions. Horizontal force coponents for finite water depth condition are approxiately siilar to the cases for deep water condition, while the first and second order forces are uch greater than those in deep water when the wave frequency is low ( 0.8 ). The agnitude of the first order can be 47% greater than that of deep water at = Fig.11 Coparison of horizontal force coponents for two water depth conditions The sae kind of coparisons for vertical force is shown in Fig.1. The first order vertical force under finite water depth condition is ore sensitive to the variations of and is significantly larger than that under deep water condition in four cases with = 0.4, 0.33, 0.4 and In case of = 0.63, the first-haronic coponent can be as large as 5% of that in deep water. Due to the wave reflection fro flat botto and re-reflection fro body surface, the first order coponents can be enhanced. Fig.10 The total forces in the x- and y-directions Figure 10 shows the total noralized forces ( Fs and Fh ), which are plotted against the nondiensional tie ( t/ T). In this figure, the steady-state tie series results for three different cases are shown, fro which a series of haronic coponents for horizontal force, vertical force and rotational oent can be obtained. We can also see that the horizontal force shows ore nonlinear Fig.1 Coparison of vertical force coponents for two water depth conditions Figure 13 shows the coparison of rotational oent coponents for two water depth conditions. The first order coponent under finite water depth condition is uch greater than that under deep water condition for cases with low frequencies ( 0.8 ). Specially, the agnitude of the first order coponent gradually increases up to 144% of the value at the

7 751 case of = This phenoenon indicates that the state of the structure becoes ore unstable and precarious in shallow water than in deep water, especially when the frequency is low. order coponents of all paraeters for sloping botto condition are close to the results for flat botto condition except in the low frequency region ( 0.33 ). Fig.13 Coparison of oent coponents for two water depth conditions 3.3 Doain with sloping botto topography The coputational doain is shown in Fig. with tan = Coputational conditions and input settings for sloping botto topography are the sae as those for flat botto topography. Figures show the coparison of horizontal force, vertical force and rotational oent coponents for sloping botto and flat botto topography. Because of the wave reflection fro sloping botto, not only the vertical force coponent but also the horizontal force and rotational oent coponents are affected. The first order coponents becoe uch larger in soe cases. Fig.14 Coparison of horizontal force coponents for two types of botto topographies As for the horizontal force coponent, the values for two types of bottos are alost the sae for in the range between 0.8 and 0.5, while the first-haronic coponent is 3% greater than that for flat botto in case of = A siilar trend for the variations of the oent coponent can be observed in Fig.16. Fro Fig.15, we can see that the first order vertical force for sloping botto is larger than that for flat botto when The second and third Fig.15 Coparison of vertical force coponents for two types of botto topographies Fig.16 Coparison of oent coponents for two types of botto topographies 4. Conclusions In the present article, the nonlinear wave-body interactions for a stationary surface-piercing body under regular waves in water of finite depth are siulated in a -D nuerical wave tank. The tank odel is based on Reynolds averaged Navier Stokes equations and k two-equation odel. The central difference schee and QUICK schee are eployed to represent the diffusion ter and convection ter, respectively. The PCG ethod is eployed for solving the algebraic equations. Boundary conditions associated with regular waves are prescribed along the left side of the coputational doain. Accordingly, artificial daping zones are allocated at the two ends of the doain to absorb the wave energy reflection and re-reflection. In order to capture the water-air free surface, VOF ethod and CICSAM schee are adopted. All these features are ipleented by utilizing the open source code naed OpenFOAM. In the solution of the fluid doain, the tie history of the wave nonlinear force on the bodies can be obtained by integrating the pressure and viscous force. By using the force tie histories, the ean and

8 75 higher-haronic force coponents are calculated. Under the finite water depth condition, the first order forces and rotational oent coponents are uch greater than those under deep water condition when the frequency is low. For the doain with sloping botto topography, the second and third order coponents of all paraeters under sloping botto conditions are close to the results under flat botto condition except in the low frequency region. Due to the wave reflection and re-reflection, the forces and rotational oent coponents vary in a coplicated way with the increase of frequency, especially under the sloping botto condition. References [1] KOO W. C., KIM M. H. and TAVASSOLI A. Fully nonlinear wave-body interactions with fully suberged dual cylinders[j]. International Journal of Offshore and Polar Engineering, 004, 14(3): [] VENGATESAN V., VARYANI K. S. and BARLTROP N. D. P. Wave force coefficients for horizontally suberged rectangular cylinders[j]. Ocean Engineering, 006, 33(11-1): [3] LIU Yong, LI Yu-cheng and TENG Bin. Wave otion over two suberged layers of horizontal thick plates[j]. Journal of Hydrodynaics, 009, 1(4): [4] SUNG H. G., HONG S. Y. and CHOI H. S. Evaluation of nonlinear wave forces on a fixed body by nuerical wave tank techniques[c]. Proceedings of the 3rd International Conference on Hydrodynaics. Seoul, Korea, 1998, [5] LI Chi-wai, LIN Peng-zhi. A nuerical study of three-diensional wave interaction with a square cylinder[j]. Ocean Engineering, 001, 8(1): [6] YOU Yun-xiang, SHI Qiang and MIAO Guo-ping. The radiation and diffraction of water waves by a botto-ounted circular cylinder in a two-layer fluid[j]. Journal of Hydrodynaics, Ser. B, 007, 19(1): 1-8. [7] TANIZAWA K., MINAMI M. Estiation of wave drift force by nuerical wave tank[c]. Proceedings of the 9th ISOPE Conference. Brest, France, 1999, [8] BAI Wei, TENG Bin and QIU Da-hong. Real tie siulation of second-order radiation of 3D bodies[j]. Journal of Hydrodynaics, Ser. A, 003, 18(4): (in Chinese). [9] KOO W. C., KIM M. H. Fully nonlinear wave-body interactions with surface-piercing bodies[j]. Ocean Engineering, 007, 34(7): [10] WANG Da-guo, ZOU Zhi-li and TANG Chun-an. Tie stepping solutions of nonlinear wave forces on a three-diensional box-shaped ship in a harbor[j]. Journal of Ship Mechanics, 007, 11(4): (in Chinese). [11] NOJIRI N., MURAYAMA K. A study on the drift force on two diensional floating body in regular waves[j]. Transactions of the West-Japan Society Naval Architect, 1975, 51: [1] MARUO H. On the increase of the resistance of a ship in rough seas[j]. Journal of Zosen Kiokai, 1960, (108): 5-13(in Japanese). [13] UBBINK O., ISSA R. I. A ethod for capturing sharp fluid interfaces on arbitrary eshes[j]. Journal of Coputational Physics, 1999, 153(1): [14] QI Peng, WANG Yong-xue and HOU Yi-un. Nuerical siulation of solitary waves overtopping a breakwater[j]. Journal of Hydrodynaics, Ser. A, 004, 19(1): (in Chinese). [15] XU Shao-kun. A nuerical wave tank based on OpenFOAM and its application[d]. Ph. D. Thesis, Tianin: Tianin University, 008(in Chinese). [16] DONG Zhi, ZHAN Jie-in. Coparison of existing ethods for wave generating and absorbing in VOFbased nuerical tank[j]. Journal of Hydrodynaics, Ser. A, 009, 4(1): 15-1(in Chinese).