水动力 波浪 文献

Article history: Received 21 February 2010 Accepted 14 June 2011 Keywords: Dynamic response Dual pontoon floating structure (DPFS) Numerical wave tank (NWT) The trend of using floating structures with cage aquaculture is becoming more popular in the open sea. The purpose of this paper is to investigate the dynamic properties of a dary idth, rtant. Weng and Chou (2007) applied a boundary element method (BEM) and a physical model to examine the dynamic Although the results offered by both articles give valuable insights into the dynamic responses of a dual pontoon structure, Contents lists available at ScienceDirect Journal of Fluids and Structures Journal of Fluids and Structures ] (]]]]) ]]]–]]] 0889-9746/$ - see front matter Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfluidstructs.2011.06.009 n Corresponding author. Tel.: þ886 7 525 5169; fax: þ886 7 525 5060. E-mail addresses: hjtang@thl.ncku.edu.tw (H.-J. Tang), cchuang@mail.nsysu.edu.tw (C.-C. Huang), zp7689@hotmail.com (W.-M. Chen). Pleas two- responses of a floating dual pontoon structure. They discovered that the clear space between pontoons had a significant influence on the responses of the structure and created an extra peak response in heave motion at a high frequency. essential for marine culture in the open sea. The hydrodynamic properties of the DPFS were investigated by Williams and Abul-Azm (1997), using a boun integral equation method. They found that the wave reflection properties of the structure depend strongly on the w draft and spacing of the pontoons and mooring line stiffness. The pre-tension in the mooring system is less impo In recent years, the development of offshore cage culture has attracted wide attention. The central spar fish cage (Fredriksson et al., 2003), tension-leg type fish cage (Lee and Wang, 2005), and gravity-type cage (Huang et al., 2007) are some examples. On the other hand, the dual pontoon floating structure (DPFS) has become popular due to its versatile features, such as using wave barriers, a catamaran (twin hull boat), or a device for raising fish. The proposed DPFS for cage aquaculture may consist of a dual pontoon with a large fish net hanging between them. The deck of each pontoon offers space for a control room, automatic feeding machine, crane, rearing or harvest equipments, etc. All of these facilities are Boundary element method (BEM) 1. Introduction e cite this article as: Tang, H.-J., et dimensional numerical wave tank. J and numerical models. A two-dimensional (2-D) fully nonlinear numerical wave tank (NWT), based on the boundary element method (BEM), is developed to calculate the wave forces on the DPFS. The wave forces on a fish net system are then evaluated using a modified Morison equation. The comparisons of dynamic behaviors between numer- ical simulations and experimental measurements on the DPFS show good agreement. Results also display that a fish net system causes the resonant response of body motions and mooring forces to be slightly lower due to the net’s damping effect. Finally, for designing the rearing space of cage aquaculture, the influences which net depth and net width have on the DPFS dynamic responses are also presented in this paper. Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved. dual pontoon floating structure (DPFS) when attached to a fish net by using physical Dynamics of dual pontoon floating structure for cage aquaculture in a two-dimensional numerical wave tank Hung-Jie Tang a, Chai-Cheng Huang b,n, Wei-Ming Chen c a Tainan Hydraulics Laboratory, National Cheng Kung University, Tainan 70955, Taiwan b Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan c Ocean Technology Department, Industrial Technology Research Institute, Hsinchu 31040, Taiwan a r t i c l e i n f o a b s t r a c t journal homepage: www.elsevier.com/locate/jfs al., Dynamics of dual pontoon floating structure for cage aquaculture in a ournal of Fluids and Structures (2011), doi:10.1016/j.jfluidstructs.2011.06.009 effects on fixed and floating structures with three-dimensional (3-D) arbitrary shapes. After the numerical damping zone H.-J. Tang et al. / Journal of Fluids and Structures ] (]]]]) ]]]–]]]2 had been proposed (Cointe, 1990; Tanizawa, 1996), researchers started to take this advantage to develop the numerical wave tank (NWT) for long period simulations. Over the last two decades, the BEM-based NWT has become one of the most popular tools in ocean engineering research, due to its highly computational efficiency, e.g. Cointe (1990, 2-D), Tanizawa (1996, 2-D), Contento (2000, 2-D), Boo (2002, 3-D), Koo and Kim (2004, 2-D), and Bai and Eatock Taylor (2006, 2007, 3-D). The review of 2-D and 3-D NWT developments and their applications can be found in detail in Kim et al. (1999) and Tanizawa (2000). The descriptions about time-stepping technique, re-gridding method, numerical damping zone, and wave–body interaction problems etc., were also presented in these works. However, the fully nonlinear simulation for a freely floating body is still considered as a very challenging problem, since it requires the treatment of the coupling between hydrodynamic forces and body motions. To precisely obtain wave forces on a body, a highly accurate calculation of the time derivative of velocity potential ft is required for the unsteady Bernoulli equation. Tanizawa (1995) introduced the acceleration potential and derived the exact body boundary condition for the acceleration field. This was known as the acceleration potential method. Koo and Kim (2004) used this method to investigate the fully nonlinear wave–body interactions for a freely floating barge-type structure in a 2-D NWT. On the other hand, Wu and Eatock Taylor (1996) proposed another technique to solve the velocity and acceleration field separately. The merit of this method is in computing faster than that of Tanizawa, while the disadvantage is that it cannot compute the pressure distribution on the body surface unless it solves another boundary value problem for the auxiliary functions, details referred to Kim et al. (1999). Bai and Eatock Taylor (2009) used this method to simulate the fully nonlinear wave interactions with fixed and freely floating flared structures in a 3-D NWT. The above BEM-based methods were shown to be robust and stable, and thus are suitable to deal with floating body problems. Recently, Yan and Ma (2007) and Ma and Yan (2009) developed a quasi arbitrary Lagrangian–Eulerian finite element method (QALE-FEM) for 2-D and 3-D floating body problems. In thus method, an improved iterative procedure, called iterative semi-implicit time integration method for floating body (ISITIMFB) is incorporated to deal with the fully nonlinear interaction problems between steep waves and floating bodies, from which the numerical results are identified in good agreement with other published data. In recent years, the fully nonlinear wave–body interactions for multiple floating structures (2-D) have been investigated by many researchers. For example, Koo and Kim (2007) studied the shielding effect and the pumping/sloshing modes of a water column with various gap distances for fixed double barges in a 2-D NWT. Koo (2009) simulated wave blocking and wave energy absorption of a pneumatic-type floating breakwater in a 2-D NWT. Huang and Tang (2009) developed a 2-D NWT to investigate the dynamic response of body motion and tension force for a moored DPFS. Sun et al. (2010) investigated the hydrodynamic interactions between waves and two parallel closely spaced rectangular barges, to characterize the general problem of liquid natural gas (LNG) offloading from a floating plant into a shuttle tanker. In the present study, we applied the 2-D NWT to investigate the dynamic response of the DPFS for cage aquaculture, based on our previous studies, Huang et al. (2008) and Huang and Tang (2009), while the calculation of wave loadings on aquaculture cage nets could be found in Loland (1991), Huang et al. (2006), and Moe et al. (2010). In this paper, the boundary integral equation (BIE) is formulated by using a linear element method. Subsequently, the nonlinear free surface is traced by the MEL technique, and the 4th order Runge–Kutta method (RK4). Meanwhile a node-regridding and smoothing technique is applied by using a cubic spline scheme to prevent free surface nodes frommoving too close to each other. Two numerical damping zones proposed by Cointe (1990) and Tanizawa (1996) are used at both ends of the numerical wave tank to absorb or dissipate the reflected and transmitted waves induced by wave–body interaction. The immediate hydrodynamic force is calculated by an acceleration potential method (Tanizawa, 1995) and a modal decomposition method (Vinje and Brevig, 1981). The wave forces on the fish net are evaluated by using a modified Morison equation (Brebbia and Walker, 1979) in order to deal with fluid and net relative motion problems. In this paper, both numerical and physical models are adopted to investigate the dynamic properties of the DPFS with and without a fish net. The first- to third-order of sway, heave, roll, and sea-side tension RAO (response amplitude operator) of the DPFS with or without a fish net are investigated. Finally, for designing the rearing space of cage aquaculture, the influences of net depth and net width on dynamic responses are also discussed. 2. Numerical model A DPFS, which consists of a pair of floating rectangular pontoons, is restrained by a linear symmetric mooring system, as shown in Fig. 1, where a is the width of each pontoon, b is the spacing between two pontoons, d is the draft, (xG, zG) is the position of the gravity center, lG is the roll moment arm, y0 is the mooring line angle, and l0 is the original length of mooring line. The floating structure is deployed in a numerical wave tank with a constant water depth, h. A numerical further application to cage aquaculture remains unknown and should be investigated. Besides, these two studies were limited to linear wave theory. To solve the nonlinear wave problem, Longuet-Higgins and Cokelet (1976) introduced the Mixed Eulerian and Lagrangian method (MEL). By this method, the fully nonlinear boundary conditions on the free water surface can be satisfied instantaneously. Hereafter, the time-domain approach of the nonlinear wave–body interaction using the BEM has become an extremely successful scheme. For instance, Isaacson (1982) used this scheme to simulate the nonlinear wave Please cite this article as: Tang, H.-J., et al., Dynamics of dual pontoon floating structure for cage aquaculture in a two-dimensional numerical wave tank. Journal of Fluids and Structures (2011), doi:10.1016/j.jfluidstructs.2011.06.009 H.-J. Tang et al. / Journal of Fluids and Structures ] (]]]]) ]]]–]]] 3 damping zone is used at each end of the wave tank to dissipate reflected and transmitted waves, where xd1 and xd2 are the entrance positions. 2.1. Governing equation The flow field is assumed to be incompressible, inviscid, and the motion irrotational. Thus, a velocity potential exists and satisfies Laplace equation, r2f¼ 0: ð1Þ Incorporating Eq. (1) into the Green second identity, the velocity potential in the fluid domain can be determined by solving the following BIE: afi ¼ Z Gj @Gij @n fj�Gij @fj @n � � dGj, ð2Þ where Gij ¼ lnrij=2p is the fundamental solution to the Laplace equation and represents a flow field generated by a concentrated unit source acting at ith source point. rij is the distance from source point (xi, zi) to field point (xj, zj) and a is the internal solid angle between two boundary elements. In this model, the linear element scheme and 6-point Gaussian quadrature integration method are applied to solve the BIE. 2.2. Inflow boundary condition Based on the continuity of velocity, a theoretical particle velocity profile can be used to specify the boundary value along the inflow boundary. For nonlinear regular waves, the second-order Stokes wave is used to prevent the mismatch between input velocity profiles and real water particle velocity, as described in Koo and Kim (2004) gAk coshkðzþhÞ2 3 Mooring line DPFS Fish net θ θ l 3L l a ab d h x z x 3L2L 2L d Fig. 1. Definition sketch of a DPFS with a fish net. @f @n ¼� s coshkh cosðkx�stÞ þ 3 4 A2ks cosh2kðzþhÞ sinh4kh cos2ðkx�stÞ 6664 7775fm on GI , ð3Þ where A, k, and s are the amplitude, wave number, and angular frequency, respectively. g is the gravitational acceleration and t is the time. The modulation function fm is used to prevent impulse-like behavior of a wave maker, and it is written as fmðtÞ ¼ 1 2 1�cos pt Tm � �� � for toTm, 1 for tZTm, 8>< >: ð4Þ where Tm is the modulation duration which depends on wave steepness. For a steeper wave, the modulation duration is usually twice as long as a regular wave period. 2.3. Free surface boundary condition One of the most popular and successful approaches to the fully nonlinear free surface simulation is the MEL method, which was first presented by Longuet-Higgins and Cokelet (1976). In this method, the kinematic and dynamic free surface boundary conditions are transformed into the Lagrangian frame. To obtain numerical solutions for wave propagation in a Please cite this article as: Tang, H.-J., et al., Dynamics of dual pontoon floating structure for cage aquaculture in a two-dimensional numerical wave tank. Journal of Fluids and Structures (2011), doi:10.1016/j.jfluidstructs.2011.06.009 H.-J. Tang et al. / Journal of Fluids and Structures ] (]]]]) ]]]–]]]4 wave tank, the scheme used numerical damping zones at both ends of the wave tank to absorb the transmitted wave energy at the end of the tank as well as to dissipate the reflected waves in front of the input boundary. Following Cointe (1990) and Tanizawa (1996), the numerical damping zones are incorporated into the free surface boundary conditions as dx dt ¼ @f @x dz dt ¼ @f @z �nðxÞðz�zeÞ df dt ¼�gzþ 1 2 9rf92�nðxÞðf�feÞ on Gf , 8>>>>>>>< >>>>>>>: ð5Þ where n(x) is the damping coefficient of numerical damping zone given by nðxÞ ¼ ads½ðxd1�xÞ=L�2 xrxd1, ads½ðx�xd2Þ=L�2 xZxd2, 0 otherwise: 8>< >: ð6Þ ad is the dimensionless parameter for the strength of the damping zone, and after several tests we found that ad set to 1 is adequate for accurate results. L is the wavelength of the input wave, xd1 and xd2 are the entrance positions of each damping zone shown in Fig. 1. ze and fe in Eq. (5) are the entrance wave elevation and potential function in the front damping zone, only existing in xrxd1. Tanizawa (1996) has applied this damping zone technique to dissipate the wave energy reflected from the structure, yet without disturbing the outgoing incident waves. For practical purposes, the nonlinear analytical solution of the second-order Stokes wave theory is adopted in the damping zone 1 to improve the computational process. In this model, the entrance potential and wave elevation are written as fe ¼ Ag s coshkðzþhÞ coshkh sinðkx�stÞþ 3 8 A2s cosh2kðzþhÞ sinh4kh sin2ðkx�stÞ, ze ¼ Acosðkx�stÞþ kA2 coshkh 4sinh3 kh ð2þcosh2khÞcos2ðkx�stÞ: 8>>>< >>>: ð7Þ Additionally, the nodal velocities in Eq. (5) are obtained by using the cubic spline scheme in the curvilinear coordinate system as described in Section 2.6. The corner problem between free surface and body surface is treated according to Grilli and Svendsen (1990) as described in Section 2.7. 2.4. Body surface boundary condition In this model, the body surface (Gs) is impermeable. Therefore, the fluid velocity is equal to the normal velocity on the body surface @f @n ¼ n1 _xGþn2 _zGþn3 _yG on Gs, ð8Þ where (n1,n2,n3)¼(nx,nz,rznx�rxnz)is the unit normal vector on the body boundary; (rx,rz)¼(x�xG,z�zG) is the position vector from body surface to gravity center. Subscript G designates the gravity center of the body; ð _xG, _zGÞ are the translational velocities in the x and z axes (sway and heave motions) and _yG is the angular velocity about the y axis (roll motion). 2.5. Rigid boundary condition At the end-wall (Gw) and bottom (Gb) of the wave tank, the boundary conditions are considered impermeable. The normal velocities are then set to zero: @f @n ¼ 0 on Gb and Gw: ð9Þ 2.6. Curvilinear coordinate system In this paper a curvilinear coordinate system and the cubic spline scheme are adopted to solve the spatial derivatives of velocity potential in Eq. (5) on the free surface boundary. The relationship between the velocity components in Cartesian and curvilinear coordinates is written as @ff @x ¼ @ff @s cosbf� @ff @n sinbf , @ff @z ¼ @ff @s sinbf þ @ff @n cosbf , 8>>>< >>>: ð10Þ Please cite this article as: Tang, H.-J., et al., Dynamics of dual pontoon floating structure for cage aquaculture in a two-dimensional numerical wave tank. Journal of Fluids and Structures (2011), doi:10.1016/j.jfluidstructs.2011.06.009 H.-J. Tang et al. / Journal of Fluids and Structures ] (]]]]) ]]]–]]] 5 where ff represents the potential function on the free surface, and bf is the angle between s, a section of free surface, and the x axis. The normal velocities of the free surface qff/qn are obtained after solving the BIE, and the angle bf is determined from the following equation: tanbf ¼ sinbf cosbf ¼ @z=@s @x=@s , ð11Þ where qff/qs, qx/qs, and qz/qs are calculated by using cubic spline interpolation in curvilinear coordinates along the free surface. Once the values of the time derivative of the potential function on the right side of Eq. (5) are known, the substantial derivative equations on the left side can be used to predict the new nodal position and its corresponding potential on the free surface boundary by employing the RK4 method as a time marching scheme. This process was repeated until the simulation reached the steady-state condition. Note that the node-regridding and smoothing technique is also applied in the present model by using the cubic spline interpolation on the curvilinear coordinate system, to prevent free surface nodes from moving too close to each other, and to prevent the saw-tooth condition occurring which may lead to the numerical instability. 2.7. Corner problem between free surface and body surface At the intersection of body surface and free surface, the discontinuity of flux occurs due to the discontinuity of normal direction. Although the cubic spline scheme is accurate to determine the tangential slope at the end-point with the nature condition (curvature equal to zero) and the Lagrangian polynomial method, the requirement of continuity of flux at the corner is still difficult to meet. To deal with this discontinuity, the double collocation node technique is often used. Grilli and Svendsen (1990) proposed that a treatment for the corner problem at the intersection based on the continuity flux is @ff @s ¼ @ff @n cosðbb�bf Þ sinðbb�bf Þ � @fb @n 1 sinðbb�bf Þ , ð12Þ where the subscripts b and f denote the body and water free surface, respectively. @fb=@n and @ff =@n are the normal velocities on the free surface and body surface, respectively; @ff =@s is the modified tangential velocity on the free surface and will be used in Eq. (10) when dealing with corner problem. In this model, the input boundary angle at the front of tank is bb ¼ p=2, while that of the wall boundary at the end of wave tank is bb ¼ 3p=2. 2.8. Wave forces on the body The hydrodynamic forces on the body can be calculated by integrating the pressure around the wetted body surface as F ¼ RGs�r ftþgzþ 12 9rf92 � � nds, M ¼ RGs�r ftþgzþ 12 9rf92 � � r � nds, 8>< >: ð13Þ where F and M are the hydrodynamic force and moment, r is the water density, n is the unit normal vector on the body surface and points into the body, and r is the position vector from gravity center to body surface. 2.9. Acceleration potential method In ord