- Email: [email protected]
Dynamic responses of floating fish cage in waves and current
Ocean Engineering 72 (2013) 297–303
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Dynamic responses of floating fish cage in waves and current Li Li, Shixiao Fu n, Yuwang Xu, Jungao Wang, Jianmin Yang State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China
art ic l e i nf o
a b s t r a c t
Article history: Received 31 May 2012 Accepted 13 July 2013 Available online 6 August 2013
This paper studies dynamic responses of the semi-immerged floater and the fish cage system consisting of the floater and nets in waves and currents. The net and floater is modeled by truss and beam element respectively, where the corresponding geometry nonlinearity in the deformations and motions is considered. A “Buoyancy Distribution” method is developed to address the instantaneous buoyancy on the floater in waves. Particularly, the motions of the flexible floater and the net volume reduction in different wave and current conditions are investigated. The effects of the interactions between the net and floater on the dynamics of the system are also studied. Additionally, the effects of the net drag forces on the dynamic responses of the fish cage system are investigated. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Floating fish cage Dynamic response Instantaneous buoyancy FEM Net volume reduction Flexible deformation
1. Introduction As the near-shore spaces of today can hardly satisfy the increasing demand for marine foods, the fish farming industry has the tendency to move to offshore area (Lader and Fredheim, 2001; Lee et al., 2009). New challenges may appear due to the severe sea loads when the fish cages move to the more exposed ocean areas. Large floating structures, such as ships and platforms, are usually treated as rigid bodies, and the wave-induced forces are normally calculated by Green function method based on the potential flow theory. However, regarding small scale floating structures, the viscous nonlinear forces become dominated, and hence the potential flow theory is no longer applicable. Such a condition makes it difficult to predict the dynamic responses of a flexible aquaculture structure in waves and currents. Many researchers have conducted both experimental and numerical studies on the dynamic responses of the fish cage system. By dividing the net into super elements, Lader and Fredheim (2006) developed a numerical model to study the net deformation in different conditions. A mass-spring model was developed by Lee et al. (2009) to investigate the dynamic behavior of the fish cages. Lader et al. (2007) experimentally investigated the wave forces on the nets by model test. Kristiansen and Faltinsen (2009) studied the nonlinear wave-induced motions of cylindrical-shaped floaters by both model tests and numerical wave tank. Berstad et al. (2005) compared the results by the numerical program Aquasim with the Norwegian Standard 9415.
n
Corresponding author. Tel.: +86 13501947087; fax: +86 21 34207050x2031. E-mail address: [email protected] (S. Fu).
0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.07.004
Lader et al. (2008) studied current-induced net deformations by full-scale measurements at sea. In this paper, dynamic responses of the individual semiimmerged floater and the whole fish cage system including the floater and nets in waves and currents are studied. The net and floater is modeled by truss and beam element respectively. A “Buoyancy Distribution” method is developed to simulate the instantaneous buoyancy on the partly-immerged floater in waves. The motions of the flexible floater and the net volume reduction in different wave and current conditions are investigated. Also studied are the interactions between the net and floater. In addition, by varying the drag coefficient Cd on the nets, we investigate the effect of the net drag forces on the dynamic responses of the whole system.
2. Basic theory For the floater and the whole fish cage system, the structural dynamic equilibrium equation can be written as: ½m½x€ þ ½c½x_ þ k ½x ¼ G þ f b þ f w þ f c ð1Þ where [m] is the total mass matrix of the floater and nets, [c] is damping matrix from the structural deformations of the floater and nets, [k] is the stiffness matrix of the deformed floater and nets. The loads applied on the structure include gravity G, buoyancy fb, wave forces fw and current forces fc. The gravity force, as G presents, should be balanced by the static buoyancy force and applied to the system in the first place, before the dynamic analysis, to find out the mean water line of the floating system.
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2.1. Hydrodynamic forces In traditional linear analysis procedures for large floating structures, buoyancy fb, wave forces fw and current forces fc in Eq. (1) are usually linearly superposed: the buoyancy fb is represented by the hydrostatic matrix; wave forces fw are calculated by diffraction and radiation theory; and current forces fc are calculated by the Morison equations. Instead of diffraction and radiation theory, here Morison equations (Faltinsen, 1990) are used to calculate all of the hydrodynamic forces on the floating fish cage system, including wave forces fw and current forces fc. All of the nonlinear interactions between those forces will be represented by hydrodynamic coefficients, the relative motion velocity and acceleration, as shown in the following equations. π 1 F ¼ C m ρ D2 u_ þ C d ρDju þ Ujðu þ UÞ 4 2
ð2Þ
taking the relative motions between the structure and water particle into consideration, the above equation can be further modified as: π 1 F ¼ C m ρ D2 ðu_ 7 ap Þ þ C d ρDju 7 vp 7 Ujðu 7 vp 7 UÞ 4 2
ð3Þ
where C m is inertia coefficient, C d is drag coefficient, D is cylinder diameter, u is wave velocity, U is current velocity, ρ is water density, the velocity and acceleration of the structure element is vp and ap respectively. 2.2. Buoyancy As Eqs. (2) and (3) present, the buoyancy force is neglected when only Morison equations serve to calculate the external loads on floating system. However, buoyancy remarkably affects the instantaneous hydrodynamic forces on the floating system and should be regarded as one of the external force components. The buoyancy force on the instantaneous wetted surface of the floater can be written as f b ¼ ρgV m
the separated beam section points n um i ui ¼ 0
um i
ð6Þ uni
where and represent the ith degree of the freedom (DOF) displacement at beam section point m and n (Dassault System, 2010), which indicates that in addition to the equilibrium of the dynamic motion equations of the whole system, um i are always forced to equal with uni at any moment. By repeatedly applying the constrain equations like Eq. (6), one can assign a same displacement, in the corresponding DOF, to all of the beam section points at a floater section. Furthermore, to make sure that the distributed beam sections have the same inertia and deforming properties as the original floater, the mass and bending stiffness of the floater should be evenly distributed to separated beams, e.g. msection ¼ ∑mi
ð7Þ
ðEI Þsection ¼ ∑ðEI Þi
ð8Þ
where msection and ðEIÞsection are the mass density and bending stiffness of the original floater section; mi and ðEIÞi are the mass density and the bending stiffness of the ith beam in the distributed beam section.
ð4Þ
where ρ is water density, V m is the instantaneous immerged volume of the structure. In standard FEM analysis software, beam element are usually adopted to simulate the deformations and motions of the floater in current and waves, where the beam section is treated as a 2D point. Whether to take into account the buoyancy in loads depends on the instantaneous position of the beam section point relative to the free surface of waves. Specifically, were the beam section point under the water surface, the buoyancy of the whole cross section would be applied; otherwise, the buoyancy is neglected. However, in real conditions, the partly-immerged floater section leads to timevarying buoyancy. Besides, such buoyancy forces of the submerged portion support the whole cage system. Therefore, the instantaneous buoyancy plays pivotal role in studying dynamic motions and deformations of the floater in waves and current. Regarding the instantaneous wetted surface effects on the buoyancy forces of the floater, a “Buoyancy Distribution” method is developed by dividing the whole beam section of the floater into several distributed and coupled beams sections as shown in Fig. 1. Fig. 1 indicates that several separated beam sections are bounded together to simulate the whole original floater cross section. Thus, the instantaneous buoyancy forces of the whole floater section f B_section should be the sum of the buoyancy on each immerged beam section ðf B_immerged_beams Þi , and can be expressed as: f B_section ¼ ∑ðf B_immerged_beams Þi
Fig. 1. Illustration of the distributed coupled beam section.
ð5Þ
To insure that the distributed beam section can move and deform as “one section”, the following constrain equations are applied to
3. Numerical examples 3.1. Floater model Fig. 2 shows the numerical model of the floater, and the main properties of a single floater model are listed in Table1. The floater is modeled by coupled distributed beams, and the nonlinear springs simulate the mooring lines on the floating fish cage system. A zero force is defined when the springs are under compression. The gravity and buoyancy of the floater are balanced to keep the floater exactly half-submerged. With Eqs. (5)–(8), the section properties of each distributed beams are calculated and then listed in Table 2 where H represents the relative distance from the center of each distributed beam section to the center of the whole floater section, R and I respectively symbolize the radius and inertia moment of each distributed beam section. The density and elastic modules of each beam are set as 512 kg/m3 and 7044.139 MPa respectively. With the combination of those distributed beams, the bending stiffness of 0.296842 N m2 and mass density 355 kg/m of the floater cross section properties can be modeled. 3.2. Fish cage model with nets and floater Fig. 3 shows the numerical model of the fish cage system. The properties of fish cage system, including the floater, nets and the sinker, are listed in Table 3. The distributed beam sections and truss elements without bending stiffness model the floater and the nets separately. At the bottom, we neglect the nets and simulate
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Fig. 3. The whole fish cage system model.
Table 3 Properties of fish cage system. Scale ratio Diameter of fish cage (m) Depth of fish cage (m) Outer diameter (m) Thickness (m) Elastic modules (MPa) Density (kg/m3) Mooring line stiffness (N/m)
Fig. 2. The floater model.
Table 1 Single floater model properties. Field diameter of fish cage (m) Floater outer diameter (m) Thickness (m) Elastic modules (MPa) Density (kg/m3) Mooring line stiffness (N/m)
40 0.3 0.048 950 953 6000 (extended)
Table 2 Cross section properties of the distribute beams. No.
H (m)
R (m)
I (m4)
10 8 6 4 2 1 3 5 7 9 11
0.135 0.12 0.09 0.06 0.03 0 0.03 0.06 0.09 0.12 0.15
0.020508372 0.027393377 0.04508354 0.049735191 0.052223756 0.053343923 0.053343923 0.052223756 0.049735191 0.04508354 0.034219738
1.38866E-07 4.4203E-07 3.24296E-06 4.80314E-06 5.83905E-06 6.35637E-06 6.35637E-06 5.83905E-06 4.80314E-06 3.24296E-06 1.07641E-06
the bottom ring with beam elements with certain weight and bending stiffness. Aimed at calculation efficiency, we utilize identical truss elements to model the net cage, which represent the properties of parallel net twines (Moe et al., 2010). To specify, the section area and tension stiffness of the truss element are set to be equal with the sum of those net twines.
4. Results and discussions ABAQUS/Standard is chosen to accomplish the nonlinear dynamic analysis. And the waves and current are propagating along the horizontal direction.
1:1 20 20 Sinker Floater 0.1 0.3 – 0.02 950 950 953 2000 6000 (extended)
Net 0.005 – 350 1120
4.1. Floater model The instantaneous deformed shapes of the floater in regular wave condition (height: 5 m, period: 8 s) are shown in Fig. 4. According to Fig. 4, the floater has not only the rigid body motions but also the structural deformations in wave. Hence the elasticity of the floater should be considered in the dynamic analysis of the system (Endresen, 2011). Time history of displacement responses of four points A, B, C and D as marked in Fig. 2 are plotted in Fig. 5. From Fig. 5, we can see that the motion frequency of the floater is the same as incident wave frequency, and the motion amplitude at vertical direction is the same as that of the incident waves. The vertical displacement is symmetric about the mean water free surface. These characters above represent that in vertical direction, the floater′s motions will basically follow the wave profile. Additionally, remarkable y direction displacements at points B and D are found, which indicates a large structural deformation on the floater in wave, whereas the y direction displacement at points A and C are always zero due to the symmetry of the structure. Dynamic responses of the single floater under different wave period and wave height are further investigated, and the corresponding results are plotted in Figs. 6 and 7. It can be seen that under the same wave period, the vertical motions change significantly with different wave heights. The larger wave height, the larger vertical motion will be found. However, under the fixed wave heights, the vertical motion amplitudes kept almost constant with different wave period. These results reveal the motions characters of the floater in waves: the floater will always ride on the waves. The effect of instantaneous buoyancy on the dynamic response of the floater is plotted in Fig. 8 where the comparison is performed between results with and without “Buoyancy Distribution” method. From Fig. 8(a), we can see that the instantaneous buoyancy will increase the amount of fluctuation under small wave height situation, which reveals the instantaneous buoyancy of the floater section change nonlinearly with the change of water surface. Yet when the wave height is pretty large compared with the
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floater section, the influences of instantaneous buoyancy become not as significant as that in small wave height conditions, as shown in Fig. 8(b). The corresponding motion trail of point A on the floater is further depicted in Fig. 9 (wave height: 5 m, wave period: 8 s). In Fig. 9, vertical motion of floater is within the range of the water particle motion; yet the horizontal motion is smaller than that of water particles, due to the floater deformations and the nonlinear spring restoring forces. These characters render the motion trail of the floater far from critical oval-shaped like
the water particles′ trail. However, we still can conclude that there are explicit similarities between the motion trails of point A and water particles, which demonstrate that the floater motions will basically follow the motions of water particles in wave conditions. 4.2. Fish cage with net and floater The net volume may reduce rapidly when the fish cage is exposed to strong currents (Berstad et al., 2005; Lader and Enerhaug, 2005). The volume reduction coefficient C v can be estimated by Cv ¼
Vd V0
ð9Þ
Fig. 6. Vertical motions of point A (different wave heights).
Fig. 4. Deformed shape of the floater in wave at different time step.
Fig. 7. Vertical motions of point A (different wave periods).
Fig. 5. Dynamic displacement responses of the single floater in wave. (a) Time history of displacement, point A. (b) Time history of displacement, point B. (c) Time history of displacement, point C. (d) Time history of displacement, point D.
L. Li et al. / Ocean Engineering 72 (2013) 297–303
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Fig. 8. Dynamic responses of point A with and without “Buoyancy Distribution” method under different waves. (a) Response under Small Wave Height and (b) Response under Large Wave Height.
Fig. 9. Motion trail of point A.
where V 0 is the initial volume of the nets and V d is the volume when the nets are exposed to current. The net deformations in different currents are shown in Fig. 10. As seen from Fig. 10(a), the net volume is significantly reduced in current velocity 0.6 m/s. The large reduction in volume occurred at high current velocity, as shown in Fig. 10(b). Fig. 11 further indicates that net volume reduction coefficient serves as a function of the current velocity. From Fig. 11, we can see that the volume of the fish cage reduces rapidly with the increase of the current velocity. The net volume is proximately half (52%) of the initial volume as the current velocity increases to 1.0 m/s. At current velocity 0.2 m/s, the net volume is 92% of the initial, whereas at current velocity 1.5 m/s, the net volume reduced significantly to 37% of the initial volume. Similar results have been reported by Lader and Enerhaug (2005), where an approximate 40% net volume reduction was observed. The instantaneous deformed shapes of fish cage system in waves (height: 5 m, period: 8 s) and current (velocity: 0.5 m/s) condition are further plotted in Fig. 12. From Fig. 12, a large geometric deformation of the fish cage system can be observed. The wave and current induce violent dynamic responses of the fish cage system. Comparing Fig. 12 and Fig. 10, we can see that in current only conditions, the net will deform in the incoming current direction, and the motions of the floater and the net will entirely be “in phase”. Nevertheless, different phase between the motions of the floater and the net will appear when wave is applied. To study the interaction between the nets and the floater, we compared the displacement of point A in different conditions— with or without the nets, and the corresponding results are plotted in Figs. 13 and 14.
Fig. 10. Net deformations in different currents (a) net deformations in current velocity of 0.6 m/s. (b) Net deformations in current velocity of 1.5 m/s. (c) Bird view of net deformation at different current velocity.
Fig. 11. Net volume under different current velocity.
As shown in Fig. 13, the vertical motion of the single floater is a little bit larger than that of the whole fish cage model, which is primarily attributable to the relative weight of the nets and sinker in the water. However, in the horizontal direction, the net cage would considerably affect the floater motion, as shown in Fig. 14.
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Fig. 15. Displacements of point A at different Cd values (wave height: 5 m, wave period: 8 s).
Table 4 Maximum stress of the floater.
Fig. 12. Instantaneous deformed shape of the cage system exposed to current and wave.
Fig. 13. Displacement of point A in z direction.
Case no.
Max. stress of the floater (107 Pa)
1 2
1.372 2.354
the dynamic responses of the fish cage system, we conduct a series of calculations with different net drag coefficients; the results are plotted in Fig. 15. According to Fig. 15, the change of Cd slightly affects vertical motions of the floater, but highly contributes to the motions of the floater in the wave and current propagation direction. The maximum stress of the floater in different wave and current conditions are listed in Table 4. The calculation cases are: (1) wave only (wave height: 5 m, wave period: 8 s); (2) wave and current combination (wave height: 5 m, wave period: 8 s, current velocity: 0.5 m/s). It is explicit in Table 4 that the floater can withstand a high stress to great extent in wave and current. Further comparing case 1 and case 2, we can conclude that the current induces a large increase in stress on the floater. Since current loads mainly act on nets, it would take risks to evaluate the strength of the floater without consideration of nets.
5. Conclusions In this paper, we investigate the dynamic behavior of a floating fish cage system in current and waves. “Buoyancy Distribution” method is developed to simulate the instantaneous buoyancy of the floater. The following conclusions can be drawn:
Fig. 14. Displacement of point A in x direction.
The reason might be that in wave and current conditions, large horizontal current-induced forces would act on nets and hence lead to a larger horizontal motion of the whole system. The marine organism would generate large amounts of fouling. Such marine fouling might adhere to the net body and increase the drag force of the nets. To study the effect of the net drag forces on
(1) In wave conditions, the floater will undergo rigid body motions and also deformations. (2) The motions of the floater in waves will increase with the wave height, and will basically follow the motion of water particles. (3) The instantaneous buoyancy would be of great importance where wave height or the floater vertical motion is small. (4) Large geometric deformations and motions are observed in both floater and the nets. The net volume reduces rapidly under the current: even 63% reduction is observed in the current velocity 1.5 m/s. (5) The interaction between the floater and the nets has a large impact on horizontal motions of the system, and hence should be considered in the design stage.
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(6) The drag forces on nets may greatly influence the dynamic motions of the floater. (7) The Cd in use to calculate the force on the nets largely affect the horizontal motions of the floater, since the nets are attached on the floater.
Acknowledgments The fanatical supports from SINTEF Fisheries and Aquaculture as well as the Natural Science Foundation of China (Grant nos. 51009088 and 51279101) are highly acknowledged. References Berstad, A.I., Tronstad, H., Sivertsen, S.A., Leite, E., 2005. Enhancement of design criteria for fish farm facilities including operations. In: Proceedings of OMAE 2005. Paper no. OMAE-67451. Dassault System, 2010. ABAQUS 6.10 Documentation.
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Endresen, P.C., 2011. Vertical Wave Loads and Response of a Floating Fish Farm with Circular Collar. Master Thesis, NTNU. Faltinsen, O.M., 1990. Sea Loads on Ships and Offshore Structures. Cambridge University Press, Cambridge, UK, pp. 223–228. Kristiansen, D., Faltinsen, O.M., 2009. Nonlinear wave-induced motions of cylindrical-shaped floaters of fish farms. Proceedings of the Institution of Mechanical Engineers, Journal of Engineering for the Maritime Environment 223 (3), 361–375. Lader, P., Fredheim, A., 2001. Modelling of Net Structures Exposed to 3D Waves and Current. In: Proceedings of OOA IV Open Ocean Aquaculture. Lader, P., Enerhaug, B., 2005. Experimental investigation of forces and geometry of a net cage in uniform flow. IEEE Journal of Ocean Engineering 30 (1), 79–84. Lader, P., Fredheim, A., 2006. Dynamic properties of a flexible net sheet in waves and current. Aquacultural Engineering 35 (3), 228–238. Lader, P., Jensen, A., Sveen, J.K., Fredheim, A., Enerhaug, B., Pal, D., 2007. Experimental investigation of wave forces on net structures. Applied Ocean Research 29 (3), 112–127. Lader, P., Dempster, T., Fredheim, A., Jensen, Ø., 2008. Current induced net deformations in full-scale sea-cages for Atlantic salmon. Aquacultural Engineering 38 (1), 52–65. Lee, C.W, Lee, G.H., Choe, M.Y., Song, D.H., Hosseini, S.A., 2009. Dynamic behavior of a submersible fish cage. In: Proceedings of OMAE 2009. Paper no. OMAE-79328. Moe, H., Fredheim, A., Hopperstad, O.S., 2010. Journal of Fluids and Structures 26 (3), 503–516.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Dynamic responses of floating fish cage in waves and current Li Li, Shixiao Fu n, Yuwang Xu, Jungao Wang, Jianmin Yang State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China
art ic l e i nf o
a b s t r a c t
Article history: Received 31 May 2012 Accepted 13 July 2013 Available online 6 August 2013
This paper studies dynamic responses of the semi-immerged floater and the fish cage system consisting of the floater and nets in waves and currents. The net and floater is modeled by truss and beam element respectively, where the corresponding geometry nonlinearity in the deformations and motions is considered. A “Buoyancy Distribution” method is developed to address the instantaneous buoyancy on the floater in waves. Particularly, the motions of the flexible floater and the net volume reduction in different wave and current conditions are investigated. The effects of the interactions between the net and floater on the dynamics of the system are also studied. Additionally, the effects of the net drag forces on the dynamic responses of the fish cage system are investigated. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Floating fish cage Dynamic response Instantaneous buoyancy FEM Net volume reduction Flexible deformation
1. Introduction As the near-shore spaces of today can hardly satisfy the increasing demand for marine foods, the fish farming industry has the tendency to move to offshore area (Lader and Fredheim, 2001; Lee et al., 2009). New challenges may appear due to the severe sea loads when the fish cages move to the more exposed ocean areas. Large floating structures, such as ships and platforms, are usually treated as rigid bodies, and the wave-induced forces are normally calculated by Green function method based on the potential flow theory. However, regarding small scale floating structures, the viscous nonlinear forces become dominated, and hence the potential flow theory is no longer applicable. Such a condition makes it difficult to predict the dynamic responses of a flexible aquaculture structure in waves and currents. Many researchers have conducted both experimental and numerical studies on the dynamic responses of the fish cage system. By dividing the net into super elements, Lader and Fredheim (2006) developed a numerical model to study the net deformation in different conditions. A mass-spring model was developed by Lee et al. (2009) to investigate the dynamic behavior of the fish cages. Lader et al. (2007) experimentally investigated the wave forces on the nets by model test. Kristiansen and Faltinsen (2009) studied the nonlinear wave-induced motions of cylindrical-shaped floaters by both model tests and numerical wave tank. Berstad et al. (2005) compared the results by the numerical program Aquasim with the Norwegian Standard 9415.
n
Corresponding author. Tel.: +86 13501947087; fax: +86 21 34207050x2031. E-mail address: [email protected] (S. Fu).
0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.07.004
Lader et al. (2008) studied current-induced net deformations by full-scale measurements at sea. In this paper, dynamic responses of the individual semiimmerged floater and the whole fish cage system including the floater and nets in waves and currents are studied. The net and floater is modeled by truss and beam element respectively. A “Buoyancy Distribution” method is developed to simulate the instantaneous buoyancy on the partly-immerged floater in waves. The motions of the flexible floater and the net volume reduction in different wave and current conditions are investigated. Also studied are the interactions between the net and floater. In addition, by varying the drag coefficient Cd on the nets, we investigate the effect of the net drag forces on the dynamic responses of the whole system.
2. Basic theory For the floater and the whole fish cage system, the structural dynamic equilibrium equation can be written as: ½m½x€ þ ½c½x_ þ k ½x ¼ G þ f b þ f w þ f c ð1Þ where [m] is the total mass matrix of the floater and nets, [c] is damping matrix from the structural deformations of the floater and nets, [k] is the stiffness matrix of the deformed floater and nets. The loads applied on the structure include gravity G, buoyancy fb, wave forces fw and current forces fc. The gravity force, as G presents, should be balanced by the static buoyancy force and applied to the system in the first place, before the dynamic analysis, to find out the mean water line of the floating system.
298
L. Li et al. / Ocean Engineering 72 (2013) 297–303
2.1. Hydrodynamic forces In traditional linear analysis procedures for large floating structures, buoyancy fb, wave forces fw and current forces fc in Eq. (1) are usually linearly superposed: the buoyancy fb is represented by the hydrostatic matrix; wave forces fw are calculated by diffraction and radiation theory; and current forces fc are calculated by the Morison equations. Instead of diffraction and radiation theory, here Morison equations (Faltinsen, 1990) are used to calculate all of the hydrodynamic forces on the floating fish cage system, including wave forces fw and current forces fc. All of the nonlinear interactions between those forces will be represented by hydrodynamic coefficients, the relative motion velocity and acceleration, as shown in the following equations. π 1 F ¼ C m ρ D2 u_ þ C d ρDju þ Ujðu þ UÞ 4 2
ð2Þ
taking the relative motions between the structure and water particle into consideration, the above equation can be further modified as: π 1 F ¼ C m ρ D2 ðu_ 7 ap Þ þ C d ρDju 7 vp 7 Ujðu 7 vp 7 UÞ 4 2
ð3Þ
where C m is inertia coefficient, C d is drag coefficient, D is cylinder diameter, u is wave velocity, U is current velocity, ρ is water density, the velocity and acceleration of the structure element is vp and ap respectively. 2.2. Buoyancy As Eqs. (2) and (3) present, the buoyancy force is neglected when only Morison equations serve to calculate the external loads on floating system. However, buoyancy remarkably affects the instantaneous hydrodynamic forces on the floating system and should be regarded as one of the external force components. The buoyancy force on the instantaneous wetted surface of the floater can be written as f b ¼ ρgV m
the separated beam section points n um i ui ¼ 0
um i
ð6Þ uni
where and represent the ith degree of the freedom (DOF) displacement at beam section point m and n (Dassault System, 2010), which indicates that in addition to the equilibrium of the dynamic motion equations of the whole system, um i are always forced to equal with uni at any moment. By repeatedly applying the constrain equations like Eq. (6), one can assign a same displacement, in the corresponding DOF, to all of the beam section points at a floater section. Furthermore, to make sure that the distributed beam sections have the same inertia and deforming properties as the original floater, the mass and bending stiffness of the floater should be evenly distributed to separated beams, e.g. msection ¼ ∑mi
ð7Þ
ðEI Þsection ¼ ∑ðEI Þi
ð8Þ
where msection and ðEIÞsection are the mass density and bending stiffness of the original floater section; mi and ðEIÞi are the mass density and the bending stiffness of the ith beam in the distributed beam section.
ð4Þ
where ρ is water density, V m is the instantaneous immerged volume of the structure. In standard FEM analysis software, beam element are usually adopted to simulate the deformations and motions of the floater in current and waves, where the beam section is treated as a 2D point. Whether to take into account the buoyancy in loads depends on the instantaneous position of the beam section point relative to the free surface of waves. Specifically, were the beam section point under the water surface, the buoyancy of the whole cross section would be applied; otherwise, the buoyancy is neglected. However, in real conditions, the partly-immerged floater section leads to timevarying buoyancy. Besides, such buoyancy forces of the submerged portion support the whole cage system. Therefore, the instantaneous buoyancy plays pivotal role in studying dynamic motions and deformations of the floater in waves and current. Regarding the instantaneous wetted surface effects on the buoyancy forces of the floater, a “Buoyancy Distribution” method is developed by dividing the whole beam section of the floater into several distributed and coupled beams sections as shown in Fig. 1. Fig. 1 indicates that several separated beam sections are bounded together to simulate the whole original floater cross section. Thus, the instantaneous buoyancy forces of the whole floater section f B_section should be the sum of the buoyancy on each immerged beam section ðf B_immerged_beams Þi , and can be expressed as: f B_section ¼ ∑ðf B_immerged_beams Þi
Fig. 1. Illustration of the distributed coupled beam section.
ð5Þ
To insure that the distributed beam section can move and deform as “one section”, the following constrain equations are applied to
3. Numerical examples 3.1. Floater model Fig. 2 shows the numerical model of the floater, and the main properties of a single floater model are listed in Table1. The floater is modeled by coupled distributed beams, and the nonlinear springs simulate the mooring lines on the floating fish cage system. A zero force is defined when the springs are under compression. The gravity and buoyancy of the floater are balanced to keep the floater exactly half-submerged. With Eqs. (5)–(8), the section properties of each distributed beams are calculated and then listed in Table 2 where H represents the relative distance from the center of each distributed beam section to the center of the whole floater section, R and I respectively symbolize the radius and inertia moment of each distributed beam section. The density and elastic modules of each beam are set as 512 kg/m3 and 7044.139 MPa respectively. With the combination of those distributed beams, the bending stiffness of 0.296842 N m2 and mass density 355 kg/m of the floater cross section properties can be modeled. 3.2. Fish cage model with nets and floater Fig. 3 shows the numerical model of the fish cage system. The properties of fish cage system, including the floater, nets and the sinker, are listed in Table 3. The distributed beam sections and truss elements without bending stiffness model the floater and the nets separately. At the bottom, we neglect the nets and simulate
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Fig. 3. The whole fish cage system model.
Table 3 Properties of fish cage system. Scale ratio Diameter of fish cage (m) Depth of fish cage (m) Outer diameter (m) Thickness (m) Elastic modules (MPa) Density (kg/m3) Mooring line stiffness (N/m)
Fig. 2. The floater model.
Table 1 Single floater model properties. Field diameter of fish cage (m) Floater outer diameter (m) Thickness (m) Elastic modules (MPa) Density (kg/m3) Mooring line stiffness (N/m)
40 0.3 0.048 950 953 6000 (extended)
Table 2 Cross section properties of the distribute beams. No.
H (m)
R (m)
I (m4)
10 8 6 4 2 1 3 5 7 9 11
0.135 0.12 0.09 0.06 0.03 0 0.03 0.06 0.09 0.12 0.15
0.020508372 0.027393377 0.04508354 0.049735191 0.052223756 0.053343923 0.053343923 0.052223756 0.049735191 0.04508354 0.034219738
1.38866E-07 4.4203E-07 3.24296E-06 4.80314E-06 5.83905E-06 6.35637E-06 6.35637E-06 5.83905E-06 4.80314E-06 3.24296E-06 1.07641E-06
the bottom ring with beam elements with certain weight and bending stiffness. Aimed at calculation efficiency, we utilize identical truss elements to model the net cage, which represent the properties of parallel net twines (Moe et al., 2010). To specify, the section area and tension stiffness of the truss element are set to be equal with the sum of those net twines.
4. Results and discussions ABAQUS/Standard is chosen to accomplish the nonlinear dynamic analysis. And the waves and current are propagating along the horizontal direction.
1:1 20 20 Sinker Floater 0.1 0.3 – 0.02 950 950 953 2000 6000 (extended)
Net 0.005 – 350 1120
4.1. Floater model The instantaneous deformed shapes of the floater in regular wave condition (height: 5 m, period: 8 s) are shown in Fig. 4. According to Fig. 4, the floater has not only the rigid body motions but also the structural deformations in wave. Hence the elasticity of the floater should be considered in the dynamic analysis of the system (Endresen, 2011). Time history of displacement responses of four points A, B, C and D as marked in Fig. 2 are plotted in Fig. 5. From Fig. 5, we can see that the motion frequency of the floater is the same as incident wave frequency, and the motion amplitude at vertical direction is the same as that of the incident waves. The vertical displacement is symmetric about the mean water free surface. These characters above represent that in vertical direction, the floater′s motions will basically follow the wave profile. Additionally, remarkable y direction displacements at points B and D are found, which indicates a large structural deformation on the floater in wave, whereas the y direction displacement at points A and C are always zero due to the symmetry of the structure. Dynamic responses of the single floater under different wave period and wave height are further investigated, and the corresponding results are plotted in Figs. 6 and 7. It can be seen that under the same wave period, the vertical motions change significantly with different wave heights. The larger wave height, the larger vertical motion will be found. However, under the fixed wave heights, the vertical motion amplitudes kept almost constant with different wave period. These results reveal the motions characters of the floater in waves: the floater will always ride on the waves. The effect of instantaneous buoyancy on the dynamic response of the floater is plotted in Fig. 8 where the comparison is performed between results with and without “Buoyancy Distribution” method. From Fig. 8(a), we can see that the instantaneous buoyancy will increase the amount of fluctuation under small wave height situation, which reveals the instantaneous buoyancy of the floater section change nonlinearly with the change of water surface. Yet when the wave height is pretty large compared with the
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floater section, the influences of instantaneous buoyancy become not as significant as that in small wave height conditions, as shown in Fig. 8(b). The corresponding motion trail of point A on the floater is further depicted in Fig. 9 (wave height: 5 m, wave period: 8 s). In Fig. 9, vertical motion of floater is within the range of the water particle motion; yet the horizontal motion is smaller than that of water particles, due to the floater deformations and the nonlinear spring restoring forces. These characters render the motion trail of the floater far from critical oval-shaped like
the water particles′ trail. However, we still can conclude that there are explicit similarities between the motion trails of point A and water particles, which demonstrate that the floater motions will basically follow the motions of water particles in wave conditions. 4.2. Fish cage with net and floater The net volume may reduce rapidly when the fish cage is exposed to strong currents (Berstad et al., 2005; Lader and Enerhaug, 2005). The volume reduction coefficient C v can be estimated by Cv ¼
Vd V0
ð9Þ
Fig. 6. Vertical motions of point A (different wave heights).
Fig. 4. Deformed shape of the floater in wave at different time step.
Fig. 7. Vertical motions of point A (different wave periods).
Fig. 5. Dynamic displacement responses of the single floater in wave. (a) Time history of displacement, point A. (b) Time history of displacement, point B. (c) Time history of displacement, point C. (d) Time history of displacement, point D.
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Fig. 8. Dynamic responses of point A with and without “Buoyancy Distribution” method under different waves. (a) Response under Small Wave Height and (b) Response under Large Wave Height.
Fig. 9. Motion trail of point A.
where V 0 is the initial volume of the nets and V d is the volume when the nets are exposed to current. The net deformations in different currents are shown in Fig. 10. As seen from Fig. 10(a), the net volume is significantly reduced in current velocity 0.6 m/s. The large reduction in volume occurred at high current velocity, as shown in Fig. 10(b). Fig. 11 further indicates that net volume reduction coefficient serves as a function of the current velocity. From Fig. 11, we can see that the volume of the fish cage reduces rapidly with the increase of the current velocity. The net volume is proximately half (52%) of the initial volume as the current velocity increases to 1.0 m/s. At current velocity 0.2 m/s, the net volume is 92% of the initial, whereas at current velocity 1.5 m/s, the net volume reduced significantly to 37% of the initial volume. Similar results have been reported by Lader and Enerhaug (2005), where an approximate 40% net volume reduction was observed. The instantaneous deformed shapes of fish cage system in waves (height: 5 m, period: 8 s) and current (velocity: 0.5 m/s) condition are further plotted in Fig. 12. From Fig. 12, a large geometric deformation of the fish cage system can be observed. The wave and current induce violent dynamic responses of the fish cage system. Comparing Fig. 12 and Fig. 10, we can see that in current only conditions, the net will deform in the incoming current direction, and the motions of the floater and the net will entirely be “in phase”. Nevertheless, different phase between the motions of the floater and the net will appear when wave is applied. To study the interaction between the nets and the floater, we compared the displacement of point A in different conditions— with or without the nets, and the corresponding results are plotted in Figs. 13 and 14.
Fig. 10. Net deformations in different currents (a) net deformations in current velocity of 0.6 m/s. (b) Net deformations in current velocity of 1.5 m/s. (c) Bird view of net deformation at different current velocity.
Fig. 11. Net volume under different current velocity.
As shown in Fig. 13, the vertical motion of the single floater is a little bit larger than that of the whole fish cage model, which is primarily attributable to the relative weight of the nets and sinker in the water. However, in the horizontal direction, the net cage would considerably affect the floater motion, as shown in Fig. 14.
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Fig. 15. Displacements of point A at different Cd values (wave height: 5 m, wave period: 8 s).
Table 4 Maximum stress of the floater.
Fig. 12. Instantaneous deformed shape of the cage system exposed to current and wave.
Fig. 13. Displacement of point A in z direction.
Case no.
Max. stress of the floater (107 Pa)
1 2
1.372 2.354
the dynamic responses of the fish cage system, we conduct a series of calculations with different net drag coefficients; the results are plotted in Fig. 15. According to Fig. 15, the change of Cd slightly affects vertical motions of the floater, but highly contributes to the motions of the floater in the wave and current propagation direction. The maximum stress of the floater in different wave and current conditions are listed in Table 4. The calculation cases are: (1) wave only (wave height: 5 m, wave period: 8 s); (2) wave and current combination (wave height: 5 m, wave period: 8 s, current velocity: 0.5 m/s). It is explicit in Table 4 that the floater can withstand a high stress to great extent in wave and current. Further comparing case 1 and case 2, we can conclude that the current induces a large increase in stress on the floater. Since current loads mainly act on nets, it would take risks to evaluate the strength of the floater without consideration of nets.
5. Conclusions In this paper, we investigate the dynamic behavior of a floating fish cage system in current and waves. “Buoyancy Distribution” method is developed to simulate the instantaneous buoyancy of the floater. The following conclusions can be drawn:
Fig. 14. Displacement of point A in x direction.
The reason might be that in wave and current conditions, large horizontal current-induced forces would act on nets and hence lead to a larger horizontal motion of the whole system. The marine organism would generate large amounts of fouling. Such marine fouling might adhere to the net body and increase the drag force of the nets. To study the effect of the net drag forces on
(1) In wave conditions, the floater will undergo rigid body motions and also deformations. (2) The motions of the floater in waves will increase with the wave height, and will basically follow the motion of water particles. (3) The instantaneous buoyancy would be of great importance where wave height or the floater vertical motion is small. (4) Large geometric deformations and motions are observed in both floater and the nets. The net volume reduces rapidly under the current: even 63% reduction is observed in the current velocity 1.5 m/s. (5) The interaction between the floater and the nets has a large impact on horizontal motions of the system, and hence should be considered in the design stage.
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(6) The drag forces on nets may greatly influence the dynamic motions of the floater. (7) The Cd in use to calculate the force on the nets largely affect the horizontal motions of the floater, since the nets are attached on the floater.
Acknowledgments The fanatical supports from SINTEF Fisheries and Aquaculture as well as the Natural Science Foundation of China (Grant nos. 51009088 and 51279101) are highly acknowledged. References Berstad, A.I., Tronstad, H., Sivertsen, S.A., Leite, E., 2005. Enhancement of design criteria for fish farm facilities including operations. In: Proceedings of OMAE 2005. Paper no. OMAE-67451. Dassault System, 2010. ABAQUS 6.10 Documentation.
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