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Dynamic simulation of a fish cage system subjected to currents and waves
ARTICLE IN PRESS Ocean Engineering 35 (2008) 1521–1532
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Dynamic simulation of a fish cage system subjected to currents and waves Chun-Woo Lee a,, Young-Bong Kim b, Gun-Ho Lee c, Moo-Youl Choe c, Mi-Kyung Lee c, Kwi-Yeon Koo d a
Division of Marine Production System Management, Pukyong National University, 599-1, Daeyeon-Dong, Nam-Gu, Busan 608-737, South Korea Division of Electronic, Computer and Telecommunication Engineering, Pukyong National University, 599-1, Daeyeon-Dong, Nam-Gu, Busan 608-737, South Korea Department of Fisheries Physics, Pukyong National University, 599-1, Daeyeon-Dong, Nam-Gu, Busan 608-737, South Korea d Institute of Marine Technology, Norwegian University of Science and Technology, Otto Nielwensveg 10, NO-7491 Trondheim, Norway b c
a r t i c l e in fo
abstract
Article history: Received 31 March 2007 Accepted 12 June 2008 Available online 25 June 2008
This paper presents a mathematical model and describes a simulation method for analyzing the performance of a fish cage system with a floating collar influenced by currents and waves. The cage system was modeled on the mass-spring model. The structure was divided into finite elements, mass points were placed at the mid-point of each element, and the mass points were connected by springs without mass. Water tank experiments using plane nets and model cages were conducted to improve the accuracy of the model, and the data obtained were used in the simulation. The computation method was applied to the dynamic simulation of the behavior of the actual cage to evaluate its practical utility. Computer-based simulation provides a method to quantitatively analyze the environmental forces acting on fish cage systems and thereby provides valuable information necessary for designing an optimal structure. & 2008 Published by Elsevier Ltd.
Keywords: Dynamic simulation Mathematical model Fish cage Floating collar Mass-spring model
1. Introduction Currently, aquaculture facilities are placed mainly in bays or protected near-shore waters. However, because of increased pollution associated with industrialization and urbanization, coastline development, and landfills and because of self-contamination of existing aquaculture facilities, bays and near-shore areas are gradually losing their value as suitable aquaculture sites. Consequently, increased attention is being focused on extending operations into the open ocean (Kim, 1999b; Fredriksson et al., 2000; Tsukrov et al., 2000), where waves and currents are much greater than in near-shore areas. To design reliable systems that can withstand these conditions, stricter design criteria and novel engineering methods specific to open ocean environments are needed. Such methods include mathematical modeling of the dynamic responses of structures placed in high-energy open ocean areas and 3D simulation of the ocean environmental loads that would act on aquaculture systems. Fish cage systems consist of netting, mooring lines, a floating collar, floats, and sinkers. Netting and ropes are the basic components of marine cage structures. Much research has been
Corresponding author. Tel.: +82 51 620 6123; fax: +82 51 622 3306.
E-mail address: [email protected] (C.-W. Lee). 0029-8018/$ - see front matter & 2008 Published by Elsevier Ltd. doi:10.1016/j.oceaneng.2008.06.009
devoted to understanding the hydrodynamic coefficients and the behavior of nets in different operating conditions (Kawakami, 1964; Kim, 1999a; Matuda, 2001). In relation to the netting properties specific to fish cages, Aarsnes et al. (1990) derived a formula to obtain the reduction ratio of fluid velocity stemming from the shielding effect of the net. Computer-based analyses of fishing gear systems have also been conducted. Bessonneau and Marichal (1998) described a modeling method for a trawl net that used a mesh grouping method, and Niedzwiedz and Hopp (1998) presented a mathematical model of a trawl system and simulated its behaviors. Takagi et al. (2002) presented a calculation model using the lumped-mass method and applied it to netting. Lee and Cha (2002) and Lee et al. (2005) proposed a mathematical model using the mass-spring model to simulate the behavior of fishing gear systems, and they also simulated a purse seine system (Kim et al., 2007). These computation methods have added to our theoretical knowledge and analytical capability in terms of modeling the performance of fishing gear. Recently, many researchers have conducted computer-aided behavior analysis of fish cage systems. Tsukrov et al. (2003) developed a consistent net element method that has been used in the numerical modeling of open ocean aquaculture systems. Tsukrov et al.’s (2005) improved modeling method for nonlinear elastic material has been applied to the development of the feed buoy mooring system. Fredriksson et al. (2003) used a stochastic
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approach to analyze the reactions of a cage designed by the linear system and compared these results with field measurements. Thomassen and Leira (2005) demonstrated the importance of the buoyancy load by measuring forces applied to the square frame without netting, and Berstad et al. (2005) conducted calculations by dividing circular cages into membranes through a simulation method to verify the efficiency of the standard Norwegian cage. Raman-Nair and Colbourne (2003) studied the dynamics of a mussel long-line system using the lumped-mass method. Lee and Pei-Wen (2000) examined the tension-leg platform of underwater net–cage systems and derived and solved analytically the equations of motion of the system subjected to flow-induced drag motion and wave-induced surge motion. Lader and Enerhaug (2005) conducted an experiment to investigate forces and net deformation using different weights attached to the model cage. Further studies include numerical modeling of the floating collar. Huang et al. (2006, 2007) studied a series of gravity cages using the lumped-mass method and net plane elements; they also analyzed the behavior of the floating collar, which they regarded as a rigid body, and estimated the volume reduction coefficient of the cage caused by the currents. Li et al. (2006) and Zhao et al. (2007) used a lumped-mass model to simulate the dynamic response of a gravity cage with a floating system to waves combined with current. Fredriksson et al. (2007) worked on finiteelement modeling of the floating collar and simulated the process of localized failure of the high-density polyethylene (HDPE) pipe. Such extensive studies of fish cages have provided an improved understanding of the various external forces that affect their components. Results of these studies provide valuable information necessary for designing an optimal structure that can tolerate the environmental forces of the open ocean. In this study, we propose a mass-spring model to describe the whole elements of a fish cage system, including the floating collar. We conducted a series of water tank experiments using model cages to validate the mathematical model and to improve the calculation accuracy. We performed a mathematical calculation to interpret the dynamic behavior of the model system when influenced by currents and waves. The simulated and experimental values were compared and analyzed. Finally, the behavior of an actual fish cage influenced by currents and waves was simulated.
2. Materials and methods 2.1. Mathematical model 2.1.1. Equation of motion Fish cage systems consist of various kinds of netting, rope and a floating collar, as well as appendages such as floats, sinkers and anchors. In this study, the system was modeled based on the mass-spring model. In this model, the nettings and ropes are regarded as flexible structures and the floating collar as an elastic structure. The structures are divided into a finite number of elements and mass points are placed at the mid-point of each element; the mass points are connected by springs without mass. Applying this method to the netting, which takes up most of the cage structure, the mesh knots in the netting can be considered as the mass points and the mesh bars can be considered as the springs connecting the mass points. Direct application of this method to a commercial system produces an enormous number of mass points. To reduce the calculation loads, we used the approximation method, in which several actual meshes are bundled together into a virtual
mathematical mesh having the same physical properties (e.g., mass, specific gravity, weight, projected area, hydrodynamic coefficients) (Lee et al., 2005; Kim et al., 2007). It is assumed that the virtual mesh and the original mesh are geometrically identical; in other words, the angle between two adjacent virtual mesh bars and the angle of attack are the same as those of the original netting. We also placed one more mass point on the mid-point of each mesh bar to represent the bending of the bar (Fig. 1). We assumed the bar of the virtual mesh to be a cylinder and the knot to be a sphere. The projected area of the virtual mesh knots is the sum of all the knots in the actual netting and the area of the virtual mesh bars is the sum of all the bars in the actual netting. For the mooring line, we divided the line into constant lengths, and mass points were placed at the mid-point of each element; the mass points were connected by springs (Fig. 2). The appendages also were considered to be mass points. In general, a floating collar consists of one big floating tube or two small tubes connected in parallel to form the circular upper structure, with a handrail installed on top of it. For convenience of analysis, we modeled one floating tube without the handrail. For the mass points of the floating collar, a rectangular parallel pipe, similar to the original 3D structure, was constructed, and mass points were placed on each corner (Fig. 2). The spring was placed along the edges and diagonally to make sure that the structure maintained its shape even if external forces were applied in several different directions. The projected area, buoyancy, and weight in air of each mass point were equal to the value obtained by dividing the total value of the floating collar by the number of mass points. Buoyancy or weight was applied to each mass point in addition to the hydrodynamic force. Whether buoyancy or weight in air was applied to each mass point was determined as follows. The computing program checked whether each mass point was in the water or the air at every calculation step; if in the water, buoyancy was applied, and if in the air, weight was applied. The equation of motion for each mass point can be described as follows: ðm þ DmÞq€ ¼ Fint þ Fext ,
(1)
where m is the mass of the mass points, Dm is the added mass, q€ is the acceleration vector, Fint is the internal forces applied between the mass points, and Fext is the external forces applied to the mass points. For the fish cage system, translational motion by the tidal current is dominated, so rotational motion is ignored. The added mass of the mass points is given as follows:
Dm ¼ rW V n C m ,
(2)
where rW is the density of seawater, Vn is the volume of the mass points, and Cm is the added mass coefficient, which is 1.5 because the structure is considered to be a sphere. For cylindrical structures like the mesh bars and floating collar, Cm is calculated as follows: C m ¼ 1 þ sin a,
(3)
where a is the angle of attack. 2.1.2. Internal force The internal force is the force that is applied to the springs connecting each mass point. It is assumed that internal forces are applied to the netting and the rope in the direction of tension and to the floating collar in the direction of both tension and compression. The length of the spring elongated or reduced in the direction of tension or compression, respectively, is assumed to be linearly proportional to the force in action. The internal force
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Fig. 1. Placement of virtual mathematical mesh using mesh grouping method (a) and vector notation of the element (b).
applied to each mass point is as follows: Fint ¼
n X
0
ki ni ðjri j li Þ,
(4)
i¼1
where ki is the stiffness of the springs comprising the structure, ni the unit vector along the line of the spring, |ri| the magnitude of 0 the position vector between the neighboring mass points, li the initial length of the spring, and n is the numbers of adjacent mass points i. The stiffness is the equivalent value of each material. It can be calculated using the following formula: k¼
EA 0
l
,
(5)
where E is the Young modulus and A is the effective area of the material. The effective area of the mesh bar or rope is 60% of the apparent cross-sectional area. Furthermore, under the same tensile load, the elongation of a rope is greater than the elongation of a solid bar of the same material and with the same metallic cross-sectional area because the ropes in a cable tighten up (Gere, 2000). Thus, the modulus of elasticity (i.e., effective modulus) of a rope is less than the modulus of the material of which it is made. In this paper, the effective modulus of the mesh bar or rope is assumed to be 60% of the modulus of the material. The effective area of the floating collar is the area of the material of the polyethylene (PE) pipe obtained by subtracting the inside diameter from the outside diameter. The stiffness of the diagonal direction spring of the floating collar is assumed to be the same as its stiffness in the straight direction.
2.1.3. External force The external force (Fext) is the force that is applied to each mass point from the outside environment and it consists of the drag force (FD), lift force (FL), and buoyancy and sinking force (FB), as follows: Fext ¼ FD þ FL þ FB .
(6)
The current is assumed to be steady and uniform, so the inertial force is neglected. The drag and lift forces are as follows: FD ¼
1 C D rw SV 2 nV 2
(7)
Fig. 2. Modeling of the fish cage net, the mooring line, and the floating collar.
FL ¼
1 C L rw SV 2 nL , 2
(8)
where CD is the drag force coefficient, S the projected area of the mass point, V the magnitude of the resultant velocity vector, nV the unit vector of the drag vector, CL the lift force coefficient, and nL is the direction of the lift force (see Fig. 1b), calculated as follows: nL ¼
ðV rÞ V , jðV rÞ Vj
(9)
where V is the resultant velocity vector. The resultant velocity vector V is composed of the motion velocity vector of the mass point Vm, the current velocity vector Vc, and the wave-induced water particle velocity vector Vw. That is V ¼ Vm Vc þ Vw .
(10)
In this study, we did not consider the influence of the mutual interference of the velocity field in the case of concurrent application of current and waves. The hydrodynamic coefficients of the mesh bar are the function of the attack angle formed by the bar and the velocity vector,
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differential equation in the time domain:
and the attack angle is obtained as follows: Vr . a ¼ cos1 jVjjrj
(11)
The drag coefficient and lift force coefficient of a cylindrical structure such as the mesh bar and the floating collar are the same as those used by Lee et al. (2005) and Kim et al. (2007). Because the mesh knot, float and sinker of the net are regarded as spheres, no lift force is applied, resistance is applied, and the drag coefficient is assumed to be 1.5 (Fredheim and Faltinsen, 2003). The buoyancy and sinking force, FB, can be written as follows: FB ¼ ðri rw ÞV n g,
(12)
where ri is the density of the structure and g is the gravity acceleration.
€ MqðtÞ ¼ Fint ðtÞ þ FD ðtÞ þ FL ðtÞ þ FB ðtÞ,
(13)
€ where M is the mass including added masses and qðtÞ is the acceleration. Eq. (13) can be transformed into two first-order differential equations: _ qðtÞ ¼ Vm ðtÞ
(14)
_ m ðtÞ ¼ M 1 ½Fint ðtÞ þ FD ðtÞ þ FL ðtÞ þ FB ðtÞ. V
(15)
In this study, to solve the behavior of the structure described in the form of the stiff system, we integrated using fourth-order Runge-Kutta method. 2.2. Water tank experiments 2.2.1. Velocity reduction ratio for the plane netting
2.1.4. Computation method After substituting the internal and external forces in Eq. (1), the equation of motion governing the motion of the fish cage system became the following non-linear second-order derivative
Table 1 Netting materials of model nets used in the experiment Net
Material
Twine diameter (mm)
Bar length (mm)
Solidity ratio
A B C
Polyester Polyethylene Polyethylene
4 5 6
55 55 55
0.15 0.19 0.23
Fig. 3. Arrangement of experimental apparatus in the circulating water tank.
In a fish–cage system, the rear netting is less affected by the force of the current flow than the front netting. This phenomenon is the result of the shielding effect, by which some of the water flow is eliminated by the obstacle created by the front netting, In this experiment, we used three types of netting with different d/l ratios (d: twine diameter, l: bar length) and measured the fluid velocities before and after the flow passed through the netting. Assuming that the attack angle is 901 when the flow is applied at right angles to the netting, we measured fluid velocities with attack angles of 301, 601, and 901. The velocity reduction ratio was obtained by subtracting the velocity after passing the netting from the original velocity before passing the netting and dividing this value by the original velocity. Table 1 lists the dimensions of the netting used in the experiment; a stainless steel frame (1 m 1 m)
Fig. 4. Arrangement of a circular model cage in the circulating water tank.
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with a 4 mm diameter was used to fix the netting. The fluid velocities were measured 1 m away from the center of the netting, and experiments were conducted with fluid velocities of 0.3 and 0.6 m/s using a circulating water tank (10.2 m L 3.2 m H 2.8 m W) at Pukyong National University (Fig. 3). 2.2.2. Deformation of the model circular cage caused by currents To check whether the accuracy of the mathematical calculation was improved when the velocity reduction ratio was applied, we constructed a model cage and conducted a water tank experiment. The same model was used for numerical calculation. The shape of the model cage was measured when a constant fluid velocity was applied to the cage. The model was a circular cage with a diameter of 1 m and depth of 0.5 m made from nylon-netting material with a mesh bar diameter of 1 mm and a bar length of 5 mm. The experiment was conducted in the circulating water tank and the deformation of the cage was measured using a digital camera and a digitizer (Fig. 4). The fluid velocities applied were 0.15 and 0.3 m/s. The parameters used in the mathematical calculation are shown in Table 2. We then compared the experimental values and the simulated values obtained from the mathematical model.
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The model was a square cage (1 m L 1 m H 1 m W; see Table 3 for the materials used to make the model). To measure the tension on the mooring lines, we installed load cells on the upper part of the front and rear mooring lines. The tests were conducted in the 3D towing tank (85 m L 4 m H 10 m W) at the National Fisheries Research & Development Institute (Fig. 5). The numerical model used the same structure and physical properties as the cage used in the experiment (see Table 4). The tension of the mooring line and the shape of the cage when waves were applied were calculated, and results of this calculation were compared with the experimental results. The cage was subjected to regular waves based on linear wave theory. The wave heights
2.2.3. Response of the model cage to waves We next measured the tension of the mooring lines and the shape changes of a cage structure when it was exposed to waves.
Table 2 Physical characteristics of calculation parameters of the model circular cage No. of mass points
Mass (kg)
Projected area (m2)
Stiffness (N/m)
Initial length (m)
2172
9 104
3.8 103
12,025
0.1
Table 3 Physical specifications of the model square cage Items
Specifications
Buoy Corner buoys Material Density (g/cm3) Diameter (cm) Number of pieces Total Buoyancy (N)
Plastic 0.23 15 4 47.82
Buoys on float line Material Density (g/cm3) Diameter (cm) No. of pieces Total Buoyancy (N)
Plastic 0.33 3.5 48 15.05
Netting Material Density (g/cm3) Twine diameter (cm) Total mass (g) Mesh size (cm) Solidity
Nylon 1.14 0.1 0.3424 0.5 0.2
Bottom weights Material Density (g/cm3) Unit mass (g/pieces) Number of pieces Total weight (N)
Lead 11.35 2.52 103 12 2.65
Fig. 5. Arrangement of a square cage in the 3D towing tank (A, B: measuring points).
Table 4 Physical characteristics of calculation parameters of the model square cage No. of mass points
Mass (kg)
Projected area (m2)
Stiffness (N/m)
Initial length (m)
2248
1.523 104
8.896 104
11,942
0.05
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were 0.1 and 0.3 m and the periods were 1.8 and 2.4 s for both the tests.
3. Results and discussion 3.1. Velocity reduction ratio Table 5 shows the fluid velocity measured at the rear of the netting when the attack angles were 901, 601, and 301, and the fluid velocities were 0.3 and 0.6 m/s. In general, the velocity reduction ratio varied depending on the solidity ratio of the netting, the angle of attack, and the velocity. The velocity reduction ratio decreased with an increase in flow velocity. The velocity reduction ratio was higher with an increase in the solidity ratio and with a decrease in the angle of attack. Netting on a slanted plane would generate lift force components and the streamlining of the flow would change; however, we did not measure streamlining in this experiment. We also did not test a wide range of netting solidity ratios; instead, we used only netting with a relatively small solidity ratio. Aarsnes et al. (1990) reported that the velocity reduction ratio was reduced by about 6% when the netting had a solidity ratio of 0.13. In our study, the average reduction ratio was 7% for the solidity
Table 5 Fluid velocity reduction ratio attack angle for different types of nettings Type
Velocity (m/s)
Attack angle (1)
Velocity reduction ratio (%)
A
0.3
90 60 30
9 9 10
0.6
90 60 30
5 6 7
0.3
90 60 30
14 15 16
0.6
90 60 30
7 9 10
0.3
90 60 30
19 19 21
0.6
90 60 30
10 11 12
B
C
ratio 0.15 (A Net) and attack angle 901 for the two velocities (Table 5). The velocity reduction ratio adopted to the numerical calculations of this paper was determined based on the solidity of the netting, and the differences caused by different angles of attack and the magnitude of the velocity were not taken into account for the convenience of calculation. That is, in calculating the velocity applied to the rear netting of the model cage, we assumed that 85% of the velocity applied to the front netting was applied uniformly to the entire rear netting based on the results of Aarsnes et al. (1990). In calculating for an actual cage, 93% of the velocity of the front netting was assumed to have been applied uniformly to the rear netting based on our experiment. It is difficult to determine the accurate velocity reduction ratio when the fluid passes through the netting or cages because of the complicated interaction between the fluid and the flexible structure. The reduced size of models used in tank experiments and the difficulty in obtaining reliable data in field experiments add to the complexity. However, velocity reduction ratio is one of the important factors that affect the deformation and loads of the cages. To precisely calculate the behavior of fish cage systems, additional detailed experiments are required to obtain more accurate velocity reduction ratios for a wide range of nettings, velocities, and cage systems rigged in various ways.
3.2. Deformation of the circular cage As shown in Fig. 6, the side shape of the model cage changed dramatically as the velocity increased. Most notably, the front netting was more deformed by the direction of the flow than the rear netting because of the shielding effect. Fig. 7 shows the results of the numerical calculation at a fluid velocity of 0.3 m/s when the fluid velocity reduction ratio was not applied (Fig. 7a) and when 15% of the velocity reduction ratio was applied (Fig. 7b). Fig. 8 shows a comparison of the experimental and calculated values in relation to the side view of the cage. Fig. 8a illustrates
Fig. 7. Simulation results using a current speed of 0.3 m/s with the reduction ratio not applied (a) or applied (b).
Fig. 6. The shape of the cage model according to a fluid velocity of 0.15 m/s (a) and 0.3 m/s (b).
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Fig. 8. Comparison of the experimental vs. the calculated side shape at a current speed of 0.3 m/s with the reduction ratio not applied (a) or applied (b).
Fig. 9. Comparison of experimented (a) and simulated (b) results of the square cage to waves (height 0.3 m, period 2.4 s).
18
3.3. Response of the model cage to waves
16
Experiment Simulation
Tension (N)
14 12 10 8 6 4 2 0
0
0.5
1
1.5
2
2.5 3 Time (s)
3.5
4
4.5
5
Fig. 10. Comparison of the tension of the mooring line between experimental and numerical simulations to waves (wave height 0.3 m and period 2.4 s).
the scenario in which the shielding effect was not considered and a 14% position calculation error occurred, whereas Fig. 8b shows the case in which the shielding effect was taken into account and a p5% calculation error occurred.
Fig. 9 shows a sample of experimental data and the numerical prediction of the cage exposed to waves (height 0.1 m, period 2.4 s). To check the accuracy of the numerical prediction, the calculated tension applied to the mooring line was compared to the model test results. Fig. 10 shows that the tension values obtained for the simulation and model tests; the data were almost identical, but the shape of the tension curves differed slightly. For the simulation, the trough and the peak of the curve were similar; for the model, the peak was sharper than the trough. The curves differed because the wave shape applied in the simulation was a perfect cosine curve, whereas the actual wave shape applied in the model test was not a complete cosine curve. The values for the locus of the buoy (A in Fig. 5) and sinker (B in Fig. 5) of the cage were shown in Fig. 11. In general, the calculated positions coincide with the measured positions. At the trough, however, the position error was pronounced. This result might reflect the difference of the wave shapes between the simulation and the model test as mentioned previously. The amplitude of the sinker was slightly smaller than that of the float in both cases due to the elongation of the twine and the damping effect of the netting. The drag and lift coefficients are important parameters that affect the loads and deformation of the cage. These coefficients depend on three main factors—the angle of attack, the solidity of the material, and the Reynolds numbers—and their influence
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must be considered. In this study, the drag and lift force coefficients of the netting, rope, and floating collar, which are regarded as cylinder-shaped, were those values related to the angle of attack. We used these hydrodynamic force coefficients
0.2 0
Depth (m)
-0.2 Exp.(float) Exp.(sinker) Simu.(float) Simu.(sinker)
-0.4 -0.6 -0.8 -1
because the influence of the angle of attack was the most dominant and the influence of the other two factors was not so obvious (Lee et al., 2005). However, the cage made large movements up and down in the vertical plane when it was exposed to waves. In such a case, we assumed that the attack angle of the side panels of the cage is the interior angle between the resultant velocity vector and the plane of the mesh (Kim et al., 2007). A more accurate calculation of the dynamic behavior of the fish cage requires refined hydrodynamic force coefficients that take the angle of attack, solidity ratio, and Reynolds number into consideration. Because this study used the hydrodynamic coefficient obtained in the experiment in the Reynolds number range 3000–5000, the hydrodynamic force probably was underestimated in the low Reynolds number range and the force likely was overestimated in the high Reynolds number range. 3.4. Verification of the mathematical model
-1.2 0
1
2
3
4
5
4
5
Time (s) 0.2 0
Depth (m)
-0.2 Exp.(float) Exp.(sinker) Simu.(float) Simu.(sinker)
-0.4 -0.6 -0.8 -1 -1.2 0
1
2
3 Time (s)
Fig. 11. Comparison of the locus of the buoy and sinker of the square cage between experimental and simulation results for different wave conditions: (a) wave height 0.1 m and period 2.4 s; and (b) wave height 0.3 m and period 2.4 s.
To validate the mathematical model proposed in this study, we used data from Lader and Enerhaug (2005). For the numerical calculation, we used the parameters of the Lader and Enerhaug’s (2005) actual experimental cage: a 1.435 m diameter and 1.435 m deep cylindrical cage; nylon knotless netting with 32 mm mesh size and a 1.8-mm-thick mesh bar; and three models with different sinkers (16 0.4 kg, 16 0.6 kg, 16 0.8 kg). For the calculation, the velocity reduction ratio of the rear netting was 15% and 2368 mass points were placed. As shown in Fig. 12, the simulated shapes of the model cage were all in close agreement with the experimental results of Lader and Enerhaug (2005). The inside volume reduction was calculated according to the method presented by Huang et al. (2006). Changes in the inside volume according to the different velocities are shown in Fig. 13. In our calculation, the inside volume reduction coefficient was lower than that reported by Lader and Enerhaug (2005). In Fig. 14, the calculated drags on the model cage net are compared with experimental values for different weights. As was pointed out in the Section 3.3, the calculated drags were slightly smaller than the experimental ones at the lower velocity range (less than 0.4 m/s), and it was opposite at the higher velocity range (more than 0.4 m/s).
Fig. 12. The deformation of the model cage as a result of the numerical simulation for different weight and current velocities.
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Inside volume reduction ratio (-)
3.5. Simulation of the actual cage 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5
Experiment Calculation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.4
0.5
0.6
Inside volume reduction ratio (-)
Velocity (m/s)
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5
Experiment Calculation
0
0.1
0.2
0.3
Inside volume reduction ratio (-)
Velocity (m/s)
To verify the practicality of the calculation method, we conducted a simulation of the motion of an actual fish cage system in an environment in which current and waves were applied. The simulated cage was a circular type used in tuna farming in the Mediterranean Sea. It had a diameter of 50 m and a depth of 20 m and was composed of a PE floating collar, PES (polyester) Raschel netting, floats, sinkers, and mooring lines. The floating collar was a 50.8 cm diameter and 46.2 mm thick tube. The total buoyancy corresponding to circumference 157 m was 174,440 N. The number of mass points placed on the floating tube was 300, and the physical characteristics of each mass point are shown in Table 6. The mesh bar length of the cage net was 55 mm and the twine diameter was 4 mm. The total number of knots was 1,701,999; they were grouped and approximated to 1412 mass points. The total sinking force of the cage was 12,828 N. Fig. 15 shows the shape of the cage when it was placed in still water (a) and when it was exposed to currents of 0.2 m/s (b), 0.4 m/s (c), 0.6 m/s (d), and 0.8 m/s (e). Fig. 16 shows the side view of the cage when waves of height 6 m, length 156 m, and a period of 10 s were applied in the same direction as the current of 0.7 m/s. Due to the stiffness of the floating structure, the floating tube did not bend in the current and maintained its original shape (Fig. 15). In response to the waves, the floating collar followed the wave-surface level with a slight time delay (see Fig. 16 or video at www.mpsl.co.kr/simucage.htm). In other
1 0.95 0.9 0.85 0.8
Table 6 Physical characteristics of calculation parameters of the actual cage for the floating collar, netting, and mooring lines
0.75 0.7 0.65 0.6
Items Experiment Calculation
0.55 0.5 0
0.1
Mass (kg)
Floating collar
0.2
0.3
0.4
0.5
0.6
Velocity (m/s) Fig. 13. The ratio of the inside volume reduction of the model cage according to velocity with weights of 0.4 kg 16 (a), 0.6 kg 16 (b), and 0.8 kg 16 (c).
4.76
Cage netting Mooring line a b
0.40 20.74
Projected area (m2)
Stiffness (N/m)
Initial length (m)
0.254
8,160,053a 39,372,206b 322,254 11,836,048
2 0.46 2 15.5
0.53 0.86
Lengthwise spring. Crosswise spring.
110 Experiment 16X0.4kg
100
Experiment 16X0.6kg
90
Experiment 16X0.8kg Calculation 16X0.4kg
80
Calculation 16X0.6kg Calculation 16X0.8kg
Drag (N)
70 60 50 40 30 20 10 0 0
0.05
0.1
0.15
0.2
0.25 0.3 0.35 Velocity (m/s)
0.4
0.45
0.5
0.55
Fig. 14. Comparison of the drag of the model cage net between experimental and numerical simulations.
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Fig. 15. Successive shapes of the cage system in still water (a) and with current velocities of 0.2 m/s (b), 0.4 m/s (c), 0.6 m/s (d), and 0.8 m/s (e). A: a referenced point for the tension values.
words, the response of the floating collar to the current or waves was qualitatively similar to those observed in the field (e.g., the pitch motion and horizontal and vertical displacements of the floating collar). The quantitative analysis of the dynamic behavior of the cage system will be presented in the next paper. The mass-spring model proposed in this paper divides the floating collar into many different mass points and the hydrodynamic forces are calculated for each different mass point. When calculating the hydrodynamic force of the floating pipe, our computing program checked that each mass point was in the water or the air at every calculation step; if in the water, the buoyancy, drag, and lift force of the relevant mass point were
calculated, and when above the water, the weight was applied. Thus, the mass-spring model can calculate the external forces no matter what state the floating collar is in. Although wave-induced hydrodynamic force coefficients of the partially submerged pipe elements can differ when there is a uniform current with full submersion (Li et al., 2007), only the coefficients obtained in full submersion and a uniform current were used in this study. The tensions on the upper side mooring line at the point A (in Fig. 15) are shown in Fig. 17. When the current was applied to the cage, the shape of the cage changed greatly and the tension on the upper side mooring line increased greatly. When waves were added to the current, the cage made large movements up and
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Fig. 16. Successive side shapes of the cage system exposed to current (0.7 m/s) and waves (height: 6 m, period: 10 s) simultaneously.
down, significantly the maximum tension on the mooring line on the upper side increased by 180% compared to when only the current was applied
4. Conclusion We used the mass-spring model to describe the behavior of a fish cage system consisting of netting, floating collar, floats, and sinkers and validated the model using water tank experiments. In addition, to prove the practical applicability of our numerical methods, we simulated the influence of the tidal current and waves independently or concurrently applied to the actual fish cage system. The mass-spring model successfully described the
motion of the whole cage structure, for both flexible materials (nettings and ropes) and elastic material (the floating collar). This mathematical calculation and simulation technique can be used to evaluate the safety of a large-scale moored cage structure exposed to diverse environmental forces. If used during the design stage (prior to construction), it will be helpful in designing a safer and more economical system. In existing systems, fouling organisms often lead to fatal damage of the structure; this problem can be simulated by adjusting the projected area and the drag coefficient. Computer-based simulation methods such as this one can provide a quantitative analysis of the environmental forces that act on fish cages and valuable information necessary for designing an optimal structure.
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80 70
Tension (KN)
60 50 40 30 20 10 0
0
50
100
150 Time (s)
200
250
300
Fig. 17. Variations in tension acting on the upper side mooring line at the point A in Fig. 15.
Acknowledgments This work was supported in part by grant no. MNF22004026-21-SB010 of the Korean Ministry of Maritime Affairs and Fisheries and grant no. R01200600011020 of the Korea Science and Engineering Foundation. The authors would like to express thanks for their financial support. References Aarsnes, J.V., Rudi, H., Loland, G., 1990. Current Force on Cage, Net Deflection, Engineering for Offshore Fish Farming. Tomas Telford, London, pp. 137–152. Berstad, A.J., 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. Bessonneau, J.S., Marichal, D., 1998. Study of the dynamics of submerged supple nets (application to trawls). Ocean Engineering 25 (1), 563–583. Fredheim, A., Faltinsen, O.M., 2003. Hydroelastic Analysis of a Fishing Net in Steady Inflow Conditions. Hydroelastic in Marine Technology, Oxford, UK, pp. 1–10. Fredriksson, D.W., Swift, M.R., Muler, E., Baldwin, K., Celikkol, B., 2000. Open ocean aquaculture engineering: system design and physical modeling. Marine Technology Society Journal 34 (1), 41–52. Fredriksson, D.W., Swift, M.R., Irish, J.D., Tsukrov, I., Celikkol, B., 2003. Fish cage and mooring system dynamics using physical and numerical models with field measurements. Aquacultural Engineering 27, 117–146. Fredriksson, D.W., DeCew, J.C., Tsukrov, I., 2007. Development of structural modeling techniques for evaluating HDPE plastic net pens used in marine aquaculture. Ocean Engineering 34, 2124–2137. Gere, J.M., 2000. Mechanics of Materials, 5th ed. Brooks/Cole/Thomson Learning, pp. 1–77. Huang, C.C., Tang, H.J., Liu, J.Y., 2006. Dynamical analysis of net cage structures for marine aquaculture: numerical simulation and model testing. Aquacultural Engineering 35, 258–270. Huang, C.C., Tang, H.J., Liu, J.Y., 2007. Modeling volume deformation in gravity-type cages with distributed bottom weights or a rigid tube-sinker. Aquacultural Engineering 37, 144–157. Kawakami, T., 1964. The theory of designing and testing fishing nets in model. In: Modern Fishing Gear of the World, vol. 2. Fishing News Books Ltd, London, pp. 471–489. Kim, D.A., 1999a. Design of Fishing Gear (in Korean). Pyounghwa Printing House, pp. 106–139.
Kim, T.H., 1999b. Offshore Cage Net System and Improved Float. National Fisheries Research & Development Institute. Kim, H.Y., Lee, C.W., Shin, J.K., Kim, H.S., Cha, B.J., Lee, G.H., 2007. Dynamic simulation of the behavior of purse seine gear and sea-trial verification. Fisheries Research 88, 109–119. Lader, P.F., Enerhaug, B., 2005. Experimental investigation of forces and geometry of a net cage in uniform flow. IEEE Journal of Oceanic Engineering 30 (1). Lee, C.W., Cha, B.J., 2002. Dynamic simulation of a midwater trawl system’s behavior. Fisheries Science 68, 1865–1868. Lee, H.H., Pei-Wen, W., 2000. Dynamic behavior of tension-leg platform with net–cage system subjected to wave forces. Ocean Engineering 28, 179–200. Lee, C.W., Lee, J.H., Cha, B.J., Kim, H.Y., Lee, J.H., 2005. Physical modeling for underwater flexible systems dynamic simulation. Ocean Engineering 32, 331–347. Li, Y.C., Zhao, Y.P., Gui, F.K., Teng, B., 2006. Numerical simulation of the hydrodynamic behaviour of submerged plane nets in current. Ocean Engineering 33, 2352–2368. Li, Y., Gui, F., Teng, B., 2007. Hydrodynamic behavior of a straight floating pipe under wave conditions. Ocean Engineering 34, 552–559. Matuda, K., 2001. Fishing Gear Physics (in Japanese). Seizando, Tokyo, pp. 21–81. Niedzwiedz, G., Hopp, M., 1998. Rope and net calculations applied to problems in marine engineering and fisheries research. Archives of Fisheries and Marine Research 46, 125–138. Raman-Nair, W., Colbourne, B., 2003. Dynamics of a mussel longline system. Aquacultural Engineering 27, 191–212. Takagi, T., Suzuki, K., Hiraishi, T., 2002. Development of the numerical simulation method of dynamic fishing net shape. Nippon Suisan Gakkaishi 68 (3), 320–326. Thomassen, P.E., Leira, B., 2005. A prototype tool for analysis and design of floating fish cages. In: Proceedings of OMAE 2005. Paper No. OMAE-67383. Tsukrov, I., Ozbay, M., Fredriksson, D.W., Swift, M.R., Baldwin, K., Celikkol, B., 2000. Open ocean aquaculture engineering numerical modeling. Marine Technology Society Journal 34 (1), 29–40. Tsukrov, I., Eroshkin, O., Fredriksson, D.W., Swift, M.R., Celikkol, B., 2003. Finite element modeling of net panels using a consistent net element. Ocean Engineering 30, 251–270. Tsukrov, I., Eroshkin, O., Paul, W., Celikkol, B., 2005. Numerical modeling of nonlinear elastic components of mooring system. IEEE Journal of Oceanic Engineering 30 (1). Zhao, Y.P., Li, Y.C., Dong, G.H., Gui, F.K., Teng, B., 2007. Numerical simulation of the effects of structure size ratio and mesh type on three-dimensional deformation of the fishing-net gravity cage in current. Aquacultural Engineering 36, 285–301.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Dynamic simulation of a fish cage system subjected to currents and waves Chun-Woo Lee a,, Young-Bong Kim b, Gun-Ho Lee c, Moo-Youl Choe c, Mi-Kyung Lee c, Kwi-Yeon Koo d a
Division of Marine Production System Management, Pukyong National University, 599-1, Daeyeon-Dong, Nam-Gu, Busan 608-737, South Korea Division of Electronic, Computer and Telecommunication Engineering, Pukyong National University, 599-1, Daeyeon-Dong, Nam-Gu, Busan 608-737, South Korea Department of Fisheries Physics, Pukyong National University, 599-1, Daeyeon-Dong, Nam-Gu, Busan 608-737, South Korea d Institute of Marine Technology, Norwegian University of Science and Technology, Otto Nielwensveg 10, NO-7491 Trondheim, Norway b c
a r t i c l e in fo
abstract
Article history: Received 31 March 2007 Accepted 12 June 2008 Available online 25 June 2008
This paper presents a mathematical model and describes a simulation method for analyzing the performance of a fish cage system with a floating collar influenced by currents and waves. The cage system was modeled on the mass-spring model. The structure was divided into finite elements, mass points were placed at the mid-point of each element, and the mass points were connected by springs without mass. Water tank experiments using plane nets and model cages were conducted to improve the accuracy of the model, and the data obtained were used in the simulation. The computation method was applied to the dynamic simulation of the behavior of the actual cage to evaluate its practical utility. Computer-based simulation provides a method to quantitatively analyze the environmental forces acting on fish cage systems and thereby provides valuable information necessary for designing an optimal structure. & 2008 Published by Elsevier Ltd.
Keywords: Dynamic simulation Mathematical model Fish cage Floating collar Mass-spring model
1. Introduction Currently, aquaculture facilities are placed mainly in bays or protected near-shore waters. However, because of increased pollution associated with industrialization and urbanization, coastline development, and landfills and because of self-contamination of existing aquaculture facilities, bays and near-shore areas are gradually losing their value as suitable aquaculture sites. Consequently, increased attention is being focused on extending operations into the open ocean (Kim, 1999b; Fredriksson et al., 2000; Tsukrov et al., 2000), where waves and currents are much greater than in near-shore areas. To design reliable systems that can withstand these conditions, stricter design criteria and novel engineering methods specific to open ocean environments are needed. Such methods include mathematical modeling of the dynamic responses of structures placed in high-energy open ocean areas and 3D simulation of the ocean environmental loads that would act on aquaculture systems. Fish cage systems consist of netting, mooring lines, a floating collar, floats, and sinkers. Netting and ropes are the basic components of marine cage structures. Much research has been
Corresponding author. Tel.: +82 51 620 6123; fax: +82 51 622 3306.
E-mail address: [email protected] (C.-W. Lee). 0029-8018/$ - see front matter & 2008 Published by Elsevier Ltd. doi:10.1016/j.oceaneng.2008.06.009
devoted to understanding the hydrodynamic coefficients and the behavior of nets in different operating conditions (Kawakami, 1964; Kim, 1999a; Matuda, 2001). In relation to the netting properties specific to fish cages, Aarsnes et al. (1990) derived a formula to obtain the reduction ratio of fluid velocity stemming from the shielding effect of the net. Computer-based analyses of fishing gear systems have also been conducted. Bessonneau and Marichal (1998) described a modeling method for a trawl net that used a mesh grouping method, and Niedzwiedz and Hopp (1998) presented a mathematical model of a trawl system and simulated its behaviors. Takagi et al. (2002) presented a calculation model using the lumped-mass method and applied it to netting. Lee and Cha (2002) and Lee et al. (2005) proposed a mathematical model using the mass-spring model to simulate the behavior of fishing gear systems, and they also simulated a purse seine system (Kim et al., 2007). These computation methods have added to our theoretical knowledge and analytical capability in terms of modeling the performance of fishing gear. Recently, many researchers have conducted computer-aided behavior analysis of fish cage systems. Tsukrov et al. (2003) developed a consistent net element method that has been used in the numerical modeling of open ocean aquaculture systems. Tsukrov et al.’s (2005) improved modeling method for nonlinear elastic material has been applied to the development of the feed buoy mooring system. Fredriksson et al. (2003) used a stochastic
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approach to analyze the reactions of a cage designed by the linear system and compared these results with field measurements. Thomassen and Leira (2005) demonstrated the importance of the buoyancy load by measuring forces applied to the square frame without netting, and Berstad et al. (2005) conducted calculations by dividing circular cages into membranes through a simulation method to verify the efficiency of the standard Norwegian cage. Raman-Nair and Colbourne (2003) studied the dynamics of a mussel long-line system using the lumped-mass method. Lee and Pei-Wen (2000) examined the tension-leg platform of underwater net–cage systems and derived and solved analytically the equations of motion of the system subjected to flow-induced drag motion and wave-induced surge motion. Lader and Enerhaug (2005) conducted an experiment to investigate forces and net deformation using different weights attached to the model cage. Further studies include numerical modeling of the floating collar. Huang et al. (2006, 2007) studied a series of gravity cages using the lumped-mass method and net plane elements; they also analyzed the behavior of the floating collar, which they regarded as a rigid body, and estimated the volume reduction coefficient of the cage caused by the currents. Li et al. (2006) and Zhao et al. (2007) used a lumped-mass model to simulate the dynamic response of a gravity cage with a floating system to waves combined with current. Fredriksson et al. (2007) worked on finiteelement modeling of the floating collar and simulated the process of localized failure of the high-density polyethylene (HDPE) pipe. Such extensive studies of fish cages have provided an improved understanding of the various external forces that affect their components. Results of these studies provide valuable information necessary for designing an optimal structure that can tolerate the environmental forces of the open ocean. In this study, we propose a mass-spring model to describe the whole elements of a fish cage system, including the floating collar. We conducted a series of water tank experiments using model cages to validate the mathematical model and to improve the calculation accuracy. We performed a mathematical calculation to interpret the dynamic behavior of the model system when influenced by currents and waves. The simulated and experimental values were compared and analyzed. Finally, the behavior of an actual fish cage influenced by currents and waves was simulated.
2. Materials and methods 2.1. Mathematical model 2.1.1. Equation of motion Fish cage systems consist of various kinds of netting, rope and a floating collar, as well as appendages such as floats, sinkers and anchors. In this study, the system was modeled based on the mass-spring model. In this model, the nettings and ropes are regarded as flexible structures and the floating collar as an elastic structure. The structures are divided into a finite number of elements and mass points are placed at the mid-point of each element; the mass points are connected by springs without mass. Applying this method to the netting, which takes up most of the cage structure, the mesh knots in the netting can be considered as the mass points and the mesh bars can be considered as the springs connecting the mass points. Direct application of this method to a commercial system produces an enormous number of mass points. To reduce the calculation loads, we used the approximation method, in which several actual meshes are bundled together into a virtual
mathematical mesh having the same physical properties (e.g., mass, specific gravity, weight, projected area, hydrodynamic coefficients) (Lee et al., 2005; Kim et al., 2007). It is assumed that the virtual mesh and the original mesh are geometrically identical; in other words, the angle between two adjacent virtual mesh bars and the angle of attack are the same as those of the original netting. We also placed one more mass point on the mid-point of each mesh bar to represent the bending of the bar (Fig. 1). We assumed the bar of the virtual mesh to be a cylinder and the knot to be a sphere. The projected area of the virtual mesh knots is the sum of all the knots in the actual netting and the area of the virtual mesh bars is the sum of all the bars in the actual netting. For the mooring line, we divided the line into constant lengths, and mass points were placed at the mid-point of each element; the mass points were connected by springs (Fig. 2). The appendages also were considered to be mass points. In general, a floating collar consists of one big floating tube or two small tubes connected in parallel to form the circular upper structure, with a handrail installed on top of it. For convenience of analysis, we modeled one floating tube without the handrail. For the mass points of the floating collar, a rectangular parallel pipe, similar to the original 3D structure, was constructed, and mass points were placed on each corner (Fig. 2). The spring was placed along the edges and diagonally to make sure that the structure maintained its shape even if external forces were applied in several different directions. The projected area, buoyancy, and weight in air of each mass point were equal to the value obtained by dividing the total value of the floating collar by the number of mass points. Buoyancy or weight was applied to each mass point in addition to the hydrodynamic force. Whether buoyancy or weight in air was applied to each mass point was determined as follows. The computing program checked whether each mass point was in the water or the air at every calculation step; if in the water, buoyancy was applied, and if in the air, weight was applied. The equation of motion for each mass point can be described as follows: ðm þ DmÞq€ ¼ Fint þ Fext ,
(1)
where m is the mass of the mass points, Dm is the added mass, q€ is the acceleration vector, Fint is the internal forces applied between the mass points, and Fext is the external forces applied to the mass points. For the fish cage system, translational motion by the tidal current is dominated, so rotational motion is ignored. The added mass of the mass points is given as follows:
Dm ¼ rW V n C m ,
(2)
where rW is the density of seawater, Vn is the volume of the mass points, and Cm is the added mass coefficient, which is 1.5 because the structure is considered to be a sphere. For cylindrical structures like the mesh bars and floating collar, Cm is calculated as follows: C m ¼ 1 þ sin a,
(3)
where a is the angle of attack. 2.1.2. Internal force The internal force is the force that is applied to the springs connecting each mass point. It is assumed that internal forces are applied to the netting and the rope in the direction of tension and to the floating collar in the direction of both tension and compression. The length of the spring elongated or reduced in the direction of tension or compression, respectively, is assumed to be linearly proportional to the force in action. The internal force
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Fig. 1. Placement of virtual mathematical mesh using mesh grouping method (a) and vector notation of the element (b).
applied to each mass point is as follows: Fint ¼
n X
0
ki ni ðjri j li Þ,
(4)
i¼1
where ki is the stiffness of the springs comprising the structure, ni the unit vector along the line of the spring, |ri| the magnitude of 0 the position vector between the neighboring mass points, li the initial length of the spring, and n is the numbers of adjacent mass points i. The stiffness is the equivalent value of each material. It can be calculated using the following formula: k¼
EA 0
l
,
(5)
where E is the Young modulus and A is the effective area of the material. The effective area of the mesh bar or rope is 60% of the apparent cross-sectional area. Furthermore, under the same tensile load, the elongation of a rope is greater than the elongation of a solid bar of the same material and with the same metallic cross-sectional area because the ropes in a cable tighten up (Gere, 2000). Thus, the modulus of elasticity (i.e., effective modulus) of a rope is less than the modulus of the material of which it is made. In this paper, the effective modulus of the mesh bar or rope is assumed to be 60% of the modulus of the material. The effective area of the floating collar is the area of the material of the polyethylene (PE) pipe obtained by subtracting the inside diameter from the outside diameter. The stiffness of the diagonal direction spring of the floating collar is assumed to be the same as its stiffness in the straight direction.
2.1.3. External force The external force (Fext) is the force that is applied to each mass point from the outside environment and it consists of the drag force (FD), lift force (FL), and buoyancy and sinking force (FB), as follows: Fext ¼ FD þ FL þ FB .
(6)
The current is assumed to be steady and uniform, so the inertial force is neglected. The drag and lift forces are as follows: FD ¼
1 C D rw SV 2 nV 2
(7)
Fig. 2. Modeling of the fish cage net, the mooring line, and the floating collar.
FL ¼
1 C L rw SV 2 nL , 2
(8)
where CD is the drag force coefficient, S the projected area of the mass point, V the magnitude of the resultant velocity vector, nV the unit vector of the drag vector, CL the lift force coefficient, and nL is the direction of the lift force (see Fig. 1b), calculated as follows: nL ¼
ðV rÞ V , jðV rÞ Vj
(9)
where V is the resultant velocity vector. The resultant velocity vector V is composed of the motion velocity vector of the mass point Vm, the current velocity vector Vc, and the wave-induced water particle velocity vector Vw. That is V ¼ Vm Vc þ Vw .
(10)
In this study, we did not consider the influence of the mutual interference of the velocity field in the case of concurrent application of current and waves. The hydrodynamic coefficients of the mesh bar are the function of the attack angle formed by the bar and the velocity vector,
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differential equation in the time domain:
and the attack angle is obtained as follows: Vr . a ¼ cos1 jVjjrj
(11)
The drag coefficient and lift force coefficient of a cylindrical structure such as the mesh bar and the floating collar are the same as those used by Lee et al. (2005) and Kim et al. (2007). Because the mesh knot, float and sinker of the net are regarded as spheres, no lift force is applied, resistance is applied, and the drag coefficient is assumed to be 1.5 (Fredheim and Faltinsen, 2003). The buoyancy and sinking force, FB, can be written as follows: FB ¼ ðri rw ÞV n g,
(12)
where ri is the density of the structure and g is the gravity acceleration.
€ MqðtÞ ¼ Fint ðtÞ þ FD ðtÞ þ FL ðtÞ þ FB ðtÞ,
(13)
€ where M is the mass including added masses and qðtÞ is the acceleration. Eq. (13) can be transformed into two first-order differential equations: _ qðtÞ ¼ Vm ðtÞ
(14)
_ m ðtÞ ¼ M 1 ½Fint ðtÞ þ FD ðtÞ þ FL ðtÞ þ FB ðtÞ. V
(15)
In this study, to solve the behavior of the structure described in the form of the stiff system, we integrated using fourth-order Runge-Kutta method. 2.2. Water tank experiments 2.2.1. Velocity reduction ratio for the plane netting
2.1.4. Computation method After substituting the internal and external forces in Eq. (1), the equation of motion governing the motion of the fish cage system became the following non-linear second-order derivative
Table 1 Netting materials of model nets used in the experiment Net
Material
Twine diameter (mm)
Bar length (mm)
Solidity ratio
A B C
Polyester Polyethylene Polyethylene
4 5 6
55 55 55
0.15 0.19 0.23
Fig. 3. Arrangement of experimental apparatus in the circulating water tank.
In a fish–cage system, the rear netting is less affected by the force of the current flow than the front netting. This phenomenon is the result of the shielding effect, by which some of the water flow is eliminated by the obstacle created by the front netting, In this experiment, we used three types of netting with different d/l ratios (d: twine diameter, l: bar length) and measured the fluid velocities before and after the flow passed through the netting. Assuming that the attack angle is 901 when the flow is applied at right angles to the netting, we measured fluid velocities with attack angles of 301, 601, and 901. The velocity reduction ratio was obtained by subtracting the velocity after passing the netting from the original velocity before passing the netting and dividing this value by the original velocity. Table 1 lists the dimensions of the netting used in the experiment; a stainless steel frame (1 m 1 m)
Fig. 4. Arrangement of a circular model cage in the circulating water tank.
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with a 4 mm diameter was used to fix the netting. The fluid velocities were measured 1 m away from the center of the netting, and experiments were conducted with fluid velocities of 0.3 and 0.6 m/s using a circulating water tank (10.2 m L 3.2 m H 2.8 m W) at Pukyong National University (Fig. 3). 2.2.2. Deformation of the model circular cage caused by currents To check whether the accuracy of the mathematical calculation was improved when the velocity reduction ratio was applied, we constructed a model cage and conducted a water tank experiment. The same model was used for numerical calculation. The shape of the model cage was measured when a constant fluid velocity was applied to the cage. The model was a circular cage with a diameter of 1 m and depth of 0.5 m made from nylon-netting material with a mesh bar diameter of 1 mm and a bar length of 5 mm. The experiment was conducted in the circulating water tank and the deformation of the cage was measured using a digital camera and a digitizer (Fig. 4). The fluid velocities applied were 0.15 and 0.3 m/s. The parameters used in the mathematical calculation are shown in Table 2. We then compared the experimental values and the simulated values obtained from the mathematical model.
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The model was a square cage (1 m L 1 m H 1 m W; see Table 3 for the materials used to make the model). To measure the tension on the mooring lines, we installed load cells on the upper part of the front and rear mooring lines. The tests were conducted in the 3D towing tank (85 m L 4 m H 10 m W) at the National Fisheries Research & Development Institute (Fig. 5). The numerical model used the same structure and physical properties as the cage used in the experiment (see Table 4). The tension of the mooring line and the shape of the cage when waves were applied were calculated, and results of this calculation were compared with the experimental results. The cage was subjected to regular waves based on linear wave theory. The wave heights
2.2.3. Response of the model cage to waves We next measured the tension of the mooring lines and the shape changes of a cage structure when it was exposed to waves.
Table 2 Physical characteristics of calculation parameters of the model circular cage No. of mass points
Mass (kg)
Projected area (m2)
Stiffness (N/m)
Initial length (m)
2172
9 104
3.8 103
12,025
0.1
Table 3 Physical specifications of the model square cage Items
Specifications
Buoy Corner buoys Material Density (g/cm3) Diameter (cm) Number of pieces Total Buoyancy (N)
Plastic 0.23 15 4 47.82
Buoys on float line Material Density (g/cm3) Diameter (cm) No. of pieces Total Buoyancy (N)
Plastic 0.33 3.5 48 15.05
Netting Material Density (g/cm3) Twine diameter (cm) Total mass (g) Mesh size (cm) Solidity
Nylon 1.14 0.1 0.3424 0.5 0.2
Bottom weights Material Density (g/cm3) Unit mass (g/pieces) Number of pieces Total weight (N)
Lead 11.35 2.52 103 12 2.65
Fig. 5. Arrangement of a square cage in the 3D towing tank (A, B: measuring points).
Table 4 Physical characteristics of calculation parameters of the model square cage No. of mass points
Mass (kg)
Projected area (m2)
Stiffness (N/m)
Initial length (m)
2248
1.523 104
8.896 104
11,942
0.05
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were 0.1 and 0.3 m and the periods were 1.8 and 2.4 s for both the tests.
3. Results and discussion 3.1. Velocity reduction ratio Table 5 shows the fluid velocity measured at the rear of the netting when the attack angles were 901, 601, and 301, and the fluid velocities were 0.3 and 0.6 m/s. In general, the velocity reduction ratio varied depending on the solidity ratio of the netting, the angle of attack, and the velocity. The velocity reduction ratio decreased with an increase in flow velocity. The velocity reduction ratio was higher with an increase in the solidity ratio and with a decrease in the angle of attack. Netting on a slanted plane would generate lift force components and the streamlining of the flow would change; however, we did not measure streamlining in this experiment. We also did not test a wide range of netting solidity ratios; instead, we used only netting with a relatively small solidity ratio. Aarsnes et al. (1990) reported that the velocity reduction ratio was reduced by about 6% when the netting had a solidity ratio of 0.13. In our study, the average reduction ratio was 7% for the solidity
Table 5 Fluid velocity reduction ratio attack angle for different types of nettings Type
Velocity (m/s)
Attack angle (1)
Velocity reduction ratio (%)
A
0.3
90 60 30
9 9 10
0.6
90 60 30
5 6 7
0.3
90 60 30
14 15 16
0.6
90 60 30
7 9 10
0.3
90 60 30
19 19 21
0.6
90 60 30
10 11 12
B
C
ratio 0.15 (A Net) and attack angle 901 for the two velocities (Table 5). The velocity reduction ratio adopted to the numerical calculations of this paper was determined based on the solidity of the netting, and the differences caused by different angles of attack and the magnitude of the velocity were not taken into account for the convenience of calculation. That is, in calculating the velocity applied to the rear netting of the model cage, we assumed that 85% of the velocity applied to the front netting was applied uniformly to the entire rear netting based on the results of Aarsnes et al. (1990). In calculating for an actual cage, 93% of the velocity of the front netting was assumed to have been applied uniformly to the rear netting based on our experiment. It is difficult to determine the accurate velocity reduction ratio when the fluid passes through the netting or cages because of the complicated interaction between the fluid and the flexible structure. The reduced size of models used in tank experiments and the difficulty in obtaining reliable data in field experiments add to the complexity. However, velocity reduction ratio is one of the important factors that affect the deformation and loads of the cages. To precisely calculate the behavior of fish cage systems, additional detailed experiments are required to obtain more accurate velocity reduction ratios for a wide range of nettings, velocities, and cage systems rigged in various ways.
3.2. Deformation of the circular cage As shown in Fig. 6, the side shape of the model cage changed dramatically as the velocity increased. Most notably, the front netting was more deformed by the direction of the flow than the rear netting because of the shielding effect. Fig. 7 shows the results of the numerical calculation at a fluid velocity of 0.3 m/s when the fluid velocity reduction ratio was not applied (Fig. 7a) and when 15% of the velocity reduction ratio was applied (Fig. 7b). Fig. 8 shows a comparison of the experimental and calculated values in relation to the side view of the cage. Fig. 8a illustrates
Fig. 7. Simulation results using a current speed of 0.3 m/s with the reduction ratio not applied (a) or applied (b).
Fig. 6. The shape of the cage model according to a fluid velocity of 0.15 m/s (a) and 0.3 m/s (b).
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Fig. 8. Comparison of the experimental vs. the calculated side shape at a current speed of 0.3 m/s with the reduction ratio not applied (a) or applied (b).
Fig. 9. Comparison of experimented (a) and simulated (b) results of the square cage to waves (height 0.3 m, period 2.4 s).
18
3.3. Response of the model cage to waves
16
Experiment Simulation
Tension (N)
14 12 10 8 6 4 2 0
0
0.5
1
1.5
2
2.5 3 Time (s)
3.5
4
4.5
5
Fig. 10. Comparison of the tension of the mooring line between experimental and numerical simulations to waves (wave height 0.3 m and period 2.4 s).
the scenario in which the shielding effect was not considered and a 14% position calculation error occurred, whereas Fig. 8b shows the case in which the shielding effect was taken into account and a p5% calculation error occurred.
Fig. 9 shows a sample of experimental data and the numerical prediction of the cage exposed to waves (height 0.1 m, period 2.4 s). To check the accuracy of the numerical prediction, the calculated tension applied to the mooring line was compared to the model test results. Fig. 10 shows that the tension values obtained for the simulation and model tests; the data were almost identical, but the shape of the tension curves differed slightly. For the simulation, the trough and the peak of the curve were similar; for the model, the peak was sharper than the trough. The curves differed because the wave shape applied in the simulation was a perfect cosine curve, whereas the actual wave shape applied in the model test was not a complete cosine curve. The values for the locus of the buoy (A in Fig. 5) and sinker (B in Fig. 5) of the cage were shown in Fig. 11. In general, the calculated positions coincide with the measured positions. At the trough, however, the position error was pronounced. This result might reflect the difference of the wave shapes between the simulation and the model test as mentioned previously. The amplitude of the sinker was slightly smaller than that of the float in both cases due to the elongation of the twine and the damping effect of the netting. The drag and lift coefficients are important parameters that affect the loads and deformation of the cage. These coefficients depend on three main factors—the angle of attack, the solidity of the material, and the Reynolds numbers—and their influence
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must be considered. In this study, the drag and lift force coefficients of the netting, rope, and floating collar, which are regarded as cylinder-shaped, were those values related to the angle of attack. We used these hydrodynamic force coefficients
0.2 0
Depth (m)
-0.2 Exp.(float) Exp.(sinker) Simu.(float) Simu.(sinker)
-0.4 -0.6 -0.8 -1
because the influence of the angle of attack was the most dominant and the influence of the other two factors was not so obvious (Lee et al., 2005). However, the cage made large movements up and down in the vertical plane when it was exposed to waves. In such a case, we assumed that the attack angle of the side panels of the cage is the interior angle between the resultant velocity vector and the plane of the mesh (Kim et al., 2007). A more accurate calculation of the dynamic behavior of the fish cage requires refined hydrodynamic force coefficients that take the angle of attack, solidity ratio, and Reynolds number into consideration. Because this study used the hydrodynamic coefficient obtained in the experiment in the Reynolds number range 3000–5000, the hydrodynamic force probably was underestimated in the low Reynolds number range and the force likely was overestimated in the high Reynolds number range. 3.4. Verification of the mathematical model
-1.2 0
1
2
3
4
5
4
5
Time (s) 0.2 0
Depth (m)
-0.2 Exp.(float) Exp.(sinker) Simu.(float) Simu.(sinker)
-0.4 -0.6 -0.8 -1 -1.2 0
1
2
3 Time (s)
Fig. 11. Comparison of the locus of the buoy and sinker of the square cage between experimental and simulation results for different wave conditions: (a) wave height 0.1 m and period 2.4 s; and (b) wave height 0.3 m and period 2.4 s.
To validate the mathematical model proposed in this study, we used data from Lader and Enerhaug (2005). For the numerical calculation, we used the parameters of the Lader and Enerhaug’s (2005) actual experimental cage: a 1.435 m diameter and 1.435 m deep cylindrical cage; nylon knotless netting with 32 mm mesh size and a 1.8-mm-thick mesh bar; and three models with different sinkers (16 0.4 kg, 16 0.6 kg, 16 0.8 kg). For the calculation, the velocity reduction ratio of the rear netting was 15% and 2368 mass points were placed. As shown in Fig. 12, the simulated shapes of the model cage were all in close agreement with the experimental results of Lader and Enerhaug (2005). The inside volume reduction was calculated according to the method presented by Huang et al. (2006). Changes in the inside volume according to the different velocities are shown in Fig. 13. In our calculation, the inside volume reduction coefficient was lower than that reported by Lader and Enerhaug (2005). In Fig. 14, the calculated drags on the model cage net are compared with experimental values for different weights. As was pointed out in the Section 3.3, the calculated drags were slightly smaller than the experimental ones at the lower velocity range (less than 0.4 m/s), and it was opposite at the higher velocity range (more than 0.4 m/s).
Fig. 12. The deformation of the model cage as a result of the numerical simulation for different weight and current velocities.
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Inside volume reduction ratio (-)
3.5. Simulation of the actual cage 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5
Experiment Calculation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.4
0.5
0.6
Inside volume reduction ratio (-)
Velocity (m/s)
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5
Experiment Calculation
0
0.1
0.2
0.3
Inside volume reduction ratio (-)
Velocity (m/s)
To verify the practicality of the calculation method, we conducted a simulation of the motion of an actual fish cage system in an environment in which current and waves were applied. The simulated cage was a circular type used in tuna farming in the Mediterranean Sea. It had a diameter of 50 m and a depth of 20 m and was composed of a PE floating collar, PES (polyester) Raschel netting, floats, sinkers, and mooring lines. The floating collar was a 50.8 cm diameter and 46.2 mm thick tube. The total buoyancy corresponding to circumference 157 m was 174,440 N. The number of mass points placed on the floating tube was 300, and the physical characteristics of each mass point are shown in Table 6. The mesh bar length of the cage net was 55 mm and the twine diameter was 4 mm. The total number of knots was 1,701,999; they were grouped and approximated to 1412 mass points. The total sinking force of the cage was 12,828 N. Fig. 15 shows the shape of the cage when it was placed in still water (a) and when it was exposed to currents of 0.2 m/s (b), 0.4 m/s (c), 0.6 m/s (d), and 0.8 m/s (e). Fig. 16 shows the side view of the cage when waves of height 6 m, length 156 m, and a period of 10 s were applied in the same direction as the current of 0.7 m/s. Due to the stiffness of the floating structure, the floating tube did not bend in the current and maintained its original shape (Fig. 15). In response to the waves, the floating collar followed the wave-surface level with a slight time delay (see Fig. 16 or video at www.mpsl.co.kr/simucage.htm). In other
1 0.95 0.9 0.85 0.8
Table 6 Physical characteristics of calculation parameters of the actual cage for the floating collar, netting, and mooring lines
0.75 0.7 0.65 0.6
Items Experiment Calculation
0.55 0.5 0
0.1
Mass (kg)
Floating collar
0.2
0.3
0.4
0.5
0.6
Velocity (m/s) Fig. 13. The ratio of the inside volume reduction of the model cage according to velocity with weights of 0.4 kg 16 (a), 0.6 kg 16 (b), and 0.8 kg 16 (c).
4.76
Cage netting Mooring line a b
0.40 20.74
Projected area (m2)
Stiffness (N/m)
Initial length (m)
0.254
8,160,053a 39,372,206b 322,254 11,836,048
2 0.46 2 15.5
0.53 0.86
Lengthwise spring. Crosswise spring.
110 Experiment 16X0.4kg
100
Experiment 16X0.6kg
90
Experiment 16X0.8kg Calculation 16X0.4kg
80
Calculation 16X0.6kg Calculation 16X0.8kg
Drag (N)
70 60 50 40 30 20 10 0 0
0.05
0.1
0.15
0.2
0.25 0.3 0.35 Velocity (m/s)
0.4
0.45
0.5
0.55
Fig. 14. Comparison of the drag of the model cage net between experimental and numerical simulations.
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Fig. 15. Successive shapes of the cage system in still water (a) and with current velocities of 0.2 m/s (b), 0.4 m/s (c), 0.6 m/s (d), and 0.8 m/s (e). A: a referenced point for the tension values.
words, the response of the floating collar to the current or waves was qualitatively similar to those observed in the field (e.g., the pitch motion and horizontal and vertical displacements of the floating collar). The quantitative analysis of the dynamic behavior of the cage system will be presented in the next paper. The mass-spring model proposed in this paper divides the floating collar into many different mass points and the hydrodynamic forces are calculated for each different mass point. When calculating the hydrodynamic force of the floating pipe, our computing program checked that each mass point was in the water or the air at every calculation step; if in the water, the buoyancy, drag, and lift force of the relevant mass point were
calculated, and when above the water, the weight was applied. Thus, the mass-spring model can calculate the external forces no matter what state the floating collar is in. Although wave-induced hydrodynamic force coefficients of the partially submerged pipe elements can differ when there is a uniform current with full submersion (Li et al., 2007), only the coefficients obtained in full submersion and a uniform current were used in this study. The tensions on the upper side mooring line at the point A (in Fig. 15) are shown in Fig. 17. When the current was applied to the cage, the shape of the cage changed greatly and the tension on the upper side mooring line increased greatly. When waves were added to the current, the cage made large movements up and
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Fig. 16. Successive side shapes of the cage system exposed to current (0.7 m/s) and waves (height: 6 m, period: 10 s) simultaneously.
down, significantly the maximum tension on the mooring line on the upper side increased by 180% compared to when only the current was applied
4. Conclusion We used the mass-spring model to describe the behavior of a fish cage system consisting of netting, floating collar, floats, and sinkers and validated the model using water tank experiments. In addition, to prove the practical applicability of our numerical methods, we simulated the influence of the tidal current and waves independently or concurrently applied to the actual fish cage system. The mass-spring model successfully described the
motion of the whole cage structure, for both flexible materials (nettings and ropes) and elastic material (the floating collar). This mathematical calculation and simulation technique can be used to evaluate the safety of a large-scale moored cage structure exposed to diverse environmental forces. If used during the design stage (prior to construction), it will be helpful in designing a safer and more economical system. In existing systems, fouling organisms often lead to fatal damage of the structure; this problem can be simulated by adjusting the projected area and the drag coefficient. Computer-based simulation methods such as this one can provide a quantitative analysis of the environmental forces that act on fish cages and valuable information necessary for designing an optimal structure.
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80 70
Tension (KN)
60 50 40 30 20 10 0
0
50
100
150 Time (s)
200
250
300
Fig. 17. Variations in tension acting on the upper side mooring line at the point A in Fig. 15.
Acknowledgments This work was supported in part by grant no. MNF22004026-21-SB010 of the Korean Ministry of Maritime Affairs and Fisheries and grant no. R01200600011020 of the Korea Science and Engineering Foundation. The authors would like to express thanks for their financial support. References Aarsnes, J.V., Rudi, H., Loland, G., 1990. Current Force on Cage, Net Deflection, Engineering for Offshore Fish Farming. Tomas Telford, London, pp. 137–152. Berstad, A.J., 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. Bessonneau, J.S., Marichal, D., 1998. Study of the dynamics of submerged supple nets (application to trawls). Ocean Engineering 25 (1), 563–583. Fredheim, A., Faltinsen, O.M., 2003. Hydroelastic Analysis of a Fishing Net in Steady Inflow Conditions. Hydroelastic in Marine Technology, Oxford, UK, pp. 1–10. Fredriksson, D.W., Swift, M.R., Muler, E., Baldwin, K., Celikkol, B., 2000. Open ocean aquaculture engineering: system design and physical modeling. Marine Technology Society Journal 34 (1), 41–52. Fredriksson, D.W., Swift, M.R., Irish, J.D., Tsukrov, I., Celikkol, B., 2003. Fish cage and mooring system dynamics using physical and numerical models with field measurements. Aquacultural Engineering 27, 117–146. Fredriksson, D.W., DeCew, J.C., Tsukrov, I., 2007. Development of structural modeling techniques for evaluating HDPE plastic net pens used in marine aquaculture. Ocean Engineering 34, 2124–2137. Gere, J.M., 2000. Mechanics of Materials, 5th ed. Brooks/Cole/Thomson Learning, pp. 1–77. Huang, C.C., Tang, H.J., Liu, J.Y., 2006. Dynamical analysis of net cage structures for marine aquaculture: numerical simulation and model testing. Aquacultural Engineering 35, 258–270. Huang, C.C., Tang, H.J., Liu, J.Y., 2007. Modeling volume deformation in gravity-type cages with distributed bottom weights or a rigid tube-sinker. Aquacultural Engineering 37, 144–157. Kawakami, T., 1964. The theory of designing and testing fishing nets in model. In: Modern Fishing Gear of the World, vol. 2. Fishing News Books Ltd, London, pp. 471–489. Kim, D.A., 1999a. Design of Fishing Gear (in Korean). Pyounghwa Printing House, pp. 106–139.
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