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Numerical study on wave attenuation inside and around a square array of biofouled net cages
Accepted Manuscript Title: Numerical study on wave attenuation inside and around a square array of biofouled net cages Authors: Chun-Wei Bi, Yun-Peng Zhao, Guo-Hai Dong, Tiao-Jian Xu, Fu-Kun Gui PII: DOI: Reference:
S0144-8609(17)30054-7 http://dx.doi.org/doi:10.1016/j.aquaeng.2017.07.006 AQUE 1912
To appear in:
Aquacultural Engineering
Received date: Revised date: Accepted date:
21-3-2017 13-7-2017 16-7-2017
Please cite this article as: Bi, Chun-Wei, Zhao, Yun-Peng, Dong, Guo-Hai, Xu, Tiao-Jian, Gui, Fu-Kun, Numerical study on wave attenuation inside and around a square array of biofouled net cages.Aquacultural Engineering http://dx.doi.org/10.1016/j.aquaeng.2017.07.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Numerical study on wave attenuation inside and around a square array of biofouled net cages
Chun-Wei Bia,*, Yun-Peng Zhaoa,*, Guo-Hai Donga, Tiao-Jian Xua, Fu-Kun Guib
a
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of
Technology, Dalian 116024, China b
Marine Science and Technology School, Zhejiang Ocean University, Zhoushan
316000, China
Highlights
Wave attenuation by a square array of biofouled net cages is numerically studied
Effect of incident angle on wave attenuation by the biofouled cage array is analyzed
The study has implication for water exchange and structural design of the net cage
Abstract In this study, waves propagating through a square array of 16 net cages with different levels of biofouling are numerically studied using a three-dimensional computational fluid dynamics (CFD) model. A porous-media fluid model is adopted to simulate both clean and biofouled netting of a cage array in waves. A numerical wave tank is built, and the oscillating-boundary method is adopted to generate waves.
*
Corresponding author. Fax: +86-0411-84708526; Tel.: +86-0411-84707116.
E-mail address: bicw@ dlut.edu.cn (C.W., Bi), [email protected] (Y.P., Zhao) 1
The flow motion is solved by the Navier-Stokes equations, and the free water surface is captured using the volume of fluid (VOF) method. The numerical model is validated by comparing the numerical data with corresponding experimental measurements of a net-cage model with clean netting. To analyze wave attenuation, a numerical analysis of wave elevation both inside and around the cage arrays is presented, which considers the effect of biofouling. Based on the results of the present study, the effect of biofouling on wave elevation is noticeable; the damping effect of the cage array increases with increasing level of biofouling. Furthermore, the incident angle of waves has a noticeable effect on the wave field inside and around the cage array. Keywords: Cage array; Biofouling; Wave attenuation; Numerical simulation
1. Introduction Currently, commercial fish farming is in a stage of dramatic expansion, where approximately half of all seafood eaten around the world is being farmed. China has become the largest exporter of aquatic product in the world due to the rapid expansion in its fish production, particularly from aquaculture (FAO, 2014). With the goal of being more socially acceptable, environmentally friendly and economically profitable, it is urgent to exploit new space for aquaculture and to develop an offshore aquaculture strategy (Aarsnes et al., 1990; Zhan et al., 2006; Klebert et al., 2013). When deployed in offshore locations, an array of net cages, or cage array for short, is subjected to intense waves and has a damping effect on the incident waves. Thus, a cage array will change the wave field in the vicinity of the region, which should be taken into account when estimating the wave loads of the net cage. Similar to other marine structures, the infrastructure of a cage array suffers from the effects of biofouling in the open sea (see Fig. 1). Biofouling, which is the accumulation of marine organisms, is prone to leading to rapid net aperture occlusion. Biofouling on netting primarily causes the following disadvantages: i) the attenuation of currents and waves through the net cage increases, which decreases the water 2
exchange and oxygen supply inside the net cage; ii) the hydrodynamic forces acting on the net cage dramatically increase, which increases the risk of structural failure and fatigue damage of the net cage and iii) the biofouled netting is easily deformed in currents/waves and thus negatively affects the effective volume for farmed fish (Braithwaite et al., 2007; Klebert et al., 2013). In the literature, Swift et al. (2006) conducted experiments both in a tow tank and in the field to measure the drag forces acting on biofouled plane nets, and a 240% increase in drag coefficient was found for the maximum biofouled plane net. Lader et al. (2015) performed field tests to investigate the growth characteristics of hydroids that grow on a net and conducted laboratory experiments to study the hydrodynamic drag on the fouled twines using fabricated models of net twines with artificial hydroid fouling. Gansel et al. (2015) studied the effect of hydroids that act on the forces that act on plane nets; a parameterization of fouling was assessed in terms of the relationship between net solidity and drag force. Bi et al. (2015a) investigated the drag force acting on a full-scale biofouled net cage using a numerical approach, and the effect of biofouling on the flow fields inside and around the net cage were evaluated. The
aforementioned
related
studies
investigated
the
hydrodynamic
characteristics of biofouled plane nets and net cages in currents. To better understand the damping effect of the net structure on wave propagation, a number of investigations has been performed in recent decades. Among many others, Chan and Lee (2001) analytically studied the wave scattering characteristics by a flexible fishing net, which were modeled as a porous-flexible barrier coupled with an equivalent hydrodynamic force. The objective of this study is to investigate how a flexible net structure affects transmitted and reflected waves. Lader et al. (2007) conducted laboratory experiments with different net panels in several regular waves. Based on the wave elevation measurements, the wave geometry, wave energy as well as crest steepness and asymmetry factors were analyzed. Dong et al. (2008) proposed a board-net floating breakwater to protect fish cages, and the transmission coefficients were examined with various lateral dimensions and number of rows of net. The 3
experimental results indicated that the board-net floating breakwater may have potential application in protecting fish and fish cages from fierce waves. Most recently, Chen and Christensen (2015) numerically studied the interaction between a moving net structure and waves based on the open source library OpenFOAM, where the net was simulated as a sheet of porous media. It was concluded that a wave transmission coefficient of 95.2% was found when a wave propagated through the net. Bi et al. (2015b) experimentally investigated a wave propagating through four tandem net cages, and the effects of different parameters of both wave and net cage on the wave transmission coefficient were discussed. Moreover, the wave fields inside and around the net
cages
were presented by
solving the three-dimensional
Reynolds-averaged Navier-Stokes equations. Although much progress has been made, to the best of our knowledge, there has been no investigation on the wave field around the net cages or cage arrays that considers the effect of biofouling. However, biofouling on nettings can increase the damping effect of the cage array on wave propagation. Thus, an investigation on a wave propagating through a cage array that considers the effect of biofouling is of great importance. As an extension to our previous work (Zhao et al., 2014; Bi et al., 2015b), a three-dimensional numerical model is established to simulate a wave propagating through cage arrays with different levels of biofouling. In this study, both clean and biofouled netting of a cage array will be simulated using the porous-media fluid model; combined with a numerical wave tank, a wave propagating through a biofouled cage array is presented.
2. Description of numerical model For computational efficiency, it is impossible to simulate the exact structure of the netting and biofouling. Therefore, both clean and biofouled netting of the net cage were simulated using the porous-media fluid model in this study. A three-dimensional numerical wave tank was built, which was primarily based on the Navier-Stokes equations with the capacity to generate regular waves. Using the same governing 4
equations, the net-cage model and the numerical wave tank can be coupled.
2.1 Governing equations review The wave motion in the numerical wave tank can be primarily described by the incompressible Navier-Stokes equations. The governing equations in tensor forms are as follows: The continuity equation
ui 0 xi
(1)
and the momentum equation ui u j 2ui 2u j ui p gi 2 2 x t x j xi xi j
Si
(2)
where t is time; µ is fluid viscosity; ρ is fluid density; p is pressure; ui and uj are the average velocity components; i, j=1, 2, 3 (x, y, z); gi is the gravitational acceleration and Si is the source term for the momentum equation. It should be noted that the definition of the source term is different for the porous-media fluid model and the damping zone in the numerical wave tank. More detailed descriptions are given in the following sections. The governing equations are solved based on ANSYS FLUENT 15.0 using the finite volume method. An implicit, unsteady formulation and pressure-based solver is chosen to achieve a converged solution. The SIMPLEC (Semi-implicit Method for Pressure-linked Equation-Consistent) algorithm is used to solve the pressure-velocity coupling problem. The discretization of pressure, momentum, turbulent kinetic energy and turbulent dissipation rate are performed using a second-order upwind scheme.
2.2 Net-cage model considering biofouling effect In this study, both clean and biofouled nettings of the net cage are simulated using the porous-media fluid model (Bi et al., 2014; Zhao et al., 2013; Patursson et al., 2010). In the field test, the biofouling scattered randomly on the netting. It is difficult 5
to perform a direct numerical simulation. For simplicity, it is assumed here that the biofouling on the netting is distributed homogeneously. The bottom ring or sinkers of the net cage ensure the netting undergoes limited deformation and maintain the geometry of the net cage. In addition, only marginally movement and deformation of the netting was observed in the physical-model experiment. Thus, deformation of both clean and biofouled netting is neglected in this study. In the numerical model, the cylindrical net cage is divided into 16 plane nets along the circumference. Each plane net can be treated as porous media at a certain attack angle (Bi et al., 2015a; Zhao et al., 2013). Herein, because the float collar and sinker system have limited effects on the wave field, only the net chamber of the net cage is simulated, as shown in Fig. 2. The porous-media fluid model is a hypothetical model that acts in the same way as the water-blocking effect for both clean and biofouled netting by setting the porous coefficients. Outside the porous-media region, the source term added in the momentum equation Si is equal to 0. Inside the porous-media region, Si is calculated by the following equation:
Si Cij
1 uu 2
(3)
where Cij is given by Cn Cij 0 0
0 Ct 0
0 0 Ct
(4)
where Cij is the porous-coefficient matrix for the porous media, Cn is the coefficient in the normal direction, and Ct is the coefficient in the tangential direction. The unit of the porous coefficient is m-1. A typical source term for a porous-media model consists of one linear term and one quadratic term based on Darcy’s law. For large porosity (e.g., an array of fixed cylinders), the quadratic term for the frictional force will completely dominate over the viscous term (the linear term) (Burcharth and Andersen, 1995). In this case, the linear term is only a fitted term that has no physical meaning 6
and thus, it can be neglected. Therefore, only the quadratic term is kept in the source term. The drag (Fd) and lift (Fl) forces of a porous medium can be described as follows:
Fd Cn
1 A u u 2
(5)
Fl Ct
1 A u u 2
(6)
where λ is the porous-media thickness, and A is the porous-media area. According to the above equations, the porous coefficients of the porous media are directly related to the forces on the plane net. Generally, there are two methods to determine the drag and lift forces acting on a plane net: i) laboratory experiments and ii) an empirical formula, such as the Morison equation:
Fd
1 Cd Au02 2
(7)
Fl
1 Cl Au02 2
(8)
where u0 is the free stream velocity, Cd is the drag coefficient and Cl is the lift coefficient. According to Tsukrov et al. (2011), the formula provided by Balash et al. (2009) has an advantage in predicting the drag coefficient of a fishing net, where the relative error was approximately 4% compared with that of the measured data. For the lift coefficient, Aarsnes et al. (1990) performed a study of plane nets and established a formula as a function of net solidity and attack angle. Thus, Eqs. (9) and (10) were chosen to calculate the drag and lift forces of a plane net by combining Eqs. (7) and (8): Cd Cdcyl (8.03Sn2 0.74Sn 0.12) (Balash et al., 2009)
(9)
Cl (0.57Sn 3.54Sn2 10.1Sn3 )sin(2 ) (Aarsnes et al., 1990)
(10)
where Cdcyl 1 10 Re2/3 for Re 2.3 105 and α is the attack angle. The porous coefficient Cn can be calculated from a curve fit between plane-net 7
drag forces and corresponding current velocities using the least-squares method when the plane net is oriented normal to the flow. The other coefficient Ct can be obtained when a plane net is oriented at an attack angle. In this case, the coefficient Ct should be transformed into Ctα using Eq. (12) (Bear, 1972). Next, Ctα can be calculated from a curve fit between the plane-net lift forces and the corresponding current velocities using the least squares method. Once Cn and Ct are obtained, the porous coefficients for the plane net at various attack angles can be calculated as follows:
Cn Ct Cn Ct cos(2 ) 2 2 C Ct Ct n sin(2 ) 2
Cn
(11) (12)
where α is the attack angle, which is defined as the angle between the flow direction and the plane net in the horizontal plane; Cnα and Ctα are the normal and tangential resistance coefficients for the plane net at an attack angle of α, respectively. Based on the parameters of the netting used in the laboratory experiment, the porous coefficients for the clean netting can be calculated using the empirical-formula method: Cn=59.0 m-1 and Ct=40.0 m-1 with a porous-media thickness of 10 mm. According to our previous research (Bi et al., 2015c), the thickness of the porous media does not obviously affect the numerical results until the thickness is greater than 10% of the width or height of the plane net. However, an increase in the thickness of the porous media can effectively reduce the number of cells, which thus reduces the computational effort greatly. Therefore, the thickness of the porous media is defined as 10 mm for both the clean and biofouling nettings. For the netting with various levels of biofouling, the relationship between the porous coefficients and features of the biofouled netting can be built using the forces acting on the biofouled netting (Bi et al., 2015a).
2.3 Numerical wave tank The numerical wave tank is capable of producing regular and irregular waves by a specified motion of the wave-generation boundary. To eliminate the effect of reflected waves, a damping zone is defined at the end of the wave tank (see Fig. 2). 8
The primary part is the computational domain, which is set to larger than 10 times that of the wavelength.
2.3.1 Wave generation In this study, regular waves are generated using an oscillating-boundary method, where the wave-generation boundary is prescribed by the following equations for a second-order Stokes wave (Madsen, 1971):
an1 n a 3 [cos t ( 1 )sin 2t ] 2 tanh kh0 2hn1 4sinh kh0 2
(13)
where a is the wave amplitude, ω is the circular frequency, k is the wavenumber, h0 is the water depth and 2kh0 1 n1 (1 ) 2 sinh 2kh0
(14)
The generated wave elevation in the time domain is as follow:
a cos(kx t )
a 2 k cosh kh0 (cosh 2kh0 2) cos 2(kx t ) 4 sinh 3 kh0
(15)
By adjusting the wave frequency and amplitude, desired waves can be obtained, which can be validated by the theoretical values of Eq. (15).
2.3.2 Wave absorption To absorb the wave energy at the end of the wave tank and eliminate wave reflection, a damping zone is defined by adding a source term in the momentum equation. The source term is as follows:
Si ( x)ui
(16)
where µ(x) is the damping coefficient, which is given by
x x0 ，x0 x xe xe x0
( x)=K
(17)
In the equation, x0 and xe are the initial position and the end of the damping 9
domain in the x-direction, respectively. K is an empirical coefficient, which is related to the wavelength and the length of the damping zone. In this study, K is set to 20, where the length of the damping zone is equal to twice the wavelength. To examine the effectiveness of the wave absorption, wave elevations in the damping zone were monitored at different positions. Nearly all the wave energy can be absorbed before the waves reach the end of the wave tank.
2.3.3 Numerical convergence study The computational grids of the numerical wave tank are hexahedral elements. The grid size is set to h0/20 and is refined at the domain around the free surface and the wavemaker to ensure computational precision. Using wave case 4 as an example, the wave heights were measured at four different positions, and a numerical convergence study was performed on the grid size and time step, as shown in Fig. 3. As the number of cells increases in the x- and z-directions, the wave becomes more stable. When the grid size is L/45 in the x-direction and H/20 in the z-direction, the wave height converges. Based on the above grids, the numerical simulation shows the best result when the time step is 0.005 s, which is T/160 in dimensionless form, where T is the wave period.
2.3.4 Validation of the numerical wave tank To validate the numerical wave tank, the time series of wave elevation from the present numerical model are compared with the analytical results calculated from Eq. (15) (see Fig. 4). Although there are deviations in wave frequency and wave height for the first several waves, stable waves are obtained after several periods, which are used for data analysis. The results show that the wave elevations are in good agreement at different positions in the numerical wave tank. The generated wave is stable in both the space and time domains, which provides an essential foundation for simulating the wave field around the net cage.
3. Experimental validation 10
Using the proposed model, numerical simulations of a wave propagating through in-line net cages with different net-cage numbers were performed for various wave cases. Validation of the numerical model was conducted by comparing the numerical results with the corresponding experimental data.
3.1 Physical model and experimental setup To validate the numerical model, laboratory experiments on a 1:50 scaled physical model were performed (Bi et al., 2015b). The net-cage model in the present experiment consists of a top frame, a cylindrical net chamber and a bottom ring (see Fig. 5). The component parameters of the net-cage model are presented in Table 1. It should be noted that the net-cage model was kept fixed by the top frame during the tests. The net chamber was positioned 5 cm above and 11 cm beneath the still-water surface. To better represent the hydrodynamic behavior, the net in the physical model was the same as that of the full-scale net cage. In this way, the net in the physical model test can satisfy the Reynolds number similarity and avoid any variation of flow pattern through it. The laboratory experiments were conducted in a wave tank at the State Key Laboratory of Coastal and Offshore Engineering (SLCOE), Dalian University of Technology, Dalian, China. The wave tank is 22 m long, 0.45 m wide and 0.6 m deep. The water depth was 0.4 m during this experiment. Both the bottom and side walls of the wave tank are made of smooth glass with negligible frictional drag. A piston-type wavemaker is equipped at one end of the wave tank, and an energy-dissipation system is equipped at the other end to eliminate wave reflection. Wave elevations inside and around the in-line net cages, up to four net cages, were recorded under different wave cases. These measurements were performed three times to assess the repeatability of the measurements. The measurement positions of the wave gauges were arranged along the axis of the wave tank, as shown in Fig. 6. The wave parameters used in the laboratory experiments are shown in Table 2. The corresponding prototypical wave height is 1~3 m and wave period is 4~7 s, which is considered in the range of 11
common waves for an aquaculture net cage.
3.2 Results and discussion The wave transmission coefficient (CT) is introduced to quantitatively analyze the damping effect of net cages on the wave field; the definition of CT is as follow: CT
Ht Hi
(18)
where Ht is the wave height of the transmission wave, and Hi is the wave height of the incident wave. Numerical simulations were performed with different numbers of in-line net cages under the wave cases, as shown in Table 2. The comparison between the wave transmission coefficient from the numerical simulation and the corresponding experimental result are presented in Fig. 7. For the wave transmission coefficient downstream from a single net cage, the numerical results are in good agreement with the corresponding experimental data under different wave periods and wave heights (see Fig. 7 a, b). As the number of net cage increases from 1 to 4, the wave transmission coefficient downstream from the net cage decreases gradually due to the increasing damping effect. Overall, the trend of the numerical results of the wave transmission coefficient is consistent with the experimental results (see Fig. 7 c). For waves with a constant wave frequency, the wave transmission coefficients show a marginal discrepancy of ~ 1.6% for various wave steepnesses (see Fig. 7 b). The effect of wave steepness on the wave transmission coefficient downstream from a single net cage was examined with various wave periods or wavelengths. According to the findings of this study, there was no statistical difference in the transmission coefficient for the three similar wave steepnesses (see Table 3). This is because the viscous dissipation due to the netting is proportional to the square of the water-particle velocity which is proportional to the wave steepness. Thus, similar wave steepnesses create similar wave attenuation.
4. Wave propagating through a biofouled cage array 12
4.1 Numerical model description A regular wave propagating through a cage array is simulated with different levels of biofouling, where the deformation of both clean and biofouled netting is neglected. The effects of biofouling on wave fields inside and around the cage array are discussed. For computational efficiency, the numerical simulations are still performed using the 1:50 scale model based on the experimental validation in section 3. The cage array in the square configuration is comprised of the 16 net cages described in Table 1. The center-to-center spacing is 0.4 m between two neighboring net cages. As discussed in section 3.2, the variations in both wave steepness and wavelength resulted in a small discrepancy in the wave fields downstream from the net cage. Furthermore, a longer wave is more common than a shorter wave in the region of a fish farm. Therefore, numerical results for the wave condition with T=1.0 s and H=4 cm is chosen for a detailed discussion. Table 4 shows the porous coefficients for the netting model with a thickness of 10 mm that correspond to both clean and biofouled nettings. The porous coefficients were calculated from both the field test data of Swift et al. (2006) and the empirical formulas proposed by Aarsnes et al. (1990), which are described in detail in our previous work (Bi et al., 2015a). In this study, two scenarios were considered: (i) cage arrays with different levels of biofouling subjected to normal incident waves and (ii) cage arrays with extremely heavy biofouling subjected to waves from various directions.
4.2 Results and discussion 4.2.1 Wave attenuation by cage arrays with different levels of biofouling Numerical results of the wave elevation of a transient field over the studied cases presented in Table 4 are shown in Figs. 8 and 9. Overall, the wave fields inside and downstream from the cage arrays with different levels of biofouling are observably changed compared with that of an undisturbed wave field. It is determined that the 13
most noticeable variation in wave elevation is produced downstream from the cage array with extremely heavy biofouling. As a wave propagates through a cage array with extremely heavy biofouling, both the wave crest and wave trough are diminished due to the damping effect of the netting. The attenuation in wave amplitude becomes more noticeable in the near field, approximately within twice the wavelength, downstream from the cage array. However, the wave amplitude is prone to increase directly behind the cage array and correspondingly decrease on both sides as the wave continues to propagate away, which indicates that wave diffraction occurs as a wave propagates through the cage array. Previous research has shown that there is attenuation in flow velocity downstream from a net cage or cage array in a current (Aarsnes et al., 1990; Bi et al., 2014; Kristiansen and Faltinsen, 2012; Patursson et al., 2010; Zhao et al., 2013). Similarly, the water-particle velocity will be attenuated by the cage array due to the damping effect of the net structure. Therefore, the wave height (or wave energy) will change as a wave propagates through the cage array. To quantitatively study the effect of biofouling on a wave propagating through the cage array, the wave height inside and downstream from one row of the cage array with different levels of biofouling is measured along the centerline. The variation of the transmission coefficient in the wave direction is fitted using the smooth spline. Overall, the damping effect of the cage array on wave propagation increases with increasing amounts of biofouling, which creates more observable attenuation in the transmission coefficient (see Fig. 10). According to the numerical results, the variation tendencies of the transmission coefficient are similar with different levels of biofouling. The transmission coefficient decreases first because of the damping effect of the fishing net and then increases as the wave propagates further away. The maximum attenuation in transmission coefficient, ~7%, was found to be approximately equal to the diameter of the net cage downstream from the edge of cage array. The minimum transmission coefficients downstream from the cage array with different levels of biofouling are presented in Fig. 11. Based on this study, the 14
minimum transmission coefficient can be represented by a linear function of drag coefficient of the corresponding biofouled netting. The best-fit equation is CT=－ 0.0844Cd+0.9819 with a satisfactory correlation coefficient of 0.9997. This result indicates that the minimum transmission coefficient decreases linearly with increasing drag coefficient of the netting, which is intuitively related to the level of biofouling. Thus, the netting with the greatest level of biofouling produces the highest damping of wave energy.
4.2.2 Wave attenuation by biofouled cage arrays with various incident angles It is important to note that waves may propagate along various directions in the open sea, which will affect the wave field inside and around the cage array. Three incident angles, 0°, 22.5° and 45°, are considered, which concerns the symmetry of the cage array. Fig. 12 shows contour plots of the instantaneous wave elevation inside and around cage arrays for discrete incident angles. Overall, the wave amplitude is diminished downstream from the cage arrays subjected to waves from various directions. Unsurprisingly, the wave field around the cage array is asymmetrical for an incident angle of 22.5° due to the asymmetrical configuration. The wave height is measured at the center of net cages to quantitatively analyze the wave field inside the cage array (see Fig. 13). As expected, the wave transmission coefficient is less than 1 for most net cages due to the damping effect of the biofouled net cages. Specifically, a maximum attenuation in wave height of 6% is found at the most downstream cages in the middle of the cage array for an incident angle of 0°. In contrast, the wave transmission coefficient is equal to and even larger than 1 at two most upstream cages for incident angles of 0° and 45°, which could be due to the interaction between incident and reflected waves. The wave elevation along the transverse direction is analyzed based on the measurements at the first wave crest and the first wave trough downstream from the cage arrays (see Fig. 14). The maximum attenuation in wave amplitude is 15
approximately 10% at both the wave crest and the wave trough over the studied incident angles. At the wave crest, the wave amplitude remains constant between the amplitude-attenuation region downstream from the cage array for an incident angle of 0°. As the incident angle increases to 22.5° and 45°, significant upwelling, which is defined as a local modification of the incident wave (Ohl et al., 2001), is indicated in the central region, and the width of the amplitude-attenuation region increases. At the wave trough, one peak of the wave amplitude is indicated in the central region for an incident angle of 0°, and two peaks are observed around the edges of the cage array for an incident angle of 22.5° and 45°. The results indicate that the incident angle has a noticeable effect on the wave field inside and around the cage array, which can be due to the fact that (i) the effective area of the cage array perpendicular to waves increases with increasing incident angle and (ii) the staggered configuration of the cage array changes for various incident angles. For the present study, the wave fields inside and around the cage array are determined by the porous coefficient which is related to the force acting on the netting. For a mesh bar of the netting, the normal drag coefficient Cdn can be calculated based on the Reynolds number as follows (Choo and Casarella, 1971; DeCew et al., 2010; Tsukrov et al., 2011):
Cdn 1.1 4 Ren0.50 30 Ren 2.3 105
(19)
where the Reynolds number Ren is given by
Ren
Ud
(20)
where U is the maximum velocity of the water particle in waves, which can be given approximately by U H
T
according linear wave theory; υ is the kinematic
viscosity of the fluid, and d is the twine diameter of the netting. For the net-cage model, the normal drag coefficient ranged from 1.26 to 1.41 in the various wave cases, whereas it ranged from 1.16 to 1.22 for the corresponding full-scale cages. The drag coefficients of the model cage and the full-scale cage are slightly different but comparable. This provides confidence that the results of the wave fields inside and around the net-cage model are representative of the full-scale 16
cages.
5. Conclusions Waves propagating through cage arrays with different levels of biofouling are investigated using a numerical model based on the joint use of the porous-media fluid model and a numerical wave tank. Experimental validation is conducted using a 1:50 scale net-cage model with clean netting, and a good agreement between the experimental and numerical results is obtained. The damping effect of the cage array increases with an increasing amount of biofouling. A maximum attenuation in transmission coefficient, approximately 7%, was found downstream from the cage array in this study. The incident angle of waves has a noticeable effect on the wave field inside and around the cage array due to the staggered configuration of the cage array and the effective area for the interaction between cage arrays and waves. The wave attenuation should be taken into account in studying the hydrodynamic characteristics of a net cage or cage array. Thus, more accurate results will be obtained for each specific net cage within a cage array, which is essential to estimate mooring line loads for aquaculture farm design.
Acknowledgements This work was financially supported by the National Natural Science Foundation of China (NSFC), project nos. 51609035, 51239002, 51579037, 51409037 and 51221961,
China
Postdoctoral
Science
Foundation
(nos.
2016M590224,
2017T100176), Fundamental Research for the Central Universities (DUT16YQ105) and Natural Science Foundation of Zhejiang Province (no. LZ16E090002).
17
References Aarsnes, J., Rudi, H., Løland, G., 1990. Current forces on cage, net deflection, Engineering for offshore fish farming. Proceedings of a conference organised by the Institution of Civil Engineers, Glasgow, UK, 17-18 October 1990. Thomas Telford, pp. 137-152. Balash, C., Colbourne, B., Bose, N., Raman-Nair, W., 2009. Aquaculture net drag force and added mass. Aquacultural Engineering 41, 14-21. Bi, C.-W., Zhao, Y.-P., Dong, G.-H., 2015a. Numerical study on the hydrodynamic characteristics of biofouled full-scale net cage. China Ocean Engineering 29, 401-414. Bi, C.-W., Zhao, Y.-P., Dong, G.-H., Cui, Y., Gui, F.-K., 2015b. Experimental and numerical investigation on the damping effect of net cages in waves. Journal of Fluids and Structures 55, 122-138. Bi, C.-W., Zhao, Y.-P., Dong, G.-H., 2015c. Development of a Coupled Fluid-Structure Model with Application to a Fishing Net in Current. Hydrodynamics Concepts and Experiments, pp: 1-22. Bi, C.-W., Zhao, Y.-P., Dong, G.-H., Zheng, Y.-N., Gui, F.-K., 2014. A numerical analysis on the hydrodynamic characteristics of net cages using coupled fluid–structure interaction model. Aquacultural Engineering 59, 1-12. Braithwaite, R.A., Carrascosa, M.C.C., McEvoy, L.A., 2007. Biofouling of salmon cage netting and the efficacy of a typical copper-based antifoulant. Aquaculture 262, 219-226. Burcharth, H.F., Andersen, O.H., 1995. On the one-dimensional steady and unsteady porous flow equations. Coastal Engineering 24(3-4), 233–257. Bear, J., 1972. Dynamics of Fluids in Porous Media. America Elsevier Publishing Company Inc., New York. Chan, A., Lee, S., 2001. Wave characteristics past a flexible fishnet. Ocean 18
engineering 28, 1517-1529. Chen, H., Christensen, E.D., 2015. Numerical simulation of wave interaction with moving net structures, ASME 2015 34th International Conference on Ocean, Offshore and Arctic Engineering. Choo, Y.I., Casarella, M.J., 1971. Hydrodynamic resistance of towed cables. Journal of Hydronautics 5(4), 126–131. DeCew, J., Tsukrov, I., Risso, A., Swift, M.R., Celikkol, B., 2010. Modeling of dynamic behavior of a single-point moored submersible fish cage under currents. Aquacut. Eng. 43(2), 38–45. Dong, G.H., Zheng, Y.N., Li, Y.C., Teng, B., Guan, C.T., Lin, D.F., 2008. Experiments on wave transmission coefficients of floating breakwaters. Ocean Engineering 35, 931-938. FAO, 2014. The state of world fisheries and aquaculture. Food and Agriculture Organization of the United Nations, Rome, Italy. Gansel, L.C., Plew, D.R., Endresen, P.C., Olsen, A.I., Misimi, E., Guenther, J., Jensen, Ø., 2015. Drag of clean and fouled net panels – measurements and parameterization of fouling. Plos One 10, 0131051. Klebert, P., Lader, P., Gansel, L., Oppedal, F., 2013. Hydrodynamic interactions on net panel and aquaculture fish cages: a review. Ocean Engineering 58, 260-274. Kristiansen, T., Faltinsen, O.M., 2012. Modelling of current loads on aquaculture net cages. Journal of Fluids and Structures 34, 218-235. Lader, P.F., Fredriksson, D.W., Guenther, J., Volent, Z., Blocher, N., Kristiansen, D., Gansel, L., 2015. Drag on hydroid-fouled nets – An experimental approach. China Ocean Engineering 29, 369-389. Lader, P.F., Olsen, A., Jensen, A., Sveen, J.K., Fredheim, A., Enerhaug, B., 2007. Experimental
investigation
of
the
interaction 19
between
waves
and
net
structures—Damping mechanism. Aquacultural Engineering 37, 100-114. Madsen, O.S., 1971. On the generation of long waves. Journal of Geophysical Research 76, 8672-8683. Ohl, C.O.G., Eatock Taylor, R., Taylor, P.H., Borthwick, A.G.L., 2001. Water wave diffraction by a cylinder array. Part 1. Regular waves. Journal of Fluid Mechanics 442, 1-32. Patursson, Ø., Swift, M.R., Tsukrov, I., Simonsen, K., Baldwin, K., Fredriksson, D.W., Celikkol, B., 2010. Development of a porous media model with application to flow through and around a net panel. Ocean Engineering 37, 314-324. Swift, M.R., Fredriksson, D.W., Unrein, A., Fullerton, B., Patursson, Ø., Baldwin, K., 2006. Drag force acting on biofouled net panels. Aquacultural Engineering 35, 292-299. Tsukrov, I., Drach, A., DeCew, J., Swift, M.R., Celikkol, B., 2011. Characterization of geometry and normal drag coefficients of copper nets. Ocean Engineering 38, 1979-1988. Zhan, J.M., Jia, X.P., Li, Y.S., Sun, M.G., Guo, G.X., Hu, Y.Z., 2006. Analytical and experimental investigation of drag on nets of fish cages. Aquacultural Engineering 35, 91-101. Zhao, Y.-P., Bi, C.-W., Dong, G.-H., Gui, F.-K., Cui, Y., Xu, T.-J., 2013. Numerical simulation of the flow field inside and around gravity cages. Aquacultural Engineering 52, 1-13. Zhao, Y.-P., Bi, C.-W., Liu, Y.-X., Dong, G.-H., Gui, F.-K., 2014. Numerical simulation of interaction between waves and net panel using porous media model. Engineering Applications of Computational Fluid Mechanics 8, 116-126.
20
Figures
Fig. 1. Image of the net cage (left), clean net (middle) and biofouled net (right)
Fig. 2. Computational domain of the numerical wave tank and computational mesh around the net cages.
21
Wave height (cm)
3.6
3.4
3.2 x=2 m x=3 m x=4 m x=5 m 3
30
45
60
Number of cells per wave length (-)
(a) H/Δz=20, T/Δt=160
Wave height (cm)
3.6
3.4
3.2 x=2 m x=3 m x=4 m x=5 m 3
0
10
20
30
40
Number of cells per wave height (-)
(b) L/Δx=45, T/Δt=160 3.8 3.6 T/200
Wave height (cm)
3.4
T/100
T/160
T/320
3.2 3
T/640
2.8 2.6 x=2 m x=3 m x=4 m
2.4 2.2 2
0
0.002
0.004
0.006
0.008
Time step (s)
(c) H/Δz=20, L/Δx=45 Fig. 3. Numerical convergence study of the numerical wave tank. 22
Fig. 4. Comparison of the wave elevation time series between the numerical and theoretical data.
Fig. 5. Images of the fishing net and the net-cage model used in the laboratory experiment.
Fig. 6. Schematic view of the experimental set-up. 23
Transmission coefficient (-)
1
0.98
0.96
0.94 Exp. Num.
0.92
0.9
0.6
0.8 Wave period (s)
1
(a) H=4 cm for one net cage
Transmission coefficient (-)
1
0.98
0.96
0.94 Exp. Num.
0.92
0.9
2
4 Wave height (cm)
6
(b) T=1.0 s for one net cage
Transmission coefficient (-)
1
0.98
0.96
0.94
0.92
0.9
Exp. Num. 0
1
2 3 Number of net cage
4
(c) T=1.0 s and H=4 cm Fig. 7. Comparisons between the experimental and numerical results of the wave transmission coefficients at measurement point 5. The experimental data are average values with standard deviations ranging from 0.001 to 0.005. 24
Fig. 8. Numerical results of the instantaneous wave elevation inside and around the cage arrays. Wave condition: T=1.0 s, H=4 cm. The five figures from top to bottom are undisturbed wave field and wave fields around cage arrays with different levels of biofouling Cases 1–5 presented in Table 4, where only half of a cage array is shown due to symmetry. Circles indicate the position of the net cages.
25
Fig. 9. An example of the wave field inside and around the cage array with the extremely heavy biofouling level at different time instants: (a) T/4, (b) T/2, (c) 3T/4 and (d) T. Only half of a cage array is shown due to symmetry.
26
Fig. 10. Numerical results of the wave transmission coefficient along the wave direction through cage arrays with different levels of biofouling. The vertical dashed lines represent the position of the cage array.
Fig. 11. The comparisons between the numerical data of the minimum transmission coefficient downstream from the cage array and the corresponding fitted curves. 27
Fig. 12. Numerical results of instantaneous wave elevation inside and around cage arrays subjected to discrete incident angles. Wave condition: T=1.0 s, H=4 cm.
28
Fig. 13. Numerical results of the wave transmission coefficient inside the net cages for various incident angles.
29
Fig. 14. Distribution of wave elevation along the transverse direction at the first wave crest (left) and the first wave trough (right) downstream from the cage arrays.
30
Tables Table 1 Specifications of the net-cage model. Component Top frame
Net chamber
Bottom frame
Parameter Ring diameter Bar diameter Material Height Mesh size Twine diameter Net solidity Material Knot factor Ring diameter Mass in air Bar diameter Material
Value 0.254 m 6.0 mm Steel 0.16 m 20.0 mm 2.6 mm 0.27 Polyethylene Knotless 0.254 m 8.0 g 1.0 mm Stainless steel
Table 2 Wave parameters used in this study. Wave case Wave no. period (s) 1 0.6 2 0.6 3 0.8 4 0.8 5 0.8 6 1.0 7 1.0 8 1.0
Wave T height (cm) 2 4 2 4 6 2 4 6
31
Wave H length (m) 0.56 0.56 0.99 0.99 0.99 1.46 1.46 1.46
Wave L steepness H/L 0.036 0.071 0.020 0.040 0.061 0.014 0.027 0.041
Table 3 Wave transmission coefficient downstream from a single net cage vs. wave steepness. Wave period T Wave height H (s) (cm) 0.6 2 0.8 4 1.0 6
Wave steepness H/L 0.036 0.040 0.041
Transmission coef. CT 0.972 0.978 0.973
Table 4 Porous coefficients of different numerical cases. Case 1 corresponds to the clean net without biofouling and Cases 2 to 5 correspond to biofouled nets with different levels of biofouling. Case no. 1 2 3 4 5
Solidity 0.121 0.547 0.502 0.743 0.566
Biofouling level No fouling Light Medium Heavy Extremely heavy
32
Cd 0.175 0.260 0.318 0.513 0.599
Cn (m−1) 19.7 30.65 38.75 70.8 88.1
Ct (m−1) 12.7 19.75 25.0 45.65 56.8
S0144-8609(17)30054-7 http://dx.doi.org/doi:10.1016/j.aquaeng.2017.07.006 AQUE 1912
To appear in:
Aquacultural Engineering
Received date: Revised date: Accepted date:
21-3-2017 13-7-2017 16-7-2017
Please cite this article as: Bi, Chun-Wei, Zhao, Yun-Peng, Dong, Guo-Hai, Xu, Tiao-Jian, Gui, Fu-Kun, Numerical study on wave attenuation inside and around a square array of biofouled net cages.Aquacultural Engineering http://dx.doi.org/10.1016/j.aquaeng.2017.07.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Numerical study on wave attenuation inside and around a square array of biofouled net cages
Chun-Wei Bia,*, Yun-Peng Zhaoa,*, Guo-Hai Donga, Tiao-Jian Xua, Fu-Kun Guib
a
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of
Technology, Dalian 116024, China b
Marine Science and Technology School, Zhejiang Ocean University, Zhoushan
316000, China
Highlights
Wave attenuation by a square array of biofouled net cages is numerically studied
Effect of incident angle on wave attenuation by the biofouled cage array is analyzed
The study has implication for water exchange and structural design of the net cage
Abstract In this study, waves propagating through a square array of 16 net cages with different levels of biofouling are numerically studied using a three-dimensional computational fluid dynamics (CFD) model. A porous-media fluid model is adopted to simulate both clean and biofouled netting of a cage array in waves. A numerical wave tank is built, and the oscillating-boundary method is adopted to generate waves.
*
Corresponding author. Fax: +86-0411-84708526; Tel.: +86-0411-84707116.
E-mail address: bicw@ dlut.edu.cn (C.W., Bi), [email protected] (Y.P., Zhao) 1
The flow motion is solved by the Navier-Stokes equations, and the free water surface is captured using the volume of fluid (VOF) method. The numerical model is validated by comparing the numerical data with corresponding experimental measurements of a net-cage model with clean netting. To analyze wave attenuation, a numerical analysis of wave elevation both inside and around the cage arrays is presented, which considers the effect of biofouling. Based on the results of the present study, the effect of biofouling on wave elevation is noticeable; the damping effect of the cage array increases with increasing level of biofouling. Furthermore, the incident angle of waves has a noticeable effect on the wave field inside and around the cage array. Keywords: Cage array; Biofouling; Wave attenuation; Numerical simulation
1. Introduction Currently, commercial fish farming is in a stage of dramatic expansion, where approximately half of all seafood eaten around the world is being farmed. China has become the largest exporter of aquatic product in the world due to the rapid expansion in its fish production, particularly from aquaculture (FAO, 2014). With the goal of being more socially acceptable, environmentally friendly and economically profitable, it is urgent to exploit new space for aquaculture and to develop an offshore aquaculture strategy (Aarsnes et al., 1990; Zhan et al., 2006; Klebert et al., 2013). When deployed in offshore locations, an array of net cages, or cage array for short, is subjected to intense waves and has a damping effect on the incident waves. Thus, a cage array will change the wave field in the vicinity of the region, which should be taken into account when estimating the wave loads of the net cage. Similar to other marine structures, the infrastructure of a cage array suffers from the effects of biofouling in the open sea (see Fig. 1). Biofouling, which is the accumulation of marine organisms, is prone to leading to rapid net aperture occlusion. Biofouling on netting primarily causes the following disadvantages: i) the attenuation of currents and waves through the net cage increases, which decreases the water 2
exchange and oxygen supply inside the net cage; ii) the hydrodynamic forces acting on the net cage dramatically increase, which increases the risk of structural failure and fatigue damage of the net cage and iii) the biofouled netting is easily deformed in currents/waves and thus negatively affects the effective volume for farmed fish (Braithwaite et al., 2007; Klebert et al., 2013). In the literature, Swift et al. (2006) conducted experiments both in a tow tank and in the field to measure the drag forces acting on biofouled plane nets, and a 240% increase in drag coefficient was found for the maximum biofouled plane net. Lader et al. (2015) performed field tests to investigate the growth characteristics of hydroids that grow on a net and conducted laboratory experiments to study the hydrodynamic drag on the fouled twines using fabricated models of net twines with artificial hydroid fouling. Gansel et al. (2015) studied the effect of hydroids that act on the forces that act on plane nets; a parameterization of fouling was assessed in terms of the relationship between net solidity and drag force. Bi et al. (2015a) investigated the drag force acting on a full-scale biofouled net cage using a numerical approach, and the effect of biofouling on the flow fields inside and around the net cage were evaluated. The
aforementioned
related
studies
investigated
the
hydrodynamic
characteristics of biofouled plane nets and net cages in currents. To better understand the damping effect of the net structure on wave propagation, a number of investigations has been performed in recent decades. Among many others, Chan and Lee (2001) analytically studied the wave scattering characteristics by a flexible fishing net, which were modeled as a porous-flexible barrier coupled with an equivalent hydrodynamic force. The objective of this study is to investigate how a flexible net structure affects transmitted and reflected waves. Lader et al. (2007) conducted laboratory experiments with different net panels in several regular waves. Based on the wave elevation measurements, the wave geometry, wave energy as well as crest steepness and asymmetry factors were analyzed. Dong et al. (2008) proposed a board-net floating breakwater to protect fish cages, and the transmission coefficients were examined with various lateral dimensions and number of rows of net. The 3
experimental results indicated that the board-net floating breakwater may have potential application in protecting fish and fish cages from fierce waves. Most recently, Chen and Christensen (2015) numerically studied the interaction between a moving net structure and waves based on the open source library OpenFOAM, where the net was simulated as a sheet of porous media. It was concluded that a wave transmission coefficient of 95.2% was found when a wave propagated through the net. Bi et al. (2015b) experimentally investigated a wave propagating through four tandem net cages, and the effects of different parameters of both wave and net cage on the wave transmission coefficient were discussed. Moreover, the wave fields inside and around the net
cages
were presented by
solving the three-dimensional
Reynolds-averaged Navier-Stokes equations. Although much progress has been made, to the best of our knowledge, there has been no investigation on the wave field around the net cages or cage arrays that considers the effect of biofouling. However, biofouling on nettings can increase the damping effect of the cage array on wave propagation. Thus, an investigation on a wave propagating through a cage array that considers the effect of biofouling is of great importance. As an extension to our previous work (Zhao et al., 2014; Bi et al., 2015b), a three-dimensional numerical model is established to simulate a wave propagating through cage arrays with different levels of biofouling. In this study, both clean and biofouled netting of a cage array will be simulated using the porous-media fluid model; combined with a numerical wave tank, a wave propagating through a biofouled cage array is presented.
2. Description of numerical model For computational efficiency, it is impossible to simulate the exact structure of the netting and biofouling. Therefore, both clean and biofouled netting of the net cage were simulated using the porous-media fluid model in this study. A three-dimensional numerical wave tank was built, which was primarily based on the Navier-Stokes equations with the capacity to generate regular waves. Using the same governing 4
equations, the net-cage model and the numerical wave tank can be coupled.
2.1 Governing equations review The wave motion in the numerical wave tank can be primarily described by the incompressible Navier-Stokes equations. The governing equations in tensor forms are as follows: The continuity equation
ui 0 xi
(1)
and the momentum equation ui u j 2ui 2u j ui p gi 2 2 x t x j xi xi j
Si
(2)
where t is time; µ is fluid viscosity; ρ is fluid density; p is pressure; ui and uj are the average velocity components; i, j=1, 2, 3 (x, y, z); gi is the gravitational acceleration and Si is the source term for the momentum equation. It should be noted that the definition of the source term is different for the porous-media fluid model and the damping zone in the numerical wave tank. More detailed descriptions are given in the following sections. The governing equations are solved based on ANSYS FLUENT 15.0 using the finite volume method. An implicit, unsteady formulation and pressure-based solver is chosen to achieve a converged solution. The SIMPLEC (Semi-implicit Method for Pressure-linked Equation-Consistent) algorithm is used to solve the pressure-velocity coupling problem. The discretization of pressure, momentum, turbulent kinetic energy and turbulent dissipation rate are performed using a second-order upwind scheme.
2.2 Net-cage model considering biofouling effect In this study, both clean and biofouled nettings of the net cage are simulated using the porous-media fluid model (Bi et al., 2014; Zhao et al., 2013; Patursson et al., 2010). In the field test, the biofouling scattered randomly on the netting. It is difficult 5
to perform a direct numerical simulation. For simplicity, it is assumed here that the biofouling on the netting is distributed homogeneously. The bottom ring or sinkers of the net cage ensure the netting undergoes limited deformation and maintain the geometry of the net cage. In addition, only marginally movement and deformation of the netting was observed in the physical-model experiment. Thus, deformation of both clean and biofouled netting is neglected in this study. In the numerical model, the cylindrical net cage is divided into 16 plane nets along the circumference. Each plane net can be treated as porous media at a certain attack angle (Bi et al., 2015a; Zhao et al., 2013). Herein, because the float collar and sinker system have limited effects on the wave field, only the net chamber of the net cage is simulated, as shown in Fig. 2. The porous-media fluid model is a hypothetical model that acts in the same way as the water-blocking effect for both clean and biofouled netting by setting the porous coefficients. Outside the porous-media region, the source term added in the momentum equation Si is equal to 0. Inside the porous-media region, Si is calculated by the following equation:
Si Cij
1 uu 2
(3)
where Cij is given by Cn Cij 0 0
0 Ct 0
0 0 Ct
(4)
where Cij is the porous-coefficient matrix for the porous media, Cn is the coefficient in the normal direction, and Ct is the coefficient in the tangential direction. The unit of the porous coefficient is m-1. A typical source term for a porous-media model consists of one linear term and one quadratic term based on Darcy’s law. For large porosity (e.g., an array of fixed cylinders), the quadratic term for the frictional force will completely dominate over the viscous term (the linear term) (Burcharth and Andersen, 1995). In this case, the linear term is only a fitted term that has no physical meaning 6
and thus, it can be neglected. Therefore, only the quadratic term is kept in the source term. The drag (Fd) and lift (Fl) forces of a porous medium can be described as follows:
Fd Cn
1 A u u 2
(5)
Fl Ct
1 A u u 2
(6)
where λ is the porous-media thickness, and A is the porous-media area. According to the above equations, the porous coefficients of the porous media are directly related to the forces on the plane net. Generally, there are two methods to determine the drag and lift forces acting on a plane net: i) laboratory experiments and ii) an empirical formula, such as the Morison equation:
Fd
1 Cd Au02 2
(7)
Fl
1 Cl Au02 2
(8)
where u0 is the free stream velocity, Cd is the drag coefficient and Cl is the lift coefficient. According to Tsukrov et al. (2011), the formula provided by Balash et al. (2009) has an advantage in predicting the drag coefficient of a fishing net, where the relative error was approximately 4% compared with that of the measured data. For the lift coefficient, Aarsnes et al. (1990) performed a study of plane nets and established a formula as a function of net solidity and attack angle. Thus, Eqs. (9) and (10) were chosen to calculate the drag and lift forces of a plane net by combining Eqs. (7) and (8): Cd Cdcyl (8.03Sn2 0.74Sn 0.12) (Balash et al., 2009)
(9)
Cl (0.57Sn 3.54Sn2 10.1Sn3 )sin(2 ) (Aarsnes et al., 1990)
(10)
where Cdcyl 1 10 Re2/3 for Re 2.3 105 and α is the attack angle. The porous coefficient Cn can be calculated from a curve fit between plane-net 7
drag forces and corresponding current velocities using the least-squares method when the plane net is oriented normal to the flow. The other coefficient Ct can be obtained when a plane net is oriented at an attack angle. In this case, the coefficient Ct should be transformed into Ctα using Eq. (12) (Bear, 1972). Next, Ctα can be calculated from a curve fit between the plane-net lift forces and the corresponding current velocities using the least squares method. Once Cn and Ct are obtained, the porous coefficients for the plane net at various attack angles can be calculated as follows:
Cn Ct Cn Ct cos(2 ) 2 2 C Ct Ct n sin(2 ) 2
Cn
(11) (12)
where α is the attack angle, which is defined as the angle between the flow direction and the plane net in the horizontal plane; Cnα and Ctα are the normal and tangential resistance coefficients for the plane net at an attack angle of α, respectively. Based on the parameters of the netting used in the laboratory experiment, the porous coefficients for the clean netting can be calculated using the empirical-formula method: Cn=59.0 m-1 and Ct=40.0 m-1 with a porous-media thickness of 10 mm. According to our previous research (Bi et al., 2015c), the thickness of the porous media does not obviously affect the numerical results until the thickness is greater than 10% of the width or height of the plane net. However, an increase in the thickness of the porous media can effectively reduce the number of cells, which thus reduces the computational effort greatly. Therefore, the thickness of the porous media is defined as 10 mm for both the clean and biofouling nettings. For the netting with various levels of biofouling, the relationship between the porous coefficients and features of the biofouled netting can be built using the forces acting on the biofouled netting (Bi et al., 2015a).
2.3 Numerical wave tank The numerical wave tank is capable of producing regular and irregular waves by a specified motion of the wave-generation boundary. To eliminate the effect of reflected waves, a damping zone is defined at the end of the wave tank (see Fig. 2). 8
The primary part is the computational domain, which is set to larger than 10 times that of the wavelength.
2.3.1 Wave generation In this study, regular waves are generated using an oscillating-boundary method, where the wave-generation boundary is prescribed by the following equations for a second-order Stokes wave (Madsen, 1971):
an1 n a 3 [cos t ( 1 )sin 2t ] 2 tanh kh0 2hn1 4sinh kh0 2
(13)
where a is the wave amplitude, ω is the circular frequency, k is the wavenumber, h0 is the water depth and 2kh0 1 n1 (1 ) 2 sinh 2kh0
(14)
The generated wave elevation in the time domain is as follow:
a cos(kx t )
a 2 k cosh kh0 (cosh 2kh0 2) cos 2(kx t ) 4 sinh 3 kh0
(15)
By adjusting the wave frequency and amplitude, desired waves can be obtained, which can be validated by the theoretical values of Eq. (15).
2.3.2 Wave absorption To absorb the wave energy at the end of the wave tank and eliminate wave reflection, a damping zone is defined by adding a source term in the momentum equation. The source term is as follows:
Si ( x)ui
(16)
where µ(x) is the damping coefficient, which is given by
x x0 ，x0 x xe xe x0
( x)=K
(17)
In the equation, x0 and xe are the initial position and the end of the damping 9
domain in the x-direction, respectively. K is an empirical coefficient, which is related to the wavelength and the length of the damping zone. In this study, K is set to 20, where the length of the damping zone is equal to twice the wavelength. To examine the effectiveness of the wave absorption, wave elevations in the damping zone were monitored at different positions. Nearly all the wave energy can be absorbed before the waves reach the end of the wave tank.
2.3.3 Numerical convergence study The computational grids of the numerical wave tank are hexahedral elements. The grid size is set to h0/20 and is refined at the domain around the free surface and the wavemaker to ensure computational precision. Using wave case 4 as an example, the wave heights were measured at four different positions, and a numerical convergence study was performed on the grid size and time step, as shown in Fig. 3. As the number of cells increases in the x- and z-directions, the wave becomes more stable. When the grid size is L/45 in the x-direction and H/20 in the z-direction, the wave height converges. Based on the above grids, the numerical simulation shows the best result when the time step is 0.005 s, which is T/160 in dimensionless form, where T is the wave period.
2.3.4 Validation of the numerical wave tank To validate the numerical wave tank, the time series of wave elevation from the present numerical model are compared with the analytical results calculated from Eq. (15) (see Fig. 4). Although there are deviations in wave frequency and wave height for the first several waves, stable waves are obtained after several periods, which are used for data analysis. The results show that the wave elevations are in good agreement at different positions in the numerical wave tank. The generated wave is stable in both the space and time domains, which provides an essential foundation for simulating the wave field around the net cage.
3. Experimental validation 10
Using the proposed model, numerical simulations of a wave propagating through in-line net cages with different net-cage numbers were performed for various wave cases. Validation of the numerical model was conducted by comparing the numerical results with the corresponding experimental data.
3.1 Physical model and experimental setup To validate the numerical model, laboratory experiments on a 1:50 scaled physical model were performed (Bi et al., 2015b). The net-cage model in the present experiment consists of a top frame, a cylindrical net chamber and a bottom ring (see Fig. 5). The component parameters of the net-cage model are presented in Table 1. It should be noted that the net-cage model was kept fixed by the top frame during the tests. The net chamber was positioned 5 cm above and 11 cm beneath the still-water surface. To better represent the hydrodynamic behavior, the net in the physical model was the same as that of the full-scale net cage. In this way, the net in the physical model test can satisfy the Reynolds number similarity and avoid any variation of flow pattern through it. The laboratory experiments were conducted in a wave tank at the State Key Laboratory of Coastal and Offshore Engineering (SLCOE), Dalian University of Technology, Dalian, China. The wave tank is 22 m long, 0.45 m wide and 0.6 m deep. The water depth was 0.4 m during this experiment. Both the bottom and side walls of the wave tank are made of smooth glass with negligible frictional drag. A piston-type wavemaker is equipped at one end of the wave tank, and an energy-dissipation system is equipped at the other end to eliminate wave reflection. Wave elevations inside and around the in-line net cages, up to four net cages, were recorded under different wave cases. These measurements were performed three times to assess the repeatability of the measurements. The measurement positions of the wave gauges were arranged along the axis of the wave tank, as shown in Fig. 6. The wave parameters used in the laboratory experiments are shown in Table 2. The corresponding prototypical wave height is 1~3 m and wave period is 4~7 s, which is considered in the range of 11
common waves for an aquaculture net cage.
3.2 Results and discussion The wave transmission coefficient (CT) is introduced to quantitatively analyze the damping effect of net cages on the wave field; the definition of CT is as follow: CT
Ht Hi
(18)
where Ht is the wave height of the transmission wave, and Hi is the wave height of the incident wave. Numerical simulations were performed with different numbers of in-line net cages under the wave cases, as shown in Table 2. The comparison between the wave transmission coefficient from the numerical simulation and the corresponding experimental result are presented in Fig. 7. For the wave transmission coefficient downstream from a single net cage, the numerical results are in good agreement with the corresponding experimental data under different wave periods and wave heights (see Fig. 7 a, b). As the number of net cage increases from 1 to 4, the wave transmission coefficient downstream from the net cage decreases gradually due to the increasing damping effect. Overall, the trend of the numerical results of the wave transmission coefficient is consistent with the experimental results (see Fig. 7 c). For waves with a constant wave frequency, the wave transmission coefficients show a marginal discrepancy of ~ 1.6% for various wave steepnesses (see Fig. 7 b). The effect of wave steepness on the wave transmission coefficient downstream from a single net cage was examined with various wave periods or wavelengths. According to the findings of this study, there was no statistical difference in the transmission coefficient for the three similar wave steepnesses (see Table 3). This is because the viscous dissipation due to the netting is proportional to the square of the water-particle velocity which is proportional to the wave steepness. Thus, similar wave steepnesses create similar wave attenuation.
4. Wave propagating through a biofouled cage array 12
4.1 Numerical model description A regular wave propagating through a cage array is simulated with different levels of biofouling, where the deformation of both clean and biofouled netting is neglected. The effects of biofouling on wave fields inside and around the cage array are discussed. For computational efficiency, the numerical simulations are still performed using the 1:50 scale model based on the experimental validation in section 3. The cage array in the square configuration is comprised of the 16 net cages described in Table 1. The center-to-center spacing is 0.4 m between two neighboring net cages. As discussed in section 3.2, the variations in both wave steepness and wavelength resulted in a small discrepancy in the wave fields downstream from the net cage. Furthermore, a longer wave is more common than a shorter wave in the region of a fish farm. Therefore, numerical results for the wave condition with T=1.0 s and H=4 cm is chosen for a detailed discussion. Table 4 shows the porous coefficients for the netting model with a thickness of 10 mm that correspond to both clean and biofouled nettings. The porous coefficients were calculated from both the field test data of Swift et al. (2006) and the empirical formulas proposed by Aarsnes et al. (1990), which are described in detail in our previous work (Bi et al., 2015a). In this study, two scenarios were considered: (i) cage arrays with different levels of biofouling subjected to normal incident waves and (ii) cage arrays with extremely heavy biofouling subjected to waves from various directions.
4.2 Results and discussion 4.2.1 Wave attenuation by cage arrays with different levels of biofouling Numerical results of the wave elevation of a transient field over the studied cases presented in Table 4 are shown in Figs. 8 and 9. Overall, the wave fields inside and downstream from the cage arrays with different levels of biofouling are observably changed compared with that of an undisturbed wave field. It is determined that the 13
most noticeable variation in wave elevation is produced downstream from the cage array with extremely heavy biofouling. As a wave propagates through a cage array with extremely heavy biofouling, both the wave crest and wave trough are diminished due to the damping effect of the netting. The attenuation in wave amplitude becomes more noticeable in the near field, approximately within twice the wavelength, downstream from the cage array. However, the wave amplitude is prone to increase directly behind the cage array and correspondingly decrease on both sides as the wave continues to propagate away, which indicates that wave diffraction occurs as a wave propagates through the cage array. Previous research has shown that there is attenuation in flow velocity downstream from a net cage or cage array in a current (Aarsnes et al., 1990; Bi et al., 2014; Kristiansen and Faltinsen, 2012; Patursson et al., 2010; Zhao et al., 2013). Similarly, the water-particle velocity will be attenuated by the cage array due to the damping effect of the net structure. Therefore, the wave height (or wave energy) will change as a wave propagates through the cage array. To quantitatively study the effect of biofouling on a wave propagating through the cage array, the wave height inside and downstream from one row of the cage array with different levels of biofouling is measured along the centerline. The variation of the transmission coefficient in the wave direction is fitted using the smooth spline. Overall, the damping effect of the cage array on wave propagation increases with increasing amounts of biofouling, which creates more observable attenuation in the transmission coefficient (see Fig. 10). According to the numerical results, the variation tendencies of the transmission coefficient are similar with different levels of biofouling. The transmission coefficient decreases first because of the damping effect of the fishing net and then increases as the wave propagates further away. The maximum attenuation in transmission coefficient, ~7%, was found to be approximately equal to the diameter of the net cage downstream from the edge of cage array. The minimum transmission coefficients downstream from the cage array with different levels of biofouling are presented in Fig. 11. Based on this study, the 14
minimum transmission coefficient can be represented by a linear function of drag coefficient of the corresponding biofouled netting. The best-fit equation is CT=－ 0.0844Cd+0.9819 with a satisfactory correlation coefficient of 0.9997. This result indicates that the minimum transmission coefficient decreases linearly with increasing drag coefficient of the netting, which is intuitively related to the level of biofouling. Thus, the netting with the greatest level of biofouling produces the highest damping of wave energy.
4.2.2 Wave attenuation by biofouled cage arrays with various incident angles It is important to note that waves may propagate along various directions in the open sea, which will affect the wave field inside and around the cage array. Three incident angles, 0°, 22.5° and 45°, are considered, which concerns the symmetry of the cage array. Fig. 12 shows contour plots of the instantaneous wave elevation inside and around cage arrays for discrete incident angles. Overall, the wave amplitude is diminished downstream from the cage arrays subjected to waves from various directions. Unsurprisingly, the wave field around the cage array is asymmetrical for an incident angle of 22.5° due to the asymmetrical configuration. The wave height is measured at the center of net cages to quantitatively analyze the wave field inside the cage array (see Fig. 13). As expected, the wave transmission coefficient is less than 1 for most net cages due to the damping effect of the biofouled net cages. Specifically, a maximum attenuation in wave height of 6% is found at the most downstream cages in the middle of the cage array for an incident angle of 0°. In contrast, the wave transmission coefficient is equal to and even larger than 1 at two most upstream cages for incident angles of 0° and 45°, which could be due to the interaction between incident and reflected waves. The wave elevation along the transverse direction is analyzed based on the measurements at the first wave crest and the first wave trough downstream from the cage arrays (see Fig. 14). The maximum attenuation in wave amplitude is 15
approximately 10% at both the wave crest and the wave trough over the studied incident angles. At the wave crest, the wave amplitude remains constant between the amplitude-attenuation region downstream from the cage array for an incident angle of 0°. As the incident angle increases to 22.5° and 45°, significant upwelling, which is defined as a local modification of the incident wave (Ohl et al., 2001), is indicated in the central region, and the width of the amplitude-attenuation region increases. At the wave trough, one peak of the wave amplitude is indicated in the central region for an incident angle of 0°, and two peaks are observed around the edges of the cage array for an incident angle of 22.5° and 45°. The results indicate that the incident angle has a noticeable effect on the wave field inside and around the cage array, which can be due to the fact that (i) the effective area of the cage array perpendicular to waves increases with increasing incident angle and (ii) the staggered configuration of the cage array changes for various incident angles. For the present study, the wave fields inside and around the cage array are determined by the porous coefficient which is related to the force acting on the netting. For a mesh bar of the netting, the normal drag coefficient Cdn can be calculated based on the Reynolds number as follows (Choo and Casarella, 1971; DeCew et al., 2010; Tsukrov et al., 2011):
Cdn 1.1 4 Ren0.50 30 Ren 2.3 105
(19)
where the Reynolds number Ren is given by
Ren
Ud
(20)
where U is the maximum velocity of the water particle in waves, which can be given approximately by U H
T
according linear wave theory; υ is the kinematic
viscosity of the fluid, and d is the twine diameter of the netting. For the net-cage model, the normal drag coefficient ranged from 1.26 to 1.41 in the various wave cases, whereas it ranged from 1.16 to 1.22 for the corresponding full-scale cages. The drag coefficients of the model cage and the full-scale cage are slightly different but comparable. This provides confidence that the results of the wave fields inside and around the net-cage model are representative of the full-scale 16
cages.
5. Conclusions Waves propagating through cage arrays with different levels of biofouling are investigated using a numerical model based on the joint use of the porous-media fluid model and a numerical wave tank. Experimental validation is conducted using a 1:50 scale net-cage model with clean netting, and a good agreement between the experimental and numerical results is obtained. The damping effect of the cage array increases with an increasing amount of biofouling. A maximum attenuation in transmission coefficient, approximately 7%, was found downstream from the cage array in this study. The incident angle of waves has a noticeable effect on the wave field inside and around the cage array due to the staggered configuration of the cage array and the effective area for the interaction between cage arrays and waves. The wave attenuation should be taken into account in studying the hydrodynamic characteristics of a net cage or cage array. Thus, more accurate results will be obtained for each specific net cage within a cage array, which is essential to estimate mooring line loads for aquaculture farm design.
Acknowledgements This work was financially supported by the National Natural Science Foundation of China (NSFC), project nos. 51609035, 51239002, 51579037, 51409037 and 51221961,
China
Postdoctoral
Science
Foundation
(nos.
2016M590224,
2017T100176), Fundamental Research for the Central Universities (DUT16YQ105) and Natural Science Foundation of Zhejiang Province (no. LZ16E090002).
17
References Aarsnes, J., Rudi, H., Løland, G., 1990. Current forces on cage, net deflection, Engineering for offshore fish farming. Proceedings of a conference organised by the Institution of Civil Engineers, Glasgow, UK, 17-18 October 1990. Thomas Telford, pp. 137-152. Balash, C., Colbourne, B., Bose, N., Raman-Nair, W., 2009. Aquaculture net drag force and added mass. Aquacultural Engineering 41, 14-21. Bi, C.-W., Zhao, Y.-P., Dong, G.-H., 2015a. Numerical study on the hydrodynamic characteristics of biofouled full-scale net cage. China Ocean Engineering 29, 401-414. Bi, C.-W., Zhao, Y.-P., Dong, G.-H., Cui, Y., Gui, F.-K., 2015b. Experimental and numerical investigation on the damping effect of net cages in waves. Journal of Fluids and Structures 55, 122-138. Bi, C.-W., Zhao, Y.-P., Dong, G.-H., 2015c. Development of a Coupled Fluid-Structure Model with Application to a Fishing Net in Current. Hydrodynamics Concepts and Experiments, pp: 1-22. Bi, C.-W., Zhao, Y.-P., Dong, G.-H., Zheng, Y.-N., Gui, F.-K., 2014. A numerical analysis on the hydrodynamic characteristics of net cages using coupled fluid–structure interaction model. Aquacultural Engineering 59, 1-12. Braithwaite, R.A., Carrascosa, M.C.C., McEvoy, L.A., 2007. Biofouling of salmon cage netting and the efficacy of a typical copper-based antifoulant. Aquaculture 262, 219-226. Burcharth, H.F., Andersen, O.H., 1995. On the one-dimensional steady and unsteady porous flow equations. Coastal Engineering 24(3-4), 233–257. Bear, J., 1972. Dynamics of Fluids in Porous Media. America Elsevier Publishing Company Inc., New York. Chan, A., Lee, S., 2001. Wave characteristics past a flexible fishnet. Ocean 18
engineering 28, 1517-1529. Chen, H., Christensen, E.D., 2015. Numerical simulation of wave interaction with moving net structures, ASME 2015 34th International Conference on Ocean, Offshore and Arctic Engineering. Choo, Y.I., Casarella, M.J., 1971. Hydrodynamic resistance of towed cables. Journal of Hydronautics 5(4), 126–131. DeCew, J., Tsukrov, I., Risso, A., Swift, M.R., Celikkol, B., 2010. Modeling of dynamic behavior of a single-point moored submersible fish cage under currents. Aquacut. Eng. 43(2), 38–45. Dong, G.H., Zheng, Y.N., Li, Y.C., Teng, B., Guan, C.T., Lin, D.F., 2008. Experiments on wave transmission coefficients of floating breakwaters. Ocean Engineering 35, 931-938. FAO, 2014. The state of world fisheries and aquaculture. Food and Agriculture Organization of the United Nations, Rome, Italy. Gansel, L.C., Plew, D.R., Endresen, P.C., Olsen, A.I., Misimi, E., Guenther, J., Jensen, Ø., 2015. Drag of clean and fouled net panels – measurements and parameterization of fouling. Plos One 10, 0131051. Klebert, P., Lader, P., Gansel, L., Oppedal, F., 2013. Hydrodynamic interactions on net panel and aquaculture fish cages: a review. Ocean Engineering 58, 260-274. Kristiansen, T., Faltinsen, O.M., 2012. Modelling of current loads on aquaculture net cages. Journal of Fluids and Structures 34, 218-235. Lader, P.F., Fredriksson, D.W., Guenther, J., Volent, Z., Blocher, N., Kristiansen, D., Gansel, L., 2015. Drag on hydroid-fouled nets – An experimental approach. China Ocean Engineering 29, 369-389. Lader, P.F., Olsen, A., Jensen, A., Sveen, J.K., Fredheim, A., Enerhaug, B., 2007. Experimental
investigation
of
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between
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structures—Damping mechanism. Aquacultural Engineering 37, 100-114. Madsen, O.S., 1971. On the generation of long waves. Journal of Geophysical Research 76, 8672-8683. Ohl, C.O.G., Eatock Taylor, R., Taylor, P.H., Borthwick, A.G.L., 2001. Water wave diffraction by a cylinder array. Part 1. Regular waves. Journal of Fluid Mechanics 442, 1-32. Patursson, Ø., Swift, M.R., Tsukrov, I., Simonsen, K., Baldwin, K., Fredriksson, D.W., Celikkol, B., 2010. Development of a porous media model with application to flow through and around a net panel. Ocean Engineering 37, 314-324. Swift, M.R., Fredriksson, D.W., Unrein, A., Fullerton, B., Patursson, Ø., Baldwin, K., 2006. Drag force acting on biofouled net panels. Aquacultural Engineering 35, 292-299. Tsukrov, I., Drach, A., DeCew, J., Swift, M.R., Celikkol, B., 2011. Characterization of geometry and normal drag coefficients of copper nets. Ocean Engineering 38, 1979-1988. Zhan, J.M., Jia, X.P., Li, Y.S., Sun, M.G., Guo, G.X., Hu, Y.Z., 2006. Analytical and experimental investigation of drag on nets of fish cages. Aquacultural Engineering 35, 91-101. Zhao, Y.-P., Bi, C.-W., Dong, G.-H., Gui, F.-K., Cui, Y., Xu, T.-J., 2013. Numerical simulation of the flow field inside and around gravity cages. Aquacultural Engineering 52, 1-13. Zhao, Y.-P., Bi, C.-W., Liu, Y.-X., Dong, G.-H., Gui, F.-K., 2014. Numerical simulation of interaction between waves and net panel using porous media model. Engineering Applications of Computational Fluid Mechanics 8, 116-126.
20
Figures
Fig. 1. Image of the net cage (left), clean net (middle) and biofouled net (right)
Fig. 2. Computational domain of the numerical wave tank and computational mesh around the net cages.
21
Wave height (cm)
3.6
3.4
3.2 x=2 m x=3 m x=4 m x=5 m 3
30
45
60
Number of cells per wave length (-)
(a) H/Δz=20, T/Δt=160
Wave height (cm)
3.6
3.4
3.2 x=2 m x=3 m x=4 m x=5 m 3
0
10
20
30
40
Number of cells per wave height (-)
(b) L/Δx=45, T/Δt=160 3.8 3.6 T/200
Wave height (cm)
3.4
T/100
T/160
T/320
3.2 3
T/640
2.8 2.6 x=2 m x=3 m x=4 m
2.4 2.2 2
0
0.002
0.004
0.006
0.008
Time step (s)
(c) H/Δz=20, L/Δx=45 Fig. 3. Numerical convergence study of the numerical wave tank. 22
Fig. 4. Comparison of the wave elevation time series between the numerical and theoretical data.
Fig. 5. Images of the fishing net and the net-cage model used in the laboratory experiment.
Fig. 6. Schematic view of the experimental set-up. 23
Transmission coefficient (-)
1
0.98
0.96
0.94 Exp. Num.
0.92
0.9
0.6
0.8 Wave period (s)
1
(a) H=4 cm for one net cage
Transmission coefficient (-)
1
0.98
0.96
0.94 Exp. Num.
0.92
0.9
2
4 Wave height (cm)
6
(b) T=1.0 s for one net cage
Transmission coefficient (-)
1
0.98
0.96
0.94
0.92
0.9
Exp. Num. 0
1
2 3 Number of net cage
4
(c) T=1.0 s and H=4 cm Fig. 7. Comparisons between the experimental and numerical results of the wave transmission coefficients at measurement point 5. The experimental data are average values with standard deviations ranging from 0.001 to 0.005. 24
Fig. 8. Numerical results of the instantaneous wave elevation inside and around the cage arrays. Wave condition: T=1.0 s, H=4 cm. The five figures from top to bottom are undisturbed wave field and wave fields around cage arrays with different levels of biofouling Cases 1–5 presented in Table 4, where only half of a cage array is shown due to symmetry. Circles indicate the position of the net cages.
25
Fig. 9. An example of the wave field inside and around the cage array with the extremely heavy biofouling level at different time instants: (a) T/4, (b) T/2, (c) 3T/4 and (d) T. Only half of a cage array is shown due to symmetry.
26
Fig. 10. Numerical results of the wave transmission coefficient along the wave direction through cage arrays with different levels of biofouling. The vertical dashed lines represent the position of the cage array.
Fig. 11. The comparisons between the numerical data of the minimum transmission coefficient downstream from the cage array and the corresponding fitted curves. 27
Fig. 12. Numerical results of instantaneous wave elevation inside and around cage arrays subjected to discrete incident angles. Wave condition: T=1.0 s, H=4 cm.
28
Fig. 13. Numerical results of the wave transmission coefficient inside the net cages for various incident angles.
29
Fig. 14. Distribution of wave elevation along the transverse direction at the first wave crest (left) and the first wave trough (right) downstream from the cage arrays.
30
Tables Table 1 Specifications of the net-cage model. Component Top frame
Net chamber
Bottom frame
Parameter Ring diameter Bar diameter Material Height Mesh size Twine diameter Net solidity Material Knot factor Ring diameter Mass in air Bar diameter Material
Value 0.254 m 6.0 mm Steel 0.16 m 20.0 mm 2.6 mm 0.27 Polyethylene Knotless 0.254 m 8.0 g 1.0 mm Stainless steel
Table 2 Wave parameters used in this study. Wave case Wave no. period (s) 1 0.6 2 0.6 3 0.8 4 0.8 5 0.8 6 1.0 7 1.0 8 1.0
Wave T height (cm) 2 4 2 4 6 2 4 6
31
Wave H length (m) 0.56 0.56 0.99 0.99 0.99 1.46 1.46 1.46
Wave L steepness H/L 0.036 0.071 0.020 0.040 0.061 0.014 0.027 0.041
Table 3 Wave transmission coefficient downstream from a single net cage vs. wave steepness. Wave period T Wave height H (s) (cm) 0.6 2 0.8 4 1.0 6
Wave steepness H/L 0.036 0.040 0.041
Transmission coef. CT 0.972 0.978 0.973
Table 4 Porous coefficients of different numerical cases. Case 1 corresponds to the clean net without biofouling and Cases 2 to 5 correspond to biofouled nets with different levels of biofouling. Case no. 1 2 3 4 5
Solidity 0.121 0.547 0.502 0.743 0.566
Biofouling level No fouling Light Medium Heavy Extremely heavy
32
Cd 0.175 0.260 0.318 0.513 0.599
Cn (m−1) 19.7 30.65 38.75 70.8 88.1
Ct (m−1) 12.7 19.75 25.0 45.65 56.8