Design development of porous collar barrier for offshore floating fish cage against wave action, debris and predators

Journal Pre-proof Design development of porous collar barrier for offshore floating fish cage against wave action, debris and predators Y.I. Chu (Conceptualization) (Data curation) (Formal analysis) (Methodology) (Project administration) (Resources) (Software) (Visualization) (Writing - original draft), C.M. Wang (Funding acquisition) (Investigation) (Supervision) (Validation) (Writing review and editing)

PII:

S0144-8609(20)30183-7

DOI:

https://doi.org/10.1016/j.aquaeng.2020.102137

Reference:

AQUE 102137

To appear in:

Aquacultural Engineering

Received Date:

25 August 2020

Revised Date:

2 November 2020

Accepted Date:

3 November 2020

Please cite this article as: Chu YI, Wang CM, Design development of porous collar barrier for offshore floating fish cage against wave action, debris and predators, Aquacultural Engineering (2020), doi: https://doi.org/10.1016/j.aquaeng.2020.102137

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Design development of porous collar barrier for offshore floating fish cage against wave action, debris and predators Y. I. Chu* [email protected] and C. M. Wang [email protected] School of Civil Engineering, The University of Queensland St Lucia, Queensland 4072, Australia (E-mail:,)

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*Corresponding author

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ABSTRACT – This paper presents a design concept of a porous collar barrier for a novel floating open-net fish cage that is integrated with a floating spar wind turbine (referred to as COSPAR fish cage). The COSPAR fish cage has an octagonal shape with each side length of 30m. The collar barrier, having an array of rectangular cut-outs with round corners, is installed at the top portion of the cage to attenuate wave transmission inside the cage as well as to protect fish from external predators and debris. Single and double collar barrier designs corresponding to single net and double net cages are studied. The wave transmission, reflection and energyloss coefficients of barriers are determined from numerical analysis based on the linear potential wave theory and the eigenfunction expansion method. Various underwater heights (2m ≤ h ≤ 8m) and porosity (0.25≤ ε ≤ 0.75) of the collar barriers are examined with the view to obtaining the barrier design for minimal transmission coefficient and energy-loss coefficient. Without a collar barrier, the single net and double net cage can only provide average wave transmission coefficients of 0.9 and 0.8, respectively. This study finds that the transmission coefficient could be reduced below 0.4 by having a single collar barrier with h = 4m and ε = 0.25. On the other hand, the transmission coefficient could be further reduced below 0.3 by a double collar barrier with the same h and ε. In addition, the double collar barrier gives lower energy-loss coefficient and better proofing against fish escape, biosecurity and predator intrusion than the single collar barrier. A double collar barrier design with porosity combination of ε1=0.25, ε2=0.5 is recommended for the COSPAR fish cage as it yields competitive wave scattering performances and saves collar material by 25% when compared with the best performing porosity combination of ε1= ε2=0.25.

Keywords: offshore fish farm, floating fish cage, porous collar barrier, wave transmission, wave reflection, energy-loss

1. Introduction Fish farming has played an important role in filling the gap between seafood supply and demand in recent years. However, at its current growth trajectory (FAO, 2018), productivity of cultured fish will not be able to keep pace with demand due to resource constraints, public and environmental opposition towards expansion of land-based and nearshore fish farms (Le 1

François, 2010). Farming in offshore sites has been identified as a potential option for increasing fish production as the sites provide more sea space and better waste dispersion. There are, however, unforeseen risks for offshore fish farming. Sea currents may be too strong, and waves may be too wild that have a negative impact on fish growth and profit for farmers. Therefore, the environmental, technical and operational challenges will require a completely new engineering approach for offshore fish farming.

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Water flow plays a significant role in determining whether fish can be farmed in offshore sites. Some farmed fish species, whose habitat is in sheltered sites in nature, are not so well adapted to living in high energy environments as they prefer calm and peaceful environment. Moreover, fish have to spend lots more energy to swim against big and irregular waves compared to regular and steady waves (Beveridge, 2004; Dominique, 2014). For example, Solstorm et al. (2015) tested post-smolts of Atlantic salmon (weighing 98.6 gm and measuring 22.3 cm in length) to water velocities corresponding to 0.04 m/s, 0.18 m/s and 0.33 m/s (slow, moderate and fast, respectively) over 6 weeks. They found that the fish subjected to fast velocity showed 5% lower weight gain as compared to the fish in moderate and slow velocities.

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Although fish can dive in deeper water where it is relatively steady and reduced water flow, they prefer to be near water surface for sunlight, oxygen saturation, lower static sea pressure, nutrients/plankton and surface air that is necessary for swim bladders. In salmon farms, fish gather at the water surface to consume dry pellets quickly which would otherwise become moist and sink fast, and out of reach from the fish. Note that dry pellets normally contain a high level of fish meal with enriched nutrients and must be kept less than 10% moisture level and supplied at water surface (Lovell, 1989; Pandey, 2018). So, it is important for a fish cage to have calm surface water in order to reduce feed wastes and keep fish growth at an acceptable level. A new fish cage design for deployment in energetic offshore sites would thus requires a proper method to reduce wave transmission inside the cage.

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Recent deployed offshore fish cages (e.g. Ocean Farm 1 and Shenlan 1), as reported in a recent review paper (Chu et al., 2020), adopted an open-net system which is ideal for natural water replenishment and waste dispersion. Nevertheless, the open-net system in more exposed and higher water flow sites has to contend with the following threats; Wild predators such as sealions, seals and sharks can get access to fish by leaping to the top or by damaging the side net. The damaged net will lead to a large number of fish escaping. Floating debris and berthing impact by farming support vessels can cause net tearing, fish injuries and escapees. Therefore, offshore fish cage designs need to provide a suitable method to protect fish from external threats.

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A floating barrier or breakwater with a sufficient draught can attenuate wave transmission through the mechanisms of either reflection or destruction of water particle orbital motions (Wang and Sun, 2010). When the floating barrier takes on closed shapes (e.g. circular or octagonal), it can also protect internal water space from external floating debris and predators. A floating impermeable barrier may be a better solution for reducing wave transmission. However, substantial wave energy absorbed by the barrier can generate more dynamic oscillatory motions of the floating barrier and increase mooring tension forces. In addition, the floating impermeable barrier in a closed shape can induce internal fluid sloshing that may be more detrimental than the incident waves to the fish. On the other hand, a floating porous barrier has advantages in reducing wave transmission to an acceptable level, keeping stable 2

floating motions and mitigating fluid sloshing effect as it has less exposed normal surface to the incident wave. Wave scattering problems (transmission, reflection and energy-loss) by using a porous barrier have been studied analytically and experimentally by coastal engineers for many years. However, most studies focus on seabed resting barriers that are more suitable for shallow water, and not so applicable to deep-water offshore sites. In recent years, however, wave effects on floating porous barriers have been given attention and there are some concrete evidences that show floating porous barriers to be effective in reducing wave transmission.

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Xiao et al. (2016) performed experiments of a ring-shaped very large floating structure (VLFS) composed of spar-type modules (see Fig. 1). A comparative study was conducted to investigate the hydrodynamic performances of different perforated-wall breakwaters vertically attached to the VLFS with porosity of 0.16, 0.2 and 0.24. The results show distinct wave attenuation, whilst vertical motions were kept small owing to low natural frequencies, and hydrodynamic response motions were negligibly affected by the porosity. Dong et al., (2008) conducted two-dimensional physical model tests in a wave-current flume to measure wave transmission coefficient of a board-net floating breakwater for use with fish cages. The experimental results show that the board-net floating breakwater can effectively protect fish and fish cages in deep-water regions (see Fig. 2). Ji et al. (2019) presented experimental results of a single-row and double-row rectangular floating breakwaters with porous plates. It reveals that porous breakwater designs can effectively attenuate incident waves with slight motion responses and small mooring forces (see Fig. 3).

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(a) (b) Fig. 1. Ring shaped VLFS with double-layered perforated-wall breakwater: (a) model in basin, (b) sketch of cross section (Xiao et al., 2016)

Fig. 2. Board-net floating breakwater (Dong et al., 2008) 3

Fig. 3. Rectangular floating breakwaters with porous plates (Ji et al., 2019)

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Recently, Chu and Wang (2020a) proposed a novel floating open-net fish cage design that combines a floating spar wind turbine and a steel fish cage. The design is referred to as COSPAR; an acronym for a COmbined Spar and PARtially porous collar barrier fish cage as shown in Fig. 4(a). The initial design of COSPAR has an octagonal shape with each side length of 30m with an impermeable collar barrier that may incur some disadvantages such as increasing added mass, drag forces, mooring tension forces and construction costs when compared to pure open-net fish cage designs (e.g. Ocean Farm 1). However, by using a rigid porous collar barrier (with an array of rectangular cut-outs and round corners) instead as shown in Fig. 4(b), one can achieve trade-offs between (a) mitigating wave transmission inside the cage and protecting fish from debris/predators and (b) increasing mooring tension forces and construction costs. It is worth noting that a notable feature of the COSPAR is the heavy mass of the spar that lowers the centre of gravity below the fish cage. This low position of centre of gravity enhances both pitching and rolling stiffnesses of COSPAR, increases the restoring moment for better hydrodynamic motion responses and reduces the mooring tension forces. It is suggested that the collar barrier height about 3m above water surface will suffice in preventing intrusion of predators and to provide enough berthing height for aquaculture supporting vessels. However, other key design parameters of the collar barrier such as underwater height, porosity and number of barriers have to be determined.

(a)

(b)

Fig. 4. (a) COSPAR fish cage design and (b) porous collar barrier 4

The aim of this study is to design such a novel collar barrier system with respect to (i) its appropriate underwater height, (ii) porosity and (iii) number of barriers (single or double) in view of obtaining minimal transmission coefficient and energy-loss coefficient against incident waves. This will ensure a calm water space inside the cage and reduce wave energy absorption by the barrier structures and fish cage. Four different design cases will be considered in this study as shown in Fig. 5: (1) single net without collar barrier (base design to represent a conventional open-net cage) as shown in Fig. 5(a), (2) single net with porous collar barrier having underwater height h as shown in Fig. 5(b), (3) double net without collar barrier as shown in Fig. 5(c), and

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(4) double net with double porous collar barriers having underwater height h as shown in Fig. 5(d).

(a)

(b)

(c)

(d)

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Fig. 5. Four design cases for net and collar barrier system

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The linear potential wave theory (Liu and Abbaspour, 1982; Nakamura, 1992) and the eigenfunction expansion method (Losada et al. 1992; Abul-Azm 1993) will be used to investigate the phenomenon of wave diffraction so as to predict coefficients of wave transmission, reflection and energy-loss for single or double barrier problems under a wide range of boundary conditions. This study considers wave diffraction only as it is assumed that the COSPAR fish cage with a porous collar barrier maintains its stationary position under small amplitude wave conditions (for a fish farming operation) due to its low natural frequency (about 0.02 Hertz) in vertical motions. Xiao et al. (2016) also observed such small motion responses from their experimental studies on the ring-shaped VLFS (see Fig. 1) and the motion responses is not affected by attaching a double layered perforated wall. Therefore, the consideration of only diffracted waves is a valid assumption for this study to gain an insight of wave interaction characteristics of the collar barrier system. In designing the collar barrier system, the range of wave periods are assumed to be from 5 to 10 seconds corresponding to the wave condition at exposed sites near the Storm Bay in Tasmania, Australia (Chu and Wang, 2020b). Extreme wave conditions are not considered in this study as the COSPAR fish cage 5

can be submerged by filling ballast tanks to avoid such strong wave action. Based on a parametric study of the wave scattering performance of the aforementioned four design cases, a suitable porous collar barrier design will be identified for the COSPAR fish cage. 2. Numerical model and formulation Numerical models are used to predict the interaction of regular waves with porous barriers. A formulation based on the linear diffraction theory with matching boundary conditions at the barriers will be used. In order to simplify the problems at hand, the following assumptions have been made:

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(1) Fluid is inviscid, incompressible and irrotational, and thus the fluid motion can be described by a velocity potential governed by the Laplace equation. (2) Seabed is horizontal and impermeable, and wave amplitudes are relatively small to the water depth. (3) Barriers are thin and rigid. (4) Problems are two-dimensional with incident waves perpendicularly approaching to the barriers. Two-dimensional numerical models for single and double floating barrier problems are shown in Fig. 6(a) and 6(b), respectively. The Cartesian coordinates system (𝑥, 𝑧) is defined with 𝑥 in the direction of wave propagation from the location of the single barrier and midway between double barrier, and 𝑧 is measured upwards from the seabed. The barrier has a thickness b and extends downwards to a distance ℎ below the still water level. The water depth is denoted by d and hence the distance the bottom edge of the barrier from the seabed is a = d - h.

(a)

(b)

Fig. 6. (a): Single floating barrier, (b): Double floating barrier

Based on the preceding assumptions, the fluid motion can be described by a velocity potential that satisfies the Laplace equation within the fluid regions and boundary conditions at the seabed, free surface and far field (Sarpkaya and Isaacson, 1981; Isaacson et al., 1999) in the form: 𝑖𝑔𝐻

Φp (𝑥, 𝑧, 𝑡) = 𝑅𝑒 [− (

2𝜔

)

1 cosh(𝑘𝑑)

𝜙𝑝 (𝑥, 𝑧)𝑒 𝑖(−𝜔𝑡) ] 6

(1)

in which 𝑖 = √−1, 𝑅𝑒[ ] is the real part of the argument, g the gravitational acceleration, 2𝜋 𝜙𝑝 the 2D spatial potential, 𝜔 = the angular wave frequency, 𝑇 the wave period, 𝑑 the water depth, 𝑘 =

2𝜋 𝐿

𝑇

the wavenumber, 𝐿 the wavelength, 𝑡 the time, the subscript 𝑝 = 1,2

for the single barrier since it has two fluid regions (see Fig. 6(a)) and 𝑝 = 1,2,3 for the double barrier since it has three fluid regions (see Fig. 6(b)). Considering frequency domain analysis, the potential 𝜙𝑝 in each region satisfies the Laplace equation:

+

𝜕𝑥 2

𝜕2 𝜙𝑝 𝜕𝑧 2

=0

(2)

and

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𝜕 2 𝜙𝑝

 Seabed condition, i.e. the vertical velocity component along the impervious seabed is zero: 𝜕𝜙𝑝 𝜕𝑧

=0

𝑎𝑡 𝑧 = 0

(3)

𝜕𝜙𝑝 𝜕𝑧

−

𝜔2 𝑔

𝜙𝑝 = 0

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 Free surface condition that combines the kinematic and dynamic free surface boundary: 𝑎𝑡 𝑧 = 𝑑

(4)

lim [

𝜕𝜙𝑝

− 𝑖𝑘𝜙𝑝 ] = 0

(5)

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|𝑥|→∞ 𝜕|𝑥|

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 Far field condition defined by the Sommerfeld radiation condition:

𝐺𝑗 =

𝜀𝑗 𝑓−𝑖𝑠𝑗

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The boundary condition along the porous barrier may be developed on the basis of the work done by Sollitt and Cross (1973), Isaacson et al. (1998, 1999), Yu (1995), Yang (1996), Koraim et al. (2011), Laju et al. (2011) and Somervell et al. (2018). By assuming the porous barrier to be a rigid homogeneous porous medium, the horizontal velocity at the opening is proportional to the pressure difference or the difference of velocity potentials across the porous barrier. The proportional constant 𝐺𝑗′ = 𝐺𝑗 /𝑏𝑗 where 𝑏𝑗 represents plate thickness, and 𝑗 = 1,2,3 … corresponds to the number of different porous surface along the single or double barrier and 𝐺𝑗 corresponds to the permeability parameters that is given by: (6)

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in which 𝜀𝑗 is the porosity of the barrier, f the friction coefficient which comes from a linearization of the velocity squared term associated with the head loss across the permeable part. The real part of 𝐺𝑗 corresponds to the resistance of the barrier, and the imaginary part corresponds to the phase difference between the velocity and the pressure due to inertial effects. In the present study, the formulation of Yu (1995) is followed where f is treated simply as a known constant. Also, in Eq. (6), 𝑠𝑗 is the inertia coefficient that is given by: 𝑠𝑗 = 1 + 𝐶𝑚 (

1−𝜀𝑗 𝜀𝑗

)

(7)

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in which 𝐶𝑚 is the added mass coefficient which is treated as a constant. In the case of porous barrier, the effect of added mass is usually small in most practical cases (Mei et al., 1974; Urashima et al., 1986). So, the added mass coefficient may be taken as zero (i.e. 𝐶𝑚 = 0) which makes 𝑠𝑗 = 1 as suggested by Isaacson et al. (1998,1999), Koraim et al. (2011) and Somervell et al. (2018). For the single porous barrier problem, the potential 𝜙𝑝 and the horizontal velocity are continuous at the interface below the porous barrier, i.e. 𝜕𝜙1

𝜙1 = 𝜙2 ;

𝜕𝑥

=

𝜕𝜙2

along 𝑥 = 0, for 0 ≤ 𝑧 ≤ 𝑎

𝜕𝑥

(8)

𝜕𝜙1 𝜕𝑥

=

𝜕𝜙2 𝜕𝑥

= −𝑖𝐺 ′𝑗 (𝜙2 − 𝜙1 )

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where a = 𝑑 − ℎ. The boundary condition along the porous barrier may be defined by using the proportional constant 𝐺𝑗′ , i.e. along 𝑥 = 0, for 𝑎 < 𝑧 < 𝑑

(9)

For double porous barriers, the potential 𝜙𝑝 and the horizontal velocity are continuous at the interface below the barriers in three fluid regions, i.e. 𝜕𝑥

𝜙 2 = 𝜙3 ;

=

𝜕𝜙2 𝜕𝑥

𝜕𝜙2

=

along 𝑥 = −𝜆, for 0 ≤ 𝑧 ≤ 𝑎

𝜕𝑥 𝜕𝜙3

along 𝑥 = 𝜆,

𝜕𝑥

(10)

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𝜕𝜙1

𝜙1 = 𝜙2 ;

for 0 ≤ 𝑧 ≤ 𝑎

(11)

𝜕𝑥 𝜕𝜙2 𝜕𝑥

=

𝜕𝜙2

=

𝜕𝜙3

𝜕𝑥 𝜕𝑥

= −𝑖𝐺 ′𝑗 (𝜙2 − 𝜙1 ) = −𝑖𝐺 ′𝑗 (𝜙3 − 𝜙2 )

along 𝑥 = −𝜆, for 𝑎 < 𝑧 < 𝑑

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𝜕𝜙1

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The boundary conditions along the porous barrier surfaces are similarly expressed as:

along 𝑥 = 𝜆,

for 𝑎 < 𝑧 < 𝑑

(12) (13)

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3. Solution by eigenfunction expansion method 3.1 Single porous barrier

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The eigenfunction expansion method is employed to obtain solutions for the potentials 𝜙1 and 𝜙2 in the single porous barrier case. These potentials must satisfy Eqs. (2) to (5) and they are assumed to be: ∞

𝜙1 (𝑥, 𝑧) = 𝜙𝐼 − ∑ 𝐴𝑛 cos(𝑘𝑛 𝑧)𝑒𝑘𝑛 𝑥 , 𝑥 ≤ 0 𝑛=0

(14) ∞

𝜙2 (𝑥, 𝑧) = 𝜙𝐼 + ∑ 𝐴𝑛 cos(𝑘𝑛 𝑧)𝑒 −𝑘𝑛 𝑥 , 𝑥 ≥ 0 𝑛=0

(15) Equations (14) and (15) represent the incident waves train combined with a superposition of a propagating mode (𝑛 = 0) and a series of non-propagating evanescent modes (𝑛 ≥ 1) which decay with respect to distance away from the barrier. For n ≥1 corresponding to 8

evanescent waves, 𝑘𝑛 are the positive real roots of the following dispersion relation: 𝜔2 = −𝑔𝑘𝑛 tan(𝑘𝑛 ℎ)

for 𝑛 ≥ 1

(16)

where 𝑘0 = −𝑖𝑘 corresponds to the imaginary root for propagating waves, with the wave number (𝑘) being given as the incident waves as the real roots of the corresponding dispersion relation: 𝜔2 = −𝑔𝑘0 tan(𝑘0 ℎ) = 𝑔𝑘 tanh(𝑘ℎ)

(17)

The incident wave potential 𝜙𝐼 is given by: 𝜙𝐼 = cosh(𝑘𝑧)𝑒 𝑖𝑘𝑥 = cos(𝑘0 𝑧) 𝑒 −𝑘0 𝑥

(18)

By matching the boundary conditions given in Eqs. (8) and (9), it can be shown that: ∑ 𝐴𝑛 cos(𝑘𝑛 𝑧) = 0

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∞

for 0 ≤ 𝑧 ≤ 𝑎

𝑛=0

(19) ∞

for 𝑎 ≤ 𝑧 ≤ 𝑑

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∑ 𝐴𝑛 (𝑘𝑛 𝑑 − 2𝑖𝐺 ′𝑗 𝑑) cos(𝑘𝑛 𝑧) = −𝑘0 𝑑 cos(𝑘0 𝑧) 𝑛=0

(20)

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In order to make the units consistent and applicable for the barrier problem, Eq. (20) has been multiplied by the water depth d.

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Equations (19) and (20) will be used to determine the variables 𝐴𝑛 , 𝑛 = 0,1,2, … which are unknown complex coefficients. Isaacson’s method (1998) is adopted to determine the variables. To begin, each equation is first multiplied by cos(𝑘𝑚 𝑧) to generate orthogonal sets. Next, they are integrated with respect to z over the appropriate domain of z (i.e. 0 ≤ 𝑧 ≤ 𝑎, or 𝑎 ≤ 𝑧 ≤ 𝑑 ), and the resulting two equations are added. This gives the following set of equations for 𝐴𝑛 : ∞

∑

𝐶𝑛𝑚 𝐴𝑛 = 𝑏𝑚 𝑓𝑜𝑟 𝑚 = 0,1, … . , ∞

𝑛=0

where

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(21)

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𝐶𝑛𝑚 = 𝑓𝑛𝑚 (0, 𝑎) + (𝑘𝑛 𝑑 − 2𝑖𝐺 ′𝑗 𝑑)𝑓𝑛𝑚 (𝑎, 𝑑) 𝑏𝑚 = −𝑘0 𝑑𝑓0𝑚 (𝑎, 𝑑)

(22) (23)

and

𝛼

𝑓𝑛𝑚 (𝛼, 𝛽) = ∫ cos(𝑘𝑛 𝑧) cos(𝑘𝑚 𝑧) 𝑑𝑧 𝛽

9

1 sin((𝑘𝑛 +𝑘𝑚 )𝑧) 2

=

[

𝑘𝑛 +𝑘𝑚 𝑧

[ +

{

+

sin((𝑘𝑛 −𝑘𝑚 )𝑧) 𝛽 𝑘𝑛 −𝑘𝑚

]

sin(2𝑘𝑛 𝑧) 𝛽

2

4𝑘𝑛

]

𝛼

for 𝑛 ≠ 𝑚 (24)

for 𝑛 = 𝑚

𝛼

}

For solution, Eq. (21) can be truncated to a finite number of terms, m = 50 that was found to furnish accurate results for the barrier problems considered. The transmission and reflection coefficients, denoted 𝐾𝑡 and 𝐾𝑟 , respectively, are defined as the ratios of wave heights, 𝐾𝑡 = 𝐻𝑡 /𝐻𝑖 and 𝐾𝑟 = 𝐻𝑟 /𝐻𝑖 , where 𝐻𝑡 , 𝐻𝑟 and 𝐻𝑖 are the transmitted, reflected and incident wave heights, respectively. These coefficients are expressed by the first term of 𝐴𝑛 (i.e. 𝐴0 ): 𝐾𝑡 = |1 + 𝐴0 | 𝐾𝑟 = |𝐴0 |

(26)

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(25)

From consideration of energy conservation, these are related to the energy-loss coefficient 𝐾𝑒 which is expressed as: 𝐾𝑒 = 1 − 𝐾𝑡2 − 𝐾𝑟2

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(27)

More detail information on this method may be obtained from the paper by Yang (1996), and Li et al. (2003).

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3.2 Double porous barriers

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The preceding method can be extended to the double porous barrier case in which an incident wave interacts with two barriers which are spaced 2λ apart (see Fig. 6(b)). As indicated in the Fig. 6(b), the flow field is divided into three regions. Region 1 is for up-wave of the barriers, Region 2 is between the barriers, and Region 3 is for down-wave of the barriers.

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The potentials 𝜙1 , 𝜙2 and 𝜙3 may be taken in a similar form as that for the single barrier problem by satisfying Eqs. (2) to (5), i.e. ∞

𝜙1 (𝑥, 𝑧) = 𝜙𝐼 + ∑ 𝐴1𝑛 cos(𝑘𝑛 𝑧)𝑒𝑘𝑛 (𝑥+𝜆) , 𝑛=0

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(28)

𝑥 ≤ −𝜆

∞

∞

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𝜙2 (𝑥, 𝑧) = ∑ 𝐴2𝑛 cos(𝑘𝑛 𝑧)𝑒 −𝑘𝑛 (𝑥+𝜆) + ∑ 𝐴3𝑛 cos(𝑘𝑛 𝑧)𝑒𝑘𝑛 (𝑥−𝜆) , 𝑛=0

−𝜆 ≤ 𝑥 ≤ 𝜆

𝑛=0

(29)

∞

𝜙3 (𝑥, 𝑧) = ∑ 𝐴4𝑛 cos(𝑘𝑛 𝑧)𝑒 −𝑘𝑛 (𝑥−𝜆) ,

𝑥≥𝜆

𝑛=0

(30) where 𝜙𝐼 is an incident wave potential given by Eq. (18). In view of the matching boundary conditions given in Eqs. (10) and (11), the following 10

equations are obtained for 0 ≤ 𝑧 ≤ 𝑎: ∞

∞

∞

∑ 𝐴1𝑛 cos(𝑘𝑛 𝑧) − ∑ 𝐴2𝑛 cos(𝑘𝑛 𝑧) − ∑ 𝐴3𝑛 cos(𝑘𝑛 𝑧)𝑒 −2𝑘𝑛 𝜆 𝑛=0

𝑛=0

𝑛=0

= − cos(𝑘0 𝑧)𝑒 𝑘0 𝜆

(31)

∞

∞

∞

∑ 𝐴1𝑛 𝑘𝑛 cos(𝑘𝑛 𝑧) + ∑ 𝐴2𝑛 𝑘𝑛 cos(𝑘𝑛 𝑧) − ∑ 𝐴3𝑛 𝑘𝑛 cos(𝑘𝑛 𝑧)𝑒 −2𝑘𝑛 𝜆 𝑛=0

𝑛=0

= 𝑘0 cos(𝑘0 𝑧)𝑒 𝑘0 𝜆

𝑛=0

(32)

∞

∞

∞

+ ∑ 𝐴3𝑛 cos(𝑘𝑛 𝑧) − ∑ 𝐴4𝑛 cos(𝑘𝑛 𝑧) = 0

𝑛=0

𝑛=0

𝑛=0

ro of

∑ 𝐴2𝑛 cos(𝑘𝑛 𝑧)𝑒

−2𝑘𝑛 𝜆

(33) ∞

∞

∑ 𝐴2𝑛 𝑘𝑛 cos(𝑘𝑛

𝑧)𝑒−2𝑘𝑛 𝜆

∞

− ∑ 𝐴3𝑛 𝑘𝑛 cos(𝑘𝑛 𝑧) − ∑ 𝐴4𝑛 𝑘𝑛 cos(𝑘𝑛 𝑧) = 0

𝑛=0

𝑛=0

𝑛=0

-p

(34)

∞

∞

∑ 𝐴1𝑛 (𝑘𝑛 −

𝑖𝐺 ′𝑗 )𝑑 cos(𝑘𝑛

𝑧) +

𝑖𝐺 ′𝑗 𝑑

∞

𝑖𝐺 ′𝑗 𝑑

𝑛=0

= (𝑘0 + 𝑖𝐺 ′𝑗 )𝑑 cos(𝑘0 𝑧)𝑒𝑘0𝜆 ∞

𝑖𝐺 ′𝑗 𝑑

∞

∞

𝑖𝐺 ′𝑗 )𝑑

𝑛=0

∞

cos(𝑘𝑛 𝑧) − ∑ 𝐴3𝑛 (𝑘𝑛 + 𝑖𝐺 ′𝑗 )𝑑 cos(𝑘𝑛 𝑧)𝑒 −2𝑘𝑛 𝜆

na

𝑛=0

−𝑖𝐺 ′𝑗 𝑑 cos(𝑘0 𝑧)𝑒𝑘0𝜆

∑ 𝐴3𝑛 cos(𝑘𝑛 𝑧)𝑒−2𝑘𝑛 𝜆

(35)

∑ 𝐴1𝑛 cos(𝑘𝑛 𝑧) + ∑ 𝐴2𝑛 (𝑘𝑛 − 𝑛=0

=

∑ 𝐴2𝑛 cos(𝑘𝑛 𝑧) +

lP

𝑛=0

re

In view of the boundary conditions along the porous surface of the barrier given in Eqs. (12) and (13), the following equations can be written for a ≤ 𝑧 ≤ 𝑑:

𝑛=0

(36)

∞

(37) ∞

∑ 𝐴2𝑛 cos(𝑘𝑛 𝑧)𝑒

Jo

𝑖𝐺 ′𝑗 𝑑

𝑛=0

𝑧)𝑒

−2𝑘𝑛 𝜆

ur

∑ 𝐴2𝑛 (𝑘𝑛 + 𝑛=0

𝑖𝐺 ′𝑗 )𝑑 cos(𝑘𝑛

−2𝑘𝑛 𝜆

+

∞

− ∑ 𝐴3𝑛 (𝑘𝑛 −

𝑖𝐺 ′𝑗 )𝑑 cos(𝑘𝑛

𝑛=0

∑ 𝐴4𝑛 cos(𝑘𝑛 𝑧) = 0 𝑛=0

∞

𝑖𝐺 ′𝑗 𝑑

𝑧) −

𝑖𝐺 ′𝑗 𝑑

∞

∑ 𝐴3𝑛 cos(𝑘𝑛 𝑧) + ∑ 𝐴4𝑛 (𝑘𝑛 − 𝑖𝐺 ′𝑗 )𝑑 cos(𝑘𝑛 𝑧) = 0 𝑛=0

𝑛=0

(38)

Note that Eqs. (35) to (38) are first multiplied by water depth d to make the units consistent. The same method of solution as used for the single barrier problem, will be applied to obtain four sets of unknown complex coefficients 𝐴1𝑛 , 𝐴2𝑛 , 𝐴3𝑛 and 𝐴4𝑛 . The four pairs of equation are obtained by adding Eqs. (31) and (35), Eqs. (32) and (36), Eqs. (33) and (37), and Eqs. (34) and (38). Each equation is multiplied by cos(𝑘𝑚 𝑧) to generate orthogonal sets, and then integrated with respect to z over the appropriate domain of z by Eq. (24). The resulting 11

equations may be written in the following matrix form: ∞

(𝑛𝑚) ∑ 𝐶11 𝑛=0 ∞ (𝑛𝑚) ∑ 𝐶21 𝑛=0 ∞ (𝑛𝑚) ∑ 𝐶31 𝑛=0 ∞ (𝑛𝑚) ∑ 𝐶41 [𝑛=0

∞

(𝑛𝑚) ∑ 𝐶12 𝑛=0 ∞ (𝑛𝑚) ∑ 𝐶22 𝑛=0 ∞ (𝑛𝑚) ∑ 𝐶32 𝑛=0 ∞ (𝑛𝑚) ∑ 𝐶42 𝑛=0

∞

(𝑛𝑚) ∑ 𝐶13 𝑛=0 ∞ (𝑛𝑚) ∑ 𝐶23 𝑛=0 ∞ (𝑛𝑚) ∑ 𝐶33 𝑛=0 ∞ (𝑛𝑚) ∑ 𝐶43 𝑛=0

(𝑛𝑚)

∑ 𝐶14 𝑛=0 ∞

(0𝑚)

𝑏1 𝐴1𝑛 (0𝑚) 𝑏 𝐴 [ 2𝑛 ] = 2(0𝑚) 𝐴3𝑛 𝑏3 𝐴4𝑛 (0𝑚) [𝑏4 ]

(𝑛𝑚)

∑ 𝐶24 𝑛=0 ∞

(𝑛𝑚)

∑ 𝐶34 𝑛=0 ∞

(𝑛𝑚)

∑ 𝐶44 𝑛=0

]

ro of

∞

(39)

(𝑛𝑚)

= 𝑓𝑛𝑚 (0, 𝑎) + (𝑘𝑛 − 𝑖𝐺 ′𝑗 )𝑑. 𝑓𝑛𝑚 (𝑎, 𝑑)

(𝑛𝑚)

= −𝑓𝑛𝑚 (0, 𝑎) + 𝑖𝐺 ′𝑗 𝑑𝑓𝑛𝑚 (𝑎, 𝑑)

(𝑛𝑚)

= 𝑒−2𝑘𝑛 𝜆 [−𝑓𝑛𝑚 (0, 𝑎) + 𝑖𝐺 ′𝑗 𝑑𝑓𝑛𝑚 (𝑎, 𝑑)]

(𝑛𝑚)

= 𝑘𝑛 𝑑𝑓𝑛𝑚 (0, 𝑎) + 𝑖𝐺 ′𝑗 𝑑𝑓𝑛𝑚 (𝑎, 𝑑)

(𝑛𝑚)

= 𝑘𝑛 𝑑𝑓𝑛𝑚 (0, 𝑎) + (𝑘𝑛 − 𝑖𝐺 ′𝑗 )𝑑. 𝑓𝑛𝑚 (𝑎, 𝑑)

(44)

(𝑛𝑚)

= 𝑒−2𝑘𝑛 𝜆 [−𝑘𝑛 𝑑𝑓𝑛𝑚 (0, 𝑎) − (𝑘𝑛 + 𝑖𝐺 ′𝑗 )𝑑𝑓𝑛𝑚 (𝑎, 𝑑)]

(45)

(𝑛𝑚)

= 𝑒−2𝑘𝑛 𝜆 [𝑓𝑛𝑚 (0, 𝑎) + (𝑘𝑛 + 𝑖𝐺 ′𝑗 )𝑑𝑓𝑛𝑚 (𝑎, 𝑑)]

(46)

(𝑛𝑚)

= 𝑓𝑛𝑚 (0, 𝑎) − (𝑘𝑛 − 𝑖𝐺 ′𝑗 )𝑑𝑓𝑛𝑚 (𝑎, 𝑑)

(47)

(𝑛𝑚)

lP

where

= −𝑓𝑛𝑚 (0, 𝑎) − 𝑖𝐺 ′𝑗 𝑑𝑓𝑛𝑚 (𝑎, 𝑑)

(48)

(𝑛𝑚)

= 𝑒−2𝑘𝑛 𝜆 [𝑘𝑛 𝑑𝑓𝑛𝑚 (0, 𝑎) + 𝑖𝐺 ′𝑗 𝑑𝑓𝑛𝑚 (𝑎, 𝑑)]

(49)

(𝑛𝑚)

= −𝑘𝑛 𝑑𝑓𝑛𝑚 (0, 𝑎) + 𝑖𝐺 ′𝑗 𝑑𝑓𝑛𝑚 (𝑎, 𝑑)

(50)

(𝑛𝑚)

= −𝑘𝑛 𝑑𝑓𝑛𝑚 (0, 𝑎) + (𝑘𝑛 − 𝑖𝐺 ′𝑗 )𝑑𝑓𝑛𝑚 (𝑎, 𝑑)

(𝑛𝑚)

= 𝐶24

(0𝑚)

= 𝑒 𝑘0 𝑎 [−𝑓0𝑚 (0, 𝑎) + (𝑘0 + 𝑖𝐺 ′𝑗 )𝑑𝑓0𝑚 (𝑎, 𝐷)]

(53)

(0𝑚)

= 𝑒 𝑘0 𝑎 [𝑘0 𝑑𝑓0𝑚 (0, 𝑎) − 𝑖𝐺 ′𝑗 𝑑𝑓𝑛𝑚 (𝑎, 𝑑)]

(54)

(0𝑚)

= 𝑏4

𝐶22 𝐶23 𝐶32 𝐶33 𝐶34 𝐶42 𝐶43 𝐶44

(𝑛𝑚)

(𝑛𝑚)

= 𝐶31

(𝑛𝑚)

= 𝐶41

=0

𝑏1

𝑏2 𝑏3

(0𝑚)

(42) (43)

(51) (52)

Jo

𝐶14

-p

𝐶21

(41)

re

𝐶13

(40)

na

𝐶12

ur

𝐶11

=0

(55)

In the computations, the number of terms used in the eigenfunction expansion is taken as m = 50 which was found to give accurate results. Microsoft Excel is used to inverse complex matrix by using the algorithm of Dudeck (2005) that decomposes the complex matrix into real matrices in order to determine the complex coefficients. The transmission and reflection 12

coefficients (𝐾𝑟 and 𝐾𝑡 ) are related to the first terms of 𝐴1𝑛 and 𝐴4𝑛 coefficients by: 𝐾𝑟 = |𝐴10 |

(56)

𝐾𝑡 = |𝐴40 |

(57)

From consideration of energy conservation, the energy-loss coefficient 𝐾𝑒 is given by Eq. (27). 4

Validation of numerical models and results by comparison with other researchers’ results

4.1 Single porous barrier model

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Consider a porous seabed resting barrier with a relative draught to water depth h/d = 1, function of wave number and water depth kd = 1.5, friction coefficient f =2.0, and added mass coefficient Cm = 0 that was studied earlier by Yu (1995) and Isaacson et al. (1998). Figure 7 shows a comparison of the variations of transmission and reflection coefficients with respect to porosity (ε) computed by the present model and those obtained numerically by Yu (1995) and Isaacson et al. (1998). It can be seen that the results are in perfect agreement; thereby verifying the present model and results.

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Fig. 7. Comparison of transmission coefficients Kt and reflection coefficients Kr for single porous barrier with h/d=1, kd=1.5, f =2.0, Cm=0

Next, a floating porous barrier is considered. The input parameters for this porous barrier are h/d = 0.5, kd = 1.9, f =2.0 and Cm = 0 which are adopted by Isaacson et al. (1998). Figure 8 compares the transmission and reflection coefficients as a function of porosity ε obtained from the present model, those numerically predicted and experimentally measured by Isaacson et al. (1998). It can be seen that the present model furnishes results that are in good agreement with those of Isaacson et al. (1998).

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Fig. 8. Comparison of transmission coefficients Kt and reflection coefficients Kr for single porous barrier case with h/d=0.5, kd=1.9, f =2.0, Cm=0

-p

4.2 Double porous barrier model

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A double porous seabed resting barrier with h/d = 1, relative distance between barriers to water depth λ/d = 0.25, porosity of first barrier ε1 = 0.1, porosity of second barrier ε2 = 0.3, f = 4.0 and Cm = 0 is considered as the same as the case of Somervell et al. (2018). It is noted that the numerical results of Somervell et al. (2018) were validated with the experimental tests. As shown in Fig. 9, the computed variations of the transmission and reflection coefficients with respect to kd obtained by the present model are in close agreement with those obtained by Somervell et al. (2018).

Fig. 9. Comparison of transmission coefficients Kt and reflection coefficients Kr for double porous barrier with λ/d =0.25, ε1=0.1, ε2=0.3, f =4.0, Cm=0, h/d=1

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Next, a floating double porous barrier model with h/d = 0.5, ε1= ε2= 0.15, f = 0.5, Cm = 0.18, and kd = 0.475 is studied. The inputs are the same those used by Isaacson et al. (1999). Figure 10 shows the comparison between the results obtained from the present model, and the numerically predicted and experimentally measured results of Isaacson et al. (1999). It can be seen that the variations of the transmission and reflection coefficients with respect to relative distance between barriers to draught λ/h show similar trends as those obtained by Isaacson et al. (1999). More interestingly, the present results cover a wider range of λ/h that approaches λ/h = 0 which captures the changing downward trend of the transmission coefficient, which is to be expected, whereas Isaacson et al. (1999) results stopped with an upward trend at λ/h =0.3.

5

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Fig. 10. Comparison of transmission coefficients Kt and reflection coefficients Kr for double porous barrier with h/d=0.5, kd=0.475, ε1= ε2=0.15, f =0.5, Cm=0.18

Parameter selection for collar barrier design

na

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The purpose of considering four different designs: (1) single net, (2) single net with collar barrier, (3) double net and (4) double net with collar barrier, is to obtain appropriate collar barrier underwater height h, porosity ε and number of barriers (single or double). For the parametric study, it is necessary to specify the constant parameters so as to establish the outcome that is affected only by a single variable input.

Jo

ur

Based on the key design parameters of the COSPAR fish cage, presented in Chu and Wang (2020b), Table 1 shows fixed parametric values for this study. For a double barrier, the spacing between two barriers is taken as 2m which should not only suffice for a walkway around the top perimeter of the fish cage, but restrain width so as to minimize water-plane area and selfweight for better hydrodynamic stability. In view of equilibrium of buoyant force and weight consideration, the thickness of collar barrier is assumed to be 0.02m. Following the design of EcoNet (developed by AKVA Group and used for Ocean Farm 1), the net solidity is taken as 0.16 (which is equivalent to a porosity of 0.84) and net thickness is 0.0025m. More accurate values of the added mass coefficient Cm, and friction coefficient f can be obtained from conducting model tests in a wave basin. However, the values highly depend on the laboratory facilities and the geometry of cut-out for porosity. For the present parametric study, we shall assume a friction coefficient f = 2, and added mass coefficient Cm = 0 that has been used by Isaacson et al. (1998,1999). Note overlapped boundaries by net thread and the porous surface of the collar barriers are only considered single permeability parameter Gj for the porous surface herein as it is not possible to combine two different permeability parameters at the same 15

boundary. This assumption is still valid with respect to small contribution of the net thread to wave scattering as presented in Sec. 6. Considering a water depth of 180m and wave periods 5 ≤ T ≤10 seconds, the range of parameter kd is determined from the dispersion relation given by Eq. (17). This range is 29.0 ≥ kd ≥ 7.2. It is assumed that the significant wave height ranges from 2 ≤ Hs ≤ 5m, and the height above water surface of the barrier is 3m. The underwater height of the barrier is to be determined in view of that the transmitted wave height has to be reduced to below 2m for fish farming. Therefore, it is desirable that the wave transmission coefficient should be below 0.4 (2m / 5m) for this site having the aforementioned environmental conditions. Results and discussion

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6

6.1 Effect of collar barrier underwater height h

lP

re

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The efficacy of the single porous barrier will be investigated by changing the barrier underwater height h. When considering both propagating and evanescent wave modes, it is found that there are no distinct differences for the transmission and reflection coefficients by changing the values of h. It may be due to the limitation of the eigenfunction expansion method when the height variable is relatively too small compared to the water depth, and so it converges numerically to the same value. As an alternative solution, a plane-wave assumption is often applied to a single barrier problem, which neglects the evanescent wave modes in order to satisfy matching boundary conditions. This simplifying assumption can still provide an adequate solution since the damping of the single barrier is small (Dalrymple et al., 1991; Park, 2000). Herein, a parametric study is conducted by adopting the plane-wave assumption that considers the propagating wave mode (𝑘0 ) only in Eqs. (19) to (24) with the view to examine the effect of collar barrier underwater height h.

Jo

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na

Figure 11 compares the variations of the transmission coefficient with respect to kd for the base single net case and single collar barrier cases with h = 2, 4, 6 and 8m. In the calculations, ε = 0.5, f = 2 and Cm = 0 are used. The results for the base single net case show relatively small decrease in wave transmission coefficient (an average of Kt = 0.9) with respect to kd. It can also be seen that the transmission coefficients are almost identical based on the original formulation and on the plane-wave assumption formulation irrespective of h due to large porosity and thin net spread thickness. On the other hand, the results associated with the different collar barrier heights show far more reduction in the transmission coefficient than the base single net case for all measured kd. Moreover, there is decreasing trends in the wave transmission coefficients with increasing h, whereas increasing trends are observed for the wave reflection coefficient and energy-loss coefficient by increasing h as shown in Figs. 12 and 13, respectively. More interestingly, cases with h ≥ 4m show more distinct reduction in the wave transmission coefficients within low region of kd (long wave periods) than the case with h = 2m. However, the differences among the curves for h = 4, 6 and 8m cases are negligible. This may be due to wave troughs not reaching more than 4m under water for the examined range of wave conditions. So, a collar barrier underwater height h = 4m will suffice in providing optimal wave scattering performance and economic feasibility for the COSPAR fish cage among examined cases. 16

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Fig. 11. Comparison of transmission coefficient Kt for single net and single net with collar barrier in variable h by plane-wave assumption

Fig. 12. Comparison of reflection coefficient Kr for single net and single net with collar barrier in variable h by plane-wave assumption

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Fig. 13. Comparison of energy-loss coefficient Ke for single net and single net with collar barrier in variable h by plane-wave assumption

6.2 Effect of porosity ε

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The single porous barrier underwater height is now set at h = 4m with given input of f = 2 and Cm = 0 to investigate the effect of various porosities ε = 0.25, 0.5 and 0.75. Figure 14 compares the transmission coefficients for the base single net case and collar barrier cases with various ε. The collar barrier cases show far more reduction in the transmission coefficients by lowering porosity ε. In contrast, the reflection coefficient increases with decreasing ε as shown in Fig. 15. It is clear that low porosity gives better performance in wave scattering by providing more surfaces to reflect incident waves and thereby reducing transmitted waves. With h = 4m, porosity ε = 0.25 case shows optimum results where the wave transmission coefficients are below 0.4 and minimal energy-loss coefficient (see Fig. 16) along measured kd for the COSPAR fish cage.

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Fig. 14. Comparison of transmission coefficient Kt for single net and single net with collar barrier in variable ε

Fig. 15. Comparison of reflection coefficient Kr for single net and single net with collar barrier in variable ε

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6.3 Effect of porosities for double collar barrier

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Fig. 16. Comparison of energy-loss coefficient Ke for single net and single net with collar barrier in variable ε

na

lP

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Double net system will be more attractive than single net for fish farm operators as it is not only a superior system with respect to biosecurity, but it can still hold the fish stock while one of the net is being cleaned, repaired, replaced. In order to support a double net system, a double collar barrier system is a natural choice. Hence numerical models of double porous barrier are investigated by setting various porosity combinations for the outer (defined by ε1) and inner (defined by ε2) barriers. In order to observe the influence of porosity combinations, we shall use h = 4m, f = 2 and Cm = 0 as inputs; making use of the findings from the preceding studies on the single collar barrier.

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Figures 17 and 18 compare transmission and reflection coefficients for the base double net and double collar barrier cases with various porosity combinations from 0.25 to 0.5. All double collar barrier cases present acceptable wave transmission coefficients which is far below the base double net case of Kt ≈ 0.8. The equal porosity combination of ε1=ε2= 0.25 shows the best performance with 0.3 ≥ Kt ≥ 0.05. On the other hand, unequal porosity combinations of ε1=0.25 & ε2=0.5 and ε1=0.5 & ε2=0.25 show almost similar wave transmission coefficients 0.35 ≥ Kt ≥ 0.10. More interestingly, from kd > 21 for the unequal porosity combinations, the variation trend of the reflection coefficient follows the trend that of the equal porosity combination which is used the same outer barrier porosity ε1. With respect to the energy-loss coefficients as shown in Fig. 19, the porosity combination of ε1=0.25 & ε2=0.5 outperforms the porosity combination of ε1=0.5 & ε2=0.25 that is controlled below 0.4 along all measured kd. Moreover, the double collar barrier design with porosity combination of ε1=0.25 & ε2=0.5 leads to a 25% reduction in collar material compared to the design associated with porosity combination of ε1=ε2=0.25. Therefore, the double collar barrier design with porosity combination of ε1=0.25 & ε2=0.5 is recommended for the COSPAR fish cage.

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Fig. 17. Comparison of transmission coefficient Kt for double net and double net with collar barrier in various ε1, ε2

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Fig. 18. Comparison of reflection coefficient Kr for double net and double net with collar barrier in various ε1, ε2

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Fig. 19. Comparison energy-loss coefficient Ke for double net and double net with collar barrier in various ε1, ε2

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Figures 20 compares hydrodynamic coefficients of wave transmission, reflection and energy-loss for shortlisted single and double collar barrier designs that are appropriate for the COSPAR fish cage. Based on these comparison studies, one may conclude that the double collar barrier is a better solution with respect to the wave scattering performance as well as restraining energy absorption within the barriers. Moreover, the double collar barrier is more convincing regarding structural robustness and keeping out external predators and floating debris. Figure 21 shows the front view of the proposed double collar barrier system with porosity combination of ε1=0.25 & ε2=0.5 that is the most competitive design with respect to wave scattering performance and saving collar barrier material to build.

Fig. 20. Comparison of hydrodynamic coefficients for single net with collar barrier vs. double net with collar barrier

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Fig. 21. Proposed double collar barrier design for COSPAR fish cage 7

Conclusion

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In this study, numerical models of single porous barrier and double porous barrier are considered and their wave interaction characteristics were investigated by using the linear potential wave theory and the eigenfunction expansion method in order to design a novel collar barrier system for the COSPAR fish cage with respect to appropriate collar barrier underwater height h, porosity ε and number of barriers (single or double) in view of minimal wave transmission coefficient and energy-loss coefficient.

ur

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With respect to the given dimensions and environmental site conditions for the COSPAR fish cage, the double net – double collar barrier design is recommended. An underwater height h = 4m and porosity combination ε1=0.25, ε2=0.5 for the double collar barrier will suffice in producing transmission coefficients below 0.35 and energy-loss coefficients below 0.4. Although, the results are slightly less than those of the best performing porosity combination ε1=ε2= 0.25, the porosity combination ε1=0.25, ε2=0.5 can save collar materials as 25% less than the porosity combination ε1=ε2= 0.25. Therefore, the double collar barrier design with the porosity combination ε1=0.25, ε2=0.5 is recommended for the COSPAR fish cage. Note that the height above water surface of the collar barrier is 3m to provide berthing space, keep out predators and prevent overtopping waves.

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Further investigations on porous collar barriers and COSPAR cage design should include model testing in a wave basin in order to calibrate the numerical model for analysis and design. This involves determining the added mass coefficient, friction coefficient and effect of hydrodynamic motion responses of the COSPAR fish cage on wave scattering performance. Author statement

Yun Il Chu: Conceptualization; Data curation; Formal analysis; Methodology; Project administration; Resources; Software; Visualization; Writing- Original draft preparation 23

Chien Ming Wang: Funding acquisition; Investigation; Supervision; Validation; Writing- Reviewing and Editing Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements

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The authors are grateful to the Australia Research Council for providing the Discovery Project DP190102983 grant and the Blue Economy CRC for supporting this study on offshore fish cages. The first author wishes to acknowledge the scholarship provided by The University of Queensland for his PhD study. References

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Table 1. Prescribed parameters

Double barrier 0.02 m 0.0025 m 0.84 2m

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Thickness of collar barrier Thread thickness of net Porosity of net Barrier distance (2λ)

Single barrier 0.02 m 0.0025 m 0.84 N.A

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Yu, X., 1995. Diffraction of water waves by porous breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering, 121(6), pp. 275-282.

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