Aquacultural Engineering 92 (2021) 102137

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Aquacultural Engineering
journal homepage: www.elsevier.com/locate/aque

Design development of porous collar barrier for offshore floating fish cage
against wave action, debris and predators
Y.I. Chu *, C.M. Wang
School of Civil Engineering, The University of Queensland, St Lucia, Queensland 4072, Australia

A R T I C L E I N F O A B S T R A C T

Keywords: This paper presents a design concept of a porous collar barrier for a novel floating open-net fish cage that is
Offshore fish farm integrated with a floating spar wind turbine (referred to as COSPAR fish cage). The COSPAR fish cage has an
Floating fish cage octagonal shape with each side length of 30m. The collar barrier, having an array of rectangular cut-outs with
Porous collar barrier
round corners, is installed at the top portion of the cage to attenuate wave transmission inside the cage as well as
Wave transmission
Wave reflection
to protect fish from external predators and debris. Single and double collar barrier designs corresponding to
Energy-loss single net and double net cages are studied. The wave transmission, reflection and energy-loss 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.

1. Introduction 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
Fish farming has played an important role in filling the gap between environments as they prefer calm and peaceful environment. Moreover,
seafood supply and demand in recent years. However, at its current fish have to spend lots more energy to swim against big and irregular
growth trajectory (FAO, 2018), productivity of cultured fish will not be waves compared to regular and steady waves (Beveridge, 2004; Domi­
able to keep pace with demand due to resource constraints, public and nique, 2014). For example, Solstorm et al. (2015) tested post-smolts of
environmental opposition towards expansion of land-based and near­ Atlantic salmon (weighing 98.6 gm and measuring 22.3 cm in length) to
shore fish farms (Le François et al., 2010). Farming in offshore sites has water velocities corresponding to 0.04 m/s, 0.18 m/s and 0.33 m/s
been identified as a potential option for increasing fish production as the (slow, moderate and fast, respectively) over 6 weeks. They found that
sites provide more sea space and better waste dispersion. There are, the fish subjected to fast velocity showed 5% lower weight gain as
however, unforeseen risks for offshore fish farming. Sea currents may be compared to the fish in moderate and slow velocities.
too strong, and waves may be too wild that have a negative impact on Although fish can dive in deeper water where it is relatively steady
fish growth and profit for farmers. Therefore, the environmental, tech­ and reduced water flow, they prefer to be near water surface for sun­
nical and operational challenges will require a completely new engi­ light, oxygen saturation, lower static sea pressure, nutrients/plankton
neering approach for offshore fish farming. and surface air that is necessary for swim bladders. In salmon farms, fish
Water flow plays a significant role in determining whether fish can gather at the water surface to consume dry pellets quickly which would

* Corresponding author.
E-mail addresses: y.chu@uq.edu.au (Y.I. Chu), cm.wang@uq.edu.au (C.M. Wang).

https://doi.org/10.1016/j.aquaeng.2020.102137
Received 25 August 2020; Received in revised form 2 November 2020; Accepted 3 November 2020
Available online 9 November 2020
0144-8609/© 2020 Elsevier B.V. All rights reserved.
Y.I. Chu and C.M. Wang Aquacultural Engineering 92 (2021) 102137

otherwise become moist and sink fast, and out of reach from the fish. show distinct wave attenuation, whilst vertical motions were kept small
Note that dry pellets normally contain a high level of fish meal with owing to low natural frequencies, and hydrodynamic response motions
enriched nutrients and must be kept less than 10 % moisture level and were negligibly affected by the porosity. Dong et al. (2008) conducted
supplied at water surface (Lovell, 1989; Pandey, 2018). So, it is two-dimensional physical model tests in a wave-current flume to mea­
important for a fish cage to have calm surface water in order to reduce sure wave transmission coefficient of a board-net floating breakwater for
feed wastes and keep fish growth at an acceptable level. A new fish cage use with fish cages. The experimental results show that the board-net
design for deployment in energetic offshore sites would thus requires a floating breakwater can effectively protect fish and fish cages in
proper method to reduce wave transmission inside the cage. deep-water regions (see Fig. 2). Ji et al. (2019) presented experimental
Recent deployed offshore fish cages (e.g. Ocean Farm 1 and Shenlan results of a single-row and double-row rectangular floating breakwaters
1), as reported in a recent review paper (Chu et al., 2020), adopted an with porous plates. It reveals that the porous breakwater designs can
open-net system which is ideal for natural water replenishment and effectively attenuate incident waves with slight motion responses and
waste dispersion. Nevertheless, the open-net system in more exposed small mooring forces (see Fig. 3).
and higher water flow sites has to contend with the following threats; Recently, Chu and Wang (2020a) proposed a novel floating open-net
Wild predators such as sealions, seals and sharks can get access to fish by fish cage design that combines a floating spar wind turbine and a steel
leaping to the top or by damaging the side net. The damaged net will fish cage. The design is referred to as COSPAR; an acronym for a
lead to a large number of fish escaping. Floating debris and berthing COmbined Spar and PARtially porous collar barrier fish cage as shown in
impact by farming support vessels can cause net tearing, fish injuries and Fig. 4(a). The initial design of COSPAR has an octagonal shape with each
escapees. Therefore, offshore fish cage designs need to provide a suitable side length of 30 m with an impermeable collar barrier that may incur
method to protect fish from external threats. some disadvantages such as increasing drag forces, mooring tension
A floating barrier or breakwater with a sufficient draught can forces and construction costs when compared to pure open-net fish cage
attenuate wave transmission through the mechanisms of either reflec­ designs (e.g. Ocean Farm 1). However, by using a rigid porous collar
tion or destruction of water particle orbital motions (Wang and Sun, barrier (with an array of rectangular cut-outs and round corners) instead
2010). When the floating barrier takes on closed shapes (e.g. circular or as shown in Fig. 4(b), one can achieve trade-offs between (a) mitigating
octagonal), it can also protect internal water space from external wave transmission inside the cage and protecting fish from debris/pre­
floating debris and predators. A floating impermeable barrier may be a dators and (b) increasing drag forces, mooring tension forces and con­
better solution for reducing wave transmission. However, substantial struction costs. It is worth noting that a notable feature of the COSPAR is
wave energy absorbed by the barrier can generate more dynamic the heavy mass of the spar that lowers the centre of gravity below the
oscillatory motions of the floating barrier and increase mooring tension fish cage. This low position of centre of gravity enhances both pitching
forces. In addition, the floating impermeable barrier in a closed shape and rolling stiffnesses of COSPAR, increases the restoring moment for
can induce internal fluid sloshing that may be more detrimental than the better hydrodynamic motion responses and reduces the mooring tension
incident waves to the fish. On the other hand, a floating porous barrier forces. It is suggested that the collar barrier height about 3 m above
has advantages in reducing wave transmission to an acceptable level, water surface will suffice in preventing intrusion of predators and to
keeping stable floating motions and mitigating fluid sloshing effect as it provide enough berthing height for aquaculture supporting vessels.
has less exposed normal surface to the incident waves. However, other key design parameters of the collar barrier such as un­
Wave scattering problems (transmission, reflection and energy-loss) derwater height, porosity and number of barriers have to be determined.
by using a porous barrier have been studied analytically and experi­ The aim of this study is to design such a novel collar barrier system
mentally by coastal engineers for many years. However, most studies with respect to (i) its appropriate underwater height, (ii) porosity and
focus on seabed resting barriers that are more suitable for shallow water, (iii) number of barriers (single or double) in view of obtaining minimal
and not so applicable to deep-water offshore sites. In recent years, transmission coefficient and energy-loss coefficient against incident
however, wave effects on floating porous barriers have been given waves. This will ensure a calm water space inside the cage and reduce
attention and there are some concrete evidences that show the floating wave energy absorption by the barrier structures and fish cage.
porous barriers to be effective in reducing wave transmission. Four different design cases will be considered in this study as shown
Xiao et al. (2016) performed experiments of a ring-shaped very large in Fig. 5:
floating structure (VLFS) composed of spar-type modules (see Fig. 1). A
comparative study was conducted to investigate the hydrodynamic (1) single net without collar barrier (base design to represent a
performances of different perforated-wall breakwaters vertically conventional open-net cage) as shown in Fig. 5(a),
attached to the VLFS with porosity of 0.16, 0.2 and 0.24. The results

Fig. 1. Ring shaped VLFS with double-layered perforated-wall breakwater.

(a) model in basin, (b) sketch of cross section (Xiao et al., 2016).

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Y.I. Chu and C.M. Wang Aquacultural Engineering 92 (2021) 102137

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

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

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

(2) single net with porous collar barrier having underwater height h small motion responses from their experimental studies on the
as shown in Fig. 5(b), ring-shaped VLFS (see Fig. 1) and the motion responses are not affected
(3) double net without collar barrier as shown in Fig. 5(c), and by attaching a double layered perforated wall. Therefore, the consid­
(4) double net with double porous collar barriers having underwater eration of only diffracted waves is a valid assumption for this study to
height h as shown in Fig. 5(d). gain an insight of wave interaction characteristics of the collar barrier
system. In designing the collar barrier system, a range of wave periods is
The linear potential wave theory (Liu and Abbaspour, 1982; Naka­ assumed to be from 5 to 10 s corresponding to wave conditions at
mura, 1992) and the eigenfunction expansion method (Losada et al., exposed sites near the Storm Bay in Tasmania, Australia (Chu and Wang,
1992; Abul-Azm, 1993) will be used to investigate the phenomenon of 2020b). Extreme wave conditions are not considered in this study as the
wave diffraction so as to predict coefficients of wave transmission, COSPAR fish cage can be submerged by filling ballast tanks to avoid such
reflection and energy-loss for single or double barrier problems under a strong wave action. Based on a parametric study of the wave scattering
wide range of boundary conditions. This study considers wave diffrac­ performance of the aforementioned four design cases, a suitable porous
tion only as it is assumed that the COSPAR fish cage with a porous collar collar barrier design will be identified for the COSPAR fish cage.
barrier maintains its stationary position under small amplitude wave
conditions (for a fish farming operation) due to its low natural frequency
(about 0.02 Hz) in vertical motions. Xiao et al. (2016) also observed such

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Y.I. Chu and C.M. Wang Aquacultural Engineering 92 (2021) 102137

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

2. Numerical model and formulation the distance the bottom edge of the barrier from the seabed is a = d - h.
Based on the preceding assumptions, the fluid motion can be
Numerical models are used to predict the interaction of regular described by a velocity potential that satisfies the Laplace equation
waves with porous barriers. A formulation based on the linear diffrac­ within the fluid regions and boundary conditions at the seabed, free
tion theory with matching boundary conditions at the barriers will be surface and far field (Sarpkaya et al., 1982; Isaacson et al., 1999) in the
used. form:
In order to simplify the problems at hand, the following assumptions [ ( ) ]
igH 1
have been made: Φp (x, z, t) = Re − ϕp (x, z)ei(− ωt) (1)
2ω cosh(kd)
(1) Fluid is inviscid, incompressible and irrotational, and thus the √̅̅̅̅̅̅̅
in which i = − 1, Re[ ] is the real part of the argument, g the gravita­
fluid motion can be described by a velocity potential governed by
the Laplace equation. tional acceleration, ϕp the 2D spatial potential, ω = 2Tπ the angular wave
(2) Seabed is horizontal and impermeable, and wave amplitudes are frequency, T the wave period, d the water depth, k = 2Lπ the wave­
relatively small to the water depth. number, L the wavelength, t the time, the subscript p = 1, 2 for the single
(3) Barriers are thin and rigid. barrier since it has two fluid regions (see Fig. 6(a)) and p = 1, 2, 3 for the
(4) Problems are two-dimensional with incident waves perpendicu­ double barrier since it has three fluid regions (see Fig. 6(b)).
larly approaching to the barriers. Considering frequency domain analysis, the potential ϕp in each re­
gion satisfies the Laplace equation:
Two-dimensional numerical models for single and double floating
barrier problems are shown in Fig. 6(a) and (b), respectively. The Car­
∂2 ϕp ∂2 ϕp
+ 2 =0 (2)
tesian coordinates system (x, z) is defined with x in the direction of ∂x2 ∂z
wave propagation from the location of the single barrier and mid-way and
between double barrier, and z is measured upwards from the seabed.
The barrier has a thickness b and extends downwards to a distance h (3) Seabed condition, i.e. the vertical velocity component along the
below the still water level. The water depth is denoted by d and hence impervious seabed is zero:

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

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Y.I. Chu and C.M. Wang Aquacultural Engineering 92 (2021) 102137

∂ϕp
= 0 at z = 0 (3) ∂ϕ1 ∂ϕ2
∂z ϕ1 = ϕ2 ; = along x = − λ, for 0 ≤ z ≤ a (10)
∂x ∂x

∂ϕ2 ∂ϕ3
(4) Free surface condition that combines the kinematic and dynamic ϕ2 = ϕ3 ; = along x = λ, for 0 ≤ z ≤ a (11)
∂x ∂x
free surface boundary:
∂ϕp ω2 The boundary conditions along the porous barrier surfaces are similarly
− ϕ = 0 at z = d (4) expressed as:
∂z g p
∂ϕ1 ∂ϕ2
= = − iG’ j (ϕ2 − ϕ1 ) along x = − λ, for a < z < d (12)
∂x ∂x
(5) Far field condition defined by the Sommerfeld radiation
condition: ∂ϕ2 ∂ϕ3
= = − iG’ j (ϕ3 − ϕ2 )along x = λ, for a < z < d (13)
[ ] ∂x ∂x
∂ϕp
lim − ikϕp = 0 (5)
|x|→∞ ∂|x|
3. Solution by eigenfunction expansion method

3.1. Single porous barrier

The boundary condition along the porous barrier may be developed

The eigenfunction expansion method is employed to obtain solutions
on the basis of the work done by Sollitt and Cross (1973), Isaacson et al.
for the potentials ϕ1 and ϕ2 in the single porous barrier case. These
(1998, 1999), Yu (1995); Yang (1996); Koraim et al. (2011); Laju et al.
potentials must satisfy Eqs. (2)–(5) and they are assumed to be:
(2011) and Somervell et al. (2018). By assuming the porous barrier to be
a rigid homogeneous porous medium, the horizontal velocity at the ∑
∞
ϕ1 (x, z) = ϕI − An cos(kn z)ekn x , x ≤ 0 (14)
opening is proportional to the pressure difference or the difference of n=0
velocity potentials across the porous barrier. The proportional constant
G’j = Gj /bj where bj represents plate thickness, and j = 1, 2, 3… corre­ ∑
∞
ϕ2 (x, z) = ϕI + An cos(kn z)e− kn x
, x≥0 (15)
sponds to the number of different porous surface along the single or n=0
double barrier and Gj corresponds to the permeability parameters that is
given by: Equations (14) and (15) represent the incident waves train combined
εj with a superposition of a propagating mode (n = 0) and a series of non-
Gj = (6) propagating evanescent modes (n ≥ 1) which decay with respect to
f − isj
distance away from the barrier. For n ≥1 corresponding to evanescent
in which εj is the porosity of the barrier, f the friction coefficient which waves, kn are the positive real roots of the following dispersion relation:
comes from a linearization of the velocity squared term associated with ω2 = − gkn tan(kn h) for n ≥ 1 (16)
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 where k0 = − ik corresponds to the imaginary root for propagating
the phase difference between the velocity and the pressure due to in­ waves, with the wave number (k) being given as the incident waves as
ertial effects. the real roots of the corresponding dispersion relation:
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 ω2 = − gk0 tan(k0 h) = gktanh(kh) (17)
coefficient that is given by:
The incident wave potential ϕI is given by:
( )
1 − εj
sj = 1 + Cm (7) ϕI = cosh(kz)eikx = cos(k0 z)e− k0 x
(18)
εj
By matching the boundary conditions given in Eqs. (8) and (9), it can be
in which Cm is the added mass coefficient which is treated as a constant.
shown that:
In the case of porous barrier, the effect of added mass is usually small in
most practical cases (Mei, 1989; Urashima et al., 1986). So, the added ∑
∞

mass coefficient may be taken as zero (i.e. Cm = 0) which makes sj = 1 as An cos(kn z) = 0 for 0 ≤ z ≤ a (19)
n=0
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 An (kn d − 2iG’ j d)cos(kn z) = − k0 dcos(k0 z) for a ≤ z ≤ d (20)
horizontal velocity are continuous at the interface below the porous
n=0

barrier, i.e. In order to make the units consistent and applicable for the barrier
∂ϕ1 ∂ϕ2 problem, Eq. (20) has been multiplied by the water depth d.
ϕ1 = ϕ2 ; = along x = 0, for 0 ≤ z ≤ a (8) Equations (19) and (20) will be used to determine the variables An ,
∂x ∂x
n = 0, 1, 2, … which are unknown complex coefficients. Isaacson’s
where a = d − h. The boundary condition along the porous barrier may method (1998) is adopted to determine the variables. To begin, each
be defined by using the proportional constant G’j , i.e. 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.
∂ϕ1 ∂ϕ2 e. 0 ≤ z ≤ a, or a ≤ z ≤ d), and the resulting two equations are added.
= = − iG’ j (ϕ2 − ϕ1 ) along x = 0, for a < z < d (9)
∂x ∂x This gives the following set of equations for An :
For double porous barriers, the potential ϕp and the horizontal velocity ∑
∞
Cnm An =bm for m = 0, 1, …., ∞ (21)
are continuous at the interface below the barriers in three fluid regions,
n=0
i.e.

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Y.I. Chu and C.M. Wang Aquacultural Engineering 92 (2021) 102137

where ∑
∞

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

, x≥λ (30)
Cnm = fnm (0, a) + kn d − 2iG’ j d fnm (a, d) (22) n=0

bm = − k0 df0m (a, d) (23) where ϕI is an incident wave potential given by Eq. (18).
In view of the matching boundary conditions given in Eqs. (10) and
and (11), the following equations are obtained for 0 ≤ z ≤ a:

⎧ ⎫
⎪ [ ]β ⎪
⎪
⎪
⎪ 1 sin((k + k )z) sin((k − k )z) ⎪
= m⎪
n m n m
∫ α ⎨ + for n ∕ ⎪
⎬
2 kn + km kn − km
fnm (α, β) = cos(kn z)cos(km z)dz = [ ]β
α
(24)
β ⎪
⎪ z sin(2kn z) ⎪
⎪
⎪
⎪
⎩ 2 + 4kn for n = m ⎪
⎪
α
⎭

For solution, Eq. (21) can be truncated to a finite number of terms, m = ∑

∞ ∑
∞ ∑
∞
2kn λ
50 that was found to furnish accurate results for the barrier problems A1n cos(kn z) − A2n cos(kn z) − A3n cos(kn z)e− = − cos(k0 z)ek0 λ
considered.
n=0 n=0 n=0

(31)
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 A1n kn cos(kn z)+ A2n kn cos(kn z)− A3n kn cos(kn z)e− 2kn λ
=k0 cos(k0 z)ek0 λ
incident wave heights, respectively. These coefficients are expressed by n=0 n=0 n=0

the first term of An (i.e. A0 ): (32)

Kt = |1 + A0 | (25) ∑
∞ ∑
∞ ∑
∞
A2n cos(kn z)e− 2kn λ
+ A3n cos(kn z) − A4n cos(kn z) = 0 (33)
(26)
n=0 n=0 n=0
Kr = |A0 |
∑
∞ ∑
∞ ∑
∞
From consideration of energy conservation, these are related to the A2n kn cos(kn z)e− 2kn λ
− A3n kn cos(kn z) − A4n kn cos(kn z) = 0 (34)
energy-loss coefficient Ke which is expressed as: n=0 n=0 n=0

Ke = 1 − Kt2 − Kr2 (27) In view of the boundary conditions along the porous surface of the
barrier given in Eqs. (12) and (13), the following equations can be
More detail information on this method may be obtained from the paper written for a ≤ z ≤ d:
by Yang (1996), and Li et al. (2003). ∑
∞ ∑
∞
A1n (kn − iG’ j )dcos(kn z) + iG’ j d A2n cos(kn z)
3.2. Double porous barriers n=0 n=0
∑
∞
+iG’ j d A3n cos(kn z)e− 2kn λ
= (k0 + iG’ j )dcos(k0 z)ek0 λ (35)
The preceding method can be extended to the double porous barrier n=0
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 ∑ ∑
∞ ∞
′ ( ′ )
iG j d A1n cos(kn z) + A2n kn − iG j d cos(kn z)
field is divided into three regions. Region 1 is for up-wave of the barriers, n=0 n=0
Region 2 is between the barriers, and Region 3 is for down-wave of the ∑
∞
( )
barriers. (36)
′ 2kn λ ′
− A3n kn + iG j dcos(kn z)e− = − iG j dcos(k0 z)ek0 λ
The potentials ϕ1 , ϕ2 and ϕ3 may be taken in a similar form as that n=0

for the single barrier problem by satisfying Eqs. (2)–(5), i.e. ∑

∞ ( ) ∑
∞ ( )
∑
∞ A2n kn + iG’j dcos(kn z)e− 2kn λ
− A3n kn − iG’j dcos(kn z)
ϕ1 (x, z) = ϕI + A1n cos(kn z)ekn (x+λ) , x ≤ − λ (28) n=0 n=0

n=0 ∑
∞
(37)
′
− iG j d A4n cos(kn z) = 0
n=0
∑
∞ ∑
∞
kn (x+λ)
ϕ2 (x, z) = A2n cos(kn z)e− + A3n cos(kn z)ekn (x− λ) , − λ ≤ x ≤ λ
n=0 n=0

(29)

∑
∞ ∑
∞ ∑
∞
i G’ j d A2n cos(kn z)e− 2kn λ
+ iG’ j d A3n cos(kn z) + A4n (kn − iG’ j )dcos(kn z) = 0 (38)
n=0 n=0 n=0

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Y.I. Chu and C.M. Wang Aquacultural Engineering 92 (2021) 102137

Note that Eqs. (35)–(38) are first multiplied by water depth d to make [ ( ) ]
b(0m) = ek0 a − f0m (0, a) + k0 + iG’ j df0m (a, D) (53)
the units consistent. The same method of solution as used for the single 1

barrier problem, will be applied to obtain four sets of unknown complex [ ]

(54)
(0m)
coefficients A1n , A2n , A3n and A4n . The four pairs of equation are ob­ b2 = ek0 a k0 df0m (0, a) − iG’ j dfnm (a, d)
tained by adding Eqs. (31) and (35), Eqs. (32) and (36), Eqs. (33) and
(55)
(0m) (0m)
(37), and Eqs. (34) and (38). Each equation is multiplied by cos(km z) to b3 = b4 =0
generate orthogonal sets, and then integrated with respect to z over the In the computations, the number of terms used in the eigenfunction
appropriate domain of z by Eq. (24). The resulting equations may be expansion is taken as m = 50 which was found to give accurate results.
written in the following matrix form: Microsoft Excel is used to inverse complex matrix by using the algorithm
⎡∑
∞
∑
∞ ∑
∞ ∑
∞ ⎤ of Dudeck (2005) that decomposes the complex matrix into real matrices
(nm) (nm) (nm) (nm)
C11 C12 C13 C14 in order to determine the complex coefficients. The transmission and
⎢ n=0 n=0 n=0 n=0 ⎥
⎢
⎢∑
⎥
⎥⎡
⎡ ⎤ reflection coefficients (Kr and Kt ) are related to the first terms of A1n and
∞
⎢ ∑
∞ ∑
∞ ∑
∞
(nm) ⎥
⎤ b(0m) A4n coefficients by:
⎢ C(nm) (nm)
C22 (nm)
C23 C24 ⎥ A1n ⎢ 1
⎥
⎢ n=0 21 ⎥⎢ ⎢ b(0m) ⎥
⎢ n=0 n=0 n=0
⎥⎢ A2n ⎥
⎥=⎢ ⎥ (56)
⎢ ∞ ⎥⎣ ⎢
2
⎥ (39) Kr = |A10 |
⎢ ∑ (nm) ∑
∞ ∑
∞ ∑∞ ⎥ A3n ⎦ ⎢ b(0m) ⎥
⎢ C31 (nm)
C32 (nm)
C33 (nm) ⎥
C34 ⎥ A4n ⎣ 3 ⎦
⎢
⎢ n=0 n=0 n=0 n=0 ⎥ (0m)
b4 Kt = |A40 | (57)
⎢ ⎥
⎣∑ ∞ ⎦
∑
∞ ∑
∞ ∑∞
(nm)
C41 (nm)
C42 (nm)
C43 (nm)
C44 From consideration of energy conservation, the energy-loss coefficient
n=0 n=0 n=0 n=0
Ke is given by Eq. (27).

where
4. Validation of numerical models and results by comparison
(nm)
C11 = fnm (0, a) + (kn − iG’ j )d.fnm (a, d) (40) with other researchers’ results

(nm)
C12 = − fnm (0, a) + iG’ j dfnm (a, d) (41) 4.1. Single porous barrier model

[ ] Consider a porous seabed resting barrier with a relative draught to

(42)
(nm) 2kn λ
C13 = e− − fnm (0, a) + iG’ j dfnm (a, d)
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
(nm)
C21 = kn dfnm (0, a) + iG’ j dfnm (a, d) (43)
studied earlier by Yu (1995) and Isaacson et al. (1998). Figure 7 shows a
comparison of the variations of transmission and reflection coefficients
(44)
(nm)
C22 = kn dfnm (0, a) + (kn − iG’ j )d.fnm (a, d) 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
(nm)
C23 = e− 2kn λ
− kn dfnm (0, a) − (kn + iG’ j )dfnm (a, d) (45) seen that the results are in perfect agreement; thereby verifying the
[ ] present model and results.
(nm)
C32 = e− 2kn λ
fnm (0, a) + (kn + iG’ j )dfnm (a, d) (46) 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
(47) are adopted by Isaacson et al. (1998). Figure 8 compares the trans­
(nm)
C33 = fnm (0, a) − (kn − iG’ j )dfnm (a, d)
mission and reflection coefficients as a function of porosity ε obtained
(nm)
C34 = − fnm (0, a) − iG’ j dfnm (a, d) (48) from the present model, those numerically predicted and experimentally
measured by Isaacson et al. (1998). It can be seen that the present model
[ ]
(nm)
C42 = e− 2kn λ
kn dfnm (0, a) + iG’ j dfnm (a, d) (49) furnishes results that are in good agreement with those of Isaacson et al.
(1998).
(50)
(nm)
C43 = − kn dfnm (0, a) + iG’ j dfnm (a, d)
4.2. Double porous barrier model
( )
(nm)
C44 = − kn dfnm (0, a) + kn − iG’ j dfnm (a, d) (51)
A double porous seabed resting barrier with h/d = 1, relative dis­
(nm)
C14 (nm)
= C24 (nm)
= C31 (nm)
= C41 =0 (52) tance 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

Fig. 7. Comparison of transmission coefficients Kt and reflection coefficients Kr Fig. 8. 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. for single porous barrier case with h/d = 0.5, kd = 1.9, f =2.0, Cm = 0.

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Y.I. Chu and C.M. Wang Aquacultural Engineering 92 (2021) 102137

taken as 2 m 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 self-weight for better hydrodynamic stability. In
view of equilibrium of buoyant force and weight consideration, the
thickness of collar barrier is assumed to be 0.02 m. 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 0.0025 m. 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
Fig. 9. Comparison of transmission coefficients Kt and reflection coefficients Kr parameter Gj for the porous surface herein as it is not possible to
for double porous barrier with λ/d = 0.25, ε1 = 0.1, ε2 = 0.3, f =4.0, Cm = 0, h/ combine two different permeability parameters at the same boundary.
d = 1.
This assumption is still valid with respect to small contribution of the net
thread to wave scattering as presented in Sec. 6.
considered as the same as the case of Somervell et al. (2018). It is noted Considering a water depth of 180 m and wave periods 5 ≤ T ≤ 10 s,
that the numerical results of Somervell et al. (2018) were validated with the range of parameter kd is determined from the dispersion relation
the experimental tests. As shown in Fig. 9, the computed variations of given by Eq. (17). This range is 29.0 ≥ kd ≥ 7.2. It is assumed that the
the transmission and reflection coefficients with respect to kd obtained significant wave height ranges from 2 ≤ Hs ≤ 5 m, and the height
by the present model are in close agreement with those obtained by above water surface of the barrier is 3 m. The underwater height of the
Somervell et al. (2018). barrier is to be determined in view of that the transmitted wave height
Next, a floating double porous barrier model with h/d = 0.5, ε1= ε2 = has to be reduced to below 2 m for fish farming. Therefore, it is desirable
0.15, f = 0.5, Cm = 0.18, and kd = 0.475 is studied. The inputs are the that the wave transmission coefficient should be below 0.4 (2 m/5 m) for
same those used by Isaacson et al. (1999). Figure 10 shows the com­ this site having the aforementioned environmental conditions.
parison between the results obtained from the present model, and the
numerically predicted and experimentally measured results of Isaacson 6. Results and discussion
et al. (1999). It can be seen that the variations of the transmission and
reflection coefficients, with respect to relative distance between barriers 6.1. Effect of collar barrier underwater height h
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 The efficacy of the single porous barrier will be investigated by
that approaches λ/h = 0 which captures the changing downward trend changing the barrier underwater height h. When considering both
of the transmission coefficient, which is to be expected, whereas Isaac­ propagating and evanescent wave modes, it is found that there are no
son et al. (1999) results stopped with an upward trend at λ/h = 0.3. distinct differences for the transmission and reflection coefficients by
changing the values of h. It may be due to the limitation of the eigen­
5. Parameter selection for collar barrier design function expansion method when the height variable is relatively too

The purpose of considering four different designs: (1) single net, (2)
single net with collar barrier, (3) double net and (4) double net with Table 1
collar barrier, is to obtain appropriate collar barrier underwater height Prescribed parameters.
h, porosity ε and number of barriers (single or double). For the para­ Single barrier Double barrier
metric study, it is necessary to specify the constant parameters so as to
Thickness of collar barrier 0.02 m 0.02 m
establish the outcome that is affected only by a single variable input. Thread thickness of net 0.0025 m 0.0025 m
Based on the key design parameters of the COSPAR fish cage, pre­ Porosity of net 0.84 0.84
sented in Chu and Wang (2020b), Table 1 shows fixed parametric values Barrier distance (2λ) N.A 2m
for this study. For a double barrier, the spacing between two barriers is

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.

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Y.I. Chu and C.M. Wang Aquacultural Engineering 92 (2021) 102137

Fig. 11. Comparison of transmission coefficient Kt for single net and single net Fig. 13. Comparison of energy-loss coefficient Ke for single net and single net
with collar barrier in variable h by plane-wave assumption. with collar barrier in variable h by plane-wave assumption.

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)–(24) with the view to examine the effect of collar
barrier underwater height h.
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 8 m. 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- Fig. 14. Comparison of transmission coefficient Kt for single net and single net
wave assumption formulation irrespective of h due to large porosity with collar barrier in variable ε.
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 co­
efficients 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 ≥ 4 m show more distinct reduction in the wave transmission
coefficients within low region of kd (long wave periods) than the case
with h = 2 m. However, the differences among the curves for h = 4, 6 and
8 m cases are negligible. This may be due to wave troughs not reaching

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

more than 4 m under water for the examined range of wave conditions.
So, a collar barrier underwater height h = 4 m will suffice in providing
optimal wave scattering performance and economic feasibility for the
COSPAR fish cage among examined cases.

6.2. Effect of porosity ε

The single porous barrier underwater height is now set at h = 4 m

Fig. 12. Comparison of reflection coefficient Kr for single net and single net with given input of f = 2 and Cm = 0 to investigate the effect of various
with collar barrier in variable h by plane-wave assumption. porosities ε = 0.25, 0.5 and 0.75. Figure 14 compares the transmission

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Y.I. Chu and C.M. Wang Aquacultural Engineering 92 (2021) 102137

Fig. 18. Comparison of reflection coefficient Kr for double net and double net
with collar barrier in various ε1, ε2.
Fig. 16. Comparison of energy-loss coefficient Ke for single net and single net
with collar barrier in variable ε.

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 = 4 m, 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.

6.3. Effect of porosities for double collar barrier

Fig. 19. Comparison energy-loss coefficient Ke for double net and double net
Double net system will be more attractive than single net for fish with collar barrier in various ε1, ε2.
farm operators as it is not only a superior system with respect to bio­
security, but it can still hold the fish stock while one of the net is being
ε2 = 0.5 and ε1 = 0.5 & ε2 = 0.25 show almost similar wave transmission
cleaned, repaired, replaced. In order to support a double net system, a
coefficients 0.35 ≥ Kt ≥ 0.10. More interestingly, from kd > 21 for the
double collar barrier system is a natural choice. Hence numerical models
unequal porosity combinations, the variation trend of the reflection
of double porous barrier are investigated by setting various porosity
coefficient follows the trend that of the equal porosity combination
combinations for the outer (defined by ε1) and inner (defined by ε2)
which is used the same outer barrier porosity ε1. With respect to the
barriers. In order to observe the influence of porosity combinations, we
energy-loss coefficients as shown in Fig. 19, the porosity combination of
shall use h = 4 m, f = 2 and Cm = 0 as inputs; making use of the findings
ε1 = 0.25 & ε2 = 0.5 outperforms the porosity combination of ε1 = 0.5 &
from the preceding studies on the single collar barrier.
ε2 = 0.25 that is controlled below 0.4 along all measured kd. Moreover,
Figures 17 and 18 compare transmission and reflection coefficients
the double collar barrier design with porosity combination of ε1 = 0.25
for the base double net and double collar barrier cases with various
& ε2 = 0.5 leads to a 25 % reduction in collar material compared to the
porosity combinations from 0.25 to 0.5. All the double collar barrier
design associated with porosity combination of ε1=ε2 = 0.25. Therefore,
cases present acceptable wave transmission coefficients which are far
the double collar barrier design with porosity combination of ε1 = 0.25
below the base double net case of Kt ≈ 0.8. The equal porosity combi­
& ε2 = 0.5 is recommended for the COSPAR fish cage.
nation of ε1 = ε2 = 0.25 shows the best performance with 0.3 ≥ Kt ≥
Figure 21 compares hydrodynamic coefficients of wave trans­
0.05. On the other hand, unequal porosity combinations of ε1 = 0.25 &
mission, 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 bar­
riers. 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 perfor­
mance and saving collar barrier material to build.

7. Conclusion

In this study, numerical models of single porous barrier and double

porous barrier are considered and their wave interaction characteristics
Fig. 17. Comparison of transmission coefficient Kt for double net and double were investigated by using the linear potential wave theory and the
net with collar barrier in various ε1, ε2.

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Y.I. Chu and C.M. Wang Aquacultural Engineering 92 (2021) 102137

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

CRediT authorship contribution statement

Y.I. Chu: Conceptualization, Data curation, Formal analysis, Meth­

odology, Project administration, Resources, Software, Visualization,
Writing - original draft. C.M. Wang: Funding acquisition, Investigation,
Supervision, Validation, Writing - review & editing.

Declaration of Competing Interest

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

The authors are grateful to the Australia Research Council for

Fig. 21. Proposed double collar barrier design for COSPAR fish cage. providing the Discovery Project DP190102983 grant and the Blue
Economy CRC for supporting this study on offshore fish cages. The first
eigenfunction expansion method in order to design a novel collar barrier author wishes to acknowledge the scholarship provided by The Uni­
system for the COSPAR fish cage with respect to appropriate collar versity of Queensland for his PhD study.
barrier underwater height h, porosity ε and number of barriers (single or
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