Modeling volume deformation in gravity-type cages with distributed bottom weights or a rigid tube-sinker

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Aquacultural Engineering 37 (2007) 144–157

Modeling volume deformation in gravity-type cages with

distributed bottom weights or a rigid tube-sinker

Chai-Cheng Huang a,*, Hung-Jie Tang a, Jin-Yuan Liu b

a Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwanb Institute of Undersea Technology, Asian Pacific Ocean Research Center, National Sun Yat-sen University, Kaohsiung 804, Taiwan

Received 26 May 2006; accepted 6 April 2007

Abstract

Fish mortality caused by the shrinkage and deformation of gravity-type cage net volume during typhoons is of great concern to

the marine cage aquaculture industry. To remedy this deformation problem, we developed a specially designed tube-sinker to

replace the weights that are commonly used at the bottom of cage nets. Herein, we describe the features of our new cage net system,

the formation of motion equations, a new approach to estimating the volume reduction coefficient, and physical model tests in a

hydrodynamic wave tank. Results of the numerical simulation and the experimental measurements show good agreement. This

study indicates that the tube-sinker not only improves the volume reduction coefficient by 10% for most wave conditions, but also

that it significantly reduces severe fluctuations in volume deformation, thereby reducing fish fatalities due to skin abrasion with the

net.

# 2007 Elsevier B.V. All rights reserved.

Keywords: Cage aquaculture; Volume reduction coefficient; Tube-sinker

1. Introduction

Fisheries represent a centuries-old method of human

exploitation of marine resources. Fishery resources are

usually abundant, but over fishing in near shore waters,

reclamation and exploitation along the coastal zone, and

dumping of pollution into the sea have diminished fish

nursery grounds at a rapid rate. In Taiwan, this decline

in near shore fishery resources has pushed the fisheries

industry to consider marine cage aquaculture, in which

fish can be raised in deeper water. However, because

Taiwan is located in a subtropical zone, every year

typhoons severely damage the cages and the resulting

net volume shrinkage and severe deformation cause

* Corresponding author. Tel.: +886 7 525 5169;

fax: +886 7 525 5060.

E-mail address: [email protected] (C.-C. Huang).

0144-8609/$ – see front matter # 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.aquaeng.2007.04.003

high fish mortality. A more flexible cage system might

alleviate this problem, but an engineering analysis is

needed to evaluate the dynamic performance and

reliability of such a system.

Kawakami (1964) proposed a relatively reliable

semi-empirical formula for a flexible net subjected to

the impact of currents. It is based on the drag force

concept, which includes the effects of the material used,

mesh size, current velocity, and the density of water.

Milne (1972) and Woods Hole Engineering Associates

(1984) proposed a series of detailed drag coefficients for

different materials under fouled/unfouled conditions.

Aarsens et al. (1990) and Loland (1991) further

decomposed the external forces into two components,

drag and lift, which take into account the angle between

current velocity and the normal direction of the net

panel. They also proposed the shielding effect in

calculating the total force on a net cage system with a

series of cage units. Lader et al. (2003) conducted a

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157 145

series of experiments to investigate forces acting on and

deformation to a net cage in a uniform current. They

concluded that the total forces on a flexible net cage and

its deformation depend on each other. Lader and

Enerhaug (2005) developed a super element approach

that can be used to predict the global forces on a flexible

net sheet. Tsukrov et al. (2003) employed a ‘‘finite

element method with consistent net element concepts’’

to model the hydrodynamic response of net panels to

environmental loading and applied it to evaluate the

performance of a tension leg fish cage. Fredriksson et al.

(2003) adopted both a finite element method and a

stochastic approach to analyze the motion response

characteristics of an in situ fish cage and the mooring

line tension response to wave forces. Their results

provided valuable information about the dynamical

processes of the fish cage under environmental forces.

Suhey et al. (2005) worked on numerical modeling of

the cage structure using a finite element method; they

showed that an inflated structure has sufficient stiffness

to be used as the structural support within a fish cage.

DeCew et al. (2005) performed an extensive set of

experiments in a wave tank using regular and random

waves to investigate the dynamic response of a modified

gravity cage system. Tsukrov et al. (2005) applied a

finite element numerical model to feed buoy mooring

systems containing nonlinear elastic components such

as feeding hoses. Their numerical simulation provided

predictions about the overall dynamics of the system

and the maximum values of tension in critical

components. Murali et al. (2005) adopted the physical

model to analyze the effect of a cage combined with a

floating breakwater, which has potential applications in

protecting minor ports and harbors, such as fisheries and

recreational harbors, where stringent tranquility

requirements are not warranted.

To date, the global demand for fishery production is

still increasing, and this demand will push fish cage-

based fisheries further towards the open sea. Use of such

non-sheltered offshore marine sites will require improve-

ments in the design, performance, and reliability of

aquaculture cage systems. This paper, following Huang

et al. (2006a), focuses on the dynamic analysis of net

cages based on the ‘‘lumped mass method and net plane

element’’ concepts. The advantage of this concept is that

the whole net cage system can be decomposed into

flexible net plane elements, line elements, bottom

weights, and rigid bodies (such as a floating collar) that

can be divided into several straight tube elements. All

elements are subjected to environmental forces that

evenly distribute to the corresponding nodes. These

nodes, or lumped mass centers, form a system of

uncoupled motion equations that can be solved by the

Runge-Kutta method. Huang et al. (2006b) applied this

numerical model to a full-scale fish cage with a single-

point mooring system deployed in the open sea of

Taiwan; the cable tension was investigated and shown to

agree well with predictions.

Huge commodity losses due to the shrinkage and

deformation of net volume and to fish fatalities from

abrasion by the cage net have been reported frequently

by local fish farmers after typhoons. The objectives of

this work were to analyze the feasibility of using a tube-

sinker to alleviate volume deformation of gravity-type

net cage systems and to predict the maximum tension

force at anchor, which is important in the design of

mooring systems. This paper is structured as follows:

Section 2 discusses features of offshore cage structures,

including the fish rearing system and the mooring

system. Section 3 describes the motion equations, and

Section 4 offers a new approach to estimating the cage

net volume reduction coefficient. Section 5 describes a

physical model study and compares the results with

numerical predictions. Section 6 is an application to a

real open sea cage case. Finally, Section 7 presents our

conclusions and suggestions.

2. Features of offshore cage structures

Most aquaculture gravity-type cages consist of

floating collars, nets, weights, buoys, mooring lines,

and anchors. Fig. 1 shows a common type of offshore

cage structure, except that most aquaculture cage

systems have a series of cages in a row instead of a

single cage. Generally speaking, an aquaculture cage

system can be decomposed into two parts, a mooring

system and a fish rearing system; the latter includes a

floating collar and a fish net.

2.1. Mooring system

The main purpose of the mooring system is to fasten

net cages at a specific location and to prevent cages from

drifting away as environmental loadings act on them.

Therefore, the strength and durability of the material

used for mooring lines are important factors. The

material most commonly used by the local fishing

industry is Nylon, PET (Polyester), and PP (Poly-

propylene). The specific gravity of Nylon is 1.14 and

PET is 1.38; both are heavier than the water’s specific

gravity, and when installed in the field these materials

tend to sink to the sea floor. The specific gravity of PP is

about 0.91, and it may float on the water surface if

disconnected from the bottom anchors.

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157146

Fig. 1. Schematic diagram of a net cage system with bottom weights.

A mooring system failure can occur if the cage

system encounters severe environmental forces, such as

those that occur during a strong typhoon. Once a cable

breaks, it may induce a ‘‘domino effect’’ so that other

mooring lines pop and the whole net cage system may

wash away instantly. To reduce the impact forces that

affect mooring lines, distance buoys (outermost buoys)

are installed in front of cages to absorb these undesired

forces.

To anchor the cage system to the sea floor three

types of anchor are commonly used in the field:

embedment anchors, pile anchors, and deadweight

anchors. Iron embedment anchors are only suitable on

sandy or muddy bottoms, whereas pile and dead-

weight anchors can be used in either rocky or sandy/

muddy bottoms. Pile anchors must be inserted deeply

into the substrate to gain enough holding capacity,

and deadweight anchors rely totally on the friction

forces with the sea bottom to resist the horizontal

tension forces acting on the mooring lines and the

weight of anchors to take care the vertical tension

forces.

2.2. Fish rearing system (or net cage system)

A rearing system is a closed space formed by a

floating collar with a fish net underneath and a cover net

on the water surface. The floating collar consists of a

concentric double tube with a set of stanchions and a

handrail on the top. Thus, the floating collar can be used

as a working platform when fish farm workers want to

inspect the growing conditions inside a net cage. The

floating collar also supports the flexible net pen and is

fastened to the mooring lines through two bridles at

each corner (Fig. 1). In the past, the floating collars were

made of PVC pipes, but recently HDPE (high density

PE) pipes have become the material of choice. A net

cage is usually composed of a cylindrical net panel

forming a closed space in which the fish can grow.

When a typhoon hits, the rearing system—especially

the net cage—can experience severe volume deforma-

tion and cause fish fatalities due to skin abrasion. To

cope with this problem, practitioners often hang much

heavier weights at the bottom of the net to maintain the

rearing space. However, after several years’ practical

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157 147

Fig. 2. Aquaculture net cage system with a tube-sinker.

field experience in cage aquaculture in the open sea, we

found that the bottom weights are not sufficient to

overcome the deformation. Thus, we developed a new

device—the tube-sinker (Fig. 2)—to overcome this

deformation problem.

3. Fundamental motion equation and

environmental forces

The external forces that act on the net cage structure

generally can be classified into four different forces:

hydrodynamic, gravitational, buoyant, and tension

forces. These external forces are described in detail

in Huang et al. (2006a).

Generally speaking, two kinds of methodologies

exist to compute the hydrodynamic forces that act on an

offshore structure, and they depend on the size of the

structure. Brebbia and Walker (1979) proposed the

following: (1) If the ratio of the characteristic length (D)

to the design wave length (L) is larger than 0.1, then the

fluid flow field may be affected by the installed structure

and the offshore structures are considered to be a large

body; in other words, a wave and structure interaction

phenomenon exists and models of such a system should

adopt a special technique, such as a boundary element

method or a finite element method, to take care the wave

diffraction and radiation effect. (2) If the ratio, D/L, is

less than 0.1, the structure is considered to be a small

body and the existing structures hardly affect the fluid

flow field; thus, the modified Morison equation can be

used to estimate the hydrodynamic forces acting on the

structures.

The net cage system has been classified as a small

body because the fish net is porous and the tube size of

the floating collar is relatively small compared to the

wave length and because of the difficulty in implement-

ing diffraction theory to a flexible net cage structure.

Aarsens et al. (1990) and Loland (1991) have pointed

out that the velocity of the flow passing through a porous

net will reduce slightly; they called this a shielding

effect (or shadow effect). The density of fish and their

swimming pattern also affect the flow field inside the

cage, but to simplify the estimation of hydrodynamic

forces these influences must be neglected. Instead, we

use a velocity reduction coefficient of 0.85 for the rear

part of the nets to account for the retardation of fluid

flow velocity that occurs after the water passes through

the previous net panel and the school of fish.

For preliminary planning and design, Eq. (1) of the

modified Morison equation (Brebbia and Walker, 1979)

is pertinent for calculating the wave forces on the small

components of cage structure:

ðmþ r 8KMÞ@R

@t¼ 1

2CDrAVRjVRj þ CMr8 @V

@t(1)

where r is the density of sea water, KM the added mass,

CD the drag coefficient, CM the inertia coefficient, A the

projected area, and 8 is the water displaced volume of

the structure member. VR is defined as:

VR ¼ V� R (2)

which is a water particle velocity V relative to the

structure velocityR. The first term on the right-hand

side of Eq. (1) is a drag force (FD), which is proportional

to the square of the relative velocity; the second term is

an inertia force (FI), which depends on the structure’s

shape/volume and the fluid acceleration at the center of

the structure. To apply Eq. (1) to the whole system, it

should also include the buoyant force (FB), the gravita-

tional force (FW), and the tension force (FT). Therefore,

the general equation of motion for describing small

body motion becomes:

ðmþ r 8KMÞ@R

@t¼ FD þ FI þ FB þ FW þ FT (3)

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157148

where the drag force and the inertia force are associated

with motion of the fluids and are regarded as the

hydrodynamic forces.

When analyzing the hydrodynamic forces on a fish

rearing system, it is common to consider the flexible net

and the rigid floating collar separately. Huang et al.

(2006a) described in detail the external force calcula-

tions for the net cage system, so here we focus on the

calculations for the mooring system.

3.1. Forces on mooring lines

The purpose of mooring lines is to keep the cage

structure from drifting away while the waves and

currents exert environmental loadings on the structure.

Most mooring lines are submerged at least 3–5 m below

the water surface (Fig. 1). To avoid entanglement with

boat propellers, bridle lines start from the connection

point beneath the innermost buoys and tilt up slightly

towards the floating collar that sits on the water surface.

Because mooring lines are actually submerged, the

external forces acting on them should include hydro-

dynamic, gravitational, buoyant, and tension forces.

A mooring line is divided into several segments

called elements that are connected by nodes (also refer

to Fig. 5). Fig. 3a shows that node i has two neighboring

Fig. 3. Modeling a lumped mass and its

elements, j � 1 and j, that experience the external forces

shown in Fig. 3b (using element j as an example).

Similarly, for element j � 1, both neighboring elements

will distribute half of the external forces to the node i, as

shown in Fig. 3c. Thus, we can develop a motion

equation for the lumped mass i.

Among the external forces acting on the line, only

environmental forces (drag and inertia forces) and

tension forces are detailed in the following sections; the

gravitational and buoyant forces are quite obvious and

we provide no further comments on them.

� D

ex

rag force

Each segment of a mooring line is subjected to a

drag force, FDj, that can be decomposed into two

components: normal force (Eq. (4)) and tangential

force (Eq. (5)):

FD jn ¼1

2rwCDnA jVR jnjVR jnj (4)

FD jt ¼1

2rwCDtA jVR jtjVR jtj (5)

where VRjt is the tangential relative velocity between

fluid particles and the centroid of line element j, defined

as VRjt = (VRj�et)et. Similarly, the normal relative velo-

ternal forces for a mooring line.

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157 149

F

city, VRjn, is defined as VRjn = VRj � VRjt. The pro-

jected area, Aj, is defined as the diameter of the mooring

line times the segment length (Aj = Dj‘j). Based on

Webster (1976), CDn and CDt can be determined by

Eqs. (6) and (7):

CDn ¼

0:0 Ren � 0:1

0:45þ 5:93

ðRenÞ0:330:1<Ren � 400

1:27 400<Ren � 105

0:3 Ren� 105

8>>>>><>>>>>:

(6)

CDt ¼1:88

ðRetÞ0:740:1<Ret � 100:55

0:062 Ret > 100:55

8<: (7)

where Ren = (jVRjnj�Dj)/n; Ret = (jVRjtj�Dj)/n; and n is

the kinematic viscosity of sea water.

� I

The inertia force of the jth segment of a mooring

line can be computed by the Morison equation and

expressed as:

FI j ¼ rw 8 jCM

@V j

@t(8)

The mid-point, or the centroid, of the segment is

used to compute the fluid particle acceleration term on

the right hand side.

� T

A mooring line segment is considered to be an

elastic material with a slender body. When subjected

to external forces, it may elongate or shrink in length.

In the case of elongation, the segment will create a

restoring force, whereas for shrinkage no restoring

force occurs. The restoring force is often regarded as a

ig. 4. Schematic diagram of a buoy under hydrodynamic forces.

tension force and is given as follows:

FT j ¼Aline js j if e j > 0

0 if e j � 0

�(9)

s j ¼ C1eC2

j (10)

where Alinej is the cross section of the line segment j; ej

is the strain (defined as ej = (‘j � ‘0)/‘0, where ‘0 is the

original segment length and ‘j is the segment length

between two ends); sj is the normal stress along the

segment; FTj is the tension force; and C1 and C2 are

elasticity coefficients. For practical application, we

usually set C2 = 1, whereas C1 is determined from

elongation tests in the laboratory.

3.2. Forces on buoys

Buoys can be classified into two categories (Fig. 1):

The outermost buoys act as a cushion to absorb the

hydrodynamic impact forces that impinge on the fish

rearing system and as boundary markers to denote the

size of the fish farming site; the innermost buoys act as

supporting floats. To avoid the environmental forces

when a typhoon is coming, the floating collar and the

fish net are submerged 6–10 m under the surface of the

water by manually opening the submerged collar valves

and letting sea water automatically flood in the floating

tubes to gain extra weight; thus, the total buoyant forces

of the innermost buoys must be sufficient to overcome

the total weight of the rearing system so that it does not

sink to the sea floor and encounter some kind of

abrasion problem. Fig. 4 illustrates a buoy under

hydrodynamic forces that tilts to the leeward side. To

compute the hydrodynamic forces, we assume that the

axis of buoy is collinear with the line element j whose

one node i � 1 is at the lower end of the element while

the other node i connects with the buoy. In other words,

the tilting position of the buoy can be determined by the

element j’s direction. The external forces are described

as follows:

� D

To simplify the estimation of hydrodynamic forces

acting on the floating buoy, we must neglect the effect

of the part of the buoy exposed to the air and only treat

the submerged portion as if it was fully submerged.

Then, according to Blevins (1984), the drag forces on

a submerged circular cylinder can be decomposed

into two parts: One is normal to the circumference of

the cylinder and the other is tangential to the cylinder.

The formulas for normal drag force and tangential

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157150

Fig. 5. Examples of schematic diagrams of node-element relationships.

force are:

FBuoyDn ¼1

2rwCDnAnjVRBuoyj2en

FBuoyDt ¼1

2rwCDtAtjVRBuoyj2et

8><>: (11)

where CDn and CDt are drag coefficients (see Eq. (12))

that are strongly associated with the flow’s attack

angle, u (Fig. 4); An is the projected area in the normal

direction; and At is the wetted area in the tangential

direction (both areas are estimated by Eq. (13)). The

relative velocity VRBuoy is estimated at the center of

the expelled water volume; the unit vector et is defined

as the tangential unit vector; and en is defined by

subtracting et from the unit vector of fluid particle

relative velocity eVRBuoy(see Eq. (14)).

CDn ¼ 1:2 sin2 u

CDt ¼ 0:083 cos u � 0:035 cos2 u

((12)

An ¼ DhB

At ¼ pDhB

�(13)

where D is the diameter of the cylinder buoy and hB is

the length of the submerged buoy.

et ¼Ri � Ri�1

jRi � Ri�1jen ¼ eVRBuoy

� et

8<: (14)

where Ri and Ri�1 denote the position vector of global

coordinates.

� I

The inertia force is similar to that of the line

element and is estimated as follows:

FBuoyI ¼ rw 8 BuoyCM

@VBuoy

@t(15)

where CM is set to 2; 8Buoy = (1/4)pD2hB is the

submerged volume of the buoy; and @VBuoy/@t is

the fluid particle local acceleration estimated at the

center of the expelled water volume. Other forces,

such as gravitational and buoyant, are easily esti-

mated, so we provide no further explanations of them.

Once all of these external forces on the buoy are

obtained, they may be incorporated into the i’s node to

form an equation of motion.

3.3. Forming a system of motion equations

The whole cage aquaculture system comprises the

soft parts, such as mooring lines and a flexible cage net,

and the relatively rigid parts, such as the floating collar

and the tube-sinker, which have been regarded as non-

deformed rigid bodies. Herein we used the lumped mass

technique to describe the dynamic motion of the fish

rearing system as well as the mooring system. Nodes are

the positions of lumped mass for the neighboring

elements that experience external forces. All nodes and

elements must be assigned a sequential number, and the

node-element relationships must be tracked (Fig. 5).

Based on these relationships, the computed external

forces for each element will evenly distribute into the

corresponding nodes, or lumped mass centers. Follow-

ing Eq. (3) and Huang et al. (2006a), the motion

equation for each node on the whole cage-net system is

written as:

ðmi þ r 8KMiÞ@2Ri

@t2

¼XNe

j¼1

ðFD þ FI þ FB þ FW þ FTÞ j (16)

where the subscript i represents the node’s sequential

number, the subscript j represents the associated neigh-

boring elements, and Ne represents the total number of

neighboring elements to the node i.

An important fact that cannot be ignored is that both

the floating collar and the tube-sinker are somewhat rigid

compared with the flexible fish net. The movement of

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157 151

Fig. 6. The relationship of a body coordinate system and a global

coordinate system.

their nodes is confined in a very small range of dis-

placement; otherwise, breakage could occur and destroy

the structure’s integrity. Thus, we treated the floating

collar and tube-sinker as rigid bodies, and their nodes

therefore were restrained by rigid body motion. In other

words, the distance between any two points within a rigid

body does not change at any instant. Thus, the description

of rigid body motion involves six degrees of freedom

(three translational and three rotational motions).

For this calculation, assume that the rigid body has a

body coordinate system 1-2-3 with an angular velocity vand its origin at a reference point, G, the center of gravity.

Its global coordinates x � y � z are shown in Fig. 6.

Let P be an arbitrary point in the body, its position

from o is defined by rp, while G is defined by rG.

Because the body coordinates’ principal axes are

symmetrical and its origin is at the mass center, G,

following D’Souza and Garg (1984), three translation

equations are derived from Newton’s second law and

three rotation equations (known as Euler’s equations of

motion) are written as follows:

xG ¼ yGv3 � zGv2 þ1

mG

XN

i¼1

Fxi

yG ¼ zGv1 � xGv3 þ1

mG

XN

i¼1

Fyi

zG ¼ xGv2 � yGv1 þ1

mG

XN

i¼1

Fzi

8>>>>>>>>>><>>>>>>>>>>:

(17)

v1 ¼ �1

I1

ðI3 � I2Þv2v3 þ1

I1

XN

i¼1

M1i

v2 ¼ �1

I2

ðI1 � I3Þv1v3 þ1

I2

XN

i¼1

M2i

v3 ¼ �1

I3

ðI2 � I1Þv1v2 þ1

I3

XN

i¼1

M3i

8>>>>>>>>>><>>>>>>>>>>:

(18)

where Fxi is the force in x direction at the lumped mass i

node on a rigid body such as the floating collar or tube-

sinker; N is the number of total nodes on the tube; mG is

the total mass of the rigid body; M1i is the moment with

respect to the principal axis 1 at the lumped mass i node;

and I1 is the principal moment of inertia of the tube

defined as:

I1 ¼1

2mGR2

I2 ¼1

2mGR2

I3 ¼ mGR2

8>>>><>>>>:

(19)

where R is the radius of the rigid body. Because the

angular acceleration, v1, is associated with body coor-

dinates, the real orientation within the global coordinates

is still unknown. A further angle’s relation between the

body coordinate system and the global coordinate system

has to be established to trace the exact orientations of the

rigid body. Although two approaches exist—Euler’s

angle and Bryant’s angle—to complete the angular velo-

city conversion, we found that Bryant’s angle (Eq. (20)),

referred to Wittenburg (1977), was more suitable to cage

structures because even the approach of f2 to 0 degree

would not create a singular problem:

f1

f2

f3

264

375 ¼

cos f3

cos f2

� sin f3

cos f2

0

sin f3 cos f3 0

� cos f3

cos f2

sin f2

sin f3

cos f2

sin f2 1

266664

377775

v1

v2

v3

264

375

(20)

where w is the angular velocity with respect to the

global coordinate system. By applying a simple finite

difference scheme to w, we obtained the actual orienta-

tion in the time domain at the global coordinates, i.e.

wðt þ DtÞ ¼ wðtÞ þ w� Dt (21)

where Dt is a time step on the order 10�4 and the system

of motion equations is solved by the fourth Runge-Kutta

method.

4. Evaluation of the net volume reduction

coefficient

To estimate cage net deformation, we consider the

net pen as a cylindrical cake and divide it into small

tetrahedral pieces. These tetrahedral volumes are

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157152

Table 1

Specifications of the physical model

Parameter Size Material properties

Floating collar Plastic

Circumference (cm) 162 Density = 0.95 g/cm3

Tube diameter (cm) 2

Total mass (g) 596

Buoy Styrofoam

Diameter (cm) 4.7

Length (cm) 2.5

Total mass (g) 5.87

Mooring line Nylon twist

Diameter (cm) 0.265 Density = 1.14 g/cm3

Unit mass (g/cm) 0.03879

Breaking strength (N) 1538

Elastic coefficient C1 (N/cm2) 75342

Fish net Nylon

Twine diameter (cm) 0.02 Density = 1.14 g/cm3

Total mass (g) 34.776 Solidity = 0.17

Mesh size (cm) 0.46

Net depth (cm) 30

Breaking strength (N) 32.7

Elastic coefficient C1 (N/cm2) 242475

Tube-sinker Plastic

Circumference (cm) 162 Density = 0.95 g/cm3

Tube diameter (cm) 1.178

Total mass (g) 226

Bottom weights Tin

Unit mass (g/piece) 28.25 Density = 2.71 g/cm3

Number of pieces 8

Total mass (g) 226

Fig. 7. Cage net volume and its triangular plane element computing

scheme.

readily estimated by the principal of scalar triple

product (see Huang et al., 2006a). However, a much

simpler way to estimate the deformed cage net volume

is through the divergence theorem (or Gauss’s theorem)

as:Z Z Z8ðr � fÞ d 8 ¼ « Sðf � nÞ dA (22)

where f is a function as long as it makes5�f equal to 1,

then the integration at the left-hand side becomes the

total volume of the net cage at any instant. n is a unit

normal vector of the triangular plane element pointing

outwards. Fig. 7 illustrates the detailed application of

this equation, where a flexible cage net has been divided

into N elements.

In this application, Gi is the geometric center of a

triangular plane element i and its global coordinates can

be computed by averaging the corner nodes’ global

positions. The f function and it outward normal unit

vector n then can be written as follows:

f i ¼xGi iþ yGi

jþ zGi k

3(23)

ni ¼R12 � R23

jR12 � R23j(24)

By substituting Eq. (23) into Eq. (22) and using the

discrete form of area integration, Eq. (22) becomes:

8 ¼XN

i¼1

ðf i � niÞDAi (25)

where 8 is the cage net volume and DAi is the area of

triangular plane element i. Its accuracy depends on how

small the plane elements are. At the very beginning, the

net shape is like a perfect circular cylinder. After several

waves pass through, the net deforms and the floating

collar tilts along with the waves and currents. By

selecting the smallest volume (Vmin) of the net during

environmental loading impact, we get the volume

reduction coefficient (Vratio) for this particular sea state,

and is written as:

V ratio ¼Vmin

V total

: (26)

5. Physical model tests

To test the capability and accuracy of the numerical

model, a physical model has been tested in the

hydrodynamic laboratory of National Sun Yat-sen

University. Based on the 1/30 model scale, we

converted the prototype cage net system into model

scale, but it was somewhat difficult to follow the Froude

similarity criteria because of the difficulty in finding the

exact mesh size of the net in the commercial market. For

example, if the actual mesh size is 0.06 m, then the

model mesh size should be 0.002 m, which is

impossible to find in the current fish net market.

Therefore, the results of the experimental tests are not

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157 153

Table 2

Experimental test cases for a net cage system with bottom weights

Cases 1 2 3 4 5 6

Depth (cm) 85 85 85 85 85 85

Current (cm/s) 14.86 14.86 14.86 14.86 14.86 14.86

Period (s) 1.58 1.77 1.97 1.77 1.77 1.77

Wave height H (cm) 9.66 9.55 11.09 12.31 13.99 15.34

Dimensionless kH 0.1717 0.1434 0.1436 0.1848 0.2101 0.2303

Steepness (H/L) 0.0249 0.0210 0.0212 0.0271 0.0308 0.0337

Table 3

Experimental test cases for a net cage system with a tube-sinker

Cases 1 2 3 4 5 6

Depth (cm) 85 85 85 85 85 85

Current (cm/s) 14.86 14.86 14.86 14.86 14.86 14.86

Period (s) 1.57 1.77 1.96 1.77 1.78 1.77

Wave height H (cm) 9.64 9.39 11.48 12.28 13.83 15.48

Dimensionless kH 0.1730 0.1410 0.1497 0.1844 0.2060 0.2324

Steepness (H/L) 0.0251 0.0207 0.0221 0.0270 0.0301 0.0341

completely applicable to the real world, and the

experimental results only serve as a check for the

numerical model. Table 1 presents the specifications of

the physical model.

We conducted the experiments in a two-dimensional

wave tank (35 m long, 1 m wide, 1.2 m high). The tank

generates waves and a uniform current at the same time.

We tested six types of waves in this study, keeping the

current (14.86 cm/s) and water depth (85 cm) constant.

Each experiment’s wave steepness (H/L) was as large as

possible to simulate the high sea situations and yet not

to over the limitation of linear wave theory. The

objectives of this experimental study is to see how good

is the performance of a net cage with a tube-sinker

system comparing to a bottom weight system during

waves and currents attack. Tables 2 and 3 provide the

Fig. 8. Schematic arrangement

detailed experimental test settings: k, in the dimension-

less parameter kH, is called the wave number and is

defined by k = 2p/L.

The model was installed at a distance of 9 m from the

wave maker, as shown in Fig. 8.

6. Numerical simulation validation and

application

6.1. Model test

Before running the computer program for this model

test, the cage system with bottom weights was divided

into 212 nodes and 194 elements; the cage system with

the tube-sinker was divided into 228 nodes and 244

elements. Although we used a video camera to take

of the experimental test.

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157154

Fig. 9. Comparisons of numerical predictions with their correspond-

ing experimental tension data (extracted from test case 6).

Fig. 10. The comparison of maximum tensions at the anchor against

the dimensionless parameter kH, where H is the wave height, k = 2p/L

is the wave number, and L is the wave length.

pictures of the side view of the deformation of the cage

net, we encountered serious hindrances to estimating

the cage net deformation rate. Although the inward

bending of the cage net was quite obvious in the model

test, this exact phenomena was not visible in the videos

in this study. (However, see Huang et al. (2006a) for

pictures of net volume deformation.) Thus, comparing

the tension records at the upstream anchor with the

corresponding numerical model results was the only

way to verify the accuracy of the numerical model. We

used a waterproof strain gauge (model KYOWA

KFWS-2N-120-C1-11L3M2R) attached to a steel plate

(1 cm � 8 cm � 0.2 cm) at the bottom anchor to obtain

tension records. Once the strain gauge experienced

tension forces transmitted through the mooring system,

a corresponding electrical voltage was detected and

recorded in the data acquisition system. Fig. 9 shows a

sample of experimental data compared with a numerical

prediction; the data are in good agreement.

Both the trough and the peak show some wiggles,

possibly caused by the damping effect of the net cage

structure movement interacting with fluid flow. How-

ever, this damping effect has little influence on the

maximum tension and neglecting this effect in the

computation of motion equations is justified. Fig. 10

shows the dimensionless maximum tensions and the

corresponding numerical model results.

Both experimental measurements and numerical

predictions agreed in most cases, except for the lowest

value of kH = 0.14. In the experimental tests, we

discovered that much better and smoother measurement

results were obtained when the upstream mooring line

was tight rather than slack. When the system was in a

slack condition, the total external forces acting on the

net cage were not transmitted smoothly to the anchor at

the sea floor, which resulted in more wiggle in the

recorded data. For this reason, we generated currents as

large as possible to push the model net cage’s upstream

mooring lines into a tightened condition.

6.2. An application to an open sea cage

To see what would happen if we deployed two types

of cages at an open sea, we selected an offshore cage site

located near Hsiao-Liu-Chiu Island, Pingtung County,

Taiwan, as our case study; this area contains a few

marine aquaculture fish farms. For this study, we used a

single net cage system under environmental loading

conditions equivalent to a typical 50-year return period

of wave conditions: 6 m wave height, 10 s wave period,

and a high tide current of 1.0 m/s. Other information

about this system is listed in Table 1. For these

prototype offshore cages, the bottom weight system was

divided into 270 nodes and 288 elements, while the

tube-sinker system was divided into 283 nodes and 312

elements. Details about the input data for the computer

program are shown in Table 4. Fig. 11 shows that both

cage types reached stable fluctuation after 60 s of

computer simulations. The system with bottom weights

had a net volume reduction coefficient in the range of

26–47%, which means that in one wave cycle (10 s) the

net volume variation goes through 21% of its total

volume. The system with the tube-sinker had a volume

reduction coefficient in the range of 36–48%; it only

goes through 12% of its volume fluctuation. These

results indicate that fewer fish inside the tube-sinker

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157 155

Table 4

Data from a typical aquaculture net cage

Parameter Size and properties Material

Mooring line Nylon twist rope

Diameter (m) 0.05

Breaking strength (kN) 413

Unit mass (kg/m) 304/200

Main rope (m) 50

Distance rope (m) 10

Frame rope (m) 28

Bridle Nylon twist rope

Diameter (m) 0.03

Breaking strength (kN) 160

Unit mass (kg/m) 109/200

Buoy HDPE (high density polyethylene)

Buoyancy (kN) 8.5

Diameter (m) 0.91

Length (m) 1.3

Total mass (kg) 41

Floating collar HDPE (high density polyethylene)

Pipe diameter (m) 0.25

Pipe thickness (m) 0.015

Inner-collar circumference (m) 50

Outer-collar circumference (m) 54

Total mass (kg) 2136

Fish net Nylon 210D/96

Net depth (m) 10

Mesh size (m) 0.0604

Twine diameter (m) 0.0021

Elastic coefficient C1 (kN/m2) 350900

Tube-sinker HDPE (high density polyethylene)

Tube diameter (m) 0.20

Tube thickness (m) 0.0182

Circumference (m) 54

Unit weight of chain (kg/m) 10 Chain inserted inside of tube

Length of chain (m) 54

Total mass (kg) 640

Bottom weights Iron

Weight per piece (kg) 40

Number of pieces 16

Total mass (kg) 640

system might die from skin abrasion or collision

problems compared to those inside the system with

bottom weights. In reality, if dead fish accumulate and

sink to the bottom of the cage, after a period of time

these relatively solid dead bodies might deteriorate the

net volume deformation. This could be one explanation

for why almost all of the fish in a cage often are dead

after a typhoon. Other causes of fatality include fish

stress due to the cage’s violent motion, exhaustion from

fighting against waves/currents, or disease. The

increased the net volume reduction coefficients pro-

vided by the tube-sinker system and its relatively mild

fluctuation might give fishes a better chance to survive

during heavy storms. Fig. 12 shows that the maximum

tension (60 kN) for the tube-sinker system was higher

than that in the bottom weight system (54 kN). This

difference occurred because the system with the tube-

sinker usually maintained a larger net volume and thus

might have induced more hydrodynamic loading on the

system; in contrast, its counterpart can fold up and

reduce its projected area during a storm. Fig. 13

illustrates the worst-case scenario, in which the net

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157156

Fig. 11. Variations in net volume during environmental loading.

Fig. 12. Variations in tension acting on the anchor.

Fig. 13. The net cage systems at the instant of minimum volume.

volume reaches its most serious shrinkage condition and

both systems tilt to the leeward side. The system with

the tube-sinker looks much larger than its counterpart.

7. Conclusions and suggestions

Fish mortality caused by the shrinkage, deformation,

and quick fluctuations of the volume of gravity-type

cage nets during typhoons is of great concern to marine

cage fish farmers. To alleviate this deformation

problem, we recommend the use of a specially designed

tube-sinker in place of the bottom weights typically

used on fish rearing systems. In our numerical model,

we adopted the concepts of lumped mass and plane

element methods for the marine cage aquaculture

system. However, we treated the floating collar and

tube-sinker as rigid bodies, so we had to incorporate the

constraints of rigid body motion into the motion

equations. We presented a new approach to estimating

the net volume reduction coefficient by Gauss’s

theorem and showed it to be a much easier computation

method. To validate the numerical approach, we carried

out a series of physical tests with a 1/30 scale model in a

hydrodynamic wave tank. Finally, we used a real open

sea case with a single cage to demonstrate the practical

application of our results. We discovered some useful

features, such as volume reduction coefficients and their

fluctuation patterns, which might offer fish farmers

good information about cage shrinkage problems. Other

conclusions and suggestions are as follows:

� T

the tensions on the mooring line at the upstream

anchor agree with each other. In other words, the

numerical model is capable of predicting real

situations that might occur in offshore cage aqua-

culture.

� T

reduction coefficient and alleviates the instant

fluctuation problem in net deformation, which means

that the fish suffer less stress during typhoons.

However, compared to the bottom weight system, this

system has increased tension on the mooring line,

which might create extra stress for other structural

parts. Therefore, a careful evaluation is still required.

� F

not easy to interpret results gathered from the field,

C.-C. Huang et al. / Aquacultural Engineering 37 (2007) 144–157 157

the data certainly will offer valuable information for

the cage aquaculture industry when moving offshore

to the rigors of the open sea. Recently, we deployed a

single-point-mooring marine cage in Taiwan Strait

near Hsiao-Liu-Chiu Island at 30 m water depth. A

load cell was installed on the mooring line and an

ADCP (Acoustic Doppler Current profiler) also was

bottom mounted nearby. Hopefully, this system is

simultaneously recording tension loading and sea

state, especially during the typhoon season. After this

field test, we hope to use the raw data to evaluate our

numerical model and offer some useful information to

the marine cage industry.

Acknowledgments

This work was supported in part by ‘‘Aim for the Top

University Plan from the Ministry of Education

(95C100303),’’ National Science Council, and the

Fisheries Agency, Council of Agriculture of Taiwan,

R.O.C. The authors would like to express profound

thanks for their financial support.

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