A numerical study on dynamic properties of the gravity cage in combined wave-current flow

ARTICLE IN PRESS

Ocean Engineering 34 (2007) 2350–2363 www.elsevier.com/locate/oceaneng

A numerical study on dynamic properties of the gravity cage in combined wave-current flow Yun-Peng Zhaoa,, Yu-Cheng Lia,b, Guo-Hai Donga, Fu-Kun Guic, Bin Tenga a

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China b R&D Center for Civil Engineering Technology, Dalian University of Technology, Dalian 116622, China c Marine Science and Technology School, Zhejiang Ocean University, Zhoushan 316000, China Received 15 December 2006; accepted 15 May 2007 Available online 23 May 2007

Abstract Recent work in the area of open-ocean aquaculture-system dynamics has focused either on the response of fish cages in waves or the steady drag response from ocean currents, not on them combined. In reality, however, the forces bearing on these open-ocean structures are a nonlinear, multidirectional combination of both waves and current profiles. In this paper, a numerical model has been developed to simulate the dynamic response of the gravity cage to waves combined with currents. When current flows are combined with regular waves, gravity-cage motion response (including heave, surge, and pitch) and mooring-line forces have been calculated. To examine the validity of simulated results, a series of physical model tests have been carried out. The results of our numerical simulation are all in close agreement with the experimental data. r 2007 Elsevier Ltd. All rights reserved. Keywords: Gravity cage; Wave–current flow; Numerical simulation; Physical model tests

1. Introduction The potential exists for the marine aquaculture engineering industry to expand if operations can be performed economically at exposed open-ocean sites. From an engineering perspective, systems need to be designed to cost-effectively withstand extreme conditions while providing a suitable growing environment. The study of the openocean sea cage and mooring system has focused primarily upon either the dynamic response of components to waves or the steady drag of these structures from ocean currents. Techniques used to investigate these mechanisms have typically included the use of scaled physical and numerical models, and, where possible, field measurements. To understand the response of the sea cage to wave forces, many research studies have been conducted. For example, Zheng et al. (2006) investigated the motions of the Corresponding author. Tel.: +86 411 84708974; fax: +86 411 84708526. E-mail addresses: [email protected] (Y.-P. Zhao), [email protected] (Y.-C. Li).

0029-8018/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2007.05.003

gravity-cage float collar in waves by the numerical method. Takashi et al. (2004) described the dynamic behavior of fishing nets using the computer-aided simulation NaLA, a system that was developed for determining net shape and load. Colbourne and Allen (2001) observed the motions and loads in aquaculture cages from full- and model-scale measurements. A comparison of motion response of cagedeck sections and underwater rings was also presented in their paper. Fredriksson et al. (2003) carried out the research on fish-cage and mooring-system dynamics, using physical and numerical models with field measurements. The field data collected have helped researchers to understand numerical and physical modeling approaches and to make improvements. Zhao et al. (2007) conducted a special investigation about the selection of wave theory in the numerical simulation of the gravity cage. The engineering design and specification of open-ocean aquaculture net-cage systems must also consider the effect of steady ocean currents. Aarsnes et al. (1990) investigated current forces and blockage characteristics for individual net types used in aquaculture by both theoretical and model test methods. Tsukrov et al. (2003) described the theoretical

ARTICLE IN PRESS Y.-P. Zhao et al. / Ocean Engineering 34 (2007) 2350–2363

models and calculation methods of nets, using the consistent net-element method. The simulated results were compared with the experimental observations and analytical results of other authors. Li et al. (2006a, b) developed a lumped mass method to simulate the hydrodynamic behaviors of both plane fishing nets and gravity cages in currents. General oceanic forces, however, typically involve a nonlinear and multidirectional combination of both waves and currents. To investigate the gravity-cage dynamics in combined wave and current flow, a numerical model was developed in the present study to simulate a typical gravity cage (Fig. 1), and a series of experiments were carried out to evaluate its validity. The research of this paper will lead to a better understanding of the hydrodynamic behavior of the gravity cage in combined wave and current flow. 2. Methods The simulation methods of the gravity cage in wave– current conditions are described in detail. First, the wave–current field is described, and then models of the net and the float collar are introduced. The model of the bottom collar is similar to that of the float collar, so it is not described in this paper.

2351

where or ¼ 2p=T r is the angular frequency of the wave in the moving frame of reference, and o ¼ 2p=T is the apparent angular frequency of the wave as noted by a stationary observer. k ¼ 2p=l represents the wave number. Based on linear wave theory, the transformation of wavelength can be calculated as follows (Li, 1990):   U 2 l=ls ¼ 1  tanh kd= tanh ks d, (2) C where subscript s denotes the relative value in still water. Based on linear wave theory and the principle of conservation of wave action flux, the transformation of wave height can be obtained (Li, 1990) by         U 0:5 lS 0:5 N S 0:5 U 2  N 0:5 H=H s ¼ 1  1þ , C C N l N (3) where N ¼ 1 þ ð2kd= sinh 2kdÞ, and N s ¼ 1 þ ð2ks d= sinh 2ks dÞ. In the stationary frame of reference, according to linear theory, the water surface Z and the horizontal and vertical components of the water-particle velocities are given by Z¼

H cos ðkx  otÞ, 2

(4)

2.1. Description of the wave–current field The interaction of waves and currents not only changes the characteristics of waves but also transform the currentflow field at the same time. As a simplification for practical engineering applications in this paper, the current is assumed to be steady and uniform over the entire water depth, and parallel to the direction of wave propagation. Many researches concern with wave–current interaction, for instance, Li (1990) and Hedges and Lee (1992). Here, several conclusions will be introduced in our paper. To a stationary observer, the waves appear to have a celerity C, while to an observer moving in the direction of wave propagation at the current velocity U the waves seem to have a celerity Cr. Consequently, based on linear wave theory, the real wave celerity C in current may be determined by C ¼ C r þ U; then o ¼ or þ kU,

(1)

ux ¼ U þ

uz ¼

H cosh kðz þ dÞ ðo  kUÞ cos ðkx  otÞ, 2 sinh kd

H sinh kðz þ dÞ ðo  kUÞ sinðkx  otÞ, 2 sinh kd

i

ξ η

V

z y z

1

Fig. 2. Schematic diagram of local coordinates for mesh twines.

Float system

Mooring line Cage net

2

T

Weight system

Fig. 1. Sketch of a typical gravity cage.

(5)

(6)

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where coordinate z is the vertical distance (positive upward) from mean-water level, and d is the water depth. Then the acceleration of water particles can be expressed as the time differential of Eqs. (5) and (6). 2.2. Model of the net By applying a lumped-mass model, the fishing net is assumed to be a connected structure with limited masses and springs. Lumped point masses are set at each knot and at the center of the mesh bar. The calculation method for the model was explained fully in our previous papers (Li et al., 2006a, b). Here, only a brief outline is described. 2.2.1. Forces and motion equations for net According to Newton’s second law, the motion equation of a lumped mass i can be expressed by Mia ¼

n * X * * ~ þ B, ~ T ij þ F D þ F I þ W

(7)

j¼1 *

where a is the acceleration of the mass point i, T ij is the vector of the tension in bar ij (j is the code for knots at the other end * of the bar ij), n is the number of adjacent knots of point i, F D is the drag force induced by waves and currents, * ~ is the gravity F I is the inertial force induced by waves, W ~ is the buoyancy force. According to the force, and B

of a mesh bar, and the local coordinates ðt; Z; xÞ passing point i are defined to simplify the procedure (Fig. 2). The Z axis lies on the plane including the t axis and V, and the velocity vector of the water particles V at point i can be divided into t (tangential) and Z (normal) components. So the local fluid force on the plane can be estimated just by defining the t and Z components of the fluid force. Under the global coordinates, the vectors drawn from i to 1 and 2 are ei1 and ei2, respectively. The unit vectors of the t, Z, x axes are et ¼ ðxt ; yt ; zt Þ, eZ ¼ ðxZ ; yZ ; zZ Þ and ex ¼ ðxx ; yx ; zx Þ. The relationship between tension and elongation, based on Wilson (1967), is given by T l  l0 ¼ C 1 C 2 ;  ¼ , (8) 2 l0 d where T is the tension force in twine, l0 is the undeformed length, l is the deformed length, d is the diameter of the mesh bar, and C1 and C2 are constants that define the elasticity of the element. C1 and C2 can be obtained by matching, referring to Gerhard (1983). For polyethylene (PE), C1 ¼ 345.37  106, and C2 ¼ 1.0121; for polyamide (PA), C1 ¼ 784.9  106, and C2 ¼ 1.6988. The units of T and d are N and m, respectively. Therefore, the t component of tension at point i can be expressed by T t ¼ T i1 ei1  et þ T i2 ei2  et ,

(9)

*

research of Gui et al. (2002), the inertial force F I on the fishing net is rather small in waves in comparison with the other external forces, so it is omitted here. The mass point at the mesh knot is assumed to be a spherical point where the fluid-force coefficient is constant in the motion direction, so the motion equation is easily set up according to Eq. (7). For the points at mesh bars, the forces and the motion equation are described in the following text. Because the points at the mesh bars are assumed to be cylindrical elements, the direction of the fluid forces acting on the point masses at each mesh bar should be considered. Therefore, the motion of the point mass i is set at the center

b

v

n

c

a d

τ

ο Fig. 4. Schematic diagram of mini-segment.

Fig. 3. Sketch of floating system of gravity cage.

ARTICLE IN PRESS Y.-P. Zhao et al. / Ocean Engineering 34 (2007) 2350–2363

where the mid-dot [  ] represents the dot product. The same expression can be applied for other tension forces (TZ, Tx) of the Z, x components. The t, Z, and x components of the relative velocity of _ respectively. Thus, point i are t_  et  V , Z_  eZ  V , and x, the t component of the drag force of point i can be represented by F dt ¼ 12rC dt Dlj_t  et  V jð_t  et  V Þ,

(10)

where Cdt represents the drag coefficient of the t component, D represents the cylindrical element diameter, and l represents the cylindrical element length. The same expression can be applied for other drag forces (FdZ, Fdx) of the Z, x components. Dividing other external forces B and W along the local coordinate axes, the t component motion equation of point i can be expressed by M i t€ i ¼ T t  F Dt þ W t þ Bt .

a given initial condition. The equations are solved by using the Runge–Kutta–Verner fifth order and sixth order method in this paper. For these calculations, the IMSL numerical library solver including Visual Fortran (Compaq) can be used. 2.2.2. Hydrodynamic coefficients of the net in combined wave–current flows For each mesh bar, the numerical procedure calculates the drag coefficient Cd using a method described by Choo and Casarella (1971) that updates the drag coefficients based on the Reynolds number as follows: 8 8p > ð1  0:87s2 Þ ð0oRen p1Þ; > < Ren s 0:90 ð0oRen p30Þ; (15) C n ¼ 1:45 þ 8:55 Ren > > 5 : 1:1 þ 4 Re0:50 ð0oRen p10 Þ n

(11)

Thus, in the same manner the motion equations of point i in the Z, x directions can also be expressed. As the displacements are expressed, using local coordinate variables in Eq. (11), we can transform these into the global coordinate system by _ T, ðx_ i ; y_ i ; z_i ÞT ¼ ½Cð_t; Z_ ; xÞ where [C] is 0 xt By ½C ¼ @ t zt

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(12)

given by 1 xZ xx yZ yx C A. zZ zx

(13)

Thus, the motion can be represented by the following system of ordinary differential equations: dx_ i ¼ f ðxi ; yi ; zi ; x_ i ; y_ i ; z_i ; x1 ; y1 ; z1 ; x2 ; y2 ; z2 ; tÞ, dt dy_ i ¼ gðxi ; yi ; zi ; x_ i ; y_ i ; z_i ; x1 ; y1 ; z1 ; x2 ; y2 ; z2 ; tÞ, dt d_zi ¼ hðxi ; yi ; zi ; x_ i ; y_ i ; z_i ; x1 ; y1 ; z1 ; x2 ; y2 ; z2 ; tÞ. ð14Þ dt The net shape at each time step can be calculated numerically by solving ordinary differential equations with

2=3 C t ¼ pmð0:55 Re1=2 n þ 0:084 Ren Þ,

(16)

where Ren ¼ V Rn D=n, s ¼ 0:07721565 þ lnð8=Ren Þ, Cn and Ct are the normal and tangential drag coefficients for the mesh bar, VRn is the normal component of the fluid velocity relative to the bar, and r is the density of water. According to many investigations, such as those of Sarpkaya et al. (1984), Gudmestad and Karunakaran (1990), Wang and Li (1999), and Wang et al. (1995), the drag force coefficient Cd in combined wave–current flows is usually less than that in waves only, but the reason for it is still not clear at present. In this paper, Cd is set as 0.6 for the knot part. 2.3. Model of the float collar 2.3.1. Forces on the float collar In general, the float-collar system of the gravity cage is usually at the water’s surface. The double floating pipes are the main parts to withstand the wave–current-induced loads. For simplicity, the float-collar system is regarded as a double-column pipe system, as shown in Fig. 3. It is assumed that the float-collar system will have no influence

v 3

n

e

urfac

es Wav γ

p r

η

G z

v

rp rG

o O Fig. 5. Sketch of simplified pipe model for calculating.

2

1 y

x Fig. 6. Schematic diagram of moving coordinates system of rigid body.

ARTICLE IN PRESS Y.-P. Zhao et al. / Ocean Engineering 34 (2007) 2350–2363

z

0.796m 1

0.43m

x

2

Transducer

Tracing point

H=1.0m

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Bottom collar Sinker

2.92m

322.3cm

3cm

Cross section

84 m 12.9°

97cm

.6c

y

CCD

3cm

2

1

Transducer

97cm

x

Plan view Fig. 7. Sketch of experimental model for gravity cage.

Table 1 Parameters of simplified floating system Outer circle

Inner circle

Mooring line

General diameter (m) Pipe diameter (m) Material

0.846 0.0153 HDPE

General diameter (m) Pipe diameter (m) Material

0.796 0.0153 HDPE

Length (m) Diameter (m)

3.152 0.0012

on waves because its size is much less than the wavelength. The coordinate system is defined as shown in Fig. 4. When calculating wave–current-induced forces on the float collar, the collar is divided into many mini-segments. The forces on the whole collar can be obtained by combining the external forces on each mini-segment. Fig. 4 is a sketch of a mini-segment of the float collar. As shown in this figure, the mini-plane is composed of abcd, and local coordinates ntv are defined in each mini-segment. Concerning the coordinate system nsv, n and s are in the normal and the tangential directions of the mini-segment, respectively. Then v is normal to the mini-plane, as shown in Fig. 4. The wave–current-induced forces on the pipe minisegment can be calculated, using the Morison equation.

According to Brebbia and Walker (1979), the Morison equation includes two terms: drag forces and inertial forces in waves and currents, as shown: ~_ ~  ð~ ~ þ rV 0~ F ¼ 12C D rAj~ u  Uj u  UÞ a  UÞ, a þ C m rV 0 ð~ (17) ~ are the velocity vectors of water particles where ~ u and U ~_ are the and the mini-segment, respectively; ~ a and U acceleration vectors of water particles and the minisegment, respectively; r is the water density; V0 is the tonnage of the mini-segment; A is the projected area normal to the wave-propagation direction; and CD and Cm are the drag and added mass coefficients, respectively. Wave–current-induced forces on the mini-segment in the ntv coordinate system can be obtained as follows: F n ¼ 12C Dn rAn  jðun  U n Þj  ðun  U n Þ þ rV 0 an þ C mt rV 0 ðan  U_ n Þ, F t ¼ 12C Dt rAt  jðut  U t Þj  ðut  U t Þ þ rV 0 at þ C mt rV 0 ðat  U_ t Þ, F v ¼ 12C Dv rAv  jðuv  U v Þj  ðuv  U v Þj þ rV 0 av þ C mt rV 0 ðav  U_ v Þ,

ð18Þ

where CDn, CDt, and CDv are the drag coefficients in their respective directions, and Cmn, Cmt, and Cmv are the added

ARTICLE IN PRESS Y.-P. Zhao et al. / Ocean Engineering 34 (2007) 2350–2363

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Table 2 Design of experimental conditions No.

1

2

3

4

5

6

7

8

9

10

11

12

Height/(cm)

20

20

20

25

25

25

30

30

30

35

35

35

Wave Period/(s)

1.2

1.4

1.6

1

Velocity/(cm s )

1.4

1.6

1.8

0

0

0

-200

-200

-400

-800

-400

-400

-600

-600

-800 0 200 X(mm)

400

600

-800 -400 -200

0 200 X(mm)

400

600

-400 -200

t=5T/8

t=4T/8

0

0

0

Z(mm)

Z(mm)

200

-600

-200 -400 -600

-800 0 200 X(mm)

400

600

400

600

-200 -400 -600

-800 -400 -200

200

t=6T/8

200

-400

0 X

200

-200

2

Z

200

-200

1.8

t=3T/8

200

-400 -200

1.6 12

200

Z(mm)

Z(mm)

1.4

t=2T/8

-600

Z(mm)

1.8

8

t=T/8

-800 -400 -200

t=7T/8

0 200 X(mm)

400

600

400

600

-400 -200

0 200 X(mm)

400

600

t=T

200

200

0

0 Z(mm)

Z(mm)

1.6

-200 -400

-200 -400 -600

-600

-800

-800 -400 -200

0

200 400 X(mm)

600

800

-400 -200

0 200 X(mm)

Fig. 8. An example of simulation. The dynamic motion of gravity cage at different time instances.

mass coefficients in their corresponding directions. For simplicity, in this paper the hydrodynamic coefficients of the mini-segments of the float collar are taken as

constants: The drag coefficients used are CDt ¼ 0.1, CDn ¼ CDv ¼ 0.6, and the added mass coefficients used are Cmn ¼ Cmv ¼ 0.2, and Cmt ¼ 0.0. For the drag

ARTICLE IN PRESS Y.-P. Zhao et al. / Ocean Engineering 34 (2007) 2350–2363

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V=8cm/s

V=12cm/s

3.5

4.0

3.0

3.5 3.0 Forces/N

Forces/N

2.5 2.0 1.5 1.0

Exp.

2.5 2.0 1.5 1.0

Cal.

0.5

Cal.

Exp.

0.5

0.0

0.0 1.2

1.4 1.6

20

1.4 1.6 1.8

1.4 1.6 1.8

25

30

2 T/s

1.6 1.8

35

1.2

1.4 1.6

20

H/cm

1.4

1.6 1.8 1.4 1.6

25

1.8 1.6 1.8

30

2 T/s

35

H/cm

Fig. 9. Comparisons on maximum values of mooring line force of calculated results vs experimental data.

Clockwise in z axis (V=8cm/s)

-20

16

-16 Pitch/deg

12 Pitch/deg.

Counter-clockwise in z axis (V=8cm/s)

8 Cal.

Exp.

4

-12 -8 Exp.

Cal.

-4 0

0 1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8

20

25

1.6 1.8

30

2 T/s

1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8

H/cm

35

1.6 1.8

2

20 25 30 35 Counter-clockwise in z axis (V=12cm/s)

Clockwise in z axis (V=12cm/s)

T/s H/cm

-20

16 Pitch/deg

Pitch/deg.

-16 12 8 Exp.

4

-12 -8 Exp.

Cal.

Cal.

-4 0

0 1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8 1.6 1.8

20

25

30

35

2

T/s H/cm

1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8

20

25

30

1.6 1.8

35

2

T/s H/cm

Fig. 10. Comparisons of maximum values of float collar pitch of calculated results vs experimental data.

coefficients, the values in this paper are lessthan those commonly used in waves-only analysis, and they are in accord with other people’s investigations of the drag coefficients in wave–current conditions, as discussed in Section 2.2.2. An, At, and Av are effective projected areas in the corresponding directions. an, at, and av represent, respectively, the acceleration vectors of water particles of the n, s and v components. Un, Ut, and Uv represent, respectively, the acceleration vectors of the mini-segment of the n, s and v components. Other parameters are the same as previously described. As to projected areas An, At, and Av, the values for the outer and inner pipes are different and have a relationship with the phase angle of

mini-segment b, the wave-surface elevation, and the space between the outer and inner pipes. But here it can be simplified to a model of a single pipe, because the space between the outer and inner pipes is much lessthan the scale of the float collar and the wavelength. Fig. 5 is the sketch of a simplified pipe model for calculation. According to Fig. 5, it is assumed that (xi, yi, zi) is the central coordinate of the mini-segment. The depth underwater of the mini-segment in the v direction is written as d n ¼ r  ðzi  ZÞ cos g,

(19)

where dn is the depth underwater of the mini-segment in the v direction, Z is the wave–surface elevation that can be

ARTICLE IN PRESS Y.-P. Zhao et al. / Ocean Engineering 34 (2007) 2350–2363

Lee side (point 2,V=8cm/s)

20 15 10 Exp.

0 1.2 1.4 1.6

20 Horizonal displacements/cm

Cal.

1.4 1.6 1.8 1.4 1.6 1.8

25 20 15 10 Exp.

Cal.

0 20

25

15 10 Exp.

5

35

Cal.

0 1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8

20

1.6 1.8

25 30 Lee side (point 2,V=12cm/s)

2 T/s

35

H/cm

25 20 15 10 Exp.

5

Cal.

0

2 T/s

1.6 1.8

30

20

H/cm

30

1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8

25

2 T/s

1.6 1.8

25 30 35 Wave side (point 1,V=12cm/s)

5

Horizonal displacements/cm

25

Horizonal displacements/cm

Horizonal displacements/cm

Wave side (point 1,V=8cm/s) 30

5

2357

1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8 1.6 1.8

20

H/cm

25

30

2

35

T/s H/cm

Fig. 11. Comparisons of maximum values of horizontal displacements of calculated results vs experimental data.

Lee side( point 2,V=8cm/s)

15 10 5

Exp.

Cal.

0 1.2

1.4 1.6 1.4

1.6 1.8 1.4 1.6

20

25

1.8

1.6

30

1.8

2

35

Vertical displacements/cm

Vertical displacements/cm

Wave side( point 1,V=8cm/s) 20

20 15 10 5

Exp.

0

T/s H/cm

1.2

1.4 1.6 1.4

20

15 10 5

Exp.

Cal.

0 1.2

1.4 1.6 1.4

1.6 1.8 1.4 1.6

20

25

30

1.8

1.6

1.8

35

2

T/s H/cm

Vertical displacements/cm

Vertical displacements/cm

Wave side( point 1,V=12cm/s) 20

Cal.

1.6 1.8 1.4 1.6

1.8

1.6

25 30 Lee side( point 2,V=12cm/s)

1.8

2 T/s

35

H/cm

25 20 15 10 Exp.

5

Cal.

0 1.2

1.4 1.6

20

1.4

1.6 1.8 1.4 1.6

25

30

1.8

1.6

1.8

2 T/s

35

H/cm

Fig. 12. Comparisons of maximum values of vertical displacements of calculated results vs experimental data.

calculated with Eq. (4), r is the radius of the pipe, and g is the inclination angle of the wave profile at point ðxi ; yi ; zi Þ. The projected chord length is given by

Then, the cross-sectional area in water r2 r2 Si ¼ fi  sin fi , 2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 d v ¼ 2  r  ðr  d n Þ ¼ 2  r2  ðzi  ZÞ2 cos2 g.

where fi is the corresponding central angle of the projected chord length, which is calculated by dn (22) cosðfi =2Þ ¼ 1  . r

(20)

(21)

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The projected areas in different directions are given by An ¼ d n l i ¼ rl i  ðzi  ZÞl i cos g;   At ¼ p1 rfi l i ¼ p2 rl i arccos ðzi ZÞr cos g , (23) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > : Av ¼ d v l i ¼ 2l i r2  ðzi  ZÞ2 cos2 g; 8 > > > <

where li is the length of the mini-segment. For the projected area in the tangential direction At, there are several methods of calculation available. Some take the crosssectional area as At. This method is often used where the length of the cylinder is limited and short. Others take the product of the cylinder diameter and length as the valid projected area in the tangential direction. This method is introduced where the length of the cylinder is relatively long, and the effect of the cross-sectional area on the tangential forces can be neglected. In this paper, we use a method different from the methods introduced above because the mini-segment is floating at the water surface, and the tangential forces play an important role in the behavior of the floating system. Here, At has a relationship with the arc area in water of the mini-segment (rfili). When the mini-segment is fully submerged underwater, the later 0.4

V1=8cm/s

0.3

V2=12cm/s

Relative error [-]

0.2 0.1

method introduced above applies. The arc area in water of the mini-segment should be divided by a constant p, as shown in Eq. (23). As for the normal projected area Av, when dnXr, let dv ¼ 2r. Using the same method as in the net model, the transformation matrix [C] for each mini-segment can be obtained. By matrix [C], the wave–current-induced forces in the ntv coordinate system (see Eq. (18)) can be transformed into the xyz coordinate system: ðF x ; F y ; F z ÞT ¼ ½C ðF n ; F t ; F v ÞT .

In addition to the wave–current-induced forces, the float collar is also subjected to the gravity, buoyancy, and mooring-line forces. The gravity of the mini-segment can be written as Gi ¼ G=N,

(25)

where G is the total gravity of the floating system, and N is the number of the mini-segments. According to Fig. 5, the buoyancy acting on a mini-segment can be calculated by r2 ðf  sin fi Þl i . (26) 2 i As for the mooring-line forces, they can be determined by the displacements of the floating system. The relationship between the mooring-line forces and elongation can be obtained directly by experimental measurement, as shown in

F f i ¼ rgS i l i ¼ rg

T ¼ 343:83  ðDS=SÞ2 þ 79:14  DS=S,

0.0

(24)

(27)

where DS is the elongation of a mooring line (m), S is the original length of the mooring line (m), and T is the tension in the mooring line (N).

-0.1 -0.2 -0.3 -0.4 1

2

3

5

4

7 8 6 Wave No.

9

10

11

12

Fig. 13. Mooring line force relative error, when V ¼ 8 and 12 cm/s combined with different wave conditions. The relative error is given by R ¼ (FsFm)/Fm, where Fs and Fm are mooring line forces from the simulations and measurements, respectively.

2.3.2. Motion equation of the float collar To obtain the motion of any mass point in the collar, the motion of the centroid must be known first, based on the theory of rigid-body kinematics. The three-dimensional motions of the float collar include surge–sway–heave translation and roll–pitch–yaw rotation. In this section, two coordinate systems are adopted, which are the fixedcoordinate system, Oxyz, and the body-coordinate system,

Clockwise in z axis 0.4 0.2

Relative error [-]

Relative error [-]

0.3

Counter-clockwise in z axis

V1=8cm/s V2=12cm/s

0.1 0.0 -0.1 -0.2 -0.3 -0.4 1 2 3 4 5 6 7 8 9 10 11 12 Wave No.

0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4

V1=8cm/s V2=12cm/s

1 2 3 4 5 6 7 8 9 10 11 12 Wave No.

Fig. 14. Relative error of float collar pitch, when V ¼ 8 and 12 cm/s combined with different wave conditions. The relative error is given by R ¼ (PsPm)/ Pm, where Ps and Pm are float collar pitch from the simulations and measurements, respectively.

ARTICLE IN PRESS Y.-P. Zhao et al. / Ocean Engineering 34 (2007) 2350–2363

G123. The body-coordinate system, G123, is rigidly attached to the float collar, and the coordinate axes 1, 2, 3 are principal axes whose origin is at the center of the mass G. At the initial moment, axes x, y, and z are parallel to axes 1, 2, and 3, respectively. * In Fig. 6, p is the arbitrary point in the rigid body, r is the position vector of point p with respect to the body* coordinate system, G123, r G is the position vector of centroid G with respect to the fixed-coordinate system, * Oxyz. The position vector r p of the point p with respect to the fixed-coordinate system, Oxyz, is then given by * rp

*

*

¼ rG þ r .

(28)

The motion of the rigid body includes two parts: translation and rotation. According to the theory of rigid-body kinematics, the absolute velocity and acceleration of point p can be expressed by: * vp * ap

*

*

*

* aG

*

*

¼ vG þ o r , ¼

*

(29) *

*

(30)

*

where v G and a G are the velocity and acceleration of * * centroid G, respectively; o and a are the angular velocity and acceleration of the rigid body under the fixed coordinates, respectively. Six degrees of freedom are required to describe the motion of the float collar in general spatial motion, resulting in six equations of motion. Three equations correspond with the translation of the mass center, and three equations for the rotation about the mass center. According to Newton’s second law, under fixed coordinates the three translational equations of motion are cos f2 cos f3 6  cos f sin f ½R ¼ 4 2 3 sin f2

cos f1 sin f3 þ sin f1 sin f2 cos f3 cos f1 cos f3  sin f1 sin f2 sin f3  sin f1 cos f2

n 1 X Fx , mG i¼1 i

y€ G ¼

n 1 X Fy , mG i¼1 i

z€G ¼

n 1 X Fz ; mG i¼1 i

_ 1 þ ðI 3  I 2 Þo2 o3 ¼ I 1o _ 2 þ ðI 1  I 3 Þo3 o1 ¼ I 2o _ 3 þ ðI 2  I 1 Þo1 o2 ¼ I 3o

n X i¼1 n X i¼1 n X

M 1i , M 2i , M 3i ,

ð32Þ

i¼1

where subscripts 1, 2, 3 represent the body-coordinate axes 1, 2, 3; o1, o2 and o3 are the components of angular * velocity vector o along the principal axes; M 1i , M 2i , M 3i are the components of the moment vector Mi (i ¼ 1,n) along the principal axes; n is the number of moment vectors; and I1, I2 and I3 are the principal moments of inertia, and can be expressed by: (33)

where m and R are the mass and radius of the float collar, respectively. Although six equations of motion have been set up, it is necessary to know the transformation relationship between fixed coordinates and body coordinates before solving the equations. Bryant angles f1, f2, f3 are applied in this section. Many books on rigid-body kinematics are available, for instance, deal with Bryant angles, Wittenburg (1977). Here, only the conclusions that are used here are introduced. If Bryant angles f1, f2, f3 are obtained, the transformation matrix [R] between fixed coordinates and body coordinates is given by 3 sin f1 sin f3  cos f1 sin f2 cos f3 sin f1 cos f3 þ cos f1 sin f2 sin f3 7 5, cos f1 cos f2

(34)

where [R] is the transformation matrix, and [R]1 ¼ [R]T. The relationship between fixed coordinates and body coordinates is given by 2 3 2 3 1 x 6 7 6 7 (35) 4 2 5 ¼ ½R4 y 5. 3 z

given by x€ G ¼

body (Bhatt and Dukkipati, 2001) are applied. In the bodycoordinate system, the three rotational equations of motion are given by

I 1 ¼ 12mR2 ; I 2 ¼ 12mR2 ; I 3 ¼ I 1 þ I 2 ¼ mR2 ,

*

þ a  r þ o ðo  r Þ,

2

2359

ð31Þ

where F xi ; F yi ; F zi ; are the components of the moment vector Fi (i ¼ 1,n) along fixed-coordinate axes xyz, n is the number of external forces, and mG is the mass of the rigid body. Axes 1, 2, 3 are principal axes originating at the center of mass G, and then Euler’s equations of motion of a rigid

The kinematics differential equations for Bryant angles are given by _ cos f cos f þ f _ sin f , o1 ¼ f 1 3 2 2 3 _ cos f , o2 ¼  f_ 1 sin f3 cos f2 þ f 2 3 _ sin f þ f _ . o3 ¼ f 1

2

3

ð36Þ

The motion of the float collar can be described by three translational displacements of the centroid and three

ARTICLE IN PRESS Y.-P. Zhao et al. / Ocean Engineering 34 (2007) 2350–2363

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Bryant angles. By solving simultaneously ordinary differential Equations (31), (32), and (36), three translational displacements of the centroid and three Bryant angles can be obtained. The equations are solved by using the Runge–Kutta–Verner fifth order and sixth order methods in this paper. By using Eqs. (28)–(30), the displacements and velocities of the arbitrary point in the float collar can be calculated. During the process of calculation, the transformation between fixed coordinates and body coordinates is carried out by the matrix [R]. In our paper, it should be noted that the forces and motions of the collar are dependent on the net by the mutual mass points that are attached to both the net and the collar.

and Offshore Engineering, Dalian University of Technology, China. The wave-current flume is 69 m long, 2 m wide, and 1.8 m high and is equipped with an irregular wave-maker and current-producing system. The mooringline forces were measured by two transducers attached to the bottom of the mooring lines. Two diodes (front and back), numbered 1 and 2, were fixed on the floating system for motion analysis. The movement of diodes was recorded by a CCD camera. The model setting is shown in Fig. 7. Detailed parameters of the float-collar system are given in Table 1. The cage net is made up of PE with a mass density of 953 kg/m3. The total number of meshes in the circumferential direction is 207, and in the depth

3. Numerical simulation and experimental data

3.5 3.0

In this part, numerical simulation has been carried out to investigate the hydrodynamic behavior of the gravity cage in combined wave-current flow. To examine the validity of the numerical model and results, a series of experiments were conducted and compared with the simulated results.

Force/N

2.5 2.0 1.5 1.0 V=12cm/s

V=8cm/s

0.5 0.0 1.2

3.1. Description of physical model tests For this paper the experiments were conducted in a wave-current flume at the State Key Laboratory of Coastal

0.4

V1=8cm/s V2=12cm/s

0.2 0.1 0.0 -0.1 -0.2

1.4

1.6

1.8

1.4

25

1.6

30

1.8

1.6

1.8

35

2

T/s H/cm

Lee side (point 2)

0.3 Relative error [-]

Relative error [-]

0.3

1.6

Fig. 17. The comparison of simulated results of mooring line force maximum with different current velocities combined with waves.

Wave side (point 1) 0.4

1.4

20

V1=8cm/s V2=12cm/s

0.2 0.1 0.0 -0.1 -0.2 -0.3

-0.3

-0.4

-0.4

1 2 3 4 5 6 7 8 9 10 11 12 Wave No.

1 2 3 4 5 6 7 8 9 10 11 12 Wave No.

Fig. 15. Relative error of horizontal displacements, when V ¼ 8 and 12 cm/s combined with different wave conditions. The relative error is given by R ¼ (HsHm)/Hm, where Hs and Hm are maximum values of horizontal displacements from the simulations and measurements, respectively.

Leeside (point 2) 0.4 0.3

V1=8cm/s V2=12cm/s

Relative error [-]

Relative error [-]

Wave side (point 1) 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4

V1=8cm/s V2=12cm/s

0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4

1 2 3 4 5 6 7 8 9 10 11 12 Wave No.

1 2 3 4 5 6 7 8 9 10 11 12 Wave No.

Fig. 16. Relative error of vertical displacements, when V ¼ 8 and 12 cm/s combined with different wave conditions. The relative error is given by R ¼ (VsVm)/Vm, where Vs and Vm are maximum values of vertical displacements from the simulations and measurements, respectively.

ARTICLE IN PRESS Y.-P. Zhao et al. / Ocean Engineering 34 (2007) 2350–2363

direction, 27. The netting is knotless, with a mesh size of 20 mm and a twine thickness of 1.18 mm. Mounted as diamond meshes, the net then forms an open vertical cylinder with a diameter of 0.796 m and a height of 0.43 m. The full-scale diameter of the net cylinder is assumed to be 16 m, and the scale thus is 1:20. The weight system of the gravity cage is composed of two parts, a bottom collar and sinkers. The bottom collar is of steel, with a mass of 25 g in water, which is attached to the bottom of the net cylinder by a thin line with a length of 5 cm. The number of sinkers is 10, and the mass of each sinker is 3.75 g in water. The shape of the sinker is spherical, with diameter 3 cm. According to gravity simulation criteria, the total size of the mass of the weight system, corresponding with the full-scale value, is 610 kg. The regular waves propagated along the positive x-direction. Two kinds of current velocities are set: V1 ¼ 8 cm/s and V2 ¼ 12 cm/s in the direction of wave propagation. The details of the wave-current conditions are shown in Table 2. 3.2. Simulated results and experimental verification Fig. 8 shows an example of the simulation of the gravity cage in combined wave–current flow. The cage is subjected to regular waves (1.6 s period and 0.25 m height) and current (8 cm/s) running in the positive x-direction. In the z axis, coordinate z ¼ 0 denotes a still water surface. Fig. 8

shows the dynamic motion of the gravity cage at different time periods. The time interval between each time period shown in Fig. 8 is approximately 0.125 times the wave period. The increased deflection of the gravity-cage net can be seen, together with dynamic loading from the waves. To verify the numerical model, quantitative comparisons were carried out between simulated and experimental results. Maximum values of mooring-line forces, the pitch of the float collar, and horizontal and vertical displacements of the float collar were selected for comparisons. Fig. 9 shows the comparisons of the maximum values of the mooring-line force between calculated and experimental results. Fig. 10 demonstrates comparisons of the pitch of the float collar between calculated and experimental results, which include two orientations: clockwise and counterclockwise in the z axis. Figs. 11 and 12 show the comparisons of the maximum values of horizontal and vertical displacements of tracing points (1 and 2) between simulated and experimental results. As shown in Figs. 9–12, the simulated results of motions and mooring-line forces of the gravity cage are all in close agreement with the experimental results. The relative errors are all in the range of 70.2 (see Figs. 13–16). The discrepancies in simulated and experimental results may come from three main sources that are difficult to avoid. First, the influence of the gravity cage on the combined wave–current field is neglected in our model, because it is difficult to examine in physical experiments and to consider in numerical modeling. Next, in this

Counter-clockwise in z axis

Clockwise in z axis

-20.00

16

-16.00 Pitch/deg

12 Pitch/deg.

2361

8 V=8cm/s

4

-12.00 -8.00 V=8cm/s.

V=12cm/s

V=12cm/s

-4.00 0.00

0 1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8 1.6 1.8

20

25

30

2

35

T/s H/cm

1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8 1.6 1.8

20

25

30

2

35

T/s H/cm

Fig. 18. The comparison of simulated results of float collar pitch maximum with different current velocities combined with waves.

Lee side (point 2 ) Horizontal displacement/cm

Horizontal displacement/cm

Wave side (point 1) 30.00 25.00 20.00 15.00 10.00 5.00

V=8cm/s

V=12cm/s

0.00 1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8 1.6 1.8 2

20

25

30

35

T /s H/cm

25.00 20.00 15.00 10.00 5.00

V=8cm/s

V=12cm/s

0.00 1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8 1.6 1.8 2

20

25

30

35

T /s H/cm

Fig. 19. The comparison of simulated results of horizontal displacement maximum with different current velocities combined with waves.

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Lee side (point 2 ) Vertical displacement/cm

Vertical displacement/cm

Wave side (point 1) 20 15 10 5

V=12cm/s.

V=8cm/s

0 1.2 1.4 1.6 1.4 1.6 1.8 1.4 1.6 1.8 1.6 1.8

20

25

30

35

2

T/s H/cm

20 15 10 5

V=8cm/s

V=12cm/s.

0 1.2 1.4 1.6

1.4 1.6 1.8

1.4 1.6 1.8

20

25

30

1.6 1.8

35

2

T/s H/cm

Fig. 20. The comparison of simulated results of vertical displacement maximum with different current velocities combined with waves.

paper the float collar is assumed to be a rigid body to simplify the motion equations, whereas a little flexibility exists in practical physical modeling tests. In addition to these discrepancies, experimental and calculated errors are unavoidable, although a great effort has been made to reduce them. According to our experimental and simulated results, it can also be found that when current velocity increases, the mooring-line force, clockwise pitch of the float collar, and horizontal displacements will become larger in combined wave–current flow (see Figs. 17, 18a, 19). The variation in current velocity has little effect on the counterclockwise pitch of the float collar and vertical displacements (see Figs. 18b, 20). 4. Conclusions This paper presents a numerical method for simulating the hydrodynamic behavior of the gravity cage in combined wave–current flow. The simulated results show good agreement with physical modeling tests under different current velocities and wave conditions. This numerical model thus has established a foundation for simulating practical net-cage systems in wave–current conditions. The potential for improvements still exists in our numerical model proposed here. For example, further study should be undertaken to investigate the influence of the net cage on the wave–current field. The determination of the hydrodynamic coefficients of floating pipe in wave–current flow is also an important field for further numerical and experimental work. Field investigations are suggested for interested groups, and such studies should result in valuable information for the cage aquaculture industry when moving outward to the turbulent open sea. Acknowledgments This study is financially supported by National 863 High Technology Project no.2006AA100301 and Program for

Changjiang Scholars and Innovative Research Team in University (PCSIRT) no. IRT0420. References Aarsnes, J.V., Rudi, H., Løland, G., 1990. Current forces on cage, net defection. In: Engineering for offshore fish farming. Thomas Telford, London, pp. 137–152. Bhatt, R.B., Dukkipati, R.V., 2001. Advanced Dynamics. Alpha Science International, Ltd., UK, pp. 213–219. Brebbia, C.A., Walker, S., 1979. Dynamic Analysis of Offshore Structures. Newnes-Butterworths, pp. 109–143. Choo, Y.I., Casarella, M.J., 1971. Hydrodynamic resistance of towed cables. Journal of Hydronautics, 126–131. Colbourne, D.B., Allen, J.H., 2001. Observations on motions and loads in aquaculture cages from full scale and model measurements. Aquaculture Engineering 24 (2), 129–148. Fredriksson, D.W., Swift, M.R., Irish, J.D., Tsukrov, I., Celikkol, B., 2003. Fish cage and mooring system dynamics using physical numerical models with field measurements. Aquaculture Engineering 27 (2), 217–270. Gerhard, K., 1983. Fibre Ropes for Fishing Gear. FAO Fishing Manuals, Fishing News Books Ltd., Farnham, UK, pp. 81–124. Gudmestad, O.T., Karunakaran, D., 1990. Wave–current interaction. In: Environmental Forces on Offshore Structure and their Pred. Kluwer, Netherlands, pp. 81–95. Gui, F.K., Li, Y.C., Zhang, H.H., 2002. The proportional criteria for model testing of force acting on fishing cage net. China Offshore Platform 17 (5), 22–25 (in Chinese). Hedges, T.S., Lee, B.W., 1992. The equivalent uniform current in wave–current computations. Coastal Engineering 16, 301–311. Li, Y.C., 1990. Wave Action on Maritime Structures. Dalian University of Technology Press, China, pp. 274–312. Li, Y.C., Zhao, Y.P., Gui, F.K., Teng, B., 2006a. Numerical simulation of the hydrodynamic behavior of submerged plane nets in current. Ocean Engineering 33 (17–18), 2352–2368. Li, Y.C., Zhao, Y.P., Gui, F.K., Teng, B., 2006b. Numerical simulation of the influences of sinker weight on the deformation and load of net of gravity sea cage in uniform flow. Acta Oceanologica Sinica 25 (3), 1–18. Sarpkaya, T., Bakmis, C., Storm, M.A., 1984. Hydrodynamic forces from combined wave and current flow on smooth and rough circular cylinder at high Reynolds number. In: Proceedings of 16th Offshore Technology Conference, Houston, pp. 455–462. Takashi, S., Tsutomu, T., Katsuya, S., Tomonori, H., Katsutaro, Y., 2004. Refined calculation model for NaLA, a fishing net shape simulator, applicable to gill nets. Fisheries Science 70, 401–411. Tsukrov, I., Eroshkin, O., Fredriksson, D., Robinson, S.M., Celikkol, B., 2003. Finite element modeling of net panels using a consistent net element. Ocean Engineering 30, 251–270.

ARTICLE IN PRESS Y.-P. Zhao et al. / Ocean Engineering 34 (2007) 2350–2363 Wang, T., Li, J.C., 1999. On wave–current interaction. Advances in Mechanics 29 (3), 331–343 (in Chinese). Wang, T., Li, J.C., Huhe, H.D., Huang, Z.H., 1995. Effect of wave–current interaction on hydrodynamic coefficients. Journal of Hydrodynamics, Series A 10 (5), 551–559 (in Chinese). Wilson, B.W., 1967. Elastic characteristics of moorings. ASCE Journal of the Waterways and Harbors Division 93 (WW4), 27–56.

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Wittenburg, J., 1977. Dynamics of Systems of Rigid Bodies. Stuttgart, Teubner. Zhao, Y.P., Li, Y.C., Dong, G.H., Gui, F.K., 2007. The selection of the wave theory selection in the simulation of gravity cage. (Proceeding of ISOPE-2007, to be published.) Zheng, Y.N., Dong, G.H., Gui, F.K., Li, Y.C., 2006. Movement response of floating circle collars of gravity cages subjected to waves. Engineering Mechanics 23 (Supp. I), 222–228 (in Chinese).