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Abstract—To mitigate marine pollution intensity at the sea bottom, an automatic rotating type of cage systems such as a single-point-mooring (SPM) cage system ...

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 35, NO. 3, JULY 2010

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Numerical Modeling for an In Situ Single-PointMooring Cage System Chai-Cheng Huang, Hung-Jie Tang, and Bo-Seng Wang

Abstract—To mitigate marine pollution intensity at the sea bottom, an automatic rotating type of cage systems such as a single-point-mooring (SPM) cage system is often regarded as biofriendly equipment for fish farming in the open sea due to spreading uneaten waste feed and fish feces into a vast area. Though the SPM cage dynamic features under regular sea state have been investigated in previous researches, the in situ sea state is by no means a regular one, thus a further exploration of the dynamic response in the random sea is critical before deploying cages into the open sea. This work developed a numerical model for irregular sea states to simulate an SPM cage system in an unsheltered open sea, considering the environmental conditions as irregular waves combined with a steady uniform current. To validate the numerical model, a full-scale physical model was tested in the field, where both sea states and mooring line tension were recorded. Results indicate that the numerical model predictions have good agreement with field measurements in both time and frequency domains, while the net-volume deformation is presented numerically to show fish net space variation in a random sea. Index Terms—Cage aquaculture, irregular waves, linear transfer function, mooring line tension, single point mooring (SPM).

I. INTRODUCTION

C

AGE aquaculture in the open sea has become an important fishery industry for many maritime countries. Environmentalists have long criticized deteriorated marine environment around the cage sites because of the accumulated waste on the seabed. Thus, the study of single-point-mooring (SPM) marine cage systems has received much attention in the past decade due to their ecofriendly features with respect to the surrounding environment. One of the advantages of SPM cage systems is that an SPM system disposes biowastes over a relatively large area compared to that of a multiple-point-mooring (MPM) system. In addition, the bow of the system points in the direction of the

Manuscript received October 29, 2009; revised March 14, 2010; accepted April 24, 2010. Date of publication August 09, 2010; date of current version September 01, 2010. This work was supported by the National Science Council under Grant NSC98–2221-E-110–087 and the Fisheries Agency, Council of Agriculture of Taiwan. Associate Editor: H. Maeda. C.-C. Huang is with the Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan (e-mail: [email protected]). H.-J. Tang is with the Tainan Hydraulic Laboratory, National Cheng Kung University, Tainan 709, Taiwan (e-mail: [email protected]). B.-S. Wang is with the Disaster Prevention Research Center, National Cheng Kung University, Tainan 709, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JOE.2010.2050351

least resistance so that it will automatically adjust the cage structure to an optimal position. Under severe weather conditions or strong current attacks, the cage may be activated to submerge below the sea surface to maintain its structure’s integrity. Fish farmers are typically concerned with two issues related to the design of a marine net-cage system: maximum tension on the mooring lines and volume reduction rate of fish nets. This study carefully examined the first issue, line tension, through numerical simulation as well as through field observation; the second issue, cage net-volume reduction, is estimated by the numerical model only. Previous studies have addressed engineering analysis of the marine cage system to a certain extent. Colbourne and Allen [1] conducted a field investigation on the mooring line tension of a full-scale cage system at a farming site. Their results indicated that the probability distribution of mooring line tension was approximated by a Rayleigh distribution, and the environmental current forces have higher influence on mooring tensions than the wave forces during the testing period. Tsukrov et al. [2] applied a finite element method with consistent net element concepts to modeling the dynamic behavior of the tension leg of a fish cage. Fredriksson et al. [3], [4] conducted a series of studies on a field cage system by developing a numerical model validated with field measurements. Their studies analyzed responses of the in situ cage dynamic motions and anchor line tensions to field external forces by a stochastic approach. Following the previous work [5], this study develops a numerical model to evaluate hydrodynamic forces on the SPM cage system under the impact of in situ random waves and currents. This study upgrades the numerical model based on a lumpedmass method [6], [7] to cope with inevitable irregular waves and currents in offshore fish farming sites. This numerical model is validated with field measurements at a farming site, while the net-volume deformation is estimated by the numerical model. The current study is organized as follows. Section II presents the mathematical background for a velocity potential function and a brief description of environmental loadings. Section III describes the in situ physical model and field observations. Section IV presents comparison results of numerical simulations and field measurements. Finally, Section V summarizes conclusions and offers suggestions for further studies. II. DESCRIPTIONS OF THE NUMERICAL MODEL A. Irregular Waves Combined With Uniform Current The current study considered the simulated domain as a 3-D space with a uniform seafloor, and adopted a superposition principle of linear velocity potential functions to represent a real

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flow field including a steady uniform current and an irregular progressive wave, given as

the elements of a net-cage system based on the lumped mass method, referring to [6]–[10]. B. A Brief Description of Environmental Loadings

(1) where and are the components of a uniform current in is a position cothe - and -directions, respectively; is the total number of wave components; is ordinate; the th wave amplitude; is the th wave angular frequency; is also referred to as the th is the th wave number; apparent wave angular frequency; is the component of a wave number in the -direction; is the component of a wave number in the -direction, where is an apparent angle, defined as the angle between the irregular wave direction and the current diis the th wave phase angle (between 0 and 2 ); rection; is gravity; is time; and is the water depth. The water surface elevation is written as (2)

In this study, a full-scale SPM cage system was deployed at a fish farming site, subjecting it to environmental loadings such as wave, current, and wind forces. Dynamic computation often neglects wind forces on the floating collar because not only is a small portion of floating collar exposed to air, but also the whole cage system may automatically submerge in water during a heavy storm. From our previous experience, mooring line tensions mainly result from external forces on the cylindrical type of net cage. To simulate hydrodynamic forces on an SPM cage system, the whole cage system is divided into many plane portions and line segments, referred to as “elements” based on the lumped-mass method, for calculating external forces on each element, and then evenly distributed to the element nodes. To simplify numerical simulation, this study considered these elements as a small body compared to the characteristic wavelength. This means that scattering effects on the flow field due to the interactions of fluid particles and the elements are negligible. Thus, it is appropriate to apply the modified Morison equation with relative motion [11] to the cage elements given as

The correspondent dispersion relation for the wave/current coexisted field is given as

(6)

(3)

where is the mass of the element; is the density of seais the added mass coefficient; is the drag cowater; is the inertia coefficient; is the efficient; projected area; is the water displaced volume of the element; is the fluid particle velocity at the element center; is the central velocity of the element; is the acceleration is the relative velocity. of the element; and The first term on the right-hand side in (6) is often regarded , whereas the second term is called the as the drag force

By taking the derivative of (1) with respect to , we obtain a 3-D velocity field, where velocity is defined as [see equation (4) at the bottom of the page]. Similarly, by taking the derivative of (4) with respect to , the local accelerations for 3-D field are obtained as shown in (5) at the bottom of the page. This work applied both velocity and acceleration fields to the Morison equation to obtain the hydrodynamic forces acting on

(4)

(5)

HUANG et al.: NUMERICAL MODELING FOR AN In Situ SINGLE-POINT-MOORING CAGE SYSTEM

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TABLE I DATA OF MOORING SYSTEM

Fig. 1. Schematic diagram for an SPM net-cage system.

inertia force . These forces are induced from a moving object subjected to marine environmental loadings. When dealing with a particular element, other external forces such as buoyant force, body weight, and tension forces deduced from contact with neighboring elements, should be also included in the calculation. Thus, from Newton’s second law, (6) may be expanded and expressed as (7) where is the buoyant force; is the gravitational force; is the tension force. Huang et al. [6], [7] described the and details of external force calculation of cage structural components (including net plane, mooring line, buoy, floating collar, and tube sinker). C. Motion Equations An SPM cage system comprises flexible parts (such as mooring lines and fish nets) and relatively rigid parts (such as the floating collar and the tube sinker), regarded as nondeformed rigid bodies. The motion equations for the cage system are also divided into two different procedures: one deals with the flexible elements, computing each element hydrodynamic force first, then evenly distributing forces to the common nodes referred to as the lumped-mass center; the other deals with rigid body parts including many elements and nodes, but having only one total mass center. Therefore, instead of dealing with each individual node’s motion equation, this work utilized a system of rigid body motion equations including three translations (surge, heave, and sway) and three rotations (roll, pitch, and yaw) for the rigid parts of a net-cage system. Huang et al. [6], [7] also described the details of the motion equations for flexible and rigid parts. III. FIELD OBSERVATIONS This study performed a field observation of an SPM cage system at Hsiao-Liu-Chiu Island, Taiwan. Fig. 1 shows the cage system settings. A 15 000-kg concrete block was used as an anchor where the mean water depth was 27 m. The total length of the mooring line was 60 m not including the chain’s length of 1.2 m connected to the anchor. A small buoy between the chain and the mooring line lifted the mooring line slightly higher to avoid abrasion with the sea bottom. The other end of the

mooring line was tied to a connection ring fastened with two bridles and one buoy line. The buoy served two functions: to indicate the mooring line’s position and to serve as a buffer for environmental impact forces. To measure mooring line tension, a load cell was deployed 5 m below the connection ring. The load cell made by KYOWA (Japan) measured tension up to 100 kN with accuracy 0.1%. Since the load cell mass of 56 kg was quite heavy compared with the mooring line, it affected the motion response of the mooring system, thus numerical simulations must account for its existence and treat it as an extra lumped mass. Fig. 1 illustrates an upgraded acoustic Doppler current profiler (ADCP) that was deployed about 20 m from the anchor (concrete block), measuring in situ wave/current data. Both devices were programmed to sample every 2 h starting at noon on August 25 until September 30, 2005 (local time), for 20 min at a sampling rate of 2 Hz. Tables I and II list the SPM net-cage system’s material data. IV. RESULTS AND DISCUSSIONS Approximately four typhoons pass by Hsiao-Liu-Chiu Island each year from May to October. These typhoons typically have great impact on marine aquaculture facilities. Those cages unable to submerge under the water surface may suffer serious damage caused by heavy sea states. Thus, designing a cage structure to sustain the impact of strong waves is an emergent task. Field testing the SPM cage system partially contributes to this task. The experimental period lasted from August 25 to September 30, 2005. In these five weeks of field testing, an intermediate scale typhoon Talim passed through the farming site on September 1, 2005. After examining the measurements, this research selected a set of field data as sample data for examining the present model. Based on these data, the comparisons of numerical simulations and field tension measurements are carried out in Sections IV-A and IV-B.

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TABLE II DATA OF REARING SYSTEM

Fig. 2. Comparison of spectral density functions of JONSWAP and field data.

and Let the lowest and highest cutoff frequency be then the frequency of each wave component is given by

,

(10) A. Field Data 1) Wave Spectrum: An upgraded ADCP with 600-kHz acoustic beams installed at the cage site simultaneously measured current profiles as well as directional waves. The device was deployed at the sea bottom and its four beams faced upward to the water surface. The mean water depth was 26.8 m with a tidal range of 0.7 m. This ADCP device provided three modes (pressure sensor, orbital velocities, and surface echo location) to evaluate water surface elevation, which then converted into wave height spectra to crosscheck the goodness of each mode’s results. This study adopted data from the surface echo location mode, which directly measured the vertical distance from the sea bottom to the water surface by acoustic speed. This research used the Joint North Sea Wave Project (JONSWAP) spectrum of Goda [12] to compare with the field spectrum derived from the local sea state. The JONSWAP spectrum is given as

(8) (9) is the peak enhancement factor; ; is the peak period; is the peak frequency of a wave is set to 0.07 as and 0.09 as . spectrum; and Herein, the significant wave height ( 1.41 m) and wave 8.10 s) were obtained by applying the zero-up period ( crossing method to the field data. For computational efficiency, the present model used the exponential frequency cutoff method into many [13] to decompose the incident wave spectrum component waves with an increasing wave frequency interval. where

where is the number of wave components and is set to 100 for the present model. The wave amplitude and angular frequency of each wave component are written as (11) where is the incident wave spectrum, obtained from the field measurements. Fig. 2 shows the comparison of spectral density function between field measurements and the prediction of the JONSWAP spectrum by Goda [12]. The result shows that these two spectral density functions have similar form, except the peak value of JONSWAP (3.41 m /Hz at 0.115 Hz) is larger than that of the field data (2.78 m /Hz above at 0.120 Hz). This verifies that the JONSWAP spectrum can be used as the engineering design wave spectrum for the net-cage system located in the study area. In other words, once the 50-year-return-period’s significant wave height and wave period are given, we may build a correspondent design random wave spectrum for the preliminary design. Establishing a general wave spectrum is always the first step for planning and designing an offshore cage system. 2) Current Speed Profile: As for the corresponding average current profiles shown in Fig. 3, each current profile was measured at each 5-min interval. The device manufacturer (RD Instruments, Poway, CA) specified that the effective current profile range is 85% of the water depth measured from the top of the device. This means that the effective water column is 26.8 m 85% 22.8 m above the device or below 4 m from the water surface. The current study selected the actual effective water column to be 18 m ( 7 to 25 m) to minimize the influence of strong winds and possible interference of breaking waves on the fluid particle speed near the water surface. Fig. 3 indicates that the current profiles look much more uniform in

HUANG et al.: NUMERICAL MODELING FOR AN In Situ SINGLE-POINT-MOORING CAGE SYSTEM

Fig. 3. Current profiles measured at local time on September 1, 2005.

Fig. 4. Field tension data measured by load cell on the mooring line.

this range. This work adopted a mean value of 0.357 m/s for the present numerical model. 3) Mooring Line Tension Measurement: The load cell was deployed on the mooring line 5 m below the connection ring (Fig. 1). The tension signal was collected in a data acquisition device tied to the floating collar. Fig. 4 shows the retrieved tension record having 2400 samples in 20 min at a sampling rate of 2 Hz. The maximum value was 6.13 kN, the minimum value was 0.94 kN, and the mean value was 2.50 kN. Fig. 5 reveals that the probability density function of field tension data is much more similar to a normal-distribution function than to Rayleigh-distribution functions. To confirm this statement, a chi-square goodness-of-fit test is applied to test this hypothesis, i.e., whether the field data are closer to a normal distribution. According to the zero-up crossing method, the total wave number is 190 waves in the 20-min recording period, which is approximately equal to

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Fig. 5. Comparison of the probability density functions against field tension data.

6.3 s for each wave. A chi-square test for both distribution hypotheses is listed in Table III, where the observation data from intervals. The degree of 0 to 6 kN are divided into , where freedom for the chi-square test is defined as is the parameters for the particular distribution, for example, a and normal distribution has two parameters such as mean while a Rayleigh distribution has one standard deviation . At the significant level 5% in a chi-square table [14], two criteria values and are utilized in the hypothesis test. Table III indicates that the normal distribution’s chi-square value such that the normal hypothesis could not be rejected. Similarly, the Rayleigh distribution’s chi-square value such that the Rayleigh distribution hypothesis also could not be rejected. However, the normal distribution’s is lower than that of the Rayleigh chi-square value distribution , thus it is appropriate to say that the normal distribution is a better fit than the Rayleigh distribution. If the density function of tension is a normal distribution, then the maximum value (5.05 kN) may be defined as a mean value (2.50 kN) plus three times the standard deviation (0.85 kN), as shown in Fig. 4. This assumption determines the maximum mooring strength for this particular typhoon event, due to the (5.05 kN) reaching the probacumulative probability of bility 0.9987. B. Numerical Results This simulation adopted the in situ wave spectrum shown in Fig. 2 as the input spectrum, and decomposed 100 wave components from 0.01 Hz to 0.5 Hz using the exponential frequency cutoff method described in (10). The uniform current speed was 0.357 m/s with an apparent angle of 83 . The total numbers of nodes and elements were 194 and 224, respectively. The total simulation time was 1200 s with a time increment of 10 s, and the sampling rate of output data was 2 Hz. 1) Comparison in Time Domain: Fig. 6(a) demonstrates the time history of the mooring line tension. The result indicates

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TABLE III CHI-SQUARE GOODNESS-OF-FIT TEST

that the simulation gradually stabilizes after 400 s. In the region from 400 to 1200 s, the mean value is 2.35 kN (field data is 2.50 is 5.63 kN (field data is 5.05 kN). kN), while the value of The discrepancies between the field data and the present simulations are 6% and 11.5%, respectively, attributed to the intrinsic

uncertainties of instruments such as load cell and ADCP, as well as the nonlinear interaction between fluid and structural components. In general, the results of the present numerical model have good agreement with the field data, and the precision of numerical simulation reaches approximately 90%.

HUANG et al.: NUMERICAL MODELING FOR AN In Situ SINGLE-POINT-MOORING CAGE SYSTEM

Fig. 6. Time-series results of numerical simulations: (a) mooring line tension; and (b) volume reduction coefficient.

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the field spectrum in peak value and frequency although the bandwidth is broader than the measurements. Another two differences are identified: one is the response magnitude between 0.01 and 0.08 Hz, and the other between 0.21 and 0.27 Hz, where the simulated results have smaller response than those of the field data. The former difference is usually attributed to slow response of the mooring line to environmental loading which may induce a second resonant peak at low frequency. This phenomenon is often referred to as the mooring effect, while the latter difference may be induced by the resonance of the cage structure itself excited at twice the predominant wave frequency (0.12 Hz). Fig. 7(b) shows the comparison of the cross-spectral density function of wave amplitude and mooring . Both curves have similar trends except that line tension the field data is slightly smaller than the numerical data. Analyzing the linear transfer function is another important way to discuss the mooring line tension response to the wave elevation. Fredriksson et al. [3] obtained the linear transfer funcand of mooring line tension, individually tions from autospectral and cross-spectral techniques, given as (12) where

Fig. 7. Comparison of tension density functions between a numerical prediction and field data.

This model used the so-called volume reduction coefficient to describe the net-volume deformation. It is the ratio of the deformed volume divided by the original volume of the fish net . The original volume in this model was calculated at the very beginning of the simulation, while the deformed volume at each time step was estimated by applying the divergence theorem to the fish net [7]. Fig. 6(b) presents the time history of the volume reduction coefficient. In this case, the minimum volume reduction coefficient is larger than 88%, which means that the net volume only reduces by 12% and still maintains enough space for fish to survive in this particular typhoon sea state. 2) Comparison in Frequency Domain: This section selects the last 1024 data points of the numerical results (see Fig. 6) to analyze the mooring tension in the frequency domain. Fig. 7(a) presents the comparison of the tension autospectral density functions , obtained from field data and numerical simulation. The numerical results show good agreement with

is an autospectral density function of tension and is an autospectral density function of wave amplitude, obtained from Fig. 2; is the cross-spectral density function between wave amplitude and mooring line tension. Fredriksson et al. [3] suggested that the cross-spectral technique should be used to provide better estimations by reducing the signal noise influence. Fig. 8 indicates that the present numerical model has a similar trend to the field data and identifies two significant differences near 0.05 and 0.24 Hz, where the results of the field data are obviously larger than the simulations. These phenomena illustrate that the field tension spectrum has clearly reflected the mooring effect and the resonance of the cage structure, while the numerical model was unable to reveal those features as in situ measurements. Nevertheless, at other frequencies including the predominant wave frequency, both numerical predictions and physical measurements show good agreement. Fig. 8 also reveals that the transfer function (cross-spectral technique) indicates much more of agreement than that of (autospectral technique) in the frequency region from 0.1 to 0.5 Hz. This coincides with the suggestion of Fredriksson et al. [3]. 3) Comparisons Between Regular and Irregular Waves: This section investigates the differences of mooring line tension and volume reduction coefficient between regular and irregular wave cases. The input conditions of regular and irregular waves 1.41 m) and are given as: the significant wave height ( wave period ( 8.10 s); a uniform current speed of 0.375 m/s; and an apparent angle between incident waves and current of 83 . Fig. 9 shows the comparison of mooring line tension and the volume reduction coefficient between regular and irregular waves during the simulation time from 800 to 1200 s. The mean values of mooring line tension are 2.31 kN for regular waves and 2.35 kN for irregular waves, which are surprisingly close.

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V. CONCLUSION AND SUGGESTIONS

Fig. 8. (a) Comparison of linear transfer functions between numerical predic( ); and (b) comparison tions and field data using the autospectral technique ( ). of linear transfer functions using the cross-spectral technique

H f

H f

Fig. 9. Comparisons of (a) mooring line tensions and (b) volume reduction coefficients between regular and irregular waves.

However, the maximum tension (6.0 kN) of irregular waves is about 1.5 times the maximum value of regular waves (4.0 kN). These instantaneously larger tensions of irregular waves in Fig. 9(a) are possibly generated by combining instantaneously higher waves and a uniform current acting on the SPM cage system according to the numerical simulation of random waves in the time domain. These hydrodynamic loadings also lead to the minimum remaining volume space (or volume reduction coefficient) of about 94% for regular waves and 91% for irregular waves, respectively, shown in Fig. 9(b). When it comes to a practical design, we usually adopt a safety factor of 1.5–2.0 for the mooring line breaking strength determined by a 50-year return period significant wave conditions to overcome the cable’s fatigue and abrasion problems [10].

Environmentalists have recently regarded the SPM cage system as an ecofriendly fish farm. This study group installed a full-scale physical model in the field to investigate dynamic features of the net-cage system. A self-established numerical model calculated the mooring line tension and net-volume reduction coefficient of the SPM cage system in regular/irregular waves with a steady uniform current. An intermediate scale typhoon passed through the farming site during the in situ test period, and the load cell as well as the ADCP recorded valuable data which were utilized to verify the numerical model. Findings from the field data show a narrowband of the frequency in the wave spectra during the passage of the typhoon and its shape matches the JONSWAP spectrum. Thus, this work suggests using the JONSWAP spectrum for the engineering design at this particular farming site. According to the chi-square goodness-of-fit test, the probability density function of the tension data is much more similar to the normal distribution than to the Rayleigh distribution. Simulated irregular wave results in the time domain demonstrate that the present model is adequate to predict the mean and maximum tension force induced from the typhoon’s impact, and the fish net-volume deformation still maintains about 90%, not as serious as anticipated, and this large survival net space may be attributed to using bottom sinkers to embrace the fish net. In the frequency domain, both autospectral and cross-spectral density functions show a similar trend for both numerical simulations and field data. The linear transfer function using cross-spectral technique fits better than the autospectral technique, even though findings show two significant differences near the frequencies close to 0.05 and 0.24 Hz. The former difference may be due to slow response of the mooring line’s elastic character, while the latter may be attributed to the resonance of the cage structure itself excited at twice the predominant wave frequency (0.12 Hz). Finally, this research conducted a comparison of mooring tension between regular waves and irregular waves. The results reveal that the maximum mooring tension for the irregular waves is about 1.5 times the maximum value determined by the regular waves in this particular scenario. For practical engineering practice, the cable breaking strength is typically determined by the design sea state such as the 50-year return period wave plus possible maximum current speed, and then multiplied by a safety factor of 1.5–2, to prevent cable fatigue and abrasion problems.

REFERENCES [1] D. B. Colbourne and J. H. Allen, “Observations on motions and loads in aquaculture cages from full scale and model scale measurements,” Aquacultural Eng., vol. 24, pp. 129–148, 2001. [2] I. Tsukrov, O. Eroshkin, D. Fredriksson, M. R. Swift, and B. Celikkol, “Finite element modeling of net panels using a consistent net element,” Ocean Eng., vol. 30, pp. 251–270, 2003. [3] D. W. Fredriksson, M. R. Swift, J. D. Irish, I. Tsukrov, and B. Celikkol, “Fish cage and mooring system dynamics using physical and numerical models with field measurements,” Aquacultural Eng., vol. 27, pp. 117–146, 2003. [4] D. W. Fredriksson, M. R. Swift, O. Eroshkin, I. Tsukrov, J. D. Irish, and B. Celikkol, “Moored fish cage dynamics in waves and currents,” IEEE J. Ocean. Eng., vol. 30, no. 1, pp. 28–36, Jan. 2005.

HUANG et al.: NUMERICAL MODELING FOR AN In Situ SINGLE-POINT-MOORING CAGE SYSTEM

[5] C. C. Huang, H. J. Tang, B. S. Wang, R. Y. Yang, L. A. Kuo, and S. J. Jan, “Numerical simulation and field study of a single-point-mooring marine cage,” in Proc. 16th Int. Offshore Polar Eng. Conf., 2006, vol. 3, pp. 292–296. [6] C. C. Huang, H. J. Tang, and J. Y. Liu, “Dynamical analysis of net cage structures for marine aquaculture: Numerical simulation and model testing,” Aquacultural Eng., vol. 35, pp. 258–270, 2006. [7] C. C. Huang, H. J. Tang, and J. Y. Liu, “Modeling volume deformation in gravity cages having distributed bottom weights or a rigid bottom ring,” Aquacultural Eng., vol. 37, pp. 144–157, 2007. [8] C. C. Huang, H. J. Tang, and J. Y. Liu, “Effects of waves and currents on gravity-type cages in the open sea,” Aquacultural Eng., vol. 38, pp. 105–116, 2008. [9] C. C. Huang, H. J. Tang, and J. Y. Pan, “Numerical modeling of an SPM cage with a frontal rigid frame,” IEEE J. Ocean. Eng., vol. 34, no. 2, pp. 113–122, Apr. 2009. [10] C. C. Huang and J. Y. Pan, “Mooring line fatigue: A risk analysis for an SPM cage system,” Aquacultural Eng., vol. 42, pp. 8–16, 2010. [11] C. A. Brebbia and S. Walker, Dynamic Analysis of Offshore Structures. London, U.K.: Newnes-Butterworths, 1979, pp. 109–143. [12] Y. Goda, “A comparative review on the functional forms of directional wave spectrum,” Coastal Eng. J., vol. 41, no. 1, pp. 1–20, 1999. [13] T. W. Hsu, S. C. Hsiao, S. H. Ou, S. K. Wang, B. D. Yang, and S. E. Chou, “An application of Boussinesq equations to Bragg reflection of irregular waves,” Ocean Eng., vol. 34, pp. 870–883, 2007. [14] E. Kreyszig, Advanced Engineering Mathematics, 6 ed. New York: Wiley, 1988, pp. 1279–1281. Chai-Cheng Huang received the B.S. degree in civil engineering from the National Cheng-Kung University, Tainan, Taiwan, in 1975, the M.E. degree in civil engineering from the National Taiwan University, Taipei, Taiwan, in 1979, and the Ph.D. degree in ocean engineering from Texas A&M University, College Station, in 1988. After working as a Research Associate for the Taiwan Power Company, he joined the Department of Marine Environment and Engineering, National Sun Yat-sen University (NSYSU), Kaohsiung, Taiwan, as a Professor. He also serves as a member of Fisheries Extension Service Committee of NSYSU. His areas of research expertise include aquaculture engineering, numerical wave tank, and ocean renewable energy. Dr. Huang is a member of the International Society of Offshore and Polar Engineers (ISOPE).

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Hung-Jie Tang received the B.S., M.E., and Ph.D. degrees in ocean engineering from the National Sun Yat-sen University (NSYSU), Kaohsiung, Taiwan, in 1999, 2001, and 2008, respectively. Currently, he is an Assistant Research Fellow at the Physical Modeling Division, Tainan Hydraulic Laboratory, National Cheng Kung University, Tainan, Taiwan. His interests are in the areas of open ocean aquaculture engineering, physical modeling, and ocean renewable energy.

Bo-Seng Wang received the B.S. and M.E. degrees in ocean engineering from the National Sun Yat-sen University (NSYSU), Kaohsiung, Taiwan, in 2003 and 2006, respectively. Currently, he is an Engineer at the Planning Division, Disaster Prevention Research Center, National Cheng Kung University, Tainan, Taiwan.

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