# Experimental Investigation on Hydrodynamic Coefficients of a Column-Stabilized Fish Cage in Waves

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{D}and C

_{M}correlated with KC number reasonably well, while absolutely no correlation was found between the Reynolds number and C

_{D}and C

_{M}. Sundar et al. [3] used the least squares method to analyze the force of the pile column under the action of regular waves, and obtained the relationship between the hydrodynamic coefficients and the KC number of the pile column with different inclination angles. Bushnell [4] studied the mutual interaction and influence of array cylinders by means of a pulsating water tunnel. It was pointed out that this interference increases with the increase of the flow orbit length, and the maximum resistance of the shielded cylinder can be reduced to half of the single cylinder. Chakrabarti [5,6] studied the variation of hydrodynamic coefficients of group piles with KC number and pile spacing under regular wave action. Kurrian et al. [7] studied the comparison of hydrodynamic forces and force coefficients for the effects of spacing between the cylinders, shape, and arrangement of arrays, and found that the hydrodynamic force and force coefficients of the cylinders have similar trends when the KC number is the same. Besides, the measurements were compared with numerical predictions by Santo et al. [8] using computational fluid dynamics (CFD), with the actual jacket represented in a three-dimensional numerical wave tank as a porous tower model, simulating a uniformly distributed Morison stress field. Good agreement was achieved, both in terms of incident surface elevation as well as total force time histories, all using a single set of C

_{D}and C

_{M}. Palm et al. [9] studied the scale effects between a model scale and a prototype scale device, and compared inviscid Euler simulations with RANS results to quantify the viscous contribution to the loads and responses of the wave energy converter (WEC). Amaechi et al. [10] developed a coupled dynamic models with both buoys and hoses using Orcaflex to investigate the effects of hose hydrodynamic loads and flow angles on the structural behavior of the hoses. Gadelho et al. [11] presented an numerical analysis on determining the hydrodynamic coefficients of an oscillating two-dimensional rigid cylindrical body, using a time domain Navier–Stokes model. However, as of now, only a very limited number of studies have addressed the determination of hydrodynamic coefficients for large-span jacket structure.

## 2. Materials and Methods

#### 2.1. Experimental Model

_{D}and C

_{M}. Other forces were not considered for the time being. Due to the large structure frame, four dynamometers were used symmetrically to obtain the horizontal wave force. The measurement range of the four dynamometers in the horizontal direction was 50 N, and the positive direction remained consistent with the direction of wave propagation. As shown in Figure 2, the hydrodynamic data collected under the action of a certain working condition, the phase of each dynamometer was always consistent, and present periodic variation consistent with the action of regular wave, which also reflected the rationality of the dynamometer arrangement of the experiment.

_{r}was 64.027. According to λ and λ

_{r}, the prototype structure of the jacket was equivalent to the model structure. The specific parameters are shown in Table 1. The outer contour of the jacket was 1.47 m long, 1.2 m wide, and 1.18 m heigh; 12 large cylindrical hollow pipes with a diameter of 32 mm and a wall thickness of 3 mm formed the jacket legs, and thin round pipes with a diameter of 25 mm and a wall thickness of 2 mm were distributed from the top to 0.62 m under the water surface as diagonal braces and cross braces. Water depth below 0.62 m was a bottom structure.

#### 2.2. Hydrodynamic Coefficient

_{d}) and the inertial force (F

_{i}) [23]. The formula is as follows:

_{D}is the velocity force coefficient, and C

_{M}is the inertial force coefficient.

_{x}and the horizontal acceleration a

_{x}of water particles at any position of the structure are:

_{m}(i), then the sum of the squared error is:

_{D}and C

_{M}can be obtained as follows:

#### 2.3. Numerical Model

_{D}and C

_{M}were obtained by the least square method described above. In addition, the origin of the global coordinate system of the numerical model must be located at stilling water surface, and the vertical axis was the Z axis, the positive direction of the Z axis was upward.

## 3. Results

#### 3.1. Horizontal Force on Various Models in Waves

#### 3.2. Hydrodynamic Coefficients of Various Models in Waves

_{max}T/D; U

_{max}: The maximum velocity of the wave; D: The maximum diameter of the jacket member) and the incident angle on the hydrodynamic coefficients of the two models in the wave are shown in this section. For the two models in each wave condition, we calculated the drag coefficient C

_{D}and the inertia coefficient C

_{M}by Equation (7) with the least square method using fifth order Stokes wave.

_{D}decreased gradually with the increase in KC number, and the step diameter was decreased as the KC number increased; C

_{D}got the maximum value in oblique wave (β = 45°), and the minimum value in lateral wave (β = 90°). The effect of KC number on C

_{M}was insignificant for the jacket, but the effect of wave incident angle was distinct; C

_{M}got the maximum value in forward wave (β = 0°), and the minimum value in lateral wave (β = 90°). For M2, it was considered that the jacket and the netting were two independent parts; that is, the horizontal wave force of the netting can be obtained by the force of M2 minus the force of M1, and the composition of the horizontal wave force of the netting was based on the study of Zhao et al. [24], whom conducted wave tests on a variety of densities of netting, and concluded that the value of the inertial force of the netting was much smaller than the value of the wave force it received, can be ignored. Therefore, it was considered that the horizontal wave force of the netting can be expressed by only the drag force. The variation of the netting drag coefficient C

_{D}with the KC number in waves is shown in Figure 11, which had consistent regularity compared with C

_{D}of jacket.

#### 3.3. Components of Horizontal Force in Waves

#### 3.3.1. Components of Horizontal Force of the Jacket Structure

#### 3.3.2. Components of Horizontal Force of the Fish Cage Structure

## 4. Discussion

#### 4.1. The Effect of Incident Angles on Wave Force

#### 4.2. The Effect of Incident Angles on Hydrodynamic Coefficients

_{D}and the inertial coefficient C

_{M}showed obvious differences at different incident angels: C

_{D}got the maximum value in the oblique wave, while got the minimum value in the lateral wave; C

_{M}got the maximum value in the forward wave, while got the minimum value in the lateral wave.

#### 4.3. The Effect of Wave Parameters on Wave Force Components

#### 4.4. Results Comparison between Calculation and Measurement

_{D}and C

_{M}obtained by the least square method above, the four groups of test conditions were taken as examples in the lateral wave. Using Equation (1), the horizontal wave forces of the two models were calculated and compared with the horizontal wave force of the test (see Figure 20 and Figure 21). The comparison results indicate that the prediction error of the present model was lower than 2%. Of course, only two periods with relatively stable experimental data were selected for comparison, which was inconsistent with the results of using the complete data, but the prediction deviation in most cases was lower than 5%, which can be acceptable. In addition, only the drag force of the netting was considered and the inertial force was ignored, although the calculation error was introduced, and the approximate solution can be achieved. In summary, the calculated values agreed very well with the time histories of the measurement values, and it can be considered that the hydrodynamic coefficients calculated by the FEM results was effective.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Wave incident angles and top view of the model (0°, 45°, and 90° were incident direction. Thereafter, they were denoted respectively as forward wave, oblique wave, and lateral wave).

**Figure 4.**Time histories of the wave height and horizontal force on M1 and M2 for A5. (

**a**)Wave height, (

**b**) force of M1 and (

**c**) force of M2.

**Figure 7.**Time histories of the wave height and horizontal forces on M1 and M2. (

**a**) A3-3 and (

**b**) A4.

**Figure 8.**Horizontal forces of the structure under the action of wave (T = 1.6 s). (

**a**) β = 0°, (

**b**) β = 45° and (

**c**) β = 90°.

**Figure 9.**Horizontal forces of the structure under the action of wave (H = 0.15 m). (

**a**) β = 0°, (

**b**) β = 45° and (

**c**) β = 90°.

**Figure 10.**Variation of hydrodynamic coefficients of jacket, along with the change of KC number. (

**a**) Relationship between C

_{D}and KC number and (

**b**) relationship between C

_{M}and KC number.

**Figure 11.**Variation of hydrodynamic coefficient (C

_{D}) of the netting, along with the change of KC number.

**Figure 12.**Composition of measured horizontal wave force of M1 (T = 1.6 s). (

**a**) β = 0°, H = 0.15 m, T = 1.6 s, (

**b**) β = 0°, H = 0.20 m, T = 1.6 s, (

**c**) β = 45°, H = 0.15 m, T = 1.6 s, (

**d**) β = 45°, H = 0.20 m, T = 1.6 s, (

**e**) β = 90°, H = 0.15 m, T = 1.6 s and (

**f**) β = 90°, H = 0.20 m, T = 1.6 s.

**Figure 13.**Composition of measured horizontal wave force of M1 (H = 0.15 m). (

**a**) β = 0°, T = 1.4 s, H = 0.15 m, (

**b**) β = 0°, T = 1.8 s, H = 0.15 m, (

**c**) β = 45°, T = 1.4 s, H = 0.15 m, T = 1.6 s, (

**d**) β = 45°, T = 1.8 s, H = 0.15 m, (

**e**) β = 90°, T = 1.4 s, H = 0.15 m and (

**f**) β = 90°, T = 1.8 s, H = 0.15 m.

**Figure 14.**Composition of measured horizontal wave force of M2 (T = 1.6 s). (

**a**) β = 0°, H = 0.15 m, T = 1.6 s, (

**b**) β = 0°, H = 0.20 m, T = 1.6 s, (

**c**) β = 45°, H = 0.15 m, T = 1.6 s, (

**d**) β = 45°, H = 0.20 m, T = 1.6 s, (

**e**) β = 90°, H = 0.15 m, T = 1.6 s and (

**f**) β = 90°, H = 0.20 m, T = 1.6 s.

**Figure 15.**Composition of measured horizontal wave force of M2 (H = 0.15 m). (

**a**) β = 0°, T = 1.4 s, H = 0.15 m, (

**b**) β = 0°, T = 1.8 s, H = 0.15 m, (

**c**) β = 45°, T = 1.4 s, H = 0.15 m, (

**d**) β = 45°, T = 1.8 s, H = 0.15 m, (

**e**) β = 90°, T = 1.4 s, H = 0.15 m and (

**f**) β = 90°, T = 1.8 s, H = 0.15 m.

**Figure 16.**Horizontal force of structure under different incident angles (T = 1.6 s). (

**a**) M1 and (

**b**) M2.

**Figure 17.**Horizontal force of structure under different incident angles (H = 0.15 m). (

**a**) M1 and (

**b**) M2.

**Figure 18.**Forces under the action of waves (T = 1.6 s). (

**a**) Drag force (C

_{D}= 1), (

**b**) Inertia force (C

_{M}= 1) and (

**c**) Total force (C

_{D}= 1, C

_{M}= 1).

**Figure 19.**Forces under the action of waves (H = 0.15 m). (

**a**) Drag force (C

_{D}= 1), (

**b**) Inertia force (C

_{M}= 1) and (

**c**) Total force (C

_{D}= 1, C

_{M}= 1).

**Figure 20.**Time histories of measured and calculated wave forces on M1 in the lateral wave (β = 90°). (

**a**) β = 90°, H = 0.15 m, T = 1.6 s, (

**b**) β = 90°, H = 0.20 m, T = 1.6 s, (

**c**) β = 90°, T = 1.4 s, H = 0.15 m and (

**d**) β = 90°, T = 1.8 s, H = 0.15 m.

**Figure 21.**Time histories of measured and calculated wave force on M2 in the lateral wave (β = 90°). (

**a**) β = 90°, H = 0.15 m, T = 1.6 s, (

**b**) β = 90°, H = 0.20 m, T = 1.6 s, (

**c**) β = 90°, T = 1.4 s, H = 0.15 m, (

**d**) β = 90°, T = 1.8 s, H = 0.15 m.

Member | Model Number | Section for Prototype | Section for Model |
---|---|---|---|

(mm) | (mm) | ||

Jacket (Outside diameter * Thickness) | P2000 × 50 | 2000 * 50 | 32 * 3 |

P1500 × 40 | 1500 * 40 | 25 * 2 | |

P1200 × 40 | 1200 * 40 | 20 * 2 |

Test Number | Wave Height H/m | Wave Period T/s | Wave Incident Angle β (°) |
---|---|---|---|

A1 | 0.05 | 1.6 | 0(A), 45(B), 90(C) |

A2 | 0.1 | 1.6 | |

A3 | 0.15 | 1.0, 1.2, 1.4, 1.6, 1.8 | |

A4 | 0.2 | 1.6 | |

A5 | 0.25 | 1.6 |

Parameter Type | Experimental Values | Numerical Values |
---|---|---|

Mesh size (mm) | 12 | 24 |

Twine diameter (mm) | 0.82 | 1.64 |

Horizontal shrinkage coefficient | 0.6 | 0.6 |

Vertical shrinkage coefficient | 0.8 | 0.8 |

Solidity ratio | 0.137 | 0.137 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, Y.-P.; Chen, Q.-P.; Bi, C.-W.; Cui, Y.
Experimental Investigation on Hydrodynamic Coefficients of a Column-Stabilized Fish Cage in Waves. *J. Mar. Sci. Eng.* **2019**, *7*, 418.
https://doi.org/10.3390/jmse7110418

**AMA Style**

Zhao Y-P, Chen Q-P, Bi C-W, Cui Y.
Experimental Investigation on Hydrodynamic Coefficients of a Column-Stabilized Fish Cage in Waves. *Journal of Marine Science and Engineering*. 2019; 7(11):418.
https://doi.org/10.3390/jmse7110418

**Chicago/Turabian Style**

Zhao, Yun-Peng, Qiu-Pan Chen, Chun-Wei Bi, and Yong Cui.
2019. "Experimental Investigation on Hydrodynamic Coefficients of a Column-Stabilized Fish Cage in Waves" *Journal of Marine Science and Engineering* 7, no. 11: 418.
https://doi.org/10.3390/jmse7110418