# Numerical Simulation of Hydrodynamic Characteristics of Layered Floating Reefs under Tidal Currents and Waves

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material and Structural Parameters of Layered Floating Reefs

_{1}is the height of the upper floating frame, L

_{2}is the length of the connecting cable, L

_{3}is the total length of the height of the lower floating frame and the auxiliary mooring rope and L

_{4}is the length of the main mooring rope. The upper and lower reefs are connected in a corresponding way by ropes; that is, the four points C, D, E and F on the upper reef are connected, respectively, to the four points A’, B’, G’ and H’ on the lower reef by connecting cables. Considering the water depth conditions, the water depth of 18 m is selected as the constant water depth suitable for the layered floating reef, and the fixed length of the main mooring rope L

_{4}is 4 m, so the total length of L

_{1}, L

_{2}and L

_{3}is 14 m. The height of the lower reef refers to the conclusion in [19] that the length of the auxiliary mooring rope is 0.59 times the height of the lower floating frame. In the following, different proportions among L

_{1}, L

_{2}and L

_{3}are used as the main research variables to carry out numerical simulations of various working conditions and to explore the influence of various structures of layered floating reefs on their hydrodynamic characteristics at specific velocity.

#### 2.2. Simulation Conditions

#### 2.2.1. Proportional Working Condition of Structure

_{4}is fixed at 4 m, and the different proportions of L

_{1}, L

_{2}and L

_{3}are changed to study the various structures on the force and movement of the layered floating reef. Set L

_{2}to be 1/2/3/4 m, respectively, and then design four groups of working conditions of k

_{1}, k

_{2}, k

_{3}and k

_{4}according to the length of L

_{2}, as shown in Table 2. Due to the fixed water depth of 18 m, L

_{1}+ L

_{3}is always (18-L

_{2}-L

_{4}) m; that is, when the length of L

_{2}is 1 m, the total length of L

_{1}and L

_{3}is 13 m, and the ratio of L

_{1}to L

_{3}is 11/2~2/11 are represented by numbers 1~10 under k

_{1}to k

_{4}working conditions. The numerical simulation was carried out under specific flow velocity and wave conditions and the force of each component of the layered floating reef is related to the motion state of the final floating reef.

#### 2.2.2. Wave Parameters

## 3. Establishment of Layered Floating Reef Model

#### 3.1. Tidal Current Conditions

#### 3.1.1. Measurement Location and Method

#### 3.1.2. Fitting Formula for Vertical Distribution of Tidal Current

_{max}is the maximum vertical velocity, u is the velocity at water depth z and u

_{*}is the friction velocity. Equation (2) can be further simplified:

#### 3.2. Wave Theory

_{x}and u

_{z}denote the velocity of the water point along the x direction and the y direction, respectively, a

_{x}and a

_{z}represent the acceleration of water particles in x direction and y direction, respectively, and h is the wave height.

_{x}, u

_{z}, a

_{x}and a

_{z}in Equations (5)–(8) are added to Equation (3).

#### 3.3. Model Construction Method

_{1}, and the floating frame is a rigid structure, with its units divided according to the finite element division method. Its movement mainly considers translation and rotation, and it is mainly affected by the netting tension, buoyancy, drag, gravity and so on, which work together as shown in Figure 1a. The lumped mass point method is used to calculate the hydrodynamic forces of the net and mooring rope. The division of the concentrated mass points of cable and mooring rope and their tension diagram are given in Figure 1b. Among them, the concentrated mass point of the mooring rope is subjected to the upper and lower tension and the clockwise tension of the four netting wires at concentrated mass points of the netting named as F

_{1}–F

_{4}, in turn. The lower floating reef is composed of lower reef, auxiliary mooring rope and main mooring rope, and the ratio of the distance from the bottom center of the lower reef floating frame to the auxiliary mooring rope knot to the height of the floating frame is designed to be 0.59 [19]. It can be seen that the force of each component unit and concentrated mass point of the floating reef can be summarized as gravity, buoyancy, tension, tidal drag force, etc., and then the mass point motion equation is established by Newton’s second law. In Reference [20], the force, translation and rotation equations of the four-prism floating frame and the tension, motion and deformation equations of the net, cable and mooring rope are given in detail, which will not be repeated here.

## 4. The Influence of the Upper and Lower Layer Height on the Force and Motion of the Floating Reef (Connecting Cable L_{2} = 2 m)

#### 4.1. Tension of Cables, Mooring Ropes and Netting

_{2}working conditions are shown in Figure 3. T

_{1}(Tension) in Figure 3a represents the tension of the connecting cable. The T

_{1}time history changes under different height conditions of nine kinds (see k

_{2}in Table 2) were drawn and the changing trend of the connecting cable tension under different structure proportions (1–9) was basically the same. After a period of false stability in the middle, it reaches a critical point of force on the cables and then slowly increases to a stable state. T

_{1max}in Figure 3d represents the maximum tension of the connecting cable, and the influence of the ratio of L

_{1}/L

_{3}on the maximum tension of the connecting cable after stabilization is analyzed at a specific height of L

_{2}(2 m) and L

_{4}(4 m), which decreases with the decrease in L

_{1}/L

_{3}.

_{2}and T

_{2max}in Figure 3b,e represent the tension and the maximum tension of the auxiliary mooring rope, respectively. Under the tidal current, the tension of the auxiliary mooring rope of the layered floating reef is greatly affected by L

_{3}. The maximum tension gradually tends to be stable when the auxiliary mooring rope is greater than 5 m, which is consistent with the optimal single-mooring conditions for floating reefs mentioned in the literature [19].

_{3}and T

_{3max}in Figure 3c,f represent the tension and the maximum tension of the main mooring rope, respectively. Under the tidal current, the tension of the main mooring rope of the layered floating reef is relatively large, and the maximum tension of the main mooring rope is significantly greater than that of the connecting cable and the auxiliary mooring rope. The tension of the main mooring rope varies little with the ratio of L

_{1}/L

_{3}, indicating that the tension of the main mooring rope is not significantly affected by the proportion of the structure.

_{4}and T

_{5}in Figure 3g,h represent the resultant force at all the fixing points of the upper netting and the lower netting, respectively. The resultant force can reflect the total hydrodynamic force of the netting. The time history changes of T

_{4}and T

_{5}under different height conditions of nine kinds of upper and lower layers were also drawn. The tension of the upper netting decreases with the increase in the ratio of L

_{1}/L

_{3}while the tension of the lower netting increases with the increase in the ratio of L

_{1}/L

_{3}. L

_{1}and L

_{3}have a greater influence on the upper and lower netting, respectively, and both have a good linear relationship. T

_{4max}and T

_{5max}in Figure 3i represent the maximum tension of the upper netting and the lower netting, respectively. The maximum tension does not exceed 1.2 kN, and the uniform distribution on the net wire is far less than the limit tension value of the net wire, indicating that the netting structure of the layered floating reef is very safe. The reason why the tension of netting shown in the results fluctuates greatly in the early stage is that the force area of the netting in the direction of the incoming flow is the largest at the beginning, and the netting is subjected to large fluid resistance, resulting in deformation and tension of the netting. As the inclination angle of the floating reef increases, the netting tension gradually increases to a stable state. As the results show, the tension of the netting is much smaller than that of the cables and mooring rope. However, under the action of tidal current, the tension of the upper netting is finally transmitted to the connecting cable through the reef floating frame, and the tension of the lower netting is transmitted to the auxiliary mooring rope.

#### 4.2. Influence of Layered Floating Reef Structure on Reef Body Movement

_{2}= 2 m) is specified, the rolling angle of the upper and lower reefs of the stratified floating reef decreases slowly due to the decrease in the ratio of the upper and lower heights. It shows that the different structural proportions of the layered floating reef have no obvious effect on the roll of the upper and lower reefs, and the roll part of the layered floating reef is mainly affected by the current. In Figure 4a, the roll angle of the layered floating reef increases with time at a specific flow rate, and the overall shape of the floating reef remains unchanged. R

_{a1}(Roll-angle), R

_{a2}and R

_{max}in Figure 4b–d represent the rolling angle of the upper and lower reefs and the maximum rolling angle of the reef, respectively. The roll angles of the upper and lower reefs initially show a parabolic increasing relationship with time, and the roll angles do not exceed 15° at most and the minimum is about 12° after stabilization. At the same time, the roll angles of the upper and lower reefs are consistent. The above results indicate that the designed layered floating reef can not only make full use of the ocean space to play its own role, but can also maintain a certain stability under the combined effect of tidal current and water depth, which effectively improves the functionality and safety of the floating reef.

## 5. Optimal Structure of Floating Reef under Tidal Current

_{1}, k

_{2}, k

_{3}and k

_{4}. Each working condition is divided into multiple groups of L

_{1}/L

_{3}structure proportions, as shown in Table 5, Table 6, Table 7 and Table 8.

_{1}), 2 m (k

_{2}), 3 m (k

_{3}) and 4 m (k

_{4}). From Table 5, Table 6, Table 7 and Table 8, it can be seen that the maximum tension of the connecting cable is mainly affected by L

_{1}, showing a linear decreasing relationship, indicating that the force on the upper reef is finally transmitted to the connecting cable. The maximum tension of the auxiliary mooring rope gradually tends to be relatively stable when L

_{3}is greater than 5 m, which is consistent with the research results of Pan et al. [19]. The maximum tension of the main mooring rope, as the supporting part of the whole layered floating reef, is less affected by different structure of the layered floating reef. The maximum tension of the netting joint is one to two orders of magnitude smaller than that of cables and mooring ropes, so the tension of the upper and lower netting is finally transmitted to the connecting cable, auxiliary mooring ropes and main mooring ropes, respectively. The inclination angles of the upper and lower reefs are basically consistent, and the maximum roll angle of the reef does not exceed 15°, indicating that the designed layered floating reef has good stability and current resistance. The maximum roll angle of the upper and lower reefs has good consistency and does not exceed 15°, which shows that the layered structure of the floating reef has better flow resistance and stability. Based on the above results, it can be concluded that the height of the upper and lower reefs of the stratified floating reef L

_{1}/L

_{3}should not be more than one to obtain better water space utilization and stability.

## 6. Response Verification of Optimal Structure of Floating Reef under Wave-Current Action

_{2}= 4 m, L

_{4}= 4 m, L

_{1}= 2 m and L

_{3}= 8 m, respectively. The effects of wave height and wave steepness on the hydrodynamic characteristics of stratified floating reefs are discussed when the wave and current are in the same direction or opposite direction. In this section, the downstream is defined as the wave-current in the same direction, and the upstream is defined as the wave-current in the opposite direction, as shown in Figure 5, Figure 6, Figure 7 and Figure 8.

#### 6.1. Influence of Wave Height on the Tension of Cable, Mooring Rope and Netting

_{1max}, T

_{2max}and T

_{3max}represent the maximum tension of the connecting cable, the auxiliary mooring rope and the main mooring rope, respectively, as shown in Figure 5a–c. All the above three increase with the increase in wave height when the wave steepness is fixed at 0.05 and the wave and current are in the same direction. The change of wave height has little effect on T

_{1max}when the wave height is fixed at 1.5 m and the wave and current are in the opposite direction. Moreover, T

_{2max}and T

_{3max}reach the minimum when the wave height is 1 m. In addition, the maximum tension of the outer cable and the auxiliary mooring rope when the wave follow the current is greater than the maximum tension when the wave is opposite the current.

_{4max}and T

_{5max}denote the maximum tension of the upper and lower layers of the layered floating reef, respectively. The direction of wave propagation is defined as positive, and the maximum tension of the fixing point of the net is defined as negative, when the wave and current are in the same direction, as shown in Figure 5d,e. The maximum tension of the upper netting fixing point is less affected by the wave height, as shown in Figure 5d. The maximum tension of the lower netting fixing point increases with the increase in wave height. The difference between the two is 0.9 kN when the wave height reaches 3 m. In addition, T

_{5max}is always greater than T

_{4max}. There are two main reasons for this effect: First, the force of the net is related to the number of net mesh, the length of the net and the size of the net line. The length of the lower net of the selected layered floating reef structure is about three times the length of the upper net. Second, the total upstream area of the lower net line is larger than the total upstream area of the upper net line, resulting in a larger wave force on the lower net.

#### 6.2. Influence of Wave Height on Reef Movement

_{1angle}and R

_{2angle}denote the maximum roll angle of the upper reef and the maximum roll angle of the lower reef, respectively. The above two variables coincide completely and increase linearly with the increase in wave height, which indicates that the stratified floating reef has good consistency and stability under the action of wave and current. The R

_{1angle}and R

_{2angle}are about 18 degrees when the wave height is 0.5 m. Nevertheless, the R

_{1angle}and R

_{2angle}are about 30 degrees when the wave height is 3 m. This phenomenon indicates that the roll angle of the reef is greatly affected by the wave height, resulting in a significant tilt change.

_{1angle}and R

_{2angle}increase with the increase in wave height under the condition of countercurrent. The R

_{1angle}and R

_{2angle}are about 13 degrees when the wave height is 0.5 m. The R

_{1angle}and R

_{2angle}are about 21 degrees when the wave height is 3 m. In general, the maximum roll angle of the reef increases with the increase in wave height under the two conditions of downstream and upstream, which is linearly increasing. The maximum roll angle of the reef under the condition of downstream is greater than that under the condition of upstream.

#### 6.3. Influence of Wave Steepness on the Tension of Cable, Mooring Rope and Netting

_{2}= 4 m, L

_{4}= 4 m, L

_{1}= 2 m and L

_{3}= 8 m of the stratified floating reef under the combined action of wave and current are selected to study the influence of wave steepness change (wave steepness λ = 0.03–0.07, increment is 0.01) on the hydrodynamic characteristics of the stratified floating reef under the combined action of wave and current.

_{1max}, T

_{2max}and T

_{3max}represent the maximum tension of the connecting cable, the auxiliary mooring rope and the main mooring rope, respectively, as shown in Figure 7a–c. The maximum tension of the cable and the mooring rope increases with the increase in the wave height when the wave steepness is fixed at 2.0 m. The curve of T

_{2max}with wave steepness is basically consistent, as shown in Figure 7a–c. It shows that the auxiliary mooring tension is almost not affected by the direction of wave current. Meanwhile, the maximum tension difference of T

_{3max}reaches the maximum when the wave steepness is 0.03, and the maximum tension of the main mooring under the downstream flow is slightly larger than that under the upstream.

_{4max}and T

_{5max}correspond to the maximum tension of the fixing point of the upper and lower layers of the layered floating reef, respectively. T

_{4max}is less affected by wave steepness while T

_{5max}increases with the increase in wave steepness. The maximum tension of the total fixing point of the netting in the lower layer is greater than the maximum tension of the downstream flow, which is similar to the influence of the wave height on the tension of the netting.

#### 6.4. Influence of Wave Steepness on Reef Movement

_{1angle}and R

_{2angle}are proportional to wave steepness, which is similar to the effect of wave height on the roll angle of stratified floating reefs. The R

_{angle}reaches a minimum of 21 degrees when the wave steepness is 0.03, and the R

_{angle}reaches a maximum of 30 degrees when the wave steepness is 0.07. This difference indicates that the wave steepness has little effect on the roll angle of the floating reef. Furthermore, the maximum negative roll angle difference of the upper and lower reefs is not more than 1 degree under the same wave steepness. It shows that the rolling angle of the stratified floating reef under the condition of countercurrent is less affected by the wave steepness, and the designed stratified floating reef also has good stability.

## 7. Conclusions

- Under the specific tidal current, the change of the proportion of the layered floating reef structure has little effect on the roll angle of the reef. The inclination angles of the upper and lower reefs are basically consistent, and the maximum roll angle of the reef does not exceed 15°, indicating that the designed layered floating reef has good stability and current resistance.
- Under the action of tidal current, when the stratified floating reef connecting cable is at a specific height of L
_{2}(1 m, 2 m, 3 m, 4 m) and L_{4}(4 m), the tension of the connecting cable decreases with the decrease in L_{1}/L_{3}ratio, showing a linear relationship. The maximum tension of the auxiliary mooring rope tends to be stable when L_{3}is greater than 5 m. The tension of the main mooring rope is not obviously affected by the proportion of layered floating reef structure. The upper net tension and the lower net tension are greatly affected by the length of L_{1}and L_{3}, respectively, and the net tension is the smallest in the overall floating reef. - The structural design of layered floating reefs suggests that the ratio of the sum of the height of the lower reef and the length of the connecting mooring line to the height of the upper reef is not greater than one. It can improve the utilization rate of water space and ensure the stability and safety of layered floating reefs under the influence of tidal current.
- The tension of layered floating reef structure is greatly affected by wave height and wave steepness. The maximum tension of connecting cable, auxiliary mooring, main mooring and upper and lower net fixing points increases with the increase in wave height and wave steepness when the wave current direction is consistent and opposite. The overall stability of the floating reef mainly depends on the force of the mooring rope. Therefore, the strength coefficient of the mooring rope should be increased when the floating reef is put into the actual sea area.
- On the one hand, the wave height has a great influence on the rolling of the upper and lower reefs. The maximum rolling angle of the upper and lower reefs increased linearly when the wave height increased. On the other hand, the wave steepness has a great influence on the rolling of the reef when the wave and current are in the same direction. On the contrary, the impact of wave steepness is smaller when the wave and current are in the opposite direction. Generally, the roll angle of the reef is larger under the combined action of wave and current. Thereby, the maximum rolling angle of the floating reef should be fully considered to avoid the mutual influence between the floating reefs.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Structural diagram of layered floating reef. (

**a**) Upper floating frame unit division and force diagram. (

**b**) Lower floating reef unit division and force diagram.

**Figure 2.**Program class library and example creation diagram of layered floating reef. (

**a**) Layered floating reef class library diagram. (

**b**) Layered floating reef example diagram.

**Figure 3.**The effect of the proportion structure of layered floating reef on the tension of cable, mooring and netting. (

**a**) Time-varying of the connecting cable tension; (

**b**) time-varying of the auxiliary mooring tension; (

**c**) time-varying of the main mooring tension; (

**d**) effect of structural proportion on the maximum tension of connecting cable; (

**e**) effect of structural proportion on the maximum tension of auxiliary mooring; (

**f**) effect of structural proportion on the maximum tension of main mooring; (

**g**) time-varying of total tension of the upper netting fixing points; (

**h**) time-varying tension of lower netting fixing points; (

**i**) effect of structural proportion on the maximum tension of upper and lower nettings.

**Figure 4.**The effect of structural proportion of layered floating reef on reef movement. (

**a**) Kinestate of the layered floating reef and force distribution of netting at the moment of maximum rolling angle; (

**b**) time-varying of the upper reef rolling angle; (

**c**) time-varying of the lower reef rolling angle; (

**d**) effect of structural proportion on the maximum rolling angle of upper and lower reefs.

**Figure 5.**The maximum tension of cable, mooring rope and netting fixing point varies with wave height. (

**a**) The maximum tension of the connecting cable varies with the wave height; (

**b**) the maximum tension of the auxiliary mooring varies with the wave height; (

**c**) the maximum tension of the main mooring varies with the wave height; (

**d**) the maximum tension of the upper netting fixing points varies with the wave height; (

**e**) the maximum tension of the lower netting fixing points varies with the wave height.

**Figure 6.**The maximum rolling angle of the reef varies with the wave height. (

**a**) The maximum rolling angle of the upper and lower reefs varies with the wave height (downstream); (

**b**) the maximum rolling angle of the upper and lower reefs varies with the wave height (upstream).

**Figure 7.**The maximum tension of cable, mooring rope and netting fixing point varies with wave steepness. (

**a**) The maximum tension of the connecting cable varies with the wave steepness; (

**b**) the maximum tension of the auxiliary mooring varies with the wave steepness; (

**c**) the maximum tension of the main mooring varies with the wave steepness; (

**d**) the maximum tension of the upper netting fixing points varies with the wave steepness; (

**e**) the maximum tension of the lower netting fixing points varies with the wave steepness.

**Figure 8.**The maximum rolling angle of the reef varies with the wave steepness. (

**a**) The maximum rolling angle of the upper and lower reefs varies with the wave steepness (downstream); (

**b**) the maximum rolling angle of the upper and lower reefs varies with the wave steepness (upstream).

Structure | Materials | Size (m) | Diameter (m) | Density (kg/m^{3}) | Coefficient of Drag Force C _{D} | Coefficient of Elasticity C _{1} | Coefficient of Elasticity C _{2} |
---|---|---|---|---|---|---|---|

Upper floating frame | HDPE | 1.69 × 1.69 × L_{1} | 0.2 | 646.57 | 0.8 | / | / |

Upper netting | PE | 6.76 × L_{1} | 0.003 | 953 | 0.6 | 345.3 × 10^{6} | 1.0121 |

Connecting cable | PE | L_{2} | 0.02 | 953 | 0.6 | 345.3 × 10^{6} | 1.0121 |

Lower floating frame | HDPE | 1.69 × 1.69 × L_{3}/1.59 | 0.2 | 646.57 | 0.8 | / | / |

Lower netting | PE | 6.76 × L_{3}/1.59 | 0.003 | 953 | 0.6 | 345.3 × 10^{6} | 1.0121 |

Auxiliary mooring rope | PE | L_{3}/1.59 × 0.59 | 0.02 | 953 | 0.6 | 345.3 × 10^{6} | 1.0121 |

Main mooring rope | PE | 4.0 (L_{4}) | 0.04 | 953 | 0.6 | 345.3 × 10^{6} | 1.0121 |

_{D}is drag coefficient; C

_{1}and C

_{2}are elastic coefficients.

Working Conditions | L_{2}/m | L_{1}/L_{3} | L_{4}/m | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||

k_{1} | 1 | 11/2 | 10/3 | 9/4 | 8/5 | 7/6 | 6/7 | 5/8 | 4/9 | 3/10 | 2/11 | 4 |

k_{2} | 2 | 10/2 | 9/3 | 8/4 | 7/5 | 6/6 | 5/7 | 4/8 | 3/9 | 2/10 | 4 | |

k_{3} | 3 | 9/2 | 8/3 | 7/4 | 6/5 | 5/6 | 4/7 | 3/8 | 2/9 | 4 | ||

k_{4} | 4 | 8/2 | 7/3 | 6/4 | 5/5 | 4/6 | 3/7 | 2/8 | 4 |

Station | Tidal Current | Surface Layer | 0.2 d | 0.4 d | 0.6 d | 0.8 d | Bottom Layer | Vertical Average | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Rate | Direction | ||||||||||||||

1# | Rising tidal current | 1.19 | 326 | 1.22 | 312 | 1.30 | 317 | 1.30 | 331 | 1.21 | 320 | 1.05 | 302 | 1.23 | 322 |

Falling tidal current | 1.10 | 153 | 1.16 | 132 | 1.10 | 158 | 1.17 | 149 | 1.18 | 145 | 0.99 | 129 | 1.13 | 140 | |

2# | Rising tidal current | 1.23 | 316 | 1.24 | 333 | 1.24 | 309 | 1.20 | 323 | 1.16 | 320 | 1.07 | 309 | 1.20 | 320 |

Falling tidal current | 1.15 | 129 | 1.14 | 128 | 1.16 | 139 | 1.15 | 130 | 1.13 | 129 | 1.04 | 148 | 1.14 | 133 | |

3# | Rising tidal current | 1.22 | 312 | 1.22 | 318 | 1.16 | 329 | 1.15 | 330 | 1.11 | 316 | 1.01 | 329 | 1.15 | 323 |

Falling tidal current | 1.14 | 140 | 1.14 | 157 | 1.15 | 157 | 1.17 | 132 | 1.14 | 129 | 1.05 | 157 | 1.14 | 145 | |

4# | Rising tidal current | 1.21 | 323 | 1.29 | 328 | 1.28 | 331 | 1.26 | 321 | 1.23 | 309 | 1.13 | 328 | 1.25 | 323 |

Falling tidal current | 1.14 | 156 | 1.20 | 133 | 1.20 | 135 | 1.22 | 133 | 1.20 | 129 | 1.11 | 132 | 1.19 | 135 | |

5# | Rising tidal current | 1.28 | 311 | 1.28 | 332 | 1.27 | 311 | 1.23 | 134 | 1.20 | 324 | 1.11 | 311 | 1.24 | 318 |

Falling tidal current | 1.16 | 134 | 1.14 | 146 | 1.16 | 135 | 1.12 | 140 | 1.08 | 152 | 1.00 | 128 | 1.12 | 141 |

**Table 4.**The least square fitting coefficient of possible maximum flow velocity distribution function.

Conditions | The Possible Maximum Flow Velocity | ||||
---|---|---|---|---|---|

1# | 2# | 3# | 4# | 5# | |

a ($\times $10^{−7}) | 42,369 | 32,749 | 40,006 | 28,090 | 32,732 |

b ($\times $10^{−4}) | 10,160 | 10,130 | 10,189 | 10,019 | 10,157 |

R^{2} ($\times $10^{−4}) | 72,484 | 83,426 | 75,442 | 73,542 | 78,808 |

**Table 5.**The working condition k

_{1}(L

_{2}= 1 m, L

_{4}= 4 m) includes layered floating reef stress condition and movement state.

Layered Floating Reef Tension and Movement | L_{1}/L_{3} | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

11/2 1 | 10/3 2 | 9/4 3 | 8/5 4 | 7/6 5 | 6/7 6 | 5/8 7 | 4/9 8 | 3/10 9 | 2/11 2 | |

Maximum tension of connecting cable | 3.0 | 2.9 | 2.7 | 2.5 | 2.3 | 2.1 | 1.9 | 1.7 | 1.5 | 1.2 |

Maximum tension of auxiliary mooring | 7.9 | 6.5 | 5.6 | 4.8 | 4.5 | 4.3 | 4.2 | 4.0 | 3.9 | 3.7 |

Maximum tension of main mooring | 16.8 | 16.7 | 16.3 | 16.3 | 15.9 | 15.5 | 15.5 | 15.0 | 14.6 | 13.8 |

Maximum tension of upper netting | 1.1 | 1.0 | 0.9 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 |

Maximum tension of lower netting | 0.1 | 0.2 | 0.2 | 0.3 | 0.4 | 0.4 | 0.5 | 0.6 | 0.6 | 0.7 |

Maximum rolling of upper reef | 14.4 | 13.9 | 13.7 | 13.5 | 13.3 | 13.1 | 12.9 | 12.7 | 12.5 | 12.2 |

Maximum rolling of lower reef | 14.4 | 13.9 | 13.7 | 13.5 | 13.3 | 13.1 | 12.9 | 12.7 | 12.5 | 12.2 |

**Table 6.**The working condition k

_{2}(L

_{2}= 2 m, L

_{4}= 4 m) includes layered floating reef stress condition and movement state.

Layered Floating Reef Tension and Movement | L_{1}/L_{3} | ||||||||
---|---|---|---|---|---|---|---|---|---|

10/2 1 | 9/3 2 | 8/4 3 | 7/5 4 | 6/6 5 | 5/7 6 | 4/8 7 | 3/9 8 | 2/10 9 | |

Maximum tension of connecting cable | 2.8 | 2.7 | 2.5 | 2.3 | 2.1 | 1.9 | 1.8 | 1.6 | 1.2 |

Maximum tension of auxiliary mooring | 7.5 | 6.2 | 5.3 | 4.6 | 4.3 | 4.1 | 4.0 | 3.8 | 3.5 |

Maximum tension of main mooring | 15.8 | 15.9 | 15.5 | 15.5 | 15.1 | 14.7 | 14.7 | 14.3 | 12.8 |

Maximum tension of upper netting | 1.0 | 0.9 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 |

Maximum tension of lower netting | 0.1 | 0.2 | 0.2 | 0.3 | 0.4 | 0.4 | 0.5 | 0.6 | 0.6 |

Maximum rolling of upper reef | 14.4 | 13.8 | 13.6 | 13.3 | 13.1 | 12.8 | 12.7 | 12.4 | 11.9 |

Maximum rolling of lower reef | 14.3 | 13.7 | 13.5 | 13.3 | 13 | 12.8 | 12.7 | 12.5 | 12.1 |

**Table 7.**The working condition k

_{3}(L

_{2}= 3 m, L

_{4}= 4 m) includes layered floating reef stress condition and movement state.

Layered Floating Reef Tension and Movement | L_{1}/L_{3} | |||||||
---|---|---|---|---|---|---|---|---|

9/2 1 | 8/3 2 | 7/4 3 | 6/5 4 | 5/6 5 | 4/7 6 | 3/8 7 | 2/9 8 | |

Maximum tension of connecting cable | 2.5 | 2.5 | 2.3 | 2.1 | 1.9 | 1.7 | 1.5 | 1.2 |

Maximum tension of auxiliary mooring | 7.1 | 5.9 | 5.0 | 4.4 | 4.1 | 3.9 | 3.7 | 3.4 |

Maximum tension of main mooring | 14.7 | 15.0 | 14.7 | 14.7 | 14.2 | 13.8 | 13.8 | 12.4 |

Maximum tension of upper netting | 0.9 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 |

Maximum tension of lower netting | 0.1 | 0.2 | 0.2 | 0.3 | 0.4 | 0.4 | 0.5 | 0.6 |

Maximum rolling of upper reef | 14.2 | 13.7 | 13.3 | 13.1 | 12.9 | 12.6 | 12.4 | 11.8 |

Maximum rolling of lower reef | 14.2 | 13.6 | 13.3 | 13.0 | 12.9 | 12.6 | 12.4 | 12.0 |

**Table 8.**The working condition k

_{4}(L

_{2}= 4 m, L

_{4}= 4 m) includes layered floating reef stress and movement state.

Layered Floating Reef Tension and Movement | L_{1}/L_{3} | ||||||
---|---|---|---|---|---|---|---|

8/2 1 | 7/3 2 | 6/4 3 | 5/5 4 | 4/6 5 | 3/7 6 | 2/8 7 | |

Maximum tension of connecting cable | 2.3 | 2.3 | 2.1 | 1.9 | 1.7 | 1.5 | 1.2 |

Maximum tension of auxiliary mooring | 6.7 | 5.5 | 4.8 | 4.1 | 3.9 | 3.7 | 3.6 |

Maximum tension of main mooring | 13.8 | 14.0 | 13.7 | 13.8 | 13.4 | 12.8 | 12.1 |

Maximum tension of upper netting | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 |

Maximum tension of lower netting | 0.1 | 0.2 | 0.2 | 0.3 | 0.4 | 0.4 | 0.5 |

Maximum rolling of upper reef | 14 | 13.4 | 13.1 | 12.8 | 12.5 | 12.2 | 11.7 |

Maximum rolling of lower reef | 13.9 | 13.4 | 13.0 | 12.8 | 12.5 | 12.3 | 11.9 |

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## Share and Cite

**MDPI and ACS Style**

Pan, Y.; Yang, L.; Xue, D.; Luo, L.
Numerical Simulation of Hydrodynamic Characteristics of Layered Floating Reefs under Tidal Currents and Waves. *Water* **2023**, *15*, 3892.
https://doi.org/10.3390/w15223892

**AMA Style**

Pan Y, Yang L, Xue D, Luo L.
Numerical Simulation of Hydrodynamic Characteristics of Layered Floating Reefs under Tidal Currents and Waves. *Water*. 2023; 15(22):3892.
https://doi.org/10.3390/w15223892

**Chicago/Turabian Style**

Pan, Yun, Lijing Yang, Dawen Xue, and Lu Luo.
2023. "Numerical Simulation of Hydrodynamic Characteristics of Layered Floating Reefs under Tidal Currents and Waves" *Water* 15, no. 22: 3892.
https://doi.org/10.3390/w15223892