# A Computation Method for the Typhoon Waves Using the Field Wave Spectrum

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation Methods

#### 2.1. Data Collection and Processing Methods

_{1/3}is 1.58 m, and the spectrum peak period T

_{p}is 13.15 s. The average wave period $\overline{T}$ is 5.96 s. An improved JONSWAP spectrum [29] and a dual peak Ochi-Hubble spectrum [15] were used to fit the field spectrum. Among them, the peak enhancement factor γ of the improved JONSWAP spectrum is 0.95, and can be expressed as follows:

_{S,j}, ω

_{p,i}and λ

_{i}for i = 1,2 are significant wave height, angular peak frequency, and spectral shape factor for low and high-frequency parts (i = 1 represents the low frequency spectrum, and i = 2 represents the high frequency spectrum), respectively. Formulation of the six parameters was given. The values of the parameters of the Ochi-Hubble dual peak spectrum (T

_{p}

_{1}and T

_{p}

_{2}, the main spectral peak period and the secondary spectral peak period, respectively) are T

_{p}

_{1}= 12.5 s, T

_{p}

_{2}= 3.57 s, R

_{p}

_{1}= 0.65, a

_{1}= 0.61, a

_{2}= 0.94, c

_{1}= 7.32, c

_{2}= 1.0, d

_{1}= 0.0, d

_{2}= 0.102.

#### 2.2. Discretization Methods of Wave Spectrum

_{n}and ω

_{n}are the amplitude and circular frequency of the constituent waves, respectively, and ε

_{n}is the initial phase uniformly distributed between 0 and 2π. The target spectrum is a random wave for S

_{η}(ω). ω represents circular frequency, and it needs to be divided into N intervals as Δω

_{i}= ω

_{i}− ω

_{i}

_{-1}= ω

_{H}/N. ω

_{H}is the circular frequency of the largest constituent wave. Obviously, the circular frequency of the smallest constituent wave is taken as 0. If ω

_{i}(randomly selected between ω

_{i}and ω

_{i}

_{−1}) is taken as the representative frequency of the i-th constituent wave, the amplitude of the i-th is shown in Equation (5). Then, by superimposing the N cosine waves of the wave energy in all N intervals, the wave surface line Equation (6) of the ocean wave and the motion Equations (7) and (8) of the two-dimensional water particle can be expressed as:

_{η}represents the ocean wave spectrum, and ε

_{i}indicates the initial phase of the i-th component wave, which is randomly generated within 0–2π.

_{x}and u

_{z}are the velocity of the water particle in the x and z directions, respectively; a

_{x}and a

_{z}are the acceleration of the water particle in the x and z directions in turn; (x, z) represents the coordinates of two-dimensional wave field in the vertical direction. d is the water depth, and k

_{i}is the wave number of the constituent waves obtained by solving the dispersion equation using the bisectional method.

_{H}is generally selected as four times the main circular frequency ω

_{p}, and the wave simulation accuracy is high. Since the circular frequency is greater than 2.5 rad/s and approaches 0, ω

_{H}is selected to be 5 times the main circular frequency, that is, 2.5 rad/s, and the corresponding value of N is also increased to 320. The equal frequency method was used to divide the circular frequency interval as Δω

_{i}= 7.81 × 10

^{−3}. The resolution of the field spectrum in Figure 2 is 0.01 Hz which is 0.063 rad/s, and the field spectrum needs to be interpolated into 320 circular frequency intervals (0–2.5 rad/s). The comparison between the wave after cubic spline interpolation and the original field spectrum is made as shown in Figure 3.

#### 2.3. Discretization Methods of Wave Spectrum and Directional Spectrum

_{j}< π, but in fact that the energy of the waves is mainly distributed in the range of π/2 on both sides of the main propagation direction, so the number of equally divided angles only on −π/2 < θ

_{j}< π/2 is M = 30. Assuming that the frequency spectrum and the directional spectrum are mutually independent. The wave spectrum considering the directional spectrum can be obtained by multiplying the two and then superimposing, and the formulas of wave spectrum are expressed in Equation (10) and directional spectrum is shown in Figure 4. With considering the directional spectrum Equation (12), the wave surface and the motion of three-dimensional water particles (Equations (13) to (14)) can be obtained by superimposing the cosine waves in M directions.

## 3. Verification of Wave Features

_{i}and ε

_{i}are randomly generated in the model, the random wave elements generated at the fixed points are tested by continuous calculation for ten times, as shown in Table 1.

_{mean}< T

_{01}(mean period of spectral calculation), while T

_{mean}≈ T

_{01}under deep water conditions. It follows that the water depth affects the calculation of the typhoon wave period, which is also related to the wave theory used. In shallow water, the linear wave theory will increase the value of the average wave period when simulated random waves.

## 4. Simulation of Typhoon Wave Features

#### 4.1. Wave Surface

#### 4.2. The Velocity of Water Particle

#### 4.3. The Acceleration of Water Particle

_{x}and a

_{z}are also given in Table 2. The maximum value of a

_{z}reaches 2.93 m/s

^{2}, which will also lead to a large wave inertia force to offshore structures. Different from the velocity of water particle, the positive and negative directions of a

_{x}are basically same.

## 5. Application to a Slender Cylinder Fastened by Mooring Cable

#### 5.1. Simulation Method of a Slender Cylinder

- (1)
- Spatial position of structure and wave conditions

- (2)
- The force and motion of slender cylinder

_{x}, J

_{y}, J

_{z}is the moment of inertia relative to the centroid of the slender cylinder in x, y, z directions. m represents the total mass of the slender cylinder. r is the radius of the slender cylinder, and h is the height. It should be noted that the slender cylinder is sometimes higher than the wave surface in the motion process, so it is necessary to compare each unit of the slender cylinder with the wave surface η calculated by Equation (3). If the z coordinate of the unit of slender cylinder is greater than η, the velocity and acceleration of the water particle are zero, i.e., the wave drag force and inertia force are zero, and the unit is not subject to the buoyancy of the sea water.

- (3)
- The tension and motion of mooring cable

#### 5.2. Simulation Results

- (1)
- The motion state of slender cylinder and the tension distribution of mooring cable

- (2)
- The centroid motion of slender cylinder

- (3)
- The rolling of slender cylinder

## 6. Conclusions

- (1)
- The field wave spectrum of Typhoon Talim is a dual peak spectrum. The spectrum width of main peak is closer to the improved JONSWAP spectrum, and the secondary peak is smaller than the calculated value by the dual peak spectrum of Ochi-Hubble.
- (2)
- There are small periodic waves on the typhoon wave surface, that is, small-period waves between the peaks and troughs of large-period waves, which may increase the resonance of floating structures and the damage of mooring system.
- (3)
- The magnitude and direction of velocity vector of the water particle on typhoon wave surface have no correlation with the position of the wave crest and trough. The wave water particle velocity calculated by the typhoon Talim simulation variations are ±1.7 m/s, and the water particle acceleration variations are ±2.5 m/s
^{2}. - (4)
- According to the Morison equation, the forces acting on a moored slender cylinder (it is a simplified marine buoy or spar platform) can be calculated under typhoon waves. The keypoint of establishing a coupling calculation model is the calculation of the velocity distribution field as well as acceleration distribution field of the water particle. Under action of the typhoon waves using the field wave spectrum, the rotation angle of moored slender cylinder is more likely to resonate with the secondary peak frequency in the wave spectrum. The response frequency of rotation angle of the JONSWAP spectrum and field spectrum is obvious difference. The numerical simulation method of typhoon wave proposed in this study can be well applied to the hydrodynamic characteristics analysis of moored structures.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**The duration curve of the wave surface simulation at the fixed point (x = 0 m, y = 0 m) and the wave features determined by upward zero-crossing method.

**Figure 6.**The 3D typhoon waves surface: (

**a**) the dual peak field spectrum; (

**b**) the single-peak JONSWAP spectrum.

**Figure 7.**The velocity of the water particle in the typhoon wave: (

**a**) The velocity vector of the water particle in the x-z plane (y = 0 m); (

**b**) Simulate the duration variations of the water particle velocity in the x-direction (U) and z-direction (W) at the fixed point (x = 0 m, y = 0 m, z = 0 m).

**Figure 8.**The acceleration of water particle in the typhoon wave: (

**a**) The acceleration vector of water particle in X-Z plane (y = 0 m); (

**b**) Simulate the duration variations of the acceleration vector of water particle in the x-direction (a

_{x}) and z-direction (a

_{z}) at a fixed point (x = 0 m, y = 0 m, z = 0 m).

**Figure 9.**The motion state of slender cylinder and the tension distribution of mooring cable: (

**a**) The single-peak of JONSWAP spectrum; (

**b**) The dual peak of field spectrum.

**Figure 10.**Time series of the centroid motion of slender cylinder: (

**a**) Y direction; (

**b**) Z direction.

**Figure 11.**Frequency curve of centroid motion of slender cylinder: (

**a**) Y direction; (

**b**) Z direction.

Statistics | Example 1 | Example 2 | Example 3 | Example 4 | Example 5 | Example 6 | Example 7 | Example 8 | Example 9 | Example 10 |
---|---|---|---|---|---|---|---|---|---|---|

H_{1/3}(m) * | 1.63 | 1.51 | 1.50 | 1.54 | 1.66 | 1.67 | 1.63 | 1.61 | 1.55 | 1.66 |

H_{error} * | 3.2% | −4.4% | −5.1% | −2.5% | 5.1% | 5.7% | 3.2% | 1.9% | −1.9% | 5.1% |

T_{mean1}(s) * | 6.67 | 6.68 | 6.78 | 6.54 | 6.71 | 6.88 | 6.73 | 6.74 | 6.06 | 6.80 |

T_{error1} * | 11.9% | 12.1% | 13.8% | 9.7% | 12.6% | 15.4% | 12.9% | 13.1% | 1.7% | 14.1% |

T_{mean2}(s) * | 6.65 | 6.67 | 6.74 | 6.52 | 6.72 | 6.90 | 6.76 | 6.74 | 6.09 | 6.76 |

T_{error2} * | 11.6% | 11.9% | 13.1% | 9.4% | 12.8% | 15.8% | 13.4% | 13.1% | 2.2% | 13.4% |

_{1/3}is the effective wave height; H

_{error}is the relative error of the effective wave height; T

_{mean1}is the wave period counted by the maximum point; T

_{mean2}is the wave period counted by the minimum point; T

_{error1}and T

_{error2}are the relative errors of the wave period respectively.

Statistical Values | U (m/s) | W (m/s) | a_{x} (m/s^{2}) | a_{z} (m/s^{2}) |
---|---|---|---|---|

+Max * | 2.06 | 1.53 | 1.88 | 2.93 |

+Mean * | 0.67 | 0.54 | 0.76 | 0.86 |

+Mean_{1/3} * | 1.18 | 0.83 | 1.25 | 1.34 |

+Mean_{1/10} * | 1.55 | 1.06 | 1.59 | 1.68 |

−Max * | −0.88 | −1.35 | −2.04 | −1.45 |

−Mean * | −0.43 | −0.52 | −0.78 | −0.47 |

−Mean_{1/3} * | −0.65 | −0.82 | −1.25 | −0.89 |

−Mean_{1/10} * | −0.80 | −1.01 | −1.58 | −1.34 |

_{1/3}is equal to sorting all statistical values first, and then taking the average value of the top 1/3, and Mean

_{1/10}is the average value of the top 1/10.

Composition | Materials | Height (m) | Radius (m) | Density (kg/m ^{3}) | Coefficient of Drag Force C _{D} | Coefficient of Inertia Force C _{I} | Damping C _{d}(Ns/m) | Elastic Modulus E (Pa) |
---|---|---|---|---|---|---|---|---|

Slender Cylinder | HDPE | 4 | 0.32 | 700 | (0.8, 0.8, 0.8) | (1.2, 1.2, 1.2) | / | / |

Mooring Cable | PE | 16 | 0.01 | 3570 | (0.8, 0.8, 0.8) | (1.2, 1.2, 1.2) | 1.0 × 10^{4} | 2.38 × 10^{9} |

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**MDPI and ACS Style**

Liu, C.; Dong, Z.; Zhou, Y.; Pan, Y.
A Computation Method for the Typhoon Waves Using the Field Wave Spectrum. *Sustainability* **2022**, *14*, 7347.
https://doi.org/10.3390/su14127347

**AMA Style**

Liu C, Dong Z, Zhou Y, Pan Y.
A Computation Method for the Typhoon Waves Using the Field Wave Spectrum. *Sustainability*. 2022; 14(12):7347.
https://doi.org/10.3390/su14127347

**Chicago/Turabian Style**

Liu, Can, Zhiyong Dong, Yang Zhou, and Yun Pan.
2022. "A Computation Method for the Typhoon Waves Using the Field Wave Spectrum" *Sustainability* 14, no. 12: 7347.
https://doi.org/10.3390/su14127347