# Investigation of the Spiral Wave Generation and Propagation on a Numerical Circular Wave Tank Model

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## Abstract

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## 1. Introduction

## 2. Governing Equations

## 3. Methodology of the Numerical Model

#### 3.1. Grid Generation

#### 3.2. Discretization of the Governing Equations

## 4. Parameters and Experimental Conditions of the Wave Tank Model

#### 4.1. Numerical Calculations

#### 4.2. Hydraulic Model Experiment

## 5. Results and Discussion

#### 5.1. Spiral Wave Generation

#### 5.2. Model Validation

#### 5.3. Wave Characteristics

#### 5.4. Comparisons for Different Conditions

#### 5.5. Cross-Shore and Longshore Velocity Distribution

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Grid arrangement in computational domain: (

**a**) Regular grid system, (

**b**) Zonal embedded grid system, (

**c**) staggered grid arrangement at a normal cell, and (

**d**) staggered grid arrangement at the interface of coarse and fine blocks cells.

**Figure 5.**Spiral wave generation snapshots at different times: (

**a**) 1.305 s, (

**b**) 3.055 s, and (

**c**) 4.105 s (wave period 2.0 s).

**Figure 6.**Water surface elevation profiles at corresponding wave gauges positions under different wave conditions: (

**a**) T-2.22 s ($r=264\mathrm{cm}$), (

**b**) T-2.22 s ($r=300\mathrm{cm}$ ), (

**c**) T-2.0 s ($r=336\mathrm{cm}$ ), and (

**d**) T-1.82 s ($r=369\mathrm{cm}$ ).

**Figure 7.**Wave heights at corresponding wave conditions at different measured positions: (

**a**) T-2.22 s ($r=264\mathrm{cm}$), (

**b**) T-2.22 s ($r=300\mathrm{cm}$), (

**c**) T-2.22 s ($r=336\mathrm{cm}$), and (

**d**) T-1.82 s ($r=369\mathrm{cm}$). Black line-numerical results and red line-experimental results.

**Figure 8.**Water surface elevation profiles at the corresponding wave gauge positions in case 1 (T-1.80 s): (

**a**) W1 ($r=234\mathrm{cm}$), (

**b**) W2 ($r=264\mathrm{cm}$), (

**c**) W3 ($r=300\mathrm{cm}$), (

**d**) W4 ($r=336\mathrm{cm}$), (

**e**) W5 ($r=369\mathrm{cm})$, and (

**f**) W6 ($r=405\mathrm{cm}$).

**Figure 9.**Variation in maximum water levels at W3 ($r=300cm$), W4 ($r=336\mathrm{cm}$), W5 ($r=369\mathrm{cm}$), and W6 ($r=405\mathrm{cm}$) for cross-section 1 of (

**a**) case 1 (T-1.80 s), (

**b**) case 2 (T-2.0 s), and (

**c**) case 3 (T-2.25 s).

**Figure 10.**Variation in maximum water levels for same radial distances ($r=369\mathrm{cm}$ and $r=405\mathrm{cm}$ ) at five different cross-sections in (

**a**,

**b**) case 1 (T-1.80 s), (

**c**,

**d**) case 2 (T-2.0 s), and (

**e**,

**f**) case 3 (T-2.25 s).

**Figure 11.**Variation in mean water levels per wave for cross-section 1 at radial distances of 300 cm (Eta 3), 336 cm (Eta 4), 426 cm (Eta 7), and 432 cm (Eta 8) in (

**a**) case 1 (T-1.80 s), (

**b**) case 2 (T-2.0 s), and (

**c**) case 3 (T-2.25 s).

**Figure 12.**Variation in mean water levels per wave for cross-sections 1–5 at radial distances of 426 cm in (

**a**) case 1 (T-1.80 s), and (

**b**) case 2 (T-2.0 s).

**Figure 13.**Variation in cross-shore velocity profiles for cross-section 1 at (

**a**) U3 ($r=300\mathrm{cm}$), (

**b**) U4 ($r=336\mathrm{cm}$), (

**c**) U5 ($r=369\mathrm{cm}$), and (

**d**) U6 ($r=405\mathrm{cm}$) in case 1 (T-1.80 s).

**Figure 14.**Cross-shore flow velocity at U4 ($r=336\mathrm{cm}$) in (

**a**) case 1 (T-1.80 s), (

**b**) case 2 (T-2.0 s), and (

**c**) case 3 (T-2.25 s).

**Figure 15.**Longshore-shore velocity profiles in cross-section 1 at (

**a**) V3 ($r=300\mathrm{cm}$), (

**b**) V4 ($r=336\mathrm{cm}$), (

**c**) V5 ($r=369\mathrm{cm}$), and (

**d**) V6 ($r=405\mathrm{cm}$) in case 2 (T-2.0 s).

Zone | M | 5 |
---|---|---|

Number of grids in radial direction | 1 | 5 |

2–4 | 5, 10, 20 | |

5 | 80 | |

Number of grids in azimuthal direction | 1–5 | 60, 120, 240, 480, 960 |

Number of grids in vertical direction | 1–5 | 88 |

Cell size | Δr | 0.03 m | Total calculation Time | 60 s | |

Δz | 0.75 cm, 1.5 cm | Time step | dt | Max. 0.005 s | |

Water depth | h | 0.18 m | Measured cross-section | 5 | |

Wave period | T | 1.80 s, 2.0 s, 2.25 s, 2.50 s | Median particle size of the beach | ${d}_{50}$ | 0.4 mm |

Wave height | H | 1.5 m | Nonlinear drag coefficient | ${C}_{D1}$ | 0.45 |

Water density | ${\rho}_{w}$ | 1000 kg/m^{3} | Linear drag coefficient | ${C}_{D2}$ | 25.0 |

Air density | ${\rho}_{a}$ | 1.2 kg/m^{3} | Kinematic viscosity of air | ${\mu}_{a}$ | 1.8 × 10−5 Pa.s |

Added mass coefficient | C_{A} | −0.04 | Kinematic viscosity of water | ${\mu}_{w}$ | 1.01 × 10−3 Pa.s |

Case | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Period | 1.82 s | 2.0 s | 2.22 s | 2.50 s |

Water depth | 18 cm | |||

Initial Terrain | 1:7 Uniform Slope |

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## Share and Cite

**MDPI and ACS Style**

Islam, M.S.; Nakamura, T.; Cho, Y.-H.; Mizutani, N.
Investigation of the Spiral Wave Generation and Propagation on a Numerical Circular Wave Tank Model. *J. Mar. Sci. Eng.* **2023**, *11*, 388.
https://doi.org/10.3390/jmse11020388

**AMA Style**

Islam MS, Nakamura T, Cho Y-H, Mizutani N.
Investigation of the Spiral Wave Generation and Propagation on a Numerical Circular Wave Tank Model. *Journal of Marine Science and Engineering*. 2023; 11(2):388.
https://doi.org/10.3390/jmse11020388

**Chicago/Turabian Style**

Islam, Mohammad Shaiful, Tomoaki Nakamura, Yong-Hwan Cho, and Norimi Mizutani.
2023. "Investigation of the Spiral Wave Generation and Propagation on a Numerical Circular Wave Tank Model" *Journal of Marine Science and Engineering* 11, no. 2: 388.
https://doi.org/10.3390/jmse11020388