# Study on the Nearshore Evolution of Regular Waves under Steady Wind

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## Abstract

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## 1. Introduction

## 2. Numerical Methods

#### 2.1. Governing Equations

#### 2.2. Turbulence Model

#### 2.3. Boundary Conditions

## 3. Experimental and Numerical Setup

#### 3.1. Experimental Setup

#### 3.2. Numerical Setup

## 4. Model Validation

## 5. Results and Discussions

#### 5.1. Wave Breaking

#### 5.2. Turbulent Flow

#### 5.3. Undertow

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 5.**Time series of horizontal (

**a**) and vertical (

**b**) velocities in the absence of wind at a depth of 0.2 m. The results are shown for case W0H07.

**Figure 6.**Time series of horizontal (

**a**) and vertical (

**b**) velocities in the presence of wind at a depth of 0.2 m. The results are shown for case W5H07.

**Figure 7.**Variation of free surface in the breaking processes. The results are shown for cases W0H07 (first row), W3H07 (second row), and W5H07 (third row), respectively.

**Figure 8.**Wave breaking processes at the initial breaking time. The results are shown for cases (

**a**) W0H05~W5H05, (

**b**) W0H07~W5H07, (

**c**) W0H09~W5H09, and (

**d**) W0H11~W5H11.

**Figure 9.**The correlation between the wave breaking locations and surf similarity parameter ${\xi}_{0}$. Symbols ○, +, ∗ denote numerical data with a slope of 1:10, 1:16, and 1:20, where the wind speed of 0 m/s, 3 m/s, and 5 m/s are labeled by blue, red, and black, respectively. Dotted lines by Equation (15) represent the calculated results for the appropriate slope.

**Figure 10.**The distribution of the cross-shore average (

**a**) wave height and (

**b**) water level. The results are shown for cases W0H07 (blue), W3H07 (red), and W5H07 (black), respectively.

**Figure 11.**The correlation between the breaker index of ${H}_{b}/{h}_{b}$ and surf similarity parameter ${\xi}_{0}$. The dotted lines are calculated by Galvin (1968) and the illustration of other symbols can be seen in the note of Figure 9.

**Figure 12.**Cross spectral analysis of ${u}^{\prime}$ and ${w}^{\prime}$ at $\left(x-{x}_{b}\right)/h=-1.25$ with three depths of $z=0.37\mathrm{m}$ (blue), $z=0.35\mathrm{m}$ (red), and $z=0.33\mathrm{m}$ (black). The results are shown for cases W0H07 (first row), W3H07 (second row), and W5H07 (third row), respectively.

**Figure 13.**The snapshots of the TKE (turbulent kinetic energy) in the breaking processes under different wind speeds. The results are shown for cases W0H07 (first row), W3H07 (second row), and W5H07 (third row), respectively.

**Figure 14.**The snapshots of the TDR (turbulent dissipation rate) in the breaking processes under different wind speeds. The results are shown for cases W0H07 (first row), W3H07 (second row), and W5H07 (third row), respectively.

**Figure 15.**Time series of (

**a**) TKE and (

**b**) TDR at $\left(x-{x}_{b}\right)/h=-0.35$ before breaking. The results are shown for cases W0H07 (blue), W3H07 (red), and W5H07 (black), respectively.

**Figure 16.**Time series of (

**a**) TKE and (

**b**) TDR at $\left(x-{x}_{b}\right)/h=0.375$ after breaking. The results are shown for cases W0H07 (blue), W3H07 (red), and W5H07 (black), respectively.

**Figure 17.**Variation of time-averaged TKE and TDR with depth under different wind speeds at (

**a**,

**b**) $\left(x-{x}_{b}\right)/h=-0.35$ before breaking and (

**c**,

**d**) $\left(x-{x}_{b}\right)/h=0.375$ after breaking. The results are shown for case W0H07 (blue), W3H07 (red), and W5H07 (black), respectively.

**Figure 18.**(

**a**) Horizontal and (

**b**) vertical time-averaged velocity profile at $x=0\mathrm{m}$ (triangle), $x=1\mathrm{m}$ (square) and $x=2\mathrm{m}$ (circle). The results are shown for cases W0H07 (blue), W3H07 (red), and W5H07 (black), respectively.

**Figure 19.**Skewness (top panel) and kurtosis (bottom panel) of horizontal and vertical velocity at (

**a**,

**b**) $x=0\mathrm{m}$, (

**c**,

**d**) $x=1\mathrm{m}$, and (

**e**,

**f**) $x=2\mathrm{m}$. The results are shown for cases W0H07, W03H07, and W05H07, respectively.

Case | W (m/s) | H_{0} (cm) | T (s) | h (m) | ξ_{0} |
---|---|---|---|---|---|

W0H05~W0H11 | 0.0 | 5, 7, 9,11 | 1.5 | 0.4 | plunging |

W3H05~W3H11 | 3.0 | 5, 7, 9,11 | 1.5 | 0.4 | plunging |

W5H05~W5H11 | 5.0 | 5, 7, 9,11 | 1.5 | 0.4 | plunging |

x (m) | h_{x} (m) | W (m/s) | Mean Flow Rate, q (m^{3}/s/m) |
---|---|---|---|

0 | 0.4 | 0, 3, 5 | −0.0011, −0.0014, −0.0016 |

1 | 0.3 | 0, 3, 5 | −0.0051, −0.0054, −0.0056 |

2 | 0.2 | 0, 3, 5 | −0.0024, −0.0029, −0.0033 |

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**MDPI and ACS Style**

Jiang, C.; Yang, Y.; Deng, B.
Study on the Nearshore Evolution of Regular Waves under Steady Wind. *Water* **2020**, *12*, 686.
https://doi.org/10.3390/w12030686

**AMA Style**

Jiang C, Yang Y, Deng B.
Study on the Nearshore Evolution of Regular Waves under Steady Wind. *Water*. 2020; 12(3):686.
https://doi.org/10.3390/w12030686

**Chicago/Turabian Style**

Jiang, Changbo, Yang Yang, and Bin Deng.
2020. "Study on the Nearshore Evolution of Regular Waves under Steady Wind" *Water* 12, no. 3: 686.
https://doi.org/10.3390/w12030686