# 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

- Longuet-Higgins, M.S.; Cartwright, D.E.; Smith, N.D. Observation of the directional spectrum of sea waves using the motion of a floating buoy. In Proceedings of the Ocean Wave Spectra, Easton, MD, USA, 1–4 May 1961; Prentice-Hall Inc.: Upper Saddle River, NJ, USA, 1963; pp. 113–136. [Google Scholar]
- Gilchrist, A.W.R. The directional spectrum of ocean waves: An experimental investigation of certain prediction of the Miles-Phillps theory of wave generation. J. Fluid Mech.
**1966**, 25, 795–816. [Google Scholar] [CrossRef][Green Version] - Fujinawa, Y. Measurements of Directional Spectrum of Wind Waves using an Array of Wave Detectors Part II. Field Observation. J. Oceanogr. Soc. Jpn.
**1975**, 31, 25–42. [Google Scholar] [CrossRef] - Jeffreys, H. On the formation of water waves by wind. R. Soc.
**1925**, 107, 189–206. [Google Scholar] - Miles, J.W. On the generation of surface waves by shear flows. J. Fluid Mech.
**1957**, 3, 185–204. [Google Scholar] [CrossRef] - Mei, C.C.; Liu, P.L. Surface waves and coastal dynamics. Annu. Rev. Fluid Mech.
**1993**, 25, 215–240. [Google Scholar] [CrossRef] - Peter, J. The Interaction of Ocean Waves and Wind; Cambridge University Press: Cambridge, UK, 2004; pp. 74–167. [Google Scholar]
- Toba, Y. Drop production by bursting of air bubbles on the sea surface. III. Study by use of a wind flume. J. Oceanogr. Soc. Jpn.
**1961**, 40, 63–64. [Google Scholar] - Kuninshi, H. An experimental study on the generation and growth of wind waves. Bull. Disaster Prev. Res. Inst. Kyoto Univ.
**1963**, 61, 1–41. [Google Scholar] - Wu, J. Laboratory studies of wind-wave interactions. J. Fluid Mech.
**1968**, 34, 99–111. [Google Scholar] [CrossRef] - Wu, T. On the formation of streaks on wind-driven water surfaces. Geophys. Res. Lett.
**2001**, 28, 3959–3962. [Google Scholar] - Lin, M.Y.; Moeng, C.H.; Tsai, W.T.; Sullivan, P.P.; Belcher, S.E. Direct numerical simulation of wind-wave generation processes. J. Fluid Mech.
**2008**, 616, 1–30. [Google Scholar] [CrossRef][Green Version] - Longo, S. Wind-generated water waves in a wind tunnel: Free surface statistics, wind friction and mean air flow properties. Coast. Eng.
**2012**, 61, 27–41. [Google Scholar] [CrossRef] - Longo, S. Turbulent flow structure in experimental laboratory wind-generated gravity waves. Coast. Eng.
**2012**, 64, 1–15. [Google Scholar] [CrossRef] - Longo, S. Study of the turbulence in the air-side and water-side boundary layers in experimental laboratory wind induced surface waves. Coast. Eng.
**2012**, 69, 67–81. [Google Scholar] [CrossRef] - Wu, Z.; Chen, J.; Jiang, C.; Liu, X.; Deng, B.; Qu, K.; He, Z.; Xie, Z. Numerical investigation of Typhoon Kai-tak (1213) using a mesoscale coupled WRF-ROMS model—Part Ⅱ: Wave effects. Ocean Eng.
**2020**, 196, 106805. [Google Scholar] [CrossRef] - Sibul, O.J.; Tickner, E.G. Model Study of Overtopping of Wind Generated Waves on Levees with Slopes 1:3 and 1:6. U.S. Beach Eros. Board Tech. Memo.
**1956**, 80, 1–27. [Google Scholar] - Kinsman, B. Wind Waves: Their Generation and Propagation on the Ocean Surface; Prentice-Hall: Englewood Cliffs, NJ, USA, 1965; pp. 543–575. [Google Scholar]
- Walker, J.R. Recreational Surf Parameters; Technical Report; Look Laboratory of Oceanographic Engineering, Hawaii University: Honolulu, HI, USA, 1974; pp. 73–130. [Google Scholar]
- Kawata, Y. Wave Breaking under Storm Condition. Coast. Eng.
**1995**, 36, 330–339. [Google Scholar] - Galloway, J.S.; Collins, M.B.; Moran, A.D. Onshore/offshore wind influence on breaking waves: An empirical study. Coast. Eng.
**1989**, 13, 305–323. [Google Scholar] [CrossRef] - Douglass, S.L. Influence of wind on breaking waves. J. Waterw. Port Coast. Ocean Eng.
**1990**, 116, 651–663. [Google Scholar] [CrossRef] - King, D.M.; Baker, C.J. Changes to wave parameters in the surf zone due to wind effects. J. Hydraul. Res.
**1996**, 34, 55–76. [Google Scholar] [CrossRef] - Feddersen, F.; Veron, F. Wind effects on shoaling wave shape. J. Phys. Oceanogr.
**2005**, 35, 1223–1228. [Google Scholar] [CrossRef][Green Version] - Kharif, C.; Giovanangeli, J.P.; Touboul, J.; Grare, L.; Pelinovsky, E. Influence of wind on extreme wave events: Experimental and numerical approaches. J. Fluid Mech.
**2008**, 594, 209–247. [Google Scholar] [CrossRef][Green Version] - Tian, Z.; Choi, W. Evolution of deep-water waves under wind forcing and wave breaking effects: Numerical simulations and experimental assessment. Eur. J. Mech. B Fluid
**2013**, 41, 11–22. [Google Scholar] [CrossRef] - Liu, K. Modeling Wind Effects on Shallow Water Waves. J. Waterw. Port Coast. Ocean Eng.
**2016**, 142, 1–8. [Google Scholar] [CrossRef] - Hasan, S.A. Numerical modelling of wind-modified focused waves in a numerical wave tank. Ocean Eng.
**2018**, 160, 276–300. [Google Scholar] [CrossRef] - Xie, Z. Numerical modelling of wind effects on breaking waves in the surf zone. Ocean Dyn.
**2017**, 67, 1251–1261. [Google Scholar] [CrossRef][Green Version] - Chen, H. Transformation of Nonlinear Waves in the Presence of Wind, Current, and Vegetation. Ph.D. Thesis, Maine University, Orono, ME, USA, 2017. [Google Scholar]
- Lafrati, A. Effects of the wind on the breaking of modulated wave trains. Eur. J. Mech. B Fluid
**2018**, 73, 1–17. [Google Scholar] - Zou, Q.; Chen, H. Numerical simulation of wind effects on the evolution of freak waves. In Proceedings of the 26th International Ocean and Polar Engineering Conference, Rhodes, Greece, 26 June–2 July 2016. [Google Scholar]
- Yang, D.; Shen, L. Direct-simulation-based study of turbulent flow over various waving boundaries. J. Fluid Mech.
**2010**, 650, 131–180. [Google Scholar] [CrossRef][Green Version] - Lakehal, D.; Meier, M.; Fulgosi, M. Interface tracking towards the direct simulation of heat and mass transfer in multiphase flows. Int. J. Heat Fluid Flow
**2002**, 23, 242–257. [Google Scholar] [CrossRef] - Fulgosi, M.; Lakehal, D.; Banerjee, S.; De Angelis, V. Direct numerical simulation of turbulence in a sheared air–water flow with a deformable interface. J. Fluid Mech.
**2003**, 482, 319–345. [Google Scholar] [CrossRef][Green Version] - Lubin, P.; Vincent, S.; Abadie, S. Three-dimensional Large Eddy Simulation of air entrainment under plunging breaking waves. Coast. Eng.
**2006**, 53, 631–655. [Google Scholar] [CrossRef][Green Version] - Hirt, C.W.; Nichols, B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Scardovelli, R.; Zaleski, S. Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech.
**1999**, 31, 567–603. [Google Scholar] [CrossRef][Green Version] - Sethian, J.A.; Smereka, P. Level set methods for fluid interfaces. Annu. Rev. Fluid Mech.
**2003**, 35, 341–372. [Google Scholar] [CrossRef][Green Version] - Zhang, Y.; Zou, Q.; Greaves, D.; Reeve, D.; Hunt-Raby, A.; Graham, D.; Lv, X. A level set immersed boundary method for water entry and exit. Commun. Comput. Phys.
**2010**, 8, 265–288. [Google Scholar] [CrossRef] - Hansen, J. A theoretical and experimental study of undertow. In Proceedings of the 19th International Conference on Coastal Engineering, Houston, TX, USA, 3–7 September 1984. [Google Scholar]
- Ting, F.; Kirby, J. Observation of undertow and turbulence in a laboratory surf zone. Coast. Eng.
**1994**, 24, 51–80. [Google Scholar] [CrossRef] - Ting, F.; Kirby, J. Dynamics of surf zone turbulence in a strong plunging breaker. Coast. Eng.
**1995**, 24, 177–204. [Google Scholar] [CrossRef] - Bakhtyar, R. Numerical simulation of surf–swash zone motions and turbulent flow. Adv. Water Resour.
**2009**, 32, 250–263. [Google Scholar] [CrossRef] - Cavallaro, L.; Scandura, P.; Foti, E. Turbulence-induced steady streaming in an oscillating boundary layer: On the reliability of turbulence closure models. Coast. Eng.
**2011**, 58, 290–304. [Google Scholar] [CrossRef] - Kranenburg, W.M.; Ribberink, J.S.; Uittenbogaard, R.E.; Hulscher, S.J.M.H. Net currents in the wave bottom boundary layer: On waveshape streaming and progressive wave streaming. J. Geophys. Res. Earth
**2012**, 117, F03005. [Google Scholar] [CrossRef][Green Version] - Blondeaux, P.; Vittori, G.; Porcile, G. Modeling the turbulent boundary layer at the bottom of sea wave. Coast. Eng.
**2018**, 141, 12–23. [Google Scholar] [CrossRef] - Rusche, H. Computational Fluid Dynamics of Dispersed Two-Phase Flows at High Phase Fractions. Ph.D. Thesis, Imperial College, London, UK, 2002. [Google Scholar]
- Bogey, C. Large eddy simulations of round free jets using explicit filtering with/without dynamic Smagorinsky model. Int. J. Heat Fluid Flow
**2006**, 27, 603–610. [Google Scholar] [CrossRef] - Higurea, P.; Lara, J.L.; Losada, I.J. Realistic wave generation and active wave absorbtion for Navior-Stokes models. Coast. Eng.
**2013**, 71, 102–118. [Google Scholar] [CrossRef] - Battjes, J.A. Surf similarity. In Proceedings of the 14th International Conference on Coastal Engineering, Honolulu, HI, USA, 24–28 June 1974. [Google Scholar]
- Inami, T. Study on Wave Reflection Coefficient and Wave Runup Height on a Slope. In Proceedings of the 26th International Ocean and Polar Engineering Conference, Rhodes, Greece, 26 June–2 July 2016. [Google Scholar]
- Wu, J. Froude number scaling of wind-stress coefficients. J. Atmos. Sci.
**1969**, 26, 408–413. [Google Scholar] [CrossRef][Green Version] - Collins, J.I.; Weir, W. Probabilities of Breaking Wave Characteristics in the surf zone. In Proceedings of the 12th International Conference on Coastal Engineering, Pasadena, CA, USA, 13–18 September 1970. [Google Scholar]
- Le Mehhaute, B. On the breaking of waves arriving at an angle to the shore. J. Hydraul Res.
**1967**, 5, 67–80. [Google Scholar] [CrossRef] - Galvin, C.J. Breaker type classification of three laboratory beaches. J. Geophys. Res. Atmos.
**1968**, 73, 3651–3659. [Google Scholar] [CrossRef] - Komori, S.; Nagaosa, R.; Murakami, Y. Turbulence structure and mass transfer across a sheared air-water interface in wind-driven turbulence. J. Fluid Mech.
**1993**, 249, 161–183. [Google Scholar] [CrossRef] - William, K.G. Lectures in Turbulence for the 21st Century; Chalmers University of Technology: Gothenburg, Sweden, 2013; pp. 59–64. [Google Scholar]
- Smagorinsky, J. General circulation experiments with the primitive equations: I. the basic experiment. Mon. Weather Rev.
**1963**, 91, 99–164. [Google Scholar] [CrossRef] - Christensen, E.D.; Walstra, D.J.; Emarat, N. Vertical variation of the flow across the surf zone. Coast. Eng.
**2002**, 45, 169–198. [Google Scholar] [CrossRef]

**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