Study on the Nearshore Evolution of Regular Waves Under Steady Wind

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

Changbo Jiang 1,2, Yang Yang 1,2,* and Bin Deng 1,2

1 School of Hydraulic Engineering, Changsha University of Science & Technology, Changsha 410114, China; [email protected] (C.J.); [email protected] (B.D.) 2 Key Laboratory of Water, Sediment Sciences & Flood Hazard Prevention of Hunan Province, Changsha 410114, China * Correspondence: [email protected]

Received: 22 January 2020; Accepted: 29 February 2020; Published: 3 March 2020

Abstract: We present a study on regular wave propagation on a sloping bed under the action of steady wind, which is of a great significance to complement and replenish the interaction mechanisms of nearshore wave and wind. Physical experiments were conducted in a wind-wave flume, and the corresponding numerical model was constructed based on the solver Waves2FOAM in OpenFOAM, with large-eddy simulation (LES) used to investigate the turbulent flow. The comparisons between the measured and calculated results of the free surface elevation and flow velocity indicated that the numerical model could predict the associated hydrodynamic characteristics of a nearshore wave regardless of the presence or absence of wind. The results showed that wind had a significant impact on nearshore wave evolution. It was found that under the same wind speed coverage constraint, wave breaking occurred ahead of time. The smaller the surf similarity ξ0 was, the higher the dispersion degree of wave breaking locations would be, and the breaker index of Hb/hb increased with wind speed under the same incident wave height. The main components of analysis for turbulent flow were the results of the cross-spectrum, the TKE (turbulent kinetic energy), and TDR (turbulent dissipation rate). The cross-spectrum illustrated that wind enhanced the degree of coherence of the residual velocity components and aggravated turbulence. The TKE indicated that in regions near the water surface, wind speed made it considerably larger and the average level rapidly decreased with depth. The TDR exhibited that the significant effect of wind was merely imposed after breaking, wherein the turbulence penetrated the deeper flow and the average level generally rose. The velocity profile on the slope showed that the wind accelerated the undertow, and the moment statistics indicated that the velocity distribution deviated gradually from the Gaussian distribution to the right.

Keywords: wind; regular wave; wave breaking; turbulent ﬂow; undertow

1. Introduction Wind is one of the driving forces of surface waves. When wind blows over the water surface, the water surface will fluctuate due to wind pressure, and the fluctuation will further develop into wind wave in the processes of continuous momentum and energy transfer in air-sea interaction. Under extreme climate, e.g., extreme wind waves and storm surges, it endangers the safety of coastal residents and buildings over the long term. In a marine dynamic environment, the mechanism of wind wave formation and propagation is a research topic that concerns many researchers. Abundant research results have been obtained ranging from field observation [1–3] to theoretical analysis [4–7], as well as experimental [8–10] and numerical study [11–16]. These studies are mainly focused on the generation of wind wave and air-sea interaction, rather than the mechanism of wind effect on waves. However, wind waves are usually inseparable from wind field when they propagate from deep water to shallow

Water 2020, 12, 686; doi:10.3390/w12030686 www.mdpi.com/journal/water Water 2020, 12, 686 2 of 21 water, and the law of wave propagation with the action of wind is more complex than that of pure wind wave or swell in coastal regions. Researchers have realized the action of wind on nearshore wave propagation since the 1950s, and managed to obtain some results [17]. Until now, research on this topic has been mainly implemented by experimental and numerical methods that investigate the relationship among different wind speed, wind direction, together with the shape, height, and location of a breaking wave. For field observation, it is difficult to obtain the temporal and spatial evolution law of waves due to the large scale of strong winds such as typhoons, and thus only some qualitative investigations were conducted where the existence of wind can change the type of breaking coastal waves [18–20]. Galloway et al. (1989) [21] conducted field measurements and found that offshore wind and onshore wind increased the number of plunging and spilling, respectively. Some researchers have carried out laboratory experiments and verified observations and findings of Galloway et al. For example, Douglass (1990) [22], King and Baker (1996) [23] had experimentally studied the influence of local wind on a nearshore breaking wave, and found that wind had a significant impact on the location, geometry, and type of breaking wave. Onshore wind was found to cause waves to break ahead of time in offshore deep water, whereas the offshore wind was found to delay the wave breaking in nearshore shallow water. Meanwhile, Feddersen and Veron (2005) [24] stated that wind could enhance the wave energy in shallow water and cause changes in wave shape. With respect to the numerical simulation, Kharif (2008) [25], Tian and Choi (2013) [26], Liu (2016) [27], and Hasan (2018) [28] applied the Jeffery’s shelter theory and Miles’ shear instability theory [4,5] to simulate the impact of wind on waves. While the core of the two theories is to predefine the wind pressure terms through empirical parameterization, it is essentially the single-phase flow and cannot truly embody the interaction processes between wind and wave. Recently, the enhanced computer performance had enabled the numerical simulation of multiphase flow, in which the two-phases flow model had been widely employed in the coupling of wind and wave [29–31]. The two-phases flow model can simultaneously perform the coupling between two different fluids, which avoids empirical parameterization, making it more reasonable compared with the single-flow model by principle and applicability [32]. The existing two-phase models can be subdivided into two kinds, one of which is solved in the air and water respectively, and the predicted air and water flow are coupled by the velocity continuity and stress balance at the free surface [12,33]. Despite that, this kind of model is not suitable for the simulation of a free surface related to deformation, breaking, and air entrapment, etc. [34,35]. The other depends on the Navier–Stokes equation where the two phases are solved simultaneously with an assumption that treats the two phases as a kind of fluid in the fixed Eulerian grids [36]. As for the complex topological deformation of free surface, e.g., the geometric VOF (volume of fluid) method [37,38] and level set method [39,40], are usually used to fulfill free surface reconstruction and transportation. Moreover, the turbulence structure and undertow in the process of wave nearshore propagation are worth researching and some researchers have already conducted studies on the topic. Hansen (1984) [41] deduced the theoretical results for the undertow, and Ting (1994, 1995) [42,43] experimentally investigated the turbulence and undertow in the surf zone for spilling and plunging waves. It was found that the average turbulence levels were much higher and the vertical variations of undertow were much smaller for plunging waves than those for spilling waves, which were numerically confirmed by Bakhtyar (2009) [44]. In addition, research on the turbulent boundary layer at the bottom of a nearshore wave has received attention. Cavallaro (2011) [45], Kranenburg (2012) [46], and Blondeaux (2018) [47] compared the precision and suitable conditions with respect to the bottom shear stress and TKE (turbulent kinetic energy), among different turbulent models. However, these studies were all conducted in such a way that the wind was not taken into account. Consequently, given the lack of experimental and numerical research, no clear conclusions have been made with respect to the wind effects on wave propagation. Likewise, the interaction between wave and wind is associated with an extremely complex physical mechanism that has not been completely solved. Therefore, it is of great significance to complement and replenish relevant studies, which is the main purpose of this paper. Water 2020, 12, 686 3 of 21

In this paper, the 2nd Stokes wave propagation on the slope under steady wind was investigated. The solver Waves2FOAM in OpenFOAM, by equipping it with the wave-making module and active sponge layer wave attenuation technology, was applied to simulate the wind and wave field. Additionally, the VOF method with an artificial compression term was used to capture air-water interface, with a large-eddy simulation (LES) for the turbulent flow. The remaining part of the paper is organized as follows. In Section2, the governing equation, turbulence model, and boundary conditions are briefly introduced. In Section3, the experimental and numerical setup are described. The accuracy of the numerical model is verified by experimental data in Section4, and the hydrodynamic characteristics of wave under wind action are discussed in Section5. A summary of the main findings are given in Section6.

2. Numerical Methods

2.1. Governing Equations On the basis of the mass and momentum conservation of ﬂuids, the continuity equation and momentum equation (Navier–Stokes equation) for incompressible viscous ﬂuid are expressed as follows:

∂ρ + (ρU) = 0 (1) ∂t ∇ · ∂ρU + (ρUUT) = p + (µ U + ρτ) g X ρ + f , (2) ∂t ∇ · −∇ ∇ · ∇ − · ∇ σ where t is the time, ρ is the fluid density, U = (u, w) is the fluid velocity vector, X = (x, z) is the 1 T 2 Cartesian coordinates, τ = µt U + ( U) kI is the stress tensor in which µt denotes the dynamic ρ ∇ ∇ − 3 viscosity, k is the turbulent kinetic energy and I is the Kronecker delta function, g is gravitational acceleration, and fσ is the surface tension. The VOF model has been adopted to capture the interface of the two-phase flow with artificial compression term [37,48]. The transport equation is expressed as:

∂αp + αpU + αp 1 αp Uc = 0 (3) ∂t ∇ · ∇ · − where αp represents the volume fraction of diﬀerent phase ﬂuids as follows:   0, air  αp =  1, water (4)   0 < αp < 1, inter f ace

Uc is the velocity ﬁeld only applicable to the compression interface, which can be expressed by the following formula:

h i αp U = min C U , max( U ) ∇ (5) c αp | | | | α ∇ p where the compression rate Cαp is expressed by a default of 1, and the curvature of the interface is α ∇ p expressed by . α ∇ p The VOF model treats the fluid of different phases as mixed phases, wherein the fluid characteristics, involving the fluid density and dynamic viscosity, are mixed characteristics that can be written in the form of the following formula: ( ρ = αpρw + (1 αp)ρa − . (6) µ = αpµw + (1 αp)µa − Water 2020, 12, 686 4 of 21

2.2. Turbulence Model The turbulence ﬂow is accompanied with the processes of nearshore wave propagation, especially in wave breaking. In this paper, LES was adopted to model the turbulent ﬂow in the processes of wind-wave interaction, which can be employed to calculate the turbulent structures in various scales and possesses a certain superiority compared with other models such as direct numerical simulation (DNS) and the Reynolds Average Navier–Stokes simulation (RANS) in terms of time cost and computation load. For details, one can refer to Bogey (2006) [49], and thus they are not repeated here.

2.3. Boundary Conditions The realization ﬂow of the inlet for wave making and autonomous wave absorption was depicted in considerable detail in Higurea (2013) [50], which will not be further covered here. The velocity function and free surface function derived from the 2nd Stokes wave are as follows: " # coshk(z + h) 3 cosh2k(z + h) u = Aσ cosθ + Ak cos2θ (7) coshkh 4 sinh4kh " # sinhk(z + h) 3 sinh2k(z + h) w = Aσ sinθ + Ak sin2θ (8) coshkh 4 sinh4kh A2k coshkh(cosh2kh + 2) η = Acosθ + cos2θ (9) 4 sinh3kh where A denotes wave amplitude. The angular frequency σ = 2π/T and T is the wave period. For the wave number k = 2π/L and L is the wave length. θ = kx ct denotes phase and the wave speed is p − c = g(H + h). h is the water depth and η is the free surface elevation. In addition, the wave absorption function is as follows:

χ3.5 e R 1 αR(χR) = 1 − (10) − e 1 − φ = φ + (1 αR)φtarget (11) calculated − where χR [0, 1] and φ denotes U or α. ∈ 3. Experimental and Numerical Setup

3.1. Experimental Setup The physical experiment was carried out in a wind wave flume located in the Hydraulics Modeling Laboratory of the Changsha University of Science and Technology. The flume is 45 m long, 0.8 m wide, and 1 m deep. One end of the flume was equipped with a flap-typed wave maker (Dalian University of Technology, China), capable of generating various kinds of waves covering a larger range of wave heights and periods, wherein the 2nd Stokes wave was used in this study. A GXF diagonal fan (Bejing Jifeng Shunda Modern Ventilation Equipment Co., Ltd., China) was installed at the other end of the flume, which could create the wind speed with a range of 0~10 m/s. The schematic layout of the physical model is illustrated in Figure1a. The flume (see Figure1b) was covered by the roof of galvanized iron sheet. To prevent the wind from escaping from the gaps between the flume walls and cover plate, sealing plasticine was used to fill the gaps. Referring to the natural beach slope angle and the setting of physical models from Sibul and Ticker (1956) [17], Battjes (1974) [51], and Inami (2016) [52], a slope beach of 1:10 (see Figure1c) was built with the PVC (polyvinyl chloride) panel at 13.25 m from the wave maker. Water 2020,, 12,, 686x FOR PEER REVIEW 65 of 22 21

Figure 1. (a) A sketch of the experimental setup. (b) Wind-wave flume. (c) Slope model. Figure 1. (a) A sketch of the experimental setup. (b) Wind-wave flume. (c) Slope model. Due to the fact that the coastal water depth and wind conditions are too complex to observe and3.2. Numerical measure inSetup the field environment, the model scale was not strictly considered in this study in viewIn of accordance any prototype. with Tothe examine experimental the rationality setup in ofFigure the magnitude1, the main between aspects theof the wind experimental speed and wavemodel elements, were reproduced a scale factorby setting of 1:20 the was relevant roughly numerical chosen boundary to calculate conditions. the relevant According variables to Figure of the prototype2, the wave and making experiment. and wind According speed conditions to the Froud were similarity set at law,the inlet. the range Meanwhile, of incident in an wave attempt heights to wasweaken 3.0~11.0 the effect cm, whichof wave was reflection, comparable the autonomous to the prototype wave of absorption 1.0~2.2 m. boundary, The maximum with experimentalthe length of windthe wave speed absorption and constant zone experimental set to 3 m to water satisfy depth the parameter were set to configuration 5 m/s and 0.4 at m, the corresponding request of at to least the prototypesone wavelength, of 22.36 was m/ sadopted and 8 m, at respectively. the inlet. Boundary conditions for other faces were: A slip wall conditionThe waveat the elevationstop boundary, were the measured no-slip wall at eleven conditions cross-shore at the bottom locations and (S1-S8). slope boundary, To monitor as well the incidentas the pressure wave in theoutflow flat-bed condition zone, the at resistance-typed the outlet. As water for turbulent level sensor eddy (AWP-24-3, viscosity, Pivotal all boundary Scientific Ltd.,conditions Canada) were was zero placed gradient. at S1, withAdditionally, an accuracy a two-dimensional of 0.1 mm and a samplingcartesian frequencycoordinate of system 50 Hz was used.defined S1 as was indicated located in atFigure a position 2, where 10.8 the m coordinate away from origin the wave was set maker, at the whereslope foot the and wave the reflection positive wasdirection relatively of axis minor. x and Therefore, z represent it could the be horizontal treated as direction the reference of wave location propagation of stable and incident the vertical wave height.upward Meanwhile,of the flume, to respectively. measure the nearshore wave shoaling and breaking, the non-intrusive water levelThe sensors computational (ULS 80D, Generaldomain Acoustics was composed Ltd., Germany), of two withsections, an accuracy which was of 0.1 discretized mm and setting with samplingstructured frequency grids made of 50up Hz,of a were total equallyof 0.5 million spaced ce fromlls. In S2 the to S8flat-bed with a zone, distance the ofcoarser 0.6 m grids, on the with slope, a whereinconstant S2size was of located0.0065 m at the× 0.0065 slope m, toe. were A velocimeter used to calculate (Vectrino, the Notek steady Ltd., evolution Norway) of wind was arranged and wave at thefields, slope which toe tocould measure effectively the flow reduce velocity. the Moreover, computat toion monitor load on the the wind precondition speed, wind of anemometersassuring the withcalculation a pitot precision. tubes type Meanwhile, (DP1000-T1, the Shanghai asymptotic Lingsheng grids with Electronic an average Instrument size of 0.005 Co., Ltd.,m, which China) varied and anin the accuracy x and of 1z mm direction,/s and setting were used sampling in thefrequency slope zone of to 8 capture Hz, were the installed wave breaking. at W1 (7.6 The m) grid and size W2 (13.1reduced m) withgradually a height from of 100.0065 cm abovem to 0.0035 the still m water in the surface. x direction, Notably, and Wu owing (1969) to [53 the] proposed contraction 10 cm of asoutlet the heightit fell to of characteristic0.001 m and wind0.0035 speed m at inthe the lower experiment and upper scale. edge In the of absence the outlet, and respectively. presence of wind, The 120physical and 240 features waves of were air separatelyand water generatedwere quite within different. 3 min To and ensure 6 min, that respectively. the global The Courant data in number the last 2was min suitable were considered for the calculation valid and convergence, used for verification. the time step had to be small enough. Therefore the adaptive time-step was employed during the simulation, with a maximum value of 0.025. In the meantime, the maximum Courant number was set at 0.2.

Water 2020, 12, 686 6 of 21

3.2. Numerical Setup In accordance with the experimental setup in Figure1, the main aspects of the experimental model were reproduced by setting the relevant numerical boundary conditions. According to Figure2, the wave making and wind speed conditions were set at the inlet. Meanwhile, in an attempt to weaken the effect of wave reflection, the autonomous wave absorption boundary, with the length of the wave absorption zone set to 3 m to satisfy the parameter configuration at the request of at least one wavelength, was adopted at the inlet. Boundary conditions for other faces were: A slip wall condition at the top boundary, the no-slip wall conditions at the bottom and slope boundary, as well as the pressure outflow condition at the outlet. As for turbulent eddy viscosity, all boundary conditions were zero gradient. Additionally, a two-dimensional cartesian coordinate system was defined as indicated in Figure2, where the coordinate origin was set at the slope foot and the positive direction of axis x and z represent the horizontal direction of wave propagation and the vertical upward of the flume,Water 2020 respectively., 12, x FOR PEER REVIEW 7 of 22

Figure 2. Sketch of the numerical setup. The computational domain was composed of two sections, which was discretized with structured 4. Model Validation grids made up of a total of 0.5 million cells. In the flat-bed zone, the coarser grids, with a constant size ofBased 0.0065 on mthe experimental0.0065 m, were and used numerical to calculate model the discussed steady evolutionabove, the ofvalidity wind andof the wave numerical fields, × whichmodel couldfor the eff hydrodynamicectively reduce characteristics the computation such load as onfree the surface precondition elevations of assuring and flow the velocity calculation was precision.verified with Meanwhile, experimental the asymptoticdata in the typical grids with cases an of averageW0H07 sizeand ofW5H07. 0.005 m, which varied in the x and zThedirection, comparison were shows used ingood the agreement slope zone with to capture the average the wave root mean breaking. square The error grid (RMSE) size reduced below gradually0.6 between from the 0.0065measured m to and 0.0035 computed m in the timex direction, series of andfree owingsurface to elevations the contraction at the ofseven outlet measured it fell to 0.001locations m and (S2-S8) 0.0035 with m atand the without lower and wind, upper as given edge in of theFigures outlet, 3 and respectively. 4, respectively. The physical At the offshore features oflocations air and (S2-S4), water were the wave quite surface different. was To symmetrical ensure that thewith global a relatively Courant stable number wave was height. suitable When for the calculationwave propagated convergence, from S5 the to timeS7, it step became had torapidly be small steeper enough. and Thereforeclose to breaking the adaptive due to time-step the shoaling was employedeffect. After during that the the wave simulation, breaking with occurred a maximum at a location value between of 0.025. S7 In and the meantime,S8, with a sharp the maximum decrease Courantof the maximum number free was surface set at 0.2. elevation at S8.

4. Model Validation Based on the experimental and numerical model discussed above, the validity of the numerical model for the hydrodynamic characteristics such as free surface elevations and flow velocity was verified with experimental data in the typical cases of W0H07 and W5H07. The comparison shows good agreement with the average root mean square error (RMSE) below 0.6 between the measured and computed time series of free surface elevations at the seven measured locations (S2-S8) with and without wind, as given in Figures3 and4, respectively. At the o ffshore locations (S2-S4), the wave surface was symmetrical with a relatively stable wave height. When the wave propagated from S5 to S7, it became rapidly steeper and close to breaking due to the shoaling effect. After that the wave breaking occurred at a location between S7 and S8, with a sharp decrease of the maximum free surface elevation at S8.

Figure 3. Time series of free surface elevation. The results are shown for case W0H07.

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Figure 2. Sketch of the numerical setup.

4. Model Validation Based on the experimental and numerical model discussed above, the validity of the numerical model for the hydrodynamic characteristics such as free surface elevations and flow velocity was verified with experimental data in the typical cases of W0H07 and W5H07. The comparison shows good agreement with the average root mean square error (RMSE) below 0.6 between the measured and computed time series of free surface elevations at the seven measured locations (S2-S8) with and without wind, as given in Figures 3 and 4, respectively. At the offshore locations (S2-S4), the wave surface was symmetrical with a relatively stable wave height. When the wave propagated from S5 to S7, it became rapidly steeper and close to breaking due to the shoaling Watereffect.2020 After, 12, 686that the wave breaking occurred at a location between S7 and S8, with a sharp decrease7 of 21 of the maximum free surface elevation at S8.

Water 2020, 12, x FOR PEER REVIEW 8 of 22 Figure 3. Time series of free surface elevation. The results are shown for case W0H07. Figure 3. Time series of free surface elevation. The results are shown for case W0H07.

Figure 4. Time series of free surface elevation. The results are shown for case W5H07. Figure 4. Time series of free surface elevation. The results are shown for case W5H07. Moreover, the measured and computed results of the two-dimensional velocity at a depth of 0.2 m Moreover, the measured and computed results of the two-dimensional velocity at a depth of 0.2 (anemometer depth) at the slope toe cross-section in the windless case of W0H07 are presented in m (anemometer depth) at the slope toe cross-section in the windless case of W0H07 are presented in Figure 5, in which the measured results were nearly horizontally and vertically identical to that of computed, and the values were basically equally distributed near zero. In contrast to the windy case of W5H07, as indicated in Figure 6, despite the fact that the measured amplitude of horizontal velocity was slightly smaller than the computed results, the average levels show the same overall downtrend, which reflects the action of wind to some extent. The vertical velocity distribution was less affected by wind with no significant differences.

Figure 5. Time series of horizontal and vertical velocities in the absence of wind at a depth of 0.2 m. The results are shown for case W0H07.

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Figure 4. Time series of free surface elevation. The results are shown for case W5H07. Water 2020, 12, 686 8 of 21 Moreover, the measured and computed results of the two-dimensional velocity at a depth of 0.2 m (anemometer depth) at the slope toe cross-section in the windless case of W0H07 are presented in FigureFigure5 ,5, in in which which the the measured measured results results were were nearly nearly horizontally horizontally and and vertically vertically identical identical to to that that of of computed,computed, and and the the values values were were basically basically equally equally distributed distributed near near zero. zero. In contrastIn contrast to theto the windy windy case case of W5H07,of W5H07, as indicated as indicated in Figure in Figure6, despite 6, despite the fact the that fact the that measured the measured amplitude amplitude of horizontal of horizontal velocity wasvelocity slightly was smaller slightly than smaller the computed than the results,computed the re averagesults, the levels average show levels the same show overall the same downtrend, overall whichdowntrend, reflects which the action reflects of wind the action to some of extent.wind to The some vertical extent. velocity The vertical distribution velocity was distribution less affected was by windless affected with no by significant wind with diff noerences. significant differences.

WaterFigure 2020, 12 5., xTime FOR seriesPEER REVIEW of horizontal (a) and vertical (b) velocities in the absence of wind at a depth of 9 of 22 Figure 5. Time series of horizontal and vertical velocities in the absence of wind at a depth of 0.2 m. 0.2 m. The results are shown for case W0H07. 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 Figure 6. Time series of horizontal and vertical velocities in the presence of wind at a depth of 0.2 m. 0.2 m. The results are shown for case W5H07. The results are shown for case W5H07. Overall, the comparisons between the measured and calculated results of the free surface elevation and ﬂowOverall, velocity the indicatescomparisons a good between model predictionthe measured regardless and calculated of the presence resultsor of absence the free of surface wind. Hence,elevation 12 scenariosand flow werevelocity simulated indicates for a subsequentgood model analysis prediction based regardless on a combination of the presence of wind or speedsabsence of wind. Hence, 12 scenarios were simulated for subsequent analysis based on a combination1/2 of wind and wave heights in Table1, wherein the surf similarity parameter ξ0 = tanα/(H0/L0) ranging ()1/2 fromspeeds 0.57~0.84 and wave satisﬁes heights the plunging in Table wave.1, wherein the surf similarity parameter ξ000=tanα //HL ranging from 0.57~0.84 satisfies the plunging wave. Table 1. Simulated scenarios (2nd Stokes). Table 1. Simulated scenarios（2nd Stokes）. Case W (m/s) H0 (cm) T (s) h (m) ξ0

W0H05~W0H11 Case0.0 5,W 7, 9,11(m/s) H0 (cm) 1.5 T (s) h (m) 0.4 ξ0 plunging W3H05~W3H11 W0H05~W0H113.0 5, 7,0.0 9,11 5, 7, 9,11 1.5 1.5 0.4 0.4 plunging plunging W5H05~W5H11 5.0 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

5. Results and Discussions In this section, we will quantitatively analyze the influence of wind on the evolution of nearshore waves taking into account wave breaking, turbulent flow, and velocity profile, which are crucial to the studies in air-sea interaction, hydrodynamics in surf-swash zone, sediment transport in coastal areas, and wave-structure interaction.

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5. Results and Discussions In this section, we will quantitatively analyze the influence of wind on the evolution of near-shore waves taking into account wave breaking, turbulent flow, and velocity profile, which are crucial to the studies in air-sea interaction, hydrodynamics in surf-swash zone, sediment transport in coastal areas, and wave-structure interaction.

5.1. Wave Breaking As described in Section4, the processes of wave propagation on the slope was accompanied with the wave distortion and wave breaking, and little is known about wave breaking under wind action up to now. The similarity shared by different cases makes it a feasible practice to focus on typical cases here. As a result, we merely present the instantaneous snapshots of wave surface during wave breaking processes in the typical cases of W0H07~W5H07, as indicated in Figure7, in which three moments are defined separately: Initial breaking, breaking, and overturning jet initiation. The figures indicate that the breaker type was almost unchanged among the three cases, yet there exists some differences, as reflected in the wave breaking ahead of time with the increase of wind speed. On the other hand, taking the initial breaking time in the absence of wind as a reference, we supplemented the compared results of all cases as evident in Figure8. As an example, Figure8a indicates that although waves with p a wind speed of 3 m/s and 5 m/s were already broken at t/ h/g = 94, it is just the beginning of initial breaking in the windless case. Such features are apparently also included in other cases (Figure8b–d) and not repeated here. It suggests that the wave breaking location would migrate towards the offshore Waterdirection 2020, 12 with, x FOR the PEER impact REVIEW ofwind, which probably caused changes of breaking height and10 depth, of 22 and even critical breaking conditions. This clearly addresses a fact that the role of wind on wave heightbreaking and in depth, time isand coincident even critical with breaking that in space conditions. throughout This clearly the breaking addresses processes. a fact that As the indicated role of windin Figure on wave8a, the breaking breaking in coursestime is coincident approached with each that other in space and even throughout overlapped the breaking for di fferent processes. wind Asspeeds. indicated It is especially in Figure true 8a, inthe the breaking case of acourse smallers approached incident wave each height. other As and the even incident overlapped wave height for differentincreased, wind this behaviorspeeds. It tended is especially to disappear true in and the thecase spatial of a smaller distribution incident of each wave course height. seemed As the to incidentdisperse, wave as shown height in increased, Figure8b–d. this Itbehavior can be inferredtended to that disappear the incident and the wave spat heightial distribution was one of of each the coursemain factors seemed aff toecting disperse, the dispersion as shown in degree Figure of 8b–8d. wave breakingIt can be inferred locations that under the theincident same wave wind height speed wascoverage one of constraint. the main factors affecting the dispersion degree of wave breaking locations under the same wind speed coverage constraint.

Figure 7. VariationVariation of free surface in the breaking proc processes.esses. The The results results are shown for cases W0H07 ((ﬁrstfirst row),row), W3H07W3H07 (second(second row),row), andand W5H07W5H07 (third(third row), row), respectively. 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.

Taking into account the critical breaking depth, the wave breaking location can be calculated by the following formula: h xL=−b (12) bxtanα

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height and depth, and even critical breaking conditions. This clearly addresses a fact that the role of wind on wave breaking in time is coincident with that in space throughout the breaking processes. As indicated in Figure 8a, the breaking courses approached each other and even overlapped for different wind speeds. It is especially true in the case of a smaller incident wave height. As the incident wave height increased, this behavior tended to disappear and the spatial distribution of each course seemed to disperse, as shown in Figure 8b–8d. It can be inferred that the incident wave height was one of the main factors affecting the dispersion degree of wave breaking locations under the same wind speed coverage constraint.

Figure 7. Variation of free surface in the breaking processes. The results are shown for cases W0H07 Water 2020, 12, 686 10 of 21 (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 Figure 8. Wave breaking processes at the initial breaking time. The results are shown for cases (a) (a) W0H05~W5H05, (b) W0H07~W5H07, (c) W0H09~W5H09, and (d) W0H11~W5H11. W0H05~W5H05, (b) W0H07~W5H07, (c) W0H09~W5H09, and (d) W0H11~W5H11. Taking into account the critical breaking depth, the wave breaking location can be calculated by Taking into account the critical breaking depth, the wave breaking location can be calculated by the following formula: the following formula: hb x = Lx (12) b − tanα =−hb xLbx (12) where the wave breaking location and breaking depthtanα are represented by xb and hb and Lx is the horizontal distance from slope toe to the shoreline. According to the proposal by Collins and Weir (1969) [54], the breaking wave height is proportional to the breaking depth, which can be expressed as:

rb = Hb/hb = 0.72 + 5.6tanα (13) as well as by Le Mehuate (1967) [55] where the relationship between the breaking wave height and the height in deep water can be expressed as:

1/7 1/4 Hb/H0 = 0.76(tanα) (H0/L0)− . (14)

Then, on the combination of Equations (12)–(14), the wave breaking locations can be predicted with the below formula:

1/7 1/4 0.76(tanα) (H0/L0)− H0 1 x = Lx . (15) b − 0.72 + 5.6tanα tanα

The correlation between the numerical dimensionless wave breaking locations (Lx x )/h and − b surf similarity parameter (ξ0) is given in Figure9. In view of ξ0 embodying the intrinsic qualities of the slope, the results of the added cases of H0 = 5 cm with the slope of 1:16 and 1:20 where ξ0 satisﬁes the spilling wave are included. Also shown are the distribution curves of breaking locations by Equation (15). Under the condition of the same wind speed and breaking parameter, the wave breaking location of the 2nd Stokes wave was farther from the shoreline than the calculated results, mainly caused by the discrepancy between numerical calculation and empirical parameterization. Likewise, the distribution of wave breaking locations was identical in essence with the above observation in Figure8, which lends support to a more accurate deduction. It could be deduced that under the same wind speed Water 2020, 12, x FOR PEER REVIEW 11 of 22

where the wave breaking location and breaking depth are represented by xb and hb and Lx is the horizontal distance from slope toe to the shoreline. According to the proposal by Collins and Weir (1969) [54], the breaking wave height is proportional to the breaking depth, which can be expressed as: + r=H/h=bbb0.72 5.6 tanα (13)

as well as by Le Mehuate (1967) [55] where the relationship between the breaking wave height and the height in deep water can be expressed as: ()()1/7− 1/4 H/H=b 0000.76 tanα H/L . (14)

Then, on the combination of Equations (12)–(14), the wave breaking locations can be predicted with the below formula:

1/7 − 0.76()tanα (H / L ) 1/4 H 1 x=L- 00 0 . (15) bx 0.72+ 5.6tanα tanα () The correlation between the numerical dimensionless wave breaking locations L-x/hxb and

surf similarity parameter ( ξ0 ) is given in Figure 9. In view of ξ0 embodying the intrinsic qualities of

the slope, the results of the added cases of H0 = 5cm with the slope of 1:16 and 1:20 where ξ0 satisfies the spilling wave are included. Also shown are the distribution curves of breaking locations by Equation (15). Under the condition of the same wind speed and breaking parameter, the wave breaking location of the 2nd Stokes wave was farther from the shoreline than the calculated results, mainly caused by the discrepancy between numerical calculation and empirical parameterization. Likewise, the distribution of wave breaking locations was identical in essence with the above Waterobservation2020, 12, 686 in Figure 8, which lends support to a more accurate deduction. It could be deduced11 of that 21

under the same wind speed coverage constraint, the smaller the ξ0 is, the higher dispersion degree coverageof wave constraint,breaking locati the smallerons will the be,ξ 0whereinis, the higher the ξ0 dispersion involves degreethe slope of wavefactor, breaking rather than locations simply will the be,incident wherein wave the ξheight.0 involves the slope factor, rather than simply the incident wave height.

Figure 9. The correlation between the wave breaking locations and surf similarity parameter ξ . Figure 9. The correlation between the wave breaking locations and surf similarity parameter ξ00 . Symbols , +, denote numerical data with a slope of 1:10, 1:16, and 1:20, where the wind speed of 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) 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. represent the calculated results for the appropriate slope. To find out the influences on wave properties along the cross-shore distance induced by the wind, To find out the influences on wave properties along the cross-shore distance induced by the we plotted the cross-shore variation of the average wave height and water level in the typical cases, wind, we plotted the cross-shore variation of the average wave height and water level in the typical asWater shown 2020, in12, Figurex FOR PEER 10. ItREVIEW is evident that wind raised the average wave height and average water12 level of 22 cases, as shown in Figure 10. It is evident that wind raised the average wave height and average water at the same cross-shore location before wave breaking. For the critical breaking height and depth, level at the same cross-shore location before wave breaking. For the critical breaking height and depth, thelatter. effect As woulda result, be itenhanced is expected wherein that the the critical variation breaker ofthe indices former like was H/hbb significantly will present greater an thanuptrend. the the effect would be enhanced wherein the variation of the former was significantly greater than the latter.We accessed As a result, the wind it is expected effect on thatthe breaker the critical index breaker of H/h indicesbb, as like shownHb/h inb will Figure present 11. With an uptrend. the absence We ξ = accessedof wind, theH/hbb wind was effect basically on the breakermaintained index at ofa constantHb/hb, as level shown with in a Figure value 11of. 1 With except the at absence 0 0.42 of wind,whereH itb /washb was a gentle basically slope maintained of 1:20 and at the a constantresults agreed level with well awith value that of 1given except by atGalvinξ0 = 0.42(1968)where [56]. it was a gentle slope of 1:20 and the results agreed well with that given by Galvin (1968) [56]. In the In the presence of wind, the overall trend of H/hbb showed an ineligible increase with wind speed, presenceimplying of that wind, the thecritical overall breaker trend index of Hb without/hb showed wind an was ineligible no longer increase suitable with for wind the windy speed, implyingcases any thatmore. the critical breaker index without wind was no longer suitable for the windy cases any more.

FigureFigure 10.10. TheThe distributiondistribution ofof thethe cross-shorecross-shore averageaverage ((aa)) wavewave heightheight andand ((bb)) waterwater level.level. TheThe resultsresults areare shownshown forfor casescases W0H07W0H07 (blue),(blue), W3H07W3H07 (red),(red), andand W5H07 W5H07 (black), (black), respectively. respectively.

ξ Figure 11. The correlation between the breaker index of H/hbb and surf similarity parameter 0 . The dotted lines are calculated by Galvin (1968) and the illustration of other symbols can be seen in the note of Figure 9.

5.2. Turbulent Flow Nearshore wave breaking is the main mode of wave energy consumption, in which the turbulence is fierce. It is generally accepted that the existence of wind will further heighten the turbulence level [57]. Hence, it is essential to quantitatively investigate the impact of wind on the turbulent flow when wave propagates on the slope. Firstly, the cross-spectrum method (for more details see appendix) was applied to examine the correlation property of horizontal and vertical residual velocity, which can effectively reflect the turbulent features. On the support of the low-pass filtering in LES, the filtered residual velocity is:  _ u=u-u'

 _ (16) ' w=w-w

Water 2020, 12, x FOR PEER REVIEW 12 of 22

latter. As a result, it is expected that the critical breaker indices like H/hbb will present an uptrend.

We accessed the wind effect on the breaker index of H/hbb, as shown in Figure 11. With the absence ξ = of wind, H/hbb was basically maintained at a constant level with a value of 1 except at 0 0.42 where it was a gentle slope of 1:20 and the results agreed well with that given by Galvin (1968) [56].

In the presence of wind, the overall trend of H/hbb showed an ineligible increase with wind speed, implying that the critical breaker index without wind was no longer suitable for the windy cases any more.

Figure 10. The distribution of the cross-shore average (a) wave height and (b) water level. The results Water 2020are, 12shown, 686 for cases W0H07 (blue), W3H07 (red), and W5H07 (black), respectively. 12 of 21

Figure 11. The correlation between the breaker index of Hb/hb and surf similarity parameter ξ0. Theξ Figure 11. The correlation between the breaker index of H/hbb and surf similarity parameter 0 . dotted lines are calculated by Galvin (1968) and the illustration of other symbols can be seen in the note The dotted lines are calculated by Galvin (1968) and the illustration of other symbols can be seen in of Figure9. the note of Figure 9. 5.2. Turbulent Flow 5.2. Turbulent Flow Nearshore wave breaking is the main mode of wave energy consumption, in which the turbulence is fierce.Nearshore It is generally wave breaking accepted is that the the main existence mode of of wind wave will energy further consumption, heighten the in turbulence which the levelturbulence [57]. Hence, is fierce. it is essentialIt is genera to quantitativelylly accepted that investigate the existence the impact of wind of wind will on further the turbulent heighten flow the whenturbulence wave propagates level [57]. onHence, the slope. it is essential to quantitatively investigate the impact of wind on the turbulentFirstly, flow the cross-spectrumwhen wave propagates method on (for the more slope. details see AppendixA) was applied to examine the correlationFirstly, the property cross-spectrum of horizontal method and (for vertical more residual details see velocity, appendix) which was can applied effectively to examine reflect the the turbulentcorrelation features. property On theof horizontal support of and the low-passvertical re filteringsidual invelocity, LES, the which filtered can residual effectively velocity reflect is: the turbulent features. On the support of the low-pass filtering in LES, the filtered residual velocity is: ( u0 = u u_  ' − (16) wu=u-u= w w 0  − _ (16)  ' where u and w are filtered velocity, the overbar andw=w-w comma indicate the time-averaged and residual quantities. As illustrated in Figure 12, in cases of W0H07~W5H07, the results of the cross-spectrum analysis of three levels below the trough at the location of (x x )/h = 1.25 show that the spectral − b − energy was mainly concentrated in the low-frequency region ( f < 2.5 Hz) where the wind input was absorbed and transported. In this region the cross spectral density, coherence, and phase lag of u0 and w0 were basically the same for different levels and the wind speed increased the coherence peak at a depth of 0.37 cm, rising from 0.76 to 0.9, which means that the wind enhanced the degree of coherence of velocity components, and aggravated turbulence. Then, the absorbed wind input spreads further, to a high-frequency region ( f > 2.5 Hz) by nonlinear interaction, which mainly reflected that the spectral density with wind was larger than that without wind. Remarkably, the phase lag tended to disperse at f = 2.5 Hz for different levels, with the degree increasingly evident as the wind speed increased. Subsequently, the generation and dissipation of TKE are worth analyzing within the processes of wave breaking. Combining the kinetic energy governing equation of LES, the following formula was used to calculate TKE in the solvable scale [58]:

1 2 2 k = u0 + w0 . (17) 2 Part of TKE in a solvable scale will dissipate in small scale ﬂuctuations, which is deﬁned as the TDR (turbulent dissipation rate) that cannot be solved directly from the LES. Via the kinetic energy equation for solvable scale [58], the TDR is found to be similar to the following form:

ε = 2 < τijSij >, (18) Water 2020, 12, x FOR PEER REVIEW 13 of 22 where u and w are filtered velocity, the overbar and comma indicate the time-averaged and residual quantities. As illustrated in Figure 12, in cases of W0H07~W5H07, the results of the cross- Water 2020, 12, 686 ()− 13 of 21 spectrum analysis of three levels below the trough at the location of x-xb /h= 1.25 show that the spectral energy was mainly concentrated in the low-frequency region ( f<2.5 Hz ) where the wind 1 ∂ui ∂uj where τij is the stress term and Sij = + denotes strain-rate tensor. A modeling method is input was absorbed and transported. In2 ∂thisxj region∂xi the cross spectral density, coherence, and phase ' ' lagproposed of u byand Smagorinsky w were basically (1963) [59 the] to same calculate for differentτij, namely levels the well-knownand the wind sub-grid speed stressincreased model: the coherence peak at a depth of 0.37 cm, rising from 0.76 to 0.9, which means that the wind enhanced 2 the degree of coherence of velocity components,τij = υSGSSij an=d (C aggravatedSGS∆) Sij S turbulence.ij Then, the absorbed wind(19) 2.5 Hz input spreads qfurther, to a high-frequency region ( f> ) by nonlinear interaction, which mainly reflectedwhere S ijthat = the2S spectralijSij is the density module with of analyticwind wa deformations larger than rate, thatυ SGSwithoutis the wind. Smagorinsky Remarkably, dynamic the phaseviscosity, lag CtendedSGS is to the disperse Smagorinsky at f= constant2.5 Hz for of different 0.16, and levels,∆ represents with the the degree Smagorinsky increasingly scale evident that is asidentical the wind to thespeed magnitude increased. of grids in the computational domain.

Figure 12. Cross spectral analysis of u0 and w0 at (x xb)/h = 1.25 with three depths of z = 0.37 m ' −' ()− −1.25 Figure(blue), z12.= 0.35Crossm (red),spectral and analysisz = 0.33 ofm (black).u and Thew resultsat x-x areb shown /h= for cases with W0H07 three (firstdepths row), of W3H07z=0.37 (secondm (blue), row), z= and0.35 W5H07 m (red), (third and row),z=0.33 respectively. m (black). The results are shown for cases W0H07 (first row), W3H07 (second row), and W5H07 (third row), respectively. In accord with the time in Figures7, 13 and 14 show the relevant snapshots of distribution of TKESubsequently, and TDR obtained the generation by Equations and (17)dissipation and (18) of during TKE are wave worth breaking. analyzing As within shown the in Figureprocesses 13, ofthe wave maximum breaking. TKE Combining occurred underthe kinetic the wave energy crest, governing near the equation windward of LES, wave the surface. following It could formula also wasbe found used thatto calculate the turbulence TKE in owingthe solvable to the scale surface [58]: roller was rapidly transported to the depths of the flow and the rear regions of wave front by the turbulence transformation mechanisms, and the TKE 1 was more distributed horizontally rather thank= vertically() u'2 +w '2 [42. ,44]. Meanwhile, the distribution of(17) TDR was relatively uniform with a constant order of2 magnitude in most regions except for the wave front and rollerPart of where TKE in the a mixingsolvable of scale air and will water dissipate with in notable small scale values fluctuations, exists, as shown which inis Figuredefined 14 as. Thethe TDRinfluence (turbulent of wind dissipation on TKE and rate) TDR that can cannot be observed be solved as welldirectly in Figures from the13 andLES. 14 Via, and the as kinetic a result, energy there equationexists some for disolvablefferences scale that [58], will the be discussedTDR is found in detail to be below. similar to the following form:

ε =<2 τijS> ij , (18)

Water 2020, 12, x FOR PEER REVIEW 14 of 22

_ _ _ ∂ ∂ u 1 ui j where τ is the stress term and S+= denotes strain-rate tensor. A modeling method ij ij ∂∂ 2 xxji  is proposed by Smagorinsky (1963) [59] to calculate τij , namely the well-known sub-grid stress model:

___ Δ 2 τij= υ SGSS=(C ij SGS )SS ij ij (19)

___ where S=2SSij ij ij is the module of analytic deformation rate, υSGS is the Smagorinsky dynamic Δ viscosity, CSGS is the Smagorinsky constant of 0.16, and represents the Smagorinsky scale that is identical to the magnitude of grids in the computational domain. In accord with the time in Figure 7, Figures 13 and 14 show the relevant snapshots of distribution of TKE and TDR obtained by Equations (17) and (18) during wave breaking. As shown in Figure 13, the maximum TKE occurred under the wave crest, near the windward wave surface. It could also be found that the turbulence owing to the surface roller was rapidly transported to the depths of the flow and the rear regions of wave front by the turbulence transformation mechanisms, and the TKE was more distributed horizontally rather than vertically [42,44]. Meanwhile, the distribution of TDR was relatively uniform with a constant order of magnitude in most regions except for the wave front and roller where the mixing of air and water with notable values exists, as shown in Figure 14. The Water 2020, 12, 686 14 of 21 influence of wind on TKE and TDR can be observed as well in Figures 13 and 14, and as a result, there exists some differences that will be discussed in detail below.

Figure 13.13. TheThe snapshots snapshots of of the the TKE TKE (turbulent (turbulent kinetic kineti energy)c energy) in the breakingin the breaking processes processes under di ﬀundererent winddifferent speeds. wind The speeds. results The are results shown are for shown cases fo W0H07r cases (ﬁrstW0H07 row), (first W3H07 row), (secondW3H07 (second row), and row), W5H07 and Water 2020, 12, x FOR PEER REVIEW 15 of 22 (thirdW5H07 row), (third respectively. row), respectively.

Figure 14. TheThe snapshots of the TDR (turbulent dissip dissipationation rate) in the breaking processes under didifferentfferent wind speeds. speeds. The The results results are are shown shown for for cases cases W0H07 W0H07 (first (first row), row), W3H07 W3H07 (second (second row), row), and andW5H07 W5H07 (third (third row), row), respectively. respectively. We plotted the time variations of TKE and TDR at a depth of 0.37 m below the trough level at We plotted the time variations of TKE and TDR at a depth of 0.37 m below the trough level at two cross-sections before and after breaking based on the above cases, as shown in Figures 15 and 16, two cross-sections before and after breaking based on the above cases, as shown in Figures 15 and 16, respectively. The results show that the TKE and TDR displayed a periodic variation throughout the respectively. The results show that the TKE and TDR displayed a periodic variation throughout the collection time, wherein the TKE was a bimodal distribution in each wave period. Before wave breaking collection time, wherein the TKE was a bimodal distribution in each wave period. Before wave ((x x )/h = 0.35), as plotted in Figure 15a, the maximum TKE occurred at an initial breaking time breaking− b ( ()x-x− /h=−0.35 ), as plotted in Figure 15a, the maximum TKE occurred at an initial and increased withb wind speed. In sharp contrast with the TKE, the TDR barely felt the impact of wind,breaking as displayedtime and inincreased Figure 15 withb. The wind maximum speed. In TDR sharp occurred contrast before with initial the TKE, breaking the TDR time, barely resulting felt inthe the impact phase of advance. wind, as Therefore,displayed in the Figure coherence 15b. The was maximum relatively TDR weak occurred as compared before to initial the case breaking in the time, resulting in the phase advance. Therefore, the coherence was relatively weak as compared to TKE. Likewise, Figure 16 shows similar results after wave breaking ((x xb)/h = 0.375), since the − ()0.375 mixingthe case of in air the and TKE. water Likewise, in the innerFigure part 16 shows of surf similar zone enhanced results after the turbulencewave breaking intensity. ( x-x Particularlyb /h= in), thesince presence the mixing of wind, of air the and average water in TKE the and inner TDR part levels of surf generally zone enhanced rose and th weree turbulence significantly intensity. larger thanParticularly that before in the breaking, presence which of wind, was likelythe average to cause TKE the and TKE TDR and TDRlevels to generally diffuse downward rose and were with moresignificantly complex larger turbulence than that transformation before breaking, mechanisms which was [likely44,60]. to Then,cause wethe plottedTKE and the TDR variation to diffuse of time-averageddownward with TKE more and complex TDR with turbulence depth in transforma Figure 17.tion As expected,mechanisms in Figure [44,60]. 17 Then,a,c, both we plotted beforeand the variation of time-averaged TKE and TDR with depth in Figure 17. As expected, in Figure 17a and c, both before and after breaking, the time-averaged TKE in regions near the water surface was considerably larger and rapidly decreased with the depth. It is also noted that wind enhanced the average TKE level. For time-averaged TDR, Figure 17b presents that before breaking it was found to be negligible even in the windy cases, while the significant effect of wind was imposed after breaking as given in Figure 16b, the turbulence penetrated to the deeper flow and the average level of TDR increased with wind speed.

Water 2020, 12, 686 15 of 21 after breaking, the time-averaged TKE in regions near the water surface was considerably larger and rapidly decreased with the depth. It is also noted that wind enhanced the average TKE level. For time-averagedWaterWater 2020 2020, 12, 12, x TDR,, FORx FOR PEER Figure PEER REVIEW REVIEW 17b presents that before breaking it was found to be negligible even16 of16 22 inof 22 the windyWater cases, 2020, 12 while, x FOR thePEER significant REVIEW effect of wind was imposed after breaking as given in Figure16 of 1622 b, the turbulence penetrated to the deeper flow and the average level of TDR increased with wind speed.

Figure 15. Time series of (a) TKE and (b) TDR at ()x-x /h=−0.35 before breaking. The results are ()b − Figure 15. Time series of (a) TKE and (b) TDR at x-xb /h= 0.35 before breaking. The results are shown for cases W0H07 (blue), W3H07 (red), and W5H07 (black), respectively. shown for cases W0H07 (blue), W3H07 (red), and W5H07 (black), respectively. Figure 15. Time series of (a) TKE and (b) TDR at (x()xb)/h =− 0.35 before breaking. The results are Figure 15. Time series of (a) TKE and (b) TDR at −x-xb /h= −0.35 before breaking. The results are shown for cases W0H07 (blue), W3H07 (red), and W5H07 (black), respectively. shown for cases W0H07 (blue), W3H07 (red), and W5H07 (black), respectively.

() Figure 16. Time series of (a) TKE and (b) TDR at x-xb /h=0.375 after breaking. The results are Figureshown 16. Time for cases series W0H07 of (a (blue),) TKE W3H07 and (b (red) TDR), and at W5H07(x()xb) (black),/h = 0.375 respectively.after breaking. The results are Figure 16. Time series of (a) TKE and (b) TDR at −x-xb /h=0.375 after breaking. The results are shown for cases W0H07 (blue), W3H07 (red), and W5H07 (black), respectively. shown for cases W0H07 (blue), W3H07 (red), and W5H07() (black), respectively. Figure 16. Time series of (a) TKE and (b) TDR at x-xb /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) Figure 17. Variation of time-averaged TKE and TDR with depth under different wind speeds at (a–b) (x xb())/h = 0.35− before breaking and (c,d) (x x()b)/h = 0.375 after breaking. The results are shown − x-xb /h=− 0.35 before breaking and (c–d)− x-xb /h=0.375 after breaking. The results are for case W0H07 (blue), W3H07 (red), and W5H07 (black), respectively. shown for case 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) ()− () x-xb /h= 0.35 before breaking and (c–d) x-xb /h=0.375 after breaking. The results are Figure 17. Variation of time-averaged TKE and TDR with depth under different wind speeds at (a–b) shown for case W0H07 (blue), W3H07 (red), and W5H07 (black), respectively. ()x-x /h=−0.35 before breaking and (c–d) ()x-x /h=0.375 after breaking. The results are b b shown for case W0H07 (blue), W3H07 (red), and W5H07 (black), respectively.

WaterWater2020 2020, 12, 12, 686, x FOR PEER REVIEW 1617 of of 21 22

5.3. Undertow 5.3. Undertow For the regular wave, the periodic motion of uprush and backrush occurred in the course of waveFor propagation the regular wave,on the the slope. periodic Since motion the large of uprush volume and of backrushwater was occurred carried inshorewards the course ofbywave wave propagationbreaking, the on longshore the slope. currents Since the were large fed volume continuo of waterusly, was and carried eventually shorewards returned by through wave breaking, the surf thezone longshore because currents of gravity, were usually fed continuously, as undertow. and Ag eventuallyain, in Figure returned 18, throughwe present the surfthe time-averaged zone because ofvelocity gravity, profiles usually below as undertow. the trough Again, level in in Figure cases 18 of, weW0H07~W5H07. present the time-averaged Given that they velocity share profiles similar belowbehaviors the trough in other level cases, in casesthe numerical of W0H07~W5H07. results of three Given cross-sections that they share ranging similar from behaviors the slope in othertoe to cases,the location the numerical in the proximity results of of three the breaking cross-sections location ranging were used from for the comparation. slope toe to theIn Figure location 18a, in even the proximityunder different of the breakingwind speeds, location the weremagnitude used for of comparation. horizontal velocity In Figure profile 18a, evenwas actually under di constantfferent wind over speeds,the depth the between magnitude the of trough horizontal level velocityand upper profile bottom was bo actuallyundary constant layer, consistent over the with depth Bakhtyar between et the al. trough(2009), level although and upper the latter bottom had boundaryno wind, layer,and maintain consistented witha larger Bakhtyar level in et the al. (2009),cross-sections although before the latterbreaking had no(e.g. wind, x=1m and maintainedand x=2m a), larger whereas level it indecreased the cross-sections sharply being before nearly breaking close (e.g. to zerox = in1 them and x = 2 m), whereas it decreased sharply being nearly close to zero in the slope toe cross-section slope toe cross-section ( x=0m), where the undertow was less significant. Meanwhile, Figure 18b (x = 0 m), where the undertow was less significant. Meanwhile, Figure 18b indicates that we cannot indicates that we cannot ignore the effect of the vertical velocity profile on undertow. The level of ignore the effect of the vertical velocity profile on undertow. The level of that tended to shift in the that tended to shift in the negative direction and the undertow intensity rose gradually with the negative direction and the undertow intensity rose gradually with the distance away from the slope distance away from the slope toe. We noticed the dependence of undertow on the wind speed, which toe. We noticed the dependence of undertow on the wind speed, which manifested specifically as: The manifested specifically as: The wind affecting the undertow intensity in various regions to different wind affecting the undertow intensity in various regions to different degrees. In the vicinity of the degrees. In the vicinity of the slope toe cross-section, we found that the undertow intensity itself was slope toe cross-section, we found that the undertow intensity itself was small whether there existed small whether there existed wind or not. The closer it was to the surf zone, the more intense the wind or not. The closer it was to the surf zone, the more intense the undertow would be as the wind undertow would be as the wind speed increased. Considering the formation mechanism of undertow speed increased. Considering the formation mechanism of undertow as well as the wind-induced as well as the wind-induced setup added to the wave setup, which is responsible for the intensified setup added to the wave setup, which is responsible for the intensified imbalance between the setup imbalance between the setup pressure gradient and the wave radiation stress, and thus contributing pressure gradient and the wave radiation stress, and thus contributing to the acceleration of undertow to the acceleration of undertow for compensating the loss volume of water carried shorewards for compensating the loss volume of water carried shorewards incurred by the effects of wave breaking incurred by the effects of wave breaking and wind. The mean flow rate per meter width below the and wind. The mean flow rate per meter width below the trough level over 6T is given in Table2, trough level over 6T is given in Table 2, the maximum is generated at x=1m, where the depth is the maximum is generated at x = 1 m, where the depth is larger with a stable return flow velocity, andlarger wind with enhances a stable the return mean flow flow velocity rate , and wind enhances the mean flow rate

x = x = FigureFigure 18. 18.(a )(a Horizontal) Horizontal and and (b) vertical(b) vertical time-averaged time-averaged velocity velocity proﬁle profile at 0atm (triangle),x=0m (triangle),1 m (square) and x = 2 m (circle). The results are shown for cases W0H07 (blue), W3H07 (red), and W5H07 x=1m (square) and x=2m (circle). The results are shown for cases W0H07 (blue), W3H07 (red), (black), respectively. and W5H07 (black), respectively. Table 2. The mean ﬂow rate below the trough level. Table 2. The mean flow rate below the trough level. 3 x (m) hx (m) W (m/s) Mean Flow Rate, q (m /s/m) 3 0x (m) 0.4hx (m) W (m/s) 0, 3,Mean 5 Flow Rate,0.0011,q (m /s/m)0.0014, 0.0016 − − − 10 0.30.4 0, 3, 5 0, 3, 5−0.0011, −0.0014,0.0051, −0.00160.0054, 0.0056 − − − 2 0.2 0, 3, 5 0.0024, 0.0029, 0.0033 1 0.3 0, 3, 5 −0.0051, −0.0054,− −0.0056− − 2 0.2 0, 3, 5 −0.0024, −0.0029, −0.0033

Water 2020, 12, x FOR PEER REVIEW 18 of 22

Water The2020, 12moment, 686 of skewness and kurtosis was used to quantitatively evaluate the undertow17 of 21 intensity. The skewness represents the deviation of velocity from the standard Gaussian distribution, whileThe the moment kurtosis of indicates skewness the and peak kurtosis level. was The used empirical to quantitatively formulas evaluateof skewness the undertow Css and intensity.kurtosis

TheCks skewnessare known represents as: the deviation of velocity from the standard Gaussian distribution, while the kurtosis indicates the peak level. The empiricaln formulas_ of skewness Css and kurtosis Cks are 1 3 known as: (xi - x) n n C= 1 Pi=1 3 ss n (xi x) 3 (20) n −_ 1i=1 2 Css = (x - x) (20) s  i 3 n i=1n   1 P ( )2  n xi x   i=1 −  n _ 1 4 n (x - x) 1nP i 4 n i=1(xi x) C=ks − 2 (21) i=n1 _ Cks = 1 2 (21) n (x - x) !2 P i 2 1n i=1 n (xi x) i=1 − _ where xxis theis the mean mean of sampleof samplexi. Then,xi . Then, the skewness the skewness and kurtosis and kurtosis with depth with atdepth three at cross-sections three cross- (sectionsx = 0, 1,( x= 2 m0,) 1, associated 2 m ) associated with thewith horizontal the horizontal and vertical and vertical velocity velocity are plotted are plotted in Figure in Figure 19. The 19. Thehorizontal horizontal skewness skewness was abovewas above the zero the line, zero and line, it graduallyand it gradually tended tended to be right to be skew right with skew the with increase the increaseof wind speedof wind except speed at except the slope at toethe whereslope toe it shifted where leftit shifted skew. Meanwhile,left skew. Meanwhile, the kurtosis the among kurtosis the amongthree cross-sections the three cross-sections was lower was than lower 3, while than the3, wh significantile the significant difference difference was mainly was mainly reflected reflected in the verticalin the vertical direction, direction, the horizontal the horizontal kurtosis kurtosis of various of various locations locations was at nearlywas at the nearly same the level, same it canlevel, also it canbe observed also be observed that the influencethat the influence of wind onof wind the kurtosis on the waskurtosis negligible. was negligible.

Figure 19. SkewnessSkewness (top (top panel) panel) and and kurtosis kurtosis (bottom (bottom panel) panel) of horizontal of horizontal and andvertical vertical velocity velocity at (a,b at) (x=a,b)0mx =, (0c,md),(x=c,d1m) x =, and1 m ,(e and,f) x= (e,2mf) x .= The2 m results. The resultsare shown are shown for cases for casesW0H07, W0H07, W03H07, W03H07, and and W05H07, respectively. W05H07, respectively. 6. Conclusions 6. Conclusions In this paper, the effect of steady wind on the 2nd Stokes wave propagation on the slope with Waves2FOAMIn this paper, was the investigated, effect of steady wherein wind specific on the 2nd aspects Stokes were wave classified propagation into three on the parts: slope Wave with Waves2FOAMbreaking, turbulent was flow,investigated, and undertow. wherein The specific main findings aspects arewere as follows:classified into three parts: Wave breaking,Under turbulent the same flow, incident and undertow. wave height, The wind main induced findings earlier are as wavefollows: breaking while the breaking locationUnder migrated the same towards incident the wave offshore height, direction. wind induced Moreover, earlier within wave the breaking same wind while speed the coveragebreaking location migrated towards the offshore direction. Moreover, within the same wind speed coverage constraint, the smaller the ξ0 was, the higher the dispersion of wave breaking locations would be. As constraint, the smaller the ξ was, the higher the dispersion of wave breaking locations would be. the critical breaker index of H0b/hb increased with wind speed, the critical breaker index without wind was no longer suitable for the windy cases any more.

Water 2020, 12, 686 18 of 21

We examined the wind effect on turbulent flow in the wave breaking processes. Wind enhanced the coherence peak and thus the turbulence intensity of wave in the low-frequency range ( f < 2.5 Hz). The spectral density was larger than that with the absence of wind in the high-frequency range ( f > 2.5 Hz), which could serve to account for the energy transfer processes of wind input. Meanwhile, both the TKE and TDR displayed periodic variation before and after wave breaking, in which the TKE showed a bimodal distribution. The TKE in regions near the water surface was considerably larger with the wind speed and average level rapidly decreasing with the depth. The TDR with depth was barely impacted by the wind before breaking, whereas a significant effect of wind was found after breaking. The turbulence penetrated to the deeper flow and the average level generally rose. The dependence of undertow on the wind speed was found in the study. The closer it was to the surf zone, the more intense the undertow would be as the wind speed increased. The wind-induced setup added to the wave setup, which could aggravate the imbalance between the setup pressure gradient and the wave radiation stress, and contribute to the acceleration of undertow for compensating the loss volume of water carried shorewards incurred by the effects of wave breaking and wind.

Author Contributions: Conceptualization, C.J.; Methodology, Y.Y.; Software, B.D.; Validation, Y.Y.; Formal Analysis, Y.Y. and B.D.; Investigation, Y.Y. and B.D; Resources, C.J.; Data Curation, B.D.; Writing—Original Draft Preparation, Y.Y.; Writing—Review & Editing, C.J., Y.Y. and B.D.; Visualization, Y.Y.; Supervision, C.J. and B.D.; Project Administration, B.D.; Funding Acquisition, C.J. and B.D. All authors have read and agreed to the published version of the manuscript. Funding: This study was supported by the National Natural Science Foundation of China (Grant No. 51839002, 51979015, and 51879015). Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A The two variables x(t) and y(t) denote diﬀerent time series within the same period. Using the Fourier transform, the spectrum of x(t) is given by:

Z + 1 ∞ i2π f t F(x(t)) = X( f ) = x(t)e− dt (A1) 2π −∞ The cross-spectrum of x(t) and y(t) is deﬁned as:

W ( f ) = X∗( f ) Y( f ) (A2) XY · where X∗( f ) is the complex conjugate of X( f ) and Y( f ) is the spectrum of y(t). The correlation between x(t) and y(t) is also examined using the coherence and phase lag which are calculated by:

 2  WXY( f )  γ2 = | |  XY WXX( f )WYY( f ) (A3)  1 im(WXY( f ))  θXY = tan− ( ) real(WXY( f ))

2 where γXY is the coherence and θXY is the phase lag.

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