# Evolution of Velocity Field and Vortex Structure during Run-Down of Solitary Wave over Very Steep Beach

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## Abstract

**:**

_{0}/h

_{0}) of 0.363, 0.263, and 0.171 were generated in a wave flume. Two flow visualization techniques and high-speed particle image velocimetry were employed. The primary topics and new findings are: (1) Mechanism of the incipient flow separation, accompanied by formation of the separated shear layer from the beach surface, is elucidated under the adverse pressure gradient, using the fine data of velocity measurements very close to the sloping boundary. (2) Occurrence of hydraulic jump subsequently followed by development of the tongue-shaped free surface and projecting jet is demonstrated through spatio-temporal variation in the Froude number. It is confirmed by a change in the Froude number from supercritical to subcritical range as the free surface rapidly rises from the onshore to offshore side. (3) A complete evolution of the primary vortex structure (including the core position, vortex size, and velocity distribution passing through the vortex core) is first introduced systematically, together with the illustration of temporal variation in the topological structure. The non-dimensional shoreward distance of the vortex core section decreases with the increase in the non-dimensional time. However, the non-dimensional size height of the primary vortex increases with increasing non-dimensional time. (4) Two universal similarity profiles for both the wall jet flow and the shear layer flow demonstrate independency of the two similarity profiles of the wave-height to water-depth ratio and the beach slope. The similarity profiles indicate the promising collapse of the data from three previous studies for 1:20, 1:10, and 1:5 sloping beaches.

## 1. Introduction

_{0}/h

_{0}) in the range of 0.036 to 0.633. Applying the Lagrangian finite-element Boussinesq wave model, Zelt [4] studied, both numerically and experimentally, the characteristics of free surface profiles in the run-up stage of non-breaking and breaking solitary waves on plane beaches with (from mild to very steep) slopes varying from 1:19.85 to 1:2.75. Grilli et al. [5,6] used a completely nonlinear wave model, based upon the equations of potential flow and a series of wave-flume experiments to elucidate the shoaling and breaking features of solitary waves on different beach slopes that ranged from 1:35.0 to 1:6.5 and from 1:100 to 1:8. In their study, the breaking criterion named “slope parameter” and prediction of breaker type were proposed, along with illustration of empirical equations for calculating the water depth and wave celerity at breaking point. Using the combined results of both numerical simulations and flume tests, Lin et al. [7] presented the free surface profiles and velocity fields during the run-up and run-down phases of a solitary wave propagating over a very steep slope of 1:1.732. On the other hand, for a mild slope of 1:20, only the numerical result was presented. In their study, variations of the free surface and velocity field in hydraulic jump during the run-down phases were briefly illustrated, without addressing the boundary layer effect which in turn results in flow separation from the bottom boundary and vortex formation beneath the separated shear layer. Later, Li and Raichlen [8] experimentally investigated the breaking, impact of plunging jet on the sloping bottom with/without splash-up, and the consequent run-up of solitary waves, with H

_{0}/h

_{0}ranging from 0.03 to 0.40, over a 1:15 plane beach. Using the particle image velocimetry (PIV) measurement technique, Jensen et al. [9] presented experimental results of velocity field and acceleration during the run-up stage of strongly nonlinear waves, for H

_{0}/h

_{0}varying from 0.12 to 0.665, propagating on a moderately steep beach having a slope of 1:5.37.

_{0}/h

_{0}in the range of 0.012 to 0.338, Hsiao et al. [10] re-examined the existing empirical formulae for wave breaking criterion, wave height variation and maximum run-up height of solitary waves on a 1:60 plane beach. The feature of the plunging solitary wave propagating over a 1:14 sloping bottom was reported by Sumer et al. [11], along with detailed illustrations of the free surface elevation and bed shear stress using hot-film probe. The mean bed shear stress on the sloping bottom was reported to increase considerably during the run-up and run-down phases. Lo et al. [12] conducted an experimental study in three different wave flumes to investigate the free surface elevation, reflected wave, maximum run-up height, mass and momentum fluxes during the run-up and run-down stages of both single and double solitary waves on four different plane beaches having slopes of 1:20, 1:12, 1:10, and 1:2.47. PIV measuring techniques were used by Pedersen et al. [13] to elucidate the disagreements between experimental and theoretical run-up heights of solitary waves traveling on a beach with a slope of 1:5.67; indicating that the former was prominently smaller than the latter, as predicted by the inviscid theory, and that the discrepancy can be attributable to the losses of onshore mass transport induced by the boundary layer flow.

_{0}/h

_{0}= 0.400) passing through submerged barriers having combination from one to three in numbers (deployed from the toe of a 1:20 sloping beach toward onshore). The relative reduction was defined therein as the ratio of the difference between the maximum run-up height without barrier (R) and the counterpart with barriers (R’) to R, i.e., (R − R’)/R. The experimental results without a barrier, associated with the maximum run-up height as well as an instantaneous velocity field and corresponding visualized image of vortex structure, were presented for the solitary wave (with H

_{0}/h

_{0}= 0.400) traveling over a 1:20 sloping beach. However, the evolution of vortex structure was not analyzed and elucidated.

^{®}) to investigate the swash flow dynamics generated by a solitary wave on a 1:3 sloping beach. More emphases were focused on the numerical results of bottom shear stress during the run-up and run-down phases and pressure gradient field around the occurrence of hydraulic jump. However, due to relatively insufficient simulation for the shoreline dynamics, the distinctions between calculated results and experimental data become more prominent as time increases, thus leading to the divergent results during the run-down phase.

_{0}, and the incident wave-height to water-depth ratio, H

_{0}/h

_{0}. As reported and further emphasized by Grilli et al. [6], the effect of S

_{0}is more significant than that of H

_{0}/h

_{0}for determining variation of free surface elevation (also wave height) evolving on the offshore and onshore sides of breaking point. In addition, according to specified terminologies employed in identifying the relative extent of the inclination of the sloping beach, S

_{0}, a gentle or mild beach was called frequently for S

_{0}≤ 1:20 (Synolakis [3]; Grilli et al. [6]; Hsiao et al. [10]; and Hwung et al. [17]); beaches having 1:15 ≤ S

_{0}≤ 1:8 are called intermediate sloping beaches (Grilli et al. [6]; Lin and Hwung [20]; and Lin et al. [14]); a moderately steep beach designated for S

_{0}= 1:5.67 and 1:5.37 (Jensen et al. [9]; and Pedersen et al. [13]); and a very steep beach was named for S

_{0}≥ 1:4 (Grilli et al. [6]; Lin et al. [7]; and Higuera et al. [19]).

_{0}= 1:20, Lin et al. [15,16] for S

_{0}= 1:10 and 1:5, and Higuera et al. [19] for S

_{0}= 1:3 demonstrated were fairly different. At this juncture, it is pertinent to mention that neither the detailed mechanism for the incipient flow separation nor a complete evolution of the vortex structure and associated velocity field during the entire run-down phase has been elucidated in these studies to date.

_{0}= 1:3, using the quantitative HSPIV measurement data and qualitative flow-visualized images.

## 2. Experiment Set-Ups and Instrumentations

#### 2.1. Wave Flume, Model of Sloping Bottom and Coordinate Systems

_{0})

^{1/2}(with g defined as gravity acceleration), where t = 0 s (also T = 0) characterizes the instant when the crest of the solitary wave is exactly above the toe of the sloping beach, i.e., at x = X = 0 cm.

#### 2.2. Flow Field Observations Using Flow Visualization Techniques (FVT)

_{2}) particles having a refractive index of 2.6 and a mean diameter of 1.8 μm were used as seeding particles. The fall velocity (or settling velocity) of titanium dioxide particles, estimated by Stoke’s law, is 4.5 × 10

^{−4}cm/s. The effect of so tiny value of the falling velocity on the velocity measured can thus be neglected. In addition, the thin-layered fluorescent dye strip was used. The dye was a water solution of both fluorescein sodium (C

_{20}H

_{10}Na

_{2}O

_{5}) and little salt having a specific gravity of 1.003, measured by a precision hydrometer. The dye strip was injected very slowly and smoothly using a needle onto the beach surface near the still-water shoreline (at x = 24.0 cm) in the spanwise direction as the water mass inside the wave flume remained quiescent. Consequently, a fluorescent dye strip with nearly constant width was formed before the start of wave maker. As a result of solitary wave being generated, the dye strip is then stretched and transported along the sloping beach.

#### 2.3. Velocity Measurements by HSPIV

_{1}–FOV

_{6}over the horizontal bottom and sloping beach is shown schematically in Figure 1a, together with the dimension, range, and pixel resolution of each FOV

_{i}(i = 1–6) being listed in Table 1. To ensure a high time-resolved HSPIV algorithm, a framing rate of 1000/2000/3000 Hz was set while capturing the images of velocity fields with FOV

_{1}/FOV

_{2}/(FOV

_{3}–FOV

_{6}).

#### 2.4. Experimental Conditions

_{0}/h

_{0}= 0.363, 0.263 and 0.171 were tested and designated as Cases 1−3, respectively (see Table 2). The solitary waves considered are of a non-breaking type over the sloping beach, which is in good agreement with that identified by the non-dimensional slope parameter (Grilli et al. [6]). Ten repeated runs were performed for the HSPIV measurements for each case. An ensemble-averaged method is used to smooth the velocity time series and to describe the spatial or temporal variation of the velocity fields. Case 1 (H

_{0}/h

_{0}= 0.363) is used to illustrate a complete evolution (including first run-up, run-down, and second run-up) of the flow field. However, the velocity data of Cases 2 and 3 (H

_{0}/h

_{0}= 0.263 and 0.171) are employed only in the analysis of similarity profile for the velocity distributions passing through the primary vortex core translating offshore over the sloping beach.

## 3. Preliminary Tests

#### 3.1. Verification for Free Surface Elevation over Horizontal Bottom

_{0}, is validated. Figure 2 presents the comparison of the temporal variation in the non-dimensional, 10-run ensemble-averaged free surface elevations η

_{0}(t’)/h

_{0}for the incident solitary wave of Case 1 measured at the section of x’ (= (x + 150.0) cm) = 0 cm with the theoretical wave profile (Dean and Dalrymple [26]), as given by

_{0})

^{1/2}, where t’ = 0 s (also T’ = 0) identifies the instant when the crest of the solitary wave is right above the section of x’ = 0 cm. The (nonlinear) wave celerity C

_{0}can be obtained theoretically from (g(H

_{0}+ h

_{0}))

^{1/2}or experimentally from the cross-correlation analysis of the time interval for the wave crest passing the first and second wave gauges (with a separation of 150.0 cm). The experimental results of two individual runs (run #1 and #2) are also plotted in Figure 2 for further comparison. Fairly good agreement between the temporal variation of these three free surface elevations (including run #1 and #2 and 10-run ensemble-averaged data) and that predicted by the theory is clearly observed for Case 1. The results demonstrate the satisfactory generation of incident solitary waves by the wave maker. It should be mentioned that, due to the wave traveling over a constant water depth at the speed of C

_{0}, almost without changing its form (Lin et al. [14,27] and Keulegan [28]), the solitary wave is identified as one of the nearly permanent waves (Mei [29]), but not the dispersive one that disperses into waves of distinct wave periods and lengths (Lin et al. [30]).

#### 3.2. Test for Viscous Effect of Two Sidewalls on Attenuation of Incident Wave Height

_{0}= 2.96, 2.90 and 2.85 cm, respectively, with a still water depth of h

_{0}= 8.0 cm (Case 1). As reported by Keulegan [28], the following two equations are used to compute the attenuation of wave height for a solitary wave traveling over a horizontal channel of finite width:

_{0}/H

_{0})

^{1/4}]

_{B}− [(h

_{0}/H

_{0})

^{1/4}]

_{A}= K

_{t}× d/h

_{0}

_{t}= K

_{1}+ K

_{2}= 1/12 × [1 + 2(h

_{0}/W)] × {ν/[g

^{1/2}× (h

_{0})

^{3/2}]}

^{1/2}

_{1}(= 1/12 × {ν/(g

^{1/2}× (h

_{0})

^{3/2})}

^{1/2}) is the coefficeint of damping due to bottom friction, and K

_{2}(= 1/12 × 2(h

_{0}/B) × {ν/(g

^{1/2}× (h

_{0})

^{3/2})}

^{1/2}) represents the coefficient of damping caused by the two sidewalls. Therefore, K

_{t}can be interpreted as the coefficient of combined damping as mentioned. In the present case, the measuring sections A and B are located at x = −250.0 and −50.0 cm, respectively, with h

_{0}= 8.0 cm, d = 200.0 cm, and W = 25.0 cm. Accordingly, the coefficeints are calculated to be K

_{1}= 3.13 × 10

^{−4}, K

_{2}= 1.75 × 10

^{−4}, and K

_{t}= 4.88 × 10

^{−4}, demonstrating that the value of K

_{2}is smaller than that of K

_{1}and highlighting that less of a role is played by the friction of two side walls. The relative attenuation of the wave height between x = −250.0 cm and x = −50.0 cm is calculated to be {(H

_{0})

_{A}− (H

_{0})

_{B}}/(H

_{0})

_{B}= 3.7%, in which the influence of two sidewalls is about 1.33% (=3.7% × K

_{2}/K

_{t}). This evidence fairly exhibits the slight effect of the finite width between the two sidewalls on the wave height attenuation and on the value of the incident wave height, H

_{0}.

#### 3.3. Verifications for Velocity Fields/Profiles Measured over Horizontal Bottom

_{1}(with a size of 2.0 cm wide × 1.0 cm high and a resolution of 1280 × 640 pixel). The former/latter represents the velocity field relatively away from/close to the bottom boundary layer before/while the wave crest reaches the section x’ = 0 cm. Following the method proposed by Chang and Liu [31], the measurement error of the velocity is described herein. Assuming the flow is two-dimensional, a continuity check is performed to compute the flux of each grid element F = |∂u/∂x + ∂v/∂y| × dA (in which Δx = Δy = 0.0282 cm for each grid element are used and dA (= Δx × Δy) is the area of a grid element) in the velocity field. Here, a representative flux, F

_{0}(= (u

_{∞})

_{max}× Δy) is defined with (u

_{∞})

_{max}is the free stream velocity outside the bottom boundary layer for t’ = 0 s. The relative error in the flux of the velocity field can be identified as F/F

_{0}. The relative errors F/F

_{0}at most positions with negligible velocity gradient (see Figure 3a) are less than 2.6%, and the counterparts at locations having significant velocity gradients near the sloping beach (see Figure 3b) are at most 3.2%. These two tiny relative errors identified thus confirm fairly precise measurements in the velocity fields using the HSPIV system.

_{0}and the maximum free stream velocity (u

_{∞})

_{max}, both taking place at t’ = 0 s (T’ = 0), are used as the representative length and velocity scales, respectively. Herewith, the time-dependent boundary layer thickness, δ(t’), is defined as the distance from the bottom surface to the position where the horizontal velocity u(t’) at the edge of boundary layer is equal to 0.99u

_{∞}(t’) at y’ = δ(t’) (Lin et al. [15,27]; and Sumer et al. [33]). It can be clearly witnessed from Figure 3c,d that the HSPIV data coincide very well with those measured by fiber-optic laser Doppler velocimetry for all of the five time instants, strongly highlighting satisfactory agreements between these two techniques of measurements.

^{−4}cm or smaller [13,34,35], which is several order of magnitudes less than the boundary layer thickness in the present study (about 0.2–0.3 cm, see Figure 3b). Therefore, the beach surface can be treated as smooth and the effect of the roughness of the beach surface on the structure of boundary layer flow can be neglected.

#### 3.4. Identification for Flow Pattern in Boundary Layer over Horizontal Bottom

_{e}= a

_{m}× (u

_{∞})

_{max}/ν, in which a

_{m}(= (u

_{∞})

_{max}× t

_{p}/2π) is the amplitude (or half of the stroke) of the water particle displacement in the free-stream region, (u

_{∞})

_{max}the maximum free stream velocity, and t

_{p}the characteristic time scale that identifies the time width in the time series of either velocity or free surface elevation of solitary wave. In the present Cases 1–3, the corresponding Reynolds numbers R

_{e}being 11,000, 8900 and 13,300 (see Lin et al. [27]) are all much smaller than 2 × 10

^{5}, indicating laminar flow in the boundary layer (Sumer et al. [33]). It should be mentioned that this study is only applicable to the flow at a laboratory scale. For example, Figure 3c,d has shown fairly identical non-dimensional profiles for different phases with a scale factor of 2.28, implying the existence of the Reynolds-number similitude for the present viscous boundary layer flows. However, scaling these cases to prototype, e.g., by a factor of 100 so that the still water depth is 8.0 m and wave heights of 2.1 and 2.9 m for Cases 1 and 2, would multiply the present R

_{e}by a factor of 1000, thus rendering the values of Reynolds numbers far from the laminar regime and thus into the turbulent regime.

#### 3.5. Repeatability Test for Instantaneous Free Surface Profile and Velocity Field over Sloping Bottom

## 4. General Description of Run-Up and Run-Down Process of Solitary Wave

## 5. Results and Discussion for Complete Evolution of Run-Down (Case 1)

#### 5.1. Start of Run-Down at t = 0.655 s

_{u}(x, t), is defined as the mean of velocities only within uniform or quasi-linear part of the whole distribution, i.e., without taking the velocities inside boundary layer flow into account. Note that the spatial and temporal variations of u

_{u}(x, t) can be used as an indicator of flow acceleration or deceleration.

_{u}(x, t) existing within each velocity profile is observed to decrease continuously from 27.5 cm/s to 20.1 cm/s with decreasing shoreward distance from x = 19.94 cm (X = 21.02 cm) to x = 13.00 cm (X = 13.70 cm), respectively. These values of u

_{u}(x, t) indicate the retreated flow decelerated spatially in the offshore direction for t = 0.655 s due to the convective acceleration (Daily and Harleman [37]) positive/negative in the onshore/offshore direction. More interestingly, as observed from the sub-figure ③ of Figure 8b for x = 17.84 cm (X = 18.80 cm), the magnitude of u

_{u}increases with increasing t for t = 0.646–0.664 s. This observation strongly reveals that the retreated flow is temporally accelerated in the offshore direction because the local acceleration (Daily and Harleman [37]) holds positive/negative value in the offshore/onshore direction.

_{r}= −U

_{d}/(gh’)

^{1/2}where U

_{d}(X, t) is the depth-averaged velocity of U(X, Y, t) along the Y axis. Figure 8a also shows the corresponding Froude number at each section of the four inclined lines (marked with the numbers from 1’ to 4’ in a circle). The Froude numbers decrease continuously from F

_{r}= 0.49 at x = 19.94 cm (i.e., X = 21.02 cm) to F

_{r}= 0.27 at x = 13.00 cm (i.e., X = 13.70 cm) and are all smaller than unity, demonstrating that the retreated flow is subcritical at t = 0.655 s (i.e., the moment for the start of run-down).

#### 5.2. Early and First-Half Middle Stages of Run-Down for 0.655 s < t < 0.921 s

_{u}(x, t), along with the spatial change of free surface profile for 0.655 s ≤ t ≤ 0.983 s (Case 1), exhibiting successive depression of the free surface elevations in the region for x > 14.6 cm (i.e., X > 15.39 cm). Further, for t = 0.722/0.813 s, prominent decrease in the magnitude of u

_{u}(x, t) in the offshore direction occurs from 36.6/52.5 cm/s at x = 20.1 cm (X = 21.19 cm) to 27.8/31.9 cm/s at x = 14.6 cm (X = 15.39 cm), respectively. This observation clearly indicates that the retreated flow is decelerated spatially in the offshore direction. More interestingly, the magnitude of u

_{u}increases with increasing t for t = 0.655–0.921 s at any specified shoreward distance between x = 10.8 cm (X = 11.38 cm) and x = 20.1 cm (X = 21.19 cm) (see Figure 9), which strongly reveals that the retreated flow is accelerated temporally in the offshore direction.

_{r}= 1.15 at X = 21.02 cm, via F

_{r}= 0.75 at X = 18.81 cm, to F

_{r}= 0.46 at X = 16.06 cm. The result demonstrates the change in the retreated flow condition from supercritical state (F

_{r}> 1.0) on the onshore side to a sub-critical one (F

_{r}< 1.0) on the offshore side. The critical section (F

_{r}= 1.0) occurs is located at X = X

_{cr}= 20.32 cm (not shown). Soon after t > 0.813 s, the water depth still keeps decreasing for X > 15.39 cm (i.e., x > 14.6 cm, see Figure 9). At the same time, the retreated flow remains decelerated spatially and accelerated temporally in the offshore direction due to the effect of gravity. As a result, a supercritical flow is formed and moves offshore toward the deeper region where a subcritical flow exists. Therefore, a critical flow should exist between the deeper and shallower regions.

#### 5.3. Incipient Flow Separation from Beach Surface at t = 0.921 s

_{u}(x) changing from 81.5 cm/s at x = 20.1 cm (X = 21.19 cm) to 25.5 cm/s at x = 10.8 cm (X = 11.38 cm) strongly implies spatial deceleration of the retreated flow and adverse pressure gradient in the offshore direction. Further, as is also seen in Figure 11a, the Froude numbers marked near the circle numbers ①–⑤ indicate that the smaller/greater the water depth, the greater/smaller the Froude number. It is pertinent to mention that the critical flow appears at the section, X = X

_{cr}= 18.04 cm (not shown) for t = 0.921 s, which is more offshore than that for t = 0.813 s (i.e., at X

_{cr}= 20.32 cm).

_{ifs}= 18.81 cm (see sub-figure 2’ of Figure 11c), which is located on the onshore side of the critical section (X

_{cr}= 18.04 cm) with supercritical flow (see Figure 11a). As, very close to the beach surface, the gradient of the velocity component U(Y) with respect to Y is first identified to be zero; and because, on the offshore and onshore sides of the incipient flow separation point, the velocities very close to the sloping beach take the positive and negative values for Y < 0.023 cm (Figure 11d), respectively. Soon after the incipient flow separation at t = 0.921 s, the fluid particles that are very close to the beach surface move away and transport toward the external stream of retreated flow, and then to displace in a direction opposite to the external stream, resulting in a very thin recirculation zone under the separated shear layer. It is worth mentioning that the instant for the critical flow taking place right at the section of incipient flow separation (i.e., the critical section at X

_{cr}= X

_{ifs}= 18.81 cm) is corresponding to t = 0.874 s (not shown).

#### 5.4. Second-Half Middle Stage I of Run-Down for 0.921 s < t < 1.007 s

_{size}. For t = 0.983 s, the size height can be estimated to be Y

_{size}= 0.21 cm at the section where the primary vortex core is located (hereafter named “core section” and defined as X

_{co}), i.e., X = X

_{co}= 17.95 cm.

_{cr}= 17.44 and 17.08 cm, respectively, which are located at more offshore positions than that (=18.04 cm) for t = 0.921 s. Further, as is also seen in Figure 8, the nature of the horizontal uniform velocity u

_{u}for t = 0.983 s is very different from those for t = 0.813 and 0.921 s. It does highlight the decrease of |u

_{u}| from x = 19.0 cm (X = 20.03 cm) down to a local minimum (=87.3 cm/s) at about x = 17.9 cm (X = 18.87 cm), then an increase of |u

_{u}| up to a local maximum (=92.3 cm/s) at x = 17.03 cm (X = 17.95 cm), and finally a prominent decrease of |u

_{u}| down to the value of 36.3 cm/s at about x = 14.0 cm (X = 14.76 cm). Such a variation feature can be attributable to the influence of the rotational and translating motion of the primary vortex (with its core at x

_{co}= 17.03 cm or X

_{co}= 17.95 cm), as evidenced in Figure 12c,d.

#### 5.5. Incipient Hydraulic Jump at t = 1.007 s

_{cr}= 16.42 cm, see section ③ in Figure 13b.

_{co}= 17.41 cm. Figure 14b presents the instantaneous streamline pattern (obtained from the corresponding velocity field), indicating the size height and core section of the primary vortex are, respectively, equal to Y

_{size}= 0.36 cm and X

_{co}= 17.41 cm, which are in good agreement with those observed from visualized image (see Figure 14a). With reference to the retreated flow, the velocity profile measured at X

_{co}=17.41 cm exhibits the “opposite (i.e., onshore) flow” distributing from the bottom surface up to the core height. Further, the Froude number at X

_{co}= 17.41 cm is still larger than unity (with F

_{r}= 1.97), identifying the retreated flow at the core section is supercritical. It should be mentioned that, for t = 1.007 s, the critical flow occurs at X

_{cr}= 16.42 cm (Figure 13a), which is located on the offshore side of the core section (at X

_{co}= 17.41 cm).

#### 5.6. Second-Half Middle Stage II of Run-Down for 1.007 s < t ≤ 1.063 s

_{size}= 0.40 cm and the core section at X

_{co}= 17.20 cm. This core section is located on the onshore side of the critical section at X

_{cr}= 16.12 cm (not shown), indicating occurrence of a supercritical flow at the core section. Two developing minor vortices rotating in the counterclockwise and clockwise directions are located on the offshore and onshore sides of the primary vortex for 15.8 cm ≤ X ≤ 16.4 cm and 17.6 cm ≤ X ≤ 18.1 cm, respectively. These two minor vortices are formed under separated shear layer with continuous energy supply from the accelerated, supercritical retreated flow. Two saddle points are seen evidently on both sides of the primary vortex, with one being located at about (X, Y) = (16.45, 0.1) cm and the other at (18.0, 0.17) cm. Moreover, a surface saddle point (Hunt et al. [43]) is identified at about (X, Y) = (17.5, 0) cm with one virtual streamline orienting upward from the bottom surface; and a stagnation point is observed at (X, Y) = (18.08, 0) cm with a streamline directing toward the beach surface. These observations exhibit, remarkably, a developing topological structure under the separated shear layer.

_{co}= 16.31 cm which is more onshore than that of the critical section (X

_{cr}= 15.25 cm). Meanwhile, the size height is about Y

_{size}= 0.57 cm, equal to half of the local water depth (i.e., h’/2 = 0.59 cm) approximately and greater than those for t = 0.983–1.017 s. Furthermore, Figure 15c shows the corresponding evolution of the secondary and finer vortices that move further offshore for 15.80 cm ≤ X ≤ 19.00 cm at t = 1.063 s. To provide more profound observation of the main structure in the primary vortex, a FOV ranging further offshore between X = 13.3 cm and X = 16.5 cm is used additionally. Figure 15d shows the primary vortex extending from X = 15.1 cm to X = 16.4 cm with the size height of Y

_{size}= 0.64 cm and the core section at X

_{co}= 15.7 cm for t = 1.063 s. The two images taken at the same t (Figure 15c,d) strongly demonstrate nearly identical flow structure existing in the overlapped region for 15.8 cm ≤ X ≤ 16.5 cm, and very high repeatability of the generated solitary wave over the sloping beach.

#### 5.7. Late Stage of Run-Down for 1.063 s < t < 1.213 s

_{size}= 0.70 cm moves toward offshore and interacts with a vortex located on its offshore side, undergoing a process of merging in 13.3 cm ≤ X ≤ 14.25 cm. The core of the primary vortex is located at X

_{co}= 14.98 cm, which is more toward offshore, as compared with that for t = 1.048 s (i.e., at X

_{co}= 16.31 cm). In addition, the external stream of the retreated flow appearing beneath the moving free surface and beyond the primary vortex is also observed. Figure 16c also presents four velocity profiles obtained at X = 16.25 cm, 15.59 cm, 14.98 cm (=X

_{co}), and 14.36 cm. Due to the existence of the hump-shaped free surface profile over the external stream, the local water depths at these four sections vary very drastically. The mean velocity of the external stream, U

_{ex}, can be obtained by partially depth-averaging over the velocity data beneath the free surface and just beyond the most outside streamline surrounding the primary vortex (see Figure 16b). Accordingly, the mean velocity of the external stream are in the order U

_{ex}= −74.7, −101.6, −54.0, and −101.7 cm/s. Nevertheless, the fully depth-averaged velocities are correspondingly equal to U

_{d}= −71.6, −49.1, −47.9, and −45.3 cm/s, demonstrating deceleration of the retreated flow in the offshore direction. In addition, the Froude numbers F

_{r}, as shown in Figure 16a (marked over the four dashed lines), are larger than unity, revealing that the retreated flow is still supercritical up to t = 1.077 s.

_{cv}, is defined as the mean velocity of the vortex core transported in the offshore direction within a specified time elapse. For t = 1.048 s, the convection velocity can be estimated from the visualized images for t = 1.040–1.057 s to be U

_{cv}= −53.1 cm/s. It is also found that the magnitude of this convection velocity U

_{cv}is, in fact, less than any one of the four mean velocities of the external stream, U

_{ex}(also shown in the top of Figure 16a). Namely, the high-speed external stream, which has higher offshore speed, translates rapidly over the vortex with distinguishable curvature on the onshore side of the primary vortex, and “impinges upon” the primary vortex due to the relative obstruction of the primary vortex moving with lower convection velocity. The high-speed external stream also acts like a free-jump flow surrounding and over the primary vortex then behaves like a free-flop flow sliding down the primary vortex.

_{co}= 16.31 cm at t = 1.048 s, via X

_{co}= 14.98 cm at t = 1.077 s, then to X

_{co}= 13.40 cm at t = 1.098 s. Soon after a short time interval, say at t = 1.134 s, the merged vortex has been carried more offshore and upward by the turbulent stream induced by the high-speed retreated flow. Very chaotic flow field in the neighborhood of the merged vortex is observed hereafter.

#### 5.8. Second Run-Up for t ≥ 1.213 s

## 6. Discussion on Characteristics of Primary Vortex Structure

#### 6.1. Variations of Critical and Core Sections as Well as Size Height for 10.20 ≤ T ≤ 12.11

_{r}= 1.0 can be summarized for t = 0.921–1.094 s (i.e., T = 10.20–12.11) for Case 1. These data are all shown in the non-dimensional form of X

_{cr}/h

_{0}versus T in Figure 18. Note that T = 10.20, the incipient flow separation occurs, and at T = 12.11, the projecting jet right impinges upon the free surface of the retreated flow. It strongly indicates that the critical section moves offshore as T increases for 10.20 ≤ T ≤ 12.11 as indicated in Figure 18 with a decrease in the value of X

_{cr}/h

_{0}. This is due to the vortex being enclosed with air core, thus leading to the two-phase flow field with a larger water depth. Later, with T ≥ 12.11, the critical section no longer exists. Figure 18 also illustrates the temporal variation of the non-dimensional size height Y

_{size}/h

_{0}, clearly indicating that the size height of the primary vortex increases linearly by increasing T for 10.55 ≤ T ≤ 12.11.

_{co}/h

_{0}is also illustrated in Figure 18. It is shown that the primary vortex core moves offshore as T increases for 10.55 ≤ T ≤ 12.11, and that the core section appears on the onshore (i.e., shallower) side of the critical section at the identical T. Remember that, soon after the incipient flow separation at T = 10.20 (t = 0.921 s, see Figure 11b–d), a very thin flow reversal zone underlying the shear layer develops for 10.20 < T < 10.55. During this interval, the core of the elongated recirculation zone is hardly recognized. However, for T ≥ 10.55, the primary vortex evolved from the elongated recirculation zone can be identified, see Figure 12a–d for T = 10.60–10.88 (t = 0.957–0.983 s).

#### 6.2. Similarity Profiles for Velocity Distributions at Core Sections for 11.00 < T < 12.11

_{M}denotes the height at which the maximum onshore velocity U

_{M}(> 0) occurs and Y

_{co}represents the core height of the primary vortex where U is equal to zero, i.e., U = U

_{co}= 0. Further, Y

_{m}represents the height where the negative maximum velocity U

_{m}(< 0) occurs, and U

_{f}(< 0) stands for the velocity nearly at the free-surface height, Y

_{f}. Note that the velocity profile, U(Y), shows the uniform feature if U

_{m}is equal to U

_{f}for Y

_{m}≤ Y ≤ Y

_{f}. However, if U

_{m}occurs at the free-surface height Y

_{f}, i.e., U

_{m}= U

_{f}, the velocity profile then exhibits the velocity gradient or non-uniform distribution for Y

_{co}≤ Y ≤ Y

_{f}. It should be emphasized that, for 0 < Y ≤ Y

_{co}, the velocity distribution is significantly influenced by the existence of the sloping beach and somehow exhibits a wall jet feature. On the other hand, the velocity profile U(Y) between Y = Y

_{co}and Y = Y

_{m}(or Y = Y

_{f}if U

_{m}= U

_{f}) has the characteristics of shear layer between Y

_{co}and Y

_{m}or Y

_{f}. Based on the interpretation mentioned above, two non-dimensional profiles for the velocity distributions U(Y) are proposed: One for the wall jet flow and the other for the shear layer flow.

_{M}and the height Y = Y

_{b}where U = U

_{M}/2 = 0 cm/s take places at the “half thickness” are taken as the characteristic velocity and length scales, respectively (Rajaratnam [45]). The data from not only Case 1 (with H

_{0}/h

_{0}= 0.363), but also Cases 2 and 3 (with H

_{0}/h

_{0}= 0.263 and 0.171, examples see Figure A1 in Appendix A) are all considered. Then, a non-dimensional profile for the velocity distributions that are close to the sloping beach, U(Y) (passing through the primary vortex core at different T), can be obtained by scaling with respect to these two characteristic scales. Figure 19b presents the non-dimensional plot for the present three cases, strongly highlighting a unique trend with a promising similarity, independent of the incident wave-height to water-depth ratio. However, to achieve a systematic summary for the effect of the beach slope, the data from Hwung et al. [17] for S

_{0}= 1:20 and Lin et al. [15,16] for S

_{0}= 1:10 and 1:5 (in which the global velocity fields were all demonstrated, but the details associated with vortex structures were not shown) are especially included in Figure 19b, via the similar rescaling process. It is surprisingly found that the trend of the data points from these three studies are almost identical to that of the present three cases, evidently indicating no effect of beach slope and wave-height to water-depth ratio on this similarity profile. Regression analysis for all of the data points illustrates that the universal similarity profile for the non-dimensional velocities, U(Y)/U

_{M}, can be uniquely expressed as

_{M}= 3.30(Y/Y

_{b}− 0.02)

^{0.65}× {1.0 − erf(1.02(Y/Y

_{b}− 0.02))} − 0.03 for 0 < Y/Y

_{b}≤ 1.4,

^{2}value of 0.982. This emphasizes that both the wave-height to water-depth ratio and the beach slope have no effect on the wall jet similarity profile. The present form of Equation (4) for the similarity profile of the onshore velocity is very similar to Verhoff’s velocity distribution equation for the traditional plane wall jet (Rajaratnam [45]; and Verhoff [46]), demonstrating the stream within the region between the core height and the sloping beach being the wall jet flow.

_{co}to Y = Y

_{m}(see Figure 19a), the representative velocity and length scales, −U

_{m}(> 0) and (Y

_{m}− Y

_{co}) are used to formulate an appropriate similarity profile. Note that (Y

_{m}− Y

_{co}) stands for the representative thickness of shear layer. Herein, the same data sets used in Figure 19b have also been incorporated in the analysis. Accordingly, the dimensionless re-scaled velocity profile, −U/U

_{m}(<0), against the dimensionless re-scaled thickness, Y* = (Y − Y

_{co})/(Y

_{m}− Y

_{co}), is plotted in Figure 19c with the non-linearly regressed fitting (R

^{2}= 0.997) as:

_{m}= −0.760 × tanh(1.869Y* − 0.484) − 0.331 for 0 ≤ Y* ≤ 1.0,

## 7. Conclusions

_{r}= 1.0), and development of the primary vortex structure in terms of variations in size height, core section, and topological structure. Some important findings are drawn as follows:

- The run-down process of the solitary wave for 7.25 ≤ T < 13.43 can be divided into the early, middle, and late stages. In the early and first-half middle stage for about 7.25 ≤ T < 10.20, formation of the very streamlined free surface characterizes the flow event, highlighting the flow being decelerated in the offshore direction and subjected to adverse pressure gradient. The section of occurrence of the critical Froude number moves offshore as T increases for T > 9.00.
- In the second-half middle stage of the run-down process for 10.20 ≤ T < 11.60, the incipient flow separation, evolution of the primary vortex structure, hydraulic jump with sudden rising of the free surface in the retreated flow, and formation of curling jet on the upper portion of free surface take place sequentially.
- As regards the late stage of run-down process with 11.60 ≤ T < 13.43, transformation of the curled free surface into the impinging jet impacting on the free surface of the retreated flow at T = 12.11, and the two-phase flow with air core and/or subsequent air bubbles being entrained into vortex occur. Right at T = 13.43, the free surface of the retreated flow reaching its offshore-most position identifies the end of the run-down process.
- The non-dimensional shoreward distance of the critical/core section moves offshore as T increases for 10.2/10.55 ≤ T ≤ 12.11, with the latter taking place on the onshore side of the former for the identical T. The magnitudes of the non-dimensional size height of the primary vortex increases linearly with increasing T for 10.55 ≤ T ≤ 12.16. As a result of entrainment of air core within vortex, the two-phase flow field develops with a larger water depth for T ≥ 12.11, and consequently the critical section no longer exists.
- A unique similarity profile for the onshore velocity U(Y) measured through the primary vortex core and obtained very close to beach surface for 0 ≤ Y ≤ Y
_{co}has been achieved by selecting the positive maximum velocity U_{M}and the specified half thickness Y_{b}as the characteristic velocity and length scales, respectively. This profile, with Equation (4), is very similar to Verhoff’s velocity distribution equation for the traditional plane wall jet, demonstrating existence of the wall jet flow right beyond the beach surface. - Further, a similarity profile for the offshore velocity U(Y) in the shear layer, obtained between the core height of the primary vortex Y
_{co}and the specified height Y_{m}, has been realized by selecting the negative maximum velocity U_{m}and the representative thickness of the shear layer (Y_{m}− Y_{co}) as the characteristic velocity and length scales, respectively. This profile, with Equation (5), identifies the feature of the shear layer flow, similar to those of traditional velocity distributions in the free and cavity shear layer flows. - The two similarity profiles obtained for wall jet flow very close to the beach surface (0 ≤ Y ≤ Y
_{co}) and for shear layer flow between the core height of the primary vortex Y_{co}and the specified height Y_{m}are obtained considering all of the three cases of the present study (H_{0}/h_{0}= 0.171, 0.263, and 0.363) as well as data of Lin et al. [15,16] for S_{0}= 1:10 and 1:5 and Hwung et al. [17] for S_{0}= 1:20. This is evidence that neither the wave-height to water-depth ratio H_{0}/h_{0}nor the beach slope S_{0}has effects on both the wall jet and shear layer similarity profiles, strongly highlighting their universal feature.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Flow visualization image and three velocity profiles U(Y) measured at and around core section of primary vortex for Case 2 at (

**a**) t = 1.007 s and (

**b**) t = 1.043 s; for Case 3 at (

**c**) t = 1.443 s and (

**d**) t = 1.457 s. Characteristic velocity and length scales U

_{M}and Y

_{b}as well as U

_{m}and Y

_{m}, together with core height Y

_{co}are shown in each subfigure ②.

## References

- British Association for the Advancement of Science. Committee on Waves. In Report of the Committee on Waves: Appointed by the British Association at Bristol in 1836 [and Consisting of Sir John Robison and John Scott Russell]; R. and J.E. Taylor: London, UK, 1838; pp. 417–496. [Google Scholar]
- United States. Beach Erosion Board. Laboratory Investigation of the Vertical Rise of Solitary Waves on Impermeable Slopes; U.S. Beach Erosion Board: Washington, DC, USA, 1953.
- Synolakis, C.E. The runup of solitary waves. J. Fluid Mech.
**1987**, 185, 523–545. [Google Scholar] [CrossRef] - Zelt, J.A. The run-up of nonbreaking and breaking solitary waves. Coast. Eng.
**1991**, 15, 205–246. [Google Scholar] [CrossRef] - Grilli, S.T.; Subramanya, R.; Svendsen, I.A.; Veeramony, J. Shoaling of solitary waves on plane beaches. J. Waterw. Port Coast. Ocean Eng.
**1994**, 120, 609–628. [Google Scholar] [CrossRef] - Grilli, S.T.; Svendsen, I.A.; Subramanya, R. Breaking criterion and characteristics for solitary waves on slopes. J. Waterw. Port Coast. Ocean Eng.
**1997**, 123, 102–112. [Google Scholar] [CrossRef] - Lin, P.; Chang, K.A.; Liu, P.L.F. Runup and rundown of solitary waves on sloping beaches. J. Waterw. Port Coast. Ocean Eng.
**1999**, 125, 247–255. [Google Scholar] [CrossRef] - Li, Y.; Raichlen, F. Energy balance model for breaking solitary wave runup. J. Waterw. Port Coast. Ocean Eng.
**2003**, 129, 47–59. [Google Scholar] [CrossRef] - Jensen, A.; Pedersen, G.K.; Wood, D.J. An experimental study of wave run-up at a steep beach. J. Fluid Mech.
**2003**, 486, 161–188. [Google Scholar] [CrossRef] - Hsiao, S.C.; Hsu, T.W.; Lin, T.C.; Chang, Y.H. On the evolution and run-up of breaking solitary waves on a mild sloping beach. Coast. Eng.
**2008**, 55, 975–988. [Google Scholar] [CrossRef] - Sumer, B.M.; Sen, M.B.; Karagali, I.; Ceren, B.; Fredsøe, J.; Sottile, M.; Zilioli, L.; Fuhrman, D.R. Flow and sediment transport induced by a plunging solitary wave. J. Geophys. Res.
**2011**, 116. [Google Scholar] [CrossRef] [Green Version] - Lo, H.Y.; Park, Y.S.; Liu, P.L.F. On the run-up and back-wash processes of single and double solitary waves—An experimental study. Coast. Eng.
**2013**, 80, 1–14. [Google Scholar] [CrossRef] - Pedersen, G.; Lindstrom, E.; Bertelsen, A.F.; Jensen, A.; Laskovski, D. Runup and boundary layers on sloping beaches. Phys. Fluids
**2013**, 25, 012102. [Google Scholar] [CrossRef] - Lin, C.; Yeh, P.H.; Hseih, S.C.; Shih, Y.N.; Lo, L.F.; Tsai, C.P. Pre-breaking internal velocity field induced by a solitary wave propagating over a 1:10 slope. Ocean Eng.
**2014**, 80, 1–12. [Google Scholar] [CrossRef] - Lin, C.; Yeh, P.H.; Kao, M.J.; Yu, M.H.; Hseih, S.C.; Chang, S.C.; Wu, T.R.; Tsai, C.P. Velocity fields in near-bottom and boundary layer flows in pre-breaking zone of solitary wave propagating over a 1:10 slope. J. Waterw. Port Coast. Ocean Eng.
**2015**, 141, 04014038. [Google Scholar] [CrossRef] - Lin, C.; Kao, M.J.; Tzeng, G.W.; Wong, W.Y.; Yang, J.; Raikar, R.V.; Wu, T.R.; Liu, P.L.F. Study on flow fields of boundary-layer separation and hydraulic jump during rundown motion of shoaling solitary wave. J. Earthq. Tsunami
**2015**, 9, 154002. [Google Scholar] [CrossRef] - Hwung, H.H.; Wu, Y.T.; Lin, C. Tsunami propagation and related new approach of mitigation. In Proceedings of the 8th Taiwan-Japan Joint Seminar on Natural Hazard Mitigation, Kyoto, Japan, 7 December 2015. [Google Scholar]
- Skene, D.M.; Bennetts, L.G.; Wright, M.; Meylan, M.H.; Maki, K.J. Water wave overwash of a step. J. Fluid Mech.
**2018**, 839, 293–312. [Google Scholar] [CrossRef] - Higuera, P.; Liu, P.L.F.; Lin, C.; Wong, W.Y.; Kao, M.J. Laboratory-scale swash flows generated by a non-breaking solitary wave on a steep slope. J. Fluid Mech.
**2018**, 847, 186–227. [Google Scholar] [CrossRef] - Lin, C.; Hwung, H.H. Observation and measurement of the bottom boundary layer flow in the prebreaking zone of shoaling waves. Ocean Eng.
**2002**, 29, 1479–1502. [Google Scholar] [CrossRef] - Goring, D.G. Tsunami: The Propagation of Long Waves onto a Shelf; Technical Report No. KH-R-38; W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology: Pasadena, CA, USA, 1978. [Google Scholar]
- Adrian, R.J.; Westerweel, J. Particle Image Velocimetry; Cambridge University Press: New York, NY, USA, 2011. [Google Scholar]
- Cowen, E.A.; Monismith, S.G. A hybrid digital particle tracking velocimetry technique. Exp. Fluids
**1997**, 22, 199–211. [Google Scholar] [CrossRef] - Mori, N.; Chang, K.A. Introduction to MPIV-PIV Toolbox in Matlab. User Reference Manual. 2009, pp. 1–15. Available online: http://www.oceanwave.jp/softwares/mpiv/index.php?Download (accessed on 1 November 2018).
- Adrian, R.J. Particle imaging techniques for experimental fluid mechanics. Annu. Rev. Fluid Mech.
**1991**, 23, 261–304. [Google Scholar] [CrossRef] - Dean, R.G.; Dalrymple, R.A. Water Wave Mechanics for Engineers and Scientists; World Scientific Publishing Co. Pte. Ltd.: Hackensack, NJ, USA, 1995. [Google Scholar]
- Lin, C.; Yu, S.M.; Wong, W.Y.; Tzeng, G.W.; Kao, M.J.; Yeh, P.H.; Raikar, R.V.; Yang, J.; Tsai, C.P. Velocity characteristics in boundary layer flow caused by solitary wave traveling over horizontal bottom. Exp. Therm. Fluid Sci.
**2016**, 76, 238–252. [Google Scholar] [CrossRef] - Keulegan, G.H. Gradual damping of solitary waves. J. Res. Nat. Bur. Stand.
**1948**, 40, 487–498. [Google Scholar] [CrossRef] - Mei, C.C. The Applied Dynamics of Ocean Surface Waves; World Scientific Publishing Co. Pte. Ltd.: Singapore, 1989. [Google Scholar]
- Lin, C.; Kao, M.J.; Wong, W.Y.; Shao, Y.P.; Fu, C.F.; Yuan, J.M.; Raikar, R.V. Effect of leading waves on velocity distribution of undular bore traveling over sloping bottom. Eur. J. Mech. Ser. B Fluids
**2018**. [Google Scholar] [CrossRef] - Chang, K.A.; Liu, P.L.F. Pseudo turbulence in PIV breaking wave measurements. Exp. Fluids
**2000**, 29, 331–338. [Google Scholar] [CrossRef] - Ho, T.C. Characteristics of Vortical Flow Fields Induced by Solitary Waves Propagating over Submerged Structures with Different Aspect Ratios. Ph.D. Thesis, Department of Civil Engineering, National Chung Hsing University, Taichung City, Taiwan, 2009. [Google Scholar]
- Sumer, B.M.; Jensen, P.M.; Sørensen, L.B.; Fredsøe, J.; Liu, P.L.F.; Carstensen, S. Coherent structures in wave boundary layers. Part 2. Solitary motion. J. Fluid Mech.
**2010**, 646, 207–231. [Google Scholar] [CrossRef] - Persson, B.N.J.; Albohr, O.; Tartaglino, U.; Volokitin, A.I.; Tossatti, E. On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion. J. Phys. Condens. Matter
**2005**, 17, 82. [Google Scholar] [CrossRef] [PubMed] - Engineering Tool Box. Available online: http://www.engineeringtoolbox.com/surface-roughness-ventilation-ducts-d_209.html (accessed on 1 November 2018).
- Tennekes, H.; Lumley, J.L. A First Course in Turbulence; the MIT Press: Cambridge, MA, USA, 1972. [Google Scholar]
- Daily, J.W.; Harleman, D.R.F. Fluid Dynamics; Addison-Wesley Publishing Company, Inc.: Boston, MA, USA, 1966. [Google Scholar]
- Henderson, F.M. Open Channel Flow; Macmillan Publishing Company Inc.: New York, NY, USA, 1966; pp. 218–219. [Google Scholar]
- Chow, V.T. Open-Channel Hydraulics; McGraw-Hill Book Company: Singapore, 1973; pp. 425–428. [Google Scholar]
- Subramanya, K. Flow in Open Channels; McGraw-Hill Book Company: New York, NY, USA, 1986; pp. 204–206. [Google Scholar]
- Schlichting, H. Boundary Layer Theory; McGraw-Hill Book Company: New York, NY, USA, 1979. [Google Scholar]
- Lin, C.; Chiu, P.H.; Hsieh, S.J. Characteristics of horseshoe vortex system near a vertical plate-base plate juncture. Exp. Therm. Fluid Sci.
**2002**, 27, 25–46. [Google Scholar] [CrossRef] - Hunt, J.C.R.; Abell, C.J.; Peterka, J.A.; Woo, H.J. Kinematical studies of the flows around free or surface-mounted obstacles—Applying topology to flow visualization. J. Fluid Mech.
**1978**, 86, 179–200. [Google Scholar] [CrossRef] - Lin, C.; Hsieh, S.C.; Lin, I.J.; Chang, K.A.; Raikar, V.R. Flow property and self-similarity in steady hydraulic jump. Exp. Fluids
**2012**, 53, 1591–1616. [Google Scholar] [CrossRef] - Rajaratnam, N. Turbulent Jet; Elsevier Scientific Publishing Company: Amsterdam, The Netherlands, 1976. [Google Scholar]
- Verhoff, A. The Two-Dimensional Turbulent Wall Jet without an External Free Stream; Report No. 626; Princeton University: Princeton, NJ, USA, 1963. [Google Scholar]
- Michalke, A. The instability of free shear layers. Prog. Aerosp. Sci.
**1972**, 12, 213–239. [Google Scholar] [CrossRef] - Kuo, C.H.; Huang, S.H.; Chang, C.W. Self-sustained oscillation induced by horizontal cover plate above cavity. J. Fluids Struct.
**2000**, 14, 25–48. [Google Scholar] [CrossRef] - Lin, C.; Ho, T.C.; Chang, S.C.; Hsieh, S.C.; Chang, K.A. Vortex shedding induced by a solitary wave propagating over a submerged vertical plate. Int. J. Heat Fluid Flow
**2005**, 26, 894–904. [Google Scholar] [CrossRef] - Lin, C.; Chang, S.C.; Chang, K.A. Laboratory observation of a solitary wave propagating over a submerged rectangular dike. J. Eng. Mech.
**2006**, 132, 545–554. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Schematic diagrams indicating sloping beach model and two coordinate systems, together with deployment of two wave gauges and six fields of view, FOV

_{1}–FOV

_{6}, for HSPIV measurements; and (

**b**) photo showing wave flume, high-speed digital camera, and part of sloping beach.

**Figure 2.**Comparison of temporal variation of free surface elevation measured at x = −150.0 cm over constant water depth with that predicted by the theory (Case 1).

**Figure 3.**(

**a**,

**b**) Spatial variations of external and near-bottom velocity fields measured over horizontal bottom at t’ = −0.135 and 0 s (i.e., T’ = −1.50 and 0), respectively; and (

**c**,

**d**) comparison of horizontal velocity profiles in boundary layer measured by HSPIV with those obtained by fiber-optic laser Doppler velocimetry (Ho [32]) for different non-dimensional times (Case 3). Note that wave height and still water depth for this study are H

_{0}= 2.74 cm and h

_{0}= 16.0 cm, and counterparts for Ho [32] are H

_{0}= 1.20 cm and h

_{0}= 7.0 cm, both having identical value of H

_{0}/h

_{0}= 0.171.

**Figure 4.**Repeatability tests of instantaneous free surface profile and velocity field of solitary wave over sloping beach at t = 1.048 s (T = 11.60): (

**a**–

**c**) instantaneous images taken for three distinct runs; (

**d**) comparison of ten instantaneous free surface profiles with ensemble-averaged one; (

**e**–

**g**) instantaneous velocity field corresponding to (

**a**–

**c**), respectively; and (

**h**) ensemble-averaged velocity field acquired from ten repeated runs (Case 1). Note that the flow structures shown are not generated by breaking wave, but by hydraulic jump with a “backward” motion (also see Figure 5b).

**Figure 5.**Temporal and spatial variations of free surface profile: (

**a**) global view for 0 s ≤ t ≤ 1.351 s (0 ≤ T ≤ 14.96); and (

**b**) magnified view for run-down process during time interval 1.007 s ≤ t ≤ 1.141 s (11.15 ≤ T ≤ 12.64) (Case 1).

**Figure 6.**Chronological sketches for (

**a**) complete evolution of run-up and run-down process; and (

**b**) detailed classification for different stages of run-down process (Case 1).

**Figure 7.**Flow visualized images: (

**a**) Close-up of green dye strip (with nearly constant width) over beach surface, while water volume inside wave flume being kept quiescent;

**(b**–

**e**) dye strip transported onshore without changing its width in spanwise direction at five different times; (

**f**–

**j**) flow structures on central vertical plane 12.5 cm away from front sidewall at five distinct times; (

**k**–

**o**) the counterparts each with same time as that shown in (

**f**–

**j**), respectively, and on vertical plane 8.0 cm away from front sidewall.

**Figure 8.**(

**a**) Spatial variation of velocity field at t = 0.655 s; (

**b**) velocity profiles, u(y*), obtained at four different shoreward distances, x, as marked in (

**a**) each with vertical line marked by circled number (Case 1). Note that the shifted height is denoted by y* = y − x/3.

**Figure 9.**Variation of horizontal uniform velocity with shoreward distance, together with a change of free surface profile for t = 0.655–0.983 s (Case 1).

**Figure 11.**(

**a**) Spatial variation of velocity field at t = 0.921 s; (

**b**) near-bottom velocity field obtained at t = 0.921 s; (

**c**) three velocity profiles selected from (a), depicting incipiency of flow separation in boundary layer at X = 18.81 cm, as shown in sub-figure 2’; and (

**d**) detailed comparison of three velocity profiles very close to the beach surface (Case 1).

**Figure 12.**(

**a**) Near-bottom velocity field at t = 0.957 s; (

**b**) close-up of near-bottom velocity field for (

**a**); (

**c**) near-bottom velocity field at t = 0.983 s; and (

**d**) close-up of near-bottom velocity field for (

**c**) (Case 1).

**Figure 13.**(

**a**) Velocity field corresponding to occurrence of hydraulic jump; and (

**b**) five velocity profiles U(Y) extracted from (

**a**) and marked with values of Froude number for t = 1.007 s (Case 1).

**Figure 14.**(

**a**) Visualized image of retreated flow moving offshore; and (

**b**) streamline pattern for t = 1.007 s (Case 1).

**Figure 15.**Visualized image (on top panel) and streamlined pattern (on bottom panel) obtained at (

**a**) t = 1.017 s; (

**b**) t = 1.048 s; (

**c**,

**d**) t = 1.063 s. Note that the range on abscissa in (

**d**) is different from that shown in (

**a**–

**c**) for Case 1.

**Figure 16.**For t = 1.077 s, (

**a**) visualized image with external stream moving offshore; (

**b**) streamlined pattern; and (

**c**) four velocity profiles U(Y) measured at four different positions, as marked by dashed lines on (

**a**).

**Figure 17.**Visualized images of interaction process between projecting jet and retreated flow at (

**a**) t = 1.085 s before jet impinging; (

**b**) t = 1.094 s for jet right impinging on retreated flow; (

**c**) t = 1.098 s for jet bifurcating into two substreams; and (

**d**) t = 1.098 s for merging of vortices (Case 1).

**Figure 18.**Variations in non-dimensional shoreward distances for critical and core sections as well as in non-dimensional size height (Case 1).

**Figure 19.**(

**a**) Schematic plot of velocity profile U(Y) passing through primary vortex core; and (

**b**) universal similarity profile of wall jet flow close to beach surface for Y ≤ Y

_{M}; (

**c**) universal similarity profile of shear layer flow between Y = Y

_{M}and Y = Y

_{m}. Note that data from the present three cases for S

_{0}= 1:3, Lin et al. [15,16] for S

_{0}= 1:5 and 1:10, and Hwung et al. [17] for S

_{0}= 1:20 have been included to highlight no effect of H

_{0}/h

_{0}and S

_{0}on these two unique profiles.

**Table 1.**A list of dimension, range, and pixel resolution for each field of view, FOV

_{i}(i = 1–6).

FOV_{i} | Dimension | Range | Pixel Resolution |
---|---|---|---|

FOV_{1} | 2.00 cm × 1.00 cm | −1.00 cm ≤ x’ ≤ 1.00 cm | 1280 × 640 |

FOV_{2} | 9.95 cm × 6.22 cm | 10.80 cm ≤ x ≤ 20.08 cm | 1280 × 800 |

FOV_{3} | 3.50 cm × 2.19 cm | 13.15 cm ≤ X ≤ 16.65 cm | 1280 × 800 |

FOV_{4} | 3.50 cm × 2.19 cm | 15.75 cm ≤ X ≤ 19.25 cm | 1280 × 800 |

FOV_{5} | 2.85 cm × 1.78 cm | 17.55 cm ≤ X ≤ 20.40 cm | 1280 × 800 |

FOV_{6} | 0.65 cm × 0.41 cm | 18.50 cm ≤ X ≤ 19.15 cm | 1280 × 800 |

Case | H_{0} (cm) | h_{0} (cm) | H_{0}/h_{0} | Breaker Type |
---|---|---|---|---|

1 | 2.90 | 8.0 | 0.363 | Non-breaking |

2 | 2.10 | 8.0 | 0.263 | Non-breaking |

3 | 2.74 | 16.0 | 0.171 | Non-breaking |

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**MDPI and ACS Style**

Lin, C.; Wong, W.-Y.; Kao, M.-J.; Tsai, C.-P.; Hwung, H.-H.; Wu, Y.-T.; Raikar, R.V.
Evolution of Velocity Field and Vortex Structure during Run-Down of Solitary Wave over Very Steep Beach. *Water* **2018**, *10*, 1713.
https://doi.org/10.3390/w10121713

**AMA Style**

Lin C, Wong W-Y, Kao M-J, Tsai C-P, Hwung H-H, Wu Y-T, Raikar RV.
Evolution of Velocity Field and Vortex Structure during Run-Down of Solitary Wave over Very Steep Beach. *Water*. 2018; 10(12):1713.
https://doi.org/10.3390/w10121713

**Chicago/Turabian Style**

Lin, Chang, Wei-Ying Wong, Ming-Jer Kao, Ching-Piao Tsai, Hwung-Hweng Hwung, Yun-Ta Wu, and Rajkumar V. Raikar.
2018. "Evolution of Velocity Field and Vortex Structure during Run-Down of Solitary Wave over Very Steep Beach" *Water* 10, no. 12: 1713.
https://doi.org/10.3390/w10121713