# Effects of Nonlinearity on Velocity, Acceleration and Pressure Gradient in Free-Stream Zone of Solitary Wave over Horizontal Bed—An Experimental Study

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## Abstract

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_{0}) to still water depth (h

_{0}) that amplifies the wave nonlinearity. The value of H

_{0}/h

_{0}tested in a wave flume ranges from 0.050 to 0.550, indicating the shift from a quasi-linear solitary wave to a highly nonlinear one. A high-speed particle image velocimetry (HSPIV) and a flow visualization technique of particle-trajectory tracking method are utilized to measure velocity fields and identify near-bed flow structures. The unsteady free-stream velocities with equal magnitude take place in a free-stream zone, FSZ). The FSZ underlies the internal flow zone, over which the external free surface of solitary wave exists and is situated beyond the boundary layer. The spatio-temporal variation of free-stream velocity, moving in phase with the free surface elevation, characterizes the pattern of pressure gradient in the FSZ and thus dominates the behavior of boundary layer flow. Accordingly, nonlinear effects on the time series as well as the maximum values of horizontal velocity, particle acceleration, and pressure gradient in the FSZs of solitary waves are presented. Before, at, and after the wave crest’s intersection with a given measurement location, favorable, zero, and adverse pressure gradients occur in the FSZ, respectively. For H

_{0}/h

_{0}= 0.179, 0.363, and 0.550, the values of the dimensionless maximum free-stream velocity are about 3.10, 5.32, and 6.20 times that (= 0.0473) for H

_{0}/h

_{0}= 0.050; and the corresponding values of the dimensionless maximum adverse pressure gradient are about 5.74, 14.54 and 19.84 times that (= 0.0061) for H

_{0}/h

_{0}= 0.050. This evidence highlights the nonlinear effect on the kinematic and hydrodynamic features of solitary waves. Finally, the effect of nonlinearity on the relationship between the dimensionless time for the maximum adverse pressure gradient in the FSZ and that for the incipient flow reversal in the bottom boundary layer is explored for the first time. It is found that the incipient flow reversal takes place immediately after the maximum adverse pressure gradient, together with a decrease in the dimensionless time for flow reversal if H

_{0}/h

_{0}increases. The fact accentuates the nonlinear effect on the incipient flow reversal right above the bed.

## 1. Introduction

_{0}/h

_{0}= 0.11–0.29. Comparisons of the experimental results with those predicted by Boussinesq, McCowan, Munk, and Grimshaw solutions were then made, aiming to examine the validity of these analytic solutions. Comparisons of the measured free surface elevations with the predicted ones by the four solutions all showed good agreement. However, comparisons between the measured horizontal and vertical velocities and the predicted ones indicated that the Boussinesq solution matched satisfactorily with the experimental data trend. Note that the nonlinear effect of solitary waves was not discussed in their study.

_{0}/h

_{0}= 0.096–0.386. Their study lacked in exploring the nonlinear effect on the kinematic and hydrodynamic features. As reported by Lin et al. [19,20], the features of (local and convective) accelerations and pressure gradient all over the internal flow as well as the relevant “similarities and Froude number similitudes” were elucidated only for a solitary wave (with H

_{0}/h

_{0}= 0.363) propagating on a horizontal bed. Further, as indicated by Lin et al. [20], fairly good predictions of horizontal and vertical velocities could be obtained by the Boussinesq solution if the linear wave celerity is employed in the computation. A series of studies on kinematic and hydrodynamic features or flow similarity and Froude number similitude was reported in Lin et al. [21,22,23,24,25,26] for the run-up and run-down motions of solitary waves traveling over 1:15–1:3 sloping beaches. Recently, Lin et al. [27] studied the characteristics of flow velocity and pressure gradient of an undular bore propagating on a horizontal bed.

## 2. Experimental Set-Up and Instrumentation

#### 2.1. Wave Flume and Coordinate System

_{s}= (h

_{0}/g)

^{1/2}a representative time scale with g being the gravity acceleration, thus leading to the dimensionless time, T [= t/t

_{s}= t × (g/h

_{0})

^{1/2}]. It should be mentioned that t = 0 specifies the time instant when the wave crest passes the SMS. The velocity components, (u, v) = (u[x, y, t], v[x, y, t]), represent the horizontal and vertical velocities of water particles along the (x, y) directions, respectively.

#### 2.2. Wave Gauge, HSPIV, and Flow Observation

_{0}at the SMS.

#### 2.3. Experimental Conditions

_{0}in the range 0.4–4.4 cm but at the same h

_{0}= 8.0 cm, thus H

_{0}/h

_{0}= 0.05–0.55 (Cases A–I). The ratio reflects a solitary wave’s transition from a quasi-linear state to a highly nonlinear one [11,15,17,18,23,24,25]. To acquire the time series of instantaneous horizontal velocity in the free-stream zone at the SMS, a symmetric 11-point smoothing scheme with distinct weightings was used to eliminate noises in the velocity data. The ensemble-average method was then employed for the 20 repeated runs to obtain the time series of ensemble-averaged horizontal velocity. Table 1 summarizes the nine experimental conditions. For all cases, the values of the linear and nonlinear wave celerity, C [= (gh

_{0})

^{1/2}] and C

_{0}(= [g(h

_{0}+ H

_{0})]

^{1/2}), as well as the counterparts of C

_{0}/C are listed in Table 1. The relationship between C

_{0}/C and H

_{0}/h

_{0}is regressed to be C

_{0}/C = (0.4546H

_{0}/h

_{0}+ 1.0). For Case E, only the flow observation using particle-trajectory tracking was conducted without performing velocity measurement by the HSPIV. These un-done counterparts were previously made throughout the full water depth and partially reported in Lin et al. [18,19,20]. In this study, elucidation of the kinematic and hydrodynamic features for Case E was achieved by mining the data banks that were set up during the experimental stage of the three studies.

## 3. Validation Tests

_{o}(t) for five runs, which are randomly selected from a total of 20 runs for Case E. Their profiles of the time series almost collapse onto each other without prominent discernible distinction. The fact still holds true for the other eight experimental cases. The evidence demonstrates the high repeatability of the solitary waves generated by the wave maker. A comparison of the time series of ensemble-averaged free surface elevation η(t) (i.e., obtained by phase-averaging over 20 repeated runs) with those predicted by solitary wave theory [31] is illustrated in Figure 3 for Case E. The trend of measured data in η(t) is in good accordance with that predicted by solitary wave theory.

_{f}= |∂u/∂x + ∂v/∂y| × dA. Note that, due to mass conservation and flow continuity, M

_{f}should be equal to zero theoretically. For the velocity field for Case E, the grid lengths are Δx = Δy = 0.1167 cm, and the corresponding area is dA = Δx × Δy = 0.0136 cm

^{2}. A representative mass flux, M

_{fr}[= (u

_{wc})

_{max}× Δy], is designated with (u

_{wc})

_{max}[= 36.94 cm/s] being the maximum horizontal velocity at wave crest for T = 0. The relative error is then defined by M

_{f}/M

_{fr}. The values of M

_{f}/M

_{fr}are found to be below 2.0%. Moreover, for a typical horizontal velocity of 20.0 or 40.0 cm/s in the flow field, the measurement error of horizontal velocity is about 0.3 or 0.46 cm/s (Lin et al. [18,22,24]), rendering the relative error to be 1.5% or 1.2%. These two pieces of evidence do indicate high precision in the velocity measurements utilizing the HSPIV system.

_{l}(t) (calculated from the time series of the ensemble-averaged horizontal velocity) and those predicted by wave theory at y = 7.04, 4.40, and 1.44 cm. Note that the details of the calculation procedure for the former are addressed later in Sec. 4.3, and the latter is directly computed by taking the time derivative of the horizontal velocity predicted by the Boussinesq solution. Not only for the entire trend but also for the individual magnitudes in these time series, fairly good agreement is thus confirmed.

## 4. Results and Discussion

_{0}or [h

_{0}+ H

_{0}] (or simply H

_{0}) and (h

_{0}/g)

^{1/2}or [(h

_{0}+ H

_{0})/g]

^{1/2}. The two scales result in the representative velocity scale being C = (gh

_{0})

^{1/2}or C

_{0}(= [g(h

_{0}+ H

_{0})]

^{1/2}), which is equivalent to the linear or nonlinear wave celerity. The length and time scales selected herein are h

_{0}and (h

_{0}/g)

^{1/2}, together with the use of H

_{0}to normalize the time series of free surface elevation, η(T). To highlight the nonlinear effect, the nonlinear wave celerity, C

_{0}(= [g(h

_{0}+ H

_{0})]

^{1/2}), is particularly used as the velocity scale. Note that C

_{0}/C = (0.4546H

_{0}/h

_{0}+ 1.0), as evidenced in Table 1, thus allowing prompt transformation between C

_{0}and C. Further, the acceleration (or pressure gradient) scale amounts to the ratio of the velocity scale to the time scale, i.e., equal to g.

#### 4.1. Elucidation of FSZs from Velocity Profiles

_{0}= 0), Figure 7a,b–h present the time series of dimensionless free surface elevation, η(T)/H

_{0}, as well as the profiles of dimensionless horizontal and vertical velocities, u(y/h

_{0}, T)/C

_{0}and v(y/h

_{0}, T)/C

_{0}, at T = −2.50, −1.39, −0.50, 0, 0.5, 1.39, and 2.50 (as marked by dashed lines with ①−⑦ in Figure 7a), respectively. The times before, at, and after the intersection of the wave crest with the SMS correspond to −6.00 ≤ T < 0, T = 0, and 0 < T ≤ 6.00, respectively, with an ascending, constant, and descending free surface. With reference to T = 0, the free surface elevations for −6.00 ≤ T < 0 are symmetric to those for 0 < T ≤ 6.00, exhibiting an even-function form with η(T)/H

_{0}= η(−T)/H

_{0}. In addition, as evidenced in Figure 7b–h, the horizontal velocities are positive with u(y/h

_{0}, T)/C

_{0}= u(y/h

_{0}, −T)/C

_{0}throughout the full water depth at the SMS. However, the vertical velocities are negative, zero, and positive with v(y/h

_{0}, T)/C

_{0}= −v(y/h

_{0}, −T)/C

_{0}, except for those very close to the bed.

_{0})/C

_{0}increases from zero at y/h

_{0}= 0 to 0.082 at y/h

_{0}= 0.022 or 0.031, characterizing the feature of the boundary layer. It then remains almost constant for y/h

_{0}= 0.031–0.35, as indicated by the two horizontal dotted lines within which the uniform horizontal velocities exist. This layer is herein termed the FSZ within which the horizontal velocities are denoted as the free-stream velocities. For y/h

_{0}> 0.35, u(y/h

_{0})/C

_{0}decreases linearly to 0.066 at y/h

_{0}= y

_{fs}/h

_{0}= 1.08 (i.e., the instantaneous free surface). It is worth mentioning that, for −6.00 ≤ T < −1.39 or 1.39 < T ≤ 6.00 and y/h

_{0}> 0.35, all the dimensionless horizontal velocity profiles exhibit similar shapes to that at T = −2.50 or 2.50. In particular, the upper portions have a non-uniform, linearly decreasing trend. This differs from the known recognition of long waves having a uniform horizontal velocity profile all over the full water depth. As seen in Figure 7c or Figure 7g at T = −1.39 or 1.39 (i.e., at phase corresponding to the “inflection point” in Figure 7a), u(y/h

_{0})/C

_{0}ranges from zero at the bed to 0.163 at y/h

_{0}= 0.024 or 0.029, then remains nearly unchanged from the lower limit of FSZ (at y/h

_{0}= 0.03) to the free surface.

_{0}= 0.026–0.35 or 0.027–0.35. Especially, for y/h

_{0}> 0.35, a distribution with a nonlinear increase in the horizontal velocity is evidenced. The fact is again contrary to the uniform horizontal profile along the full water depth, assumed frequently in the theoretical prediction or numerical simulation of long-wave propagation. The maximum values of dimensionless horizontal velocity, [u(y/h

_{0}, T)]

_{max}/C

_{0}, occur at T = 0 with the FSZ located at y/h

_{0}= 0.027–0.35. Note that the extreme value of [u(y/h

_{0}, 0)]

_{max}/C

_{0}, taking place right at the wave crest (i.e., y/h

_{0}= y

_{crest}/h

_{0}= 1.363), is equal to 0.357, greater than those (= 0.251) in the FSZ.

_{0}, T)/C

_{0}, linearly varies from zero at the bed to a certain positive or negative maximum, [v(y

_{fs}/h

_{0}, T)]

_{max+}/C

_{0}or [v(y

_{fs}/h

_{0}, T)]

_{max−}/C

_{0}, at the instantaneous free surface. The profile with all positive or negative maxima, [v(y/h

_{0})]

_{max+}/C

_{0}or [v(y/h

_{0})]

_{max−}/C

_{0}(= −[v(y/h

_{0})]

_{max+}/C

_{0}< 0), occurs at T = −1.39 or 1.39 (i.e., at the phase of the inflection point in Figure 7a). Further, at a given y/h

_{0}and −6.00 ≤ T ≤ −1.39 or 1.39 ≤ T ≤ 6.00, v(T)/C

_{0}increases from nearly zero to [v(y/h

_{0})]

_{max+}/C

_{0}or decreasing from [v(y/h

_{0})]

_{max−}/C

_{0}to almost zero. As a contrast, for −1.39 ≤ T ≤ 1.39, it does decrease from the maximum, [v(y/h

_{0})]

_{max+}/C

_{0}, via zero at T = 0, to [v(y/h

_{0})]

_{max−}/C

_{0}. It should be emphasized that, at the SMS and with increasing y/h

_{0}values, the “magnitudes” of vertical velocity remain increasing in the FSZ, in which the unsteady free-stream velocities are almost identical.

_{o}(T) exist in the FSZ situated inside the range within the two horizontal dotted lines. Further, the boundary layer is located inside the zone between the lower horizontal dotted line and the bed (i.e., y/h

_{0}= 0). The boundary layer thickness is defined herewith as a specified height measured upwards from the bed to the height where the horizontal velocity u(T) at the edge of the boundary layer is equal to 0.99u

_{o}(T), as shown at the lower horizontal dotted lines in Figure 8a–g. Note that, as primarily evidenced in Figure 8a to Figure 8g, the boundary layer thickness does increase slightly as T varies from −6.00 to 6.00. More details relevant to the kinematic features and unique similarity profiles in the boundary layer flows can be referred to in Lin et al. [18].

_{0}obtained at eight heights for y/h

_{0}= 0.03–0.55. For any one of the time series, horizontal velocity increases from near zero to a maximum of −6.00 ≤ T < 0, indicative of the temporal acceleration in the horizontal direction. For 0 < T ≤ 6.00, u(T)/C

_{0}decreases from its maximum to near zero, suggestive of the temporal deceleration in the horizontal direction. The two features highlight the temporal acceleration equal to zero for T = 0, corresponding to which the maximum horizontal velocity occurs. For y/h

_{0}= 0.05–0.35 and T = 0, the maximum u/C

_{0}values are almost the same (about 0.251), smaller than those at y/h

_{0}= 0.39–0.55 in the internal zone and larger than those at y/h

_{0}= 0.03–0.04 in the bottom boundary layer. Further, the temporal variation in u(T)/C

_{0}at y/h

_{0}= 0.05 or 0.35 collapses completely onto those at y/h

_{0}= 0.06–0.31. These evidences strongly indicate that the FSZ exists for y/h

_{0}= 0.05–0.35, in which the magnitudes of free-stream velocities are nearly identical and the horizontal velocity profile is uniform. For the cases examined excluding Case E, the FSZs are located between y/h

_{0}= (0.035~0.055) and (0.335~0.366), which are almost equivalent to y/h

_{0}= 0.05–0.350 for Case E. For ease of data analysis, the flow velocities that represent kinematic features of FSZs are mined from the velocity fields only between y/h

_{0}= 0.06 and 0.330.

_{0}/h

_{0}= 0.050–0.550, Table 1) are analyzed to explore the effect of nonlinearity (or H

_{0}/h

_{0}) not only on the dimensionless free surface elevations; but also on the dimensionless free-stream velocities, horizontal accelerations and pressure gradients in the FSZs of solitary waves.

#### 4.2. Nonlinear Effect on Free Surface Elevation and Free-Stream Velocity

_{0}and T for Cases A–I (H

_{0}/h

_{0}= 0.050–0.550), along with a comparison of the measured data with the solitary wave theory [31] for each case. It is found that the experimental data are in good agreement with the analytic results for all cases. With an increase in H

_{0}/h

_{0}, the dimensionless free surface elevation becomes more focused around T = 0 with a narrower bell shape, exhibiting the H

_{0}/h

_{0}(or nonlinearity) effect on the free surface profile. In other words, for a solitary wave with a larger H

_{0}/h

_{0}, a shorter time is taken to generate a complete wave motion. For example, for η(T)/H

_{0}= 5%, the corresponding dimensionless times are examined to be

**|**T

**|**= 11.47, 5.85, 4.12, and 2.85 for Case A (H

_{0}/h

_{0}= 0.050), Case C (H

_{0}/h

_{0}= 0.179), Case E (H

_{0}/h

_{0}= 0.363), and Case I (H

_{0}/h

_{0}= 0.550). The fact demonstrates the larger the H

_{0}/h

_{0}, the narrower the temporal range of wave shape. From the physical point of view, this trend shows that the change of ascending or descending free surface elevation per unit time becomes greater in magnitude. Namely, the free-surface slope ∂[η(T)/H

_{0}]/∂T gets larger at a specified T at a greater H

_{0}/h

_{0}, except those at η(T) = 0 and η(0) = H

_{0}. It is noted that, as shown in Figure 11 for −14.0 ≤ T ≤ 14.0, the averaged free surface elevation [η(T)/H

_{0}]

_{mean}decreases with an increase in H

_{0}/h

_{0}, rendering the relationship expressed by

_{0}]

_{mean}= 0.0737(H

_{0}/h

_{0})

^{−0.545}.

_{0}]

_{max}= 1.0 for all experimental cases and the larger value of H

_{0}involved in the greater magnitude (i.e., higher nonlinearity) of H

_{0}/h

_{0}at the same h

_{0}(= 8.0 cm).

_{o}(T)/C

_{0}, for Cases A–I (H

_{0}/h

_{0}= 0.050–0.550). The nonlinear wave celerity C

_{0}(= [g(H

_{0}+ h

_{0})]

^{1/2}) is used herewith as the velocity scale. It is found that, at a larger H

_{0}/h

_{0}, the time series of u

_{o}(T)/C

_{0}becomes more concentrated around T = 0 with a sharper bell shape. Less time is thus taken to achieve a complete variation of the free-stream velocities in the FSZ. The fact exhibits the H

_{0}/h

_{0}(or nonlinearity) effect on the temporal distribution of free-stream velocity. Taking u

_{o}(T)/C

_{0}= 0.5% for instance, the dimensionless times are identified to be about

**|**T

**|**= 9.56, 6.77, 5.58, and 4.65 for Case A (H

_{0}/h

_{0}= 0.050), Case C (H

_{0}/h

_{0}= 0.179), Case E (H

_{0}/h

_{0}= 0.363), and Case I (H

_{0}/h

_{0}= 0.550), respectively. These data do testify to the relatively narrower temporal range of free-stream velocity induced by a solitary wave with a greater value of H

_{0}/h

_{0}. At a specified dimensionless time, say

**|**T

**|**< 3.95, this trend does show that the change rate of free-stream velocity, ∂[u

_{o}(T)/C

_{0}]/∂T (equivalent to the dimensionless local acceleration in the FSZ), becomes larger in magnitude. For T < 0 or T > 0, ∂[u

_{o}(T)/C

_{0}]/∂T is positive or negative, indicative of flow acceleration or deceleration in the FSZ for each case.

_{o}/C

_{0}]

_{max}, occurs with zero acceleration at T = 0. As evidenced in Figure 13, [u

_{o}/C

_{0}]

_{max}becomes larger with an increasing H

_{0}/h

_{0}. Its overall nonlinear form is regressed with an R

^{2}value of 0.992 as:

_{o}/C

_{0}]

_{max}= −0.8208(H

_{0}/h

_{0})

^{2}+ 0.9789(H

_{0}/h

_{0})

_{0}/h

_{0}= 0.179), E (H

_{0}/h

_{0}= 0.363), and I (H

_{0}/h

_{0}= 0.550), the values of [u

_{o}/C

_{0}]

_{max}are about 3.10, 5.32, and 6.20 times that (= 0.0473) for Case A (H

_{0}/h

_{0}= 0.050), clearly substantiating the nonlinear effect on [u

_{o}/C

_{0}]

_{max}. Further, as illustrated in Figure 14 for −14.0 ≤ T ≤ 14.0, the averaged free-stream velocity [u

_{o}(T)/C

_{0}]

_{mean}increases if H

_{0}/h

_{0}increases. The relationship between the former and the latter is written as

_{o}(T)/C

_{0}]

_{mean}= −0.2962(H

_{0}/h

_{0})

^{4}+ 0.6725(H

_{0}/h

_{0})

^{3}− 0.5646(H

_{0}/h

_{0})

^{2}+ 0.2123(H

_{0}/h

_{0}) + 0.0088

#### 4.3. Nonlinear Effect on Local and Convective Accelerations in FSZ

_{l}(x, y, t) = ∂u(x, y, t)/∂t

_{c}(x, y, t) = u(x, y, t) × ∂u(x, y, t)/∂x + v(x, y, t) × ∂u(x, y, t)/∂y

_{o}(x, y, t) [= u

_{o}(x, t)] are uniform in the vertical direction, thus leading to ∂u

_{o}(x, t)/∂y = 0. Therefore, at the SMS and in the FSZ, the local and convective accelerations in the horizontal direction are simplified as:

_{lo}(0, t) = ∂u

_{o}(0, t)/∂t ≈ Δu

_{o}(0, t)/Δt = A

_{lo}(t)

_{co}(0, t) = u

_{o}(0, t) × ∂u

_{o}(0, t)/∂x ≈ u

_{o}(0, t) × Δu

_{o}(0, t)/Δx = A

_{co}(t).

_{lo}(t) and A

_{co}(t) can be computed, using a central difference scheme, from the instantaneous free-stream velocities obtained at the SMS and in its close neighborhood. From a theoretical point of view, the time and spatial interval, Δt and Δx, should be as small as possible (i.e., approaching zero). However, as reported by Lin et al. [19,20,25,27], even the use of the smallest time or spatial interval in the differential computation would result in failure to find a convergent A

_{lo}(t) or A

_{co}(t) value. The reason is attributable to the rapid temporal or spatial fluctuations in the “image-based (or pixel-based)” PIV/HSPIV measurements.

_{lo}(t) or A

_{co}(t) would only vary less than 4.0% of its average, is selected. Detailed illustrations for the trial-and-error calculation of A

_{lo}(t) and A

_{co}(t) with different Δt or Δx values, together with the determination of promising time or spatial interval, can be referred to Lin et al. [25,27]. In this study, the promising time or interval used is equal to (Δt)

_{promising}= 0.01–0.0156 s, (i.e., 25 times (Δt)

_{framing}[= 1/2500–1/1600 s = 0.0004–0.000625 s]) or (Δx)

_{promising}= 0.10712–0.248 cm (i.e., 13–31 times the grid size (Δx)

_{gs}[= (Δy)

_{gs}= 0.00824 or 0.008 cm] used in the HSPIV measurements). It should be emphasized that previous investigations [25,27,35] did indicate (Δt)

_{promising}or (Δx)

_{promising}much larger than Δt

_{framing}or (Δx)

_{gs}, but not as small as (Δt)

_{framing}or (Δx)

_{gs}. Like the data processing used in Jensen et al. [35], a symmetric 7-point smoothing scheme with distinct weightings was utilized to smoothen the time series of A

_{lo}(t) or A

_{co}(t).

_{lo}/g, for Cases A–I (H

_{0}/h

_{0}= 0.050–0.550). Note that A

_{lo}(T)/g ≈ −A

_{lo}(−T)/g, exhibits the odd-function feature. As addressed in Figure 8, the free-stream velocity increases from near zero to a maximum for −6.00 ≤ T < 0 or decreases from the maximum to about zero for 0 < T ≤ 6.00, underscoring the temporal acceleration or deceleration in the FSZ. Accordingly, for all cases at the SMS, A

_{lo}/g is positive for −6.00 ≤ T < 0 and negative for 0 < T ≤ 6.00. Note that A

_{lo}/g = 0 for T = 0, at which the wave crest moves right over the SMS and the free-stream velocity arrives at its maximum. It is interesting to mention that the averaged value of the dimensionless local acceleration, [A

_{lo}(T)/g]

_{mean}, is equal to almost zero for −6.00 ≤ T ≤ 6.00, which is attributable to the time series of A

_{lo}(T)/g featured with an odd-function form.

_{lo}/g, A

_{lo+}/g, and A

_{lo}

^{_}/g (≈ −A

_{lo+}/g < 0), highlight almost equal magnitude at the dimensionless characteristic time T = T

_{Alo+}(< 0) and T = T

_{Alo−}(> 0), respectively. As illustrated in Figure 7 and Figure 8, the translation of free-stream velocity is in phase with the free-surface motion of a solitary wave. Therefore, T

_{Alo+}and T

_{Alo−}correspond, in fact, to the inflection points in the time series of both η(T)/H

_{0}and u

_{o}(T)/C

_{0}. Namely, the slopes of η(T)/H

_{0}and u

_{o}(T)/C

_{0}(i.e., ∂[η(T)/H

_{0}]/∂T and ∂[u

_{o}(T)/C

_{0}]/∂T) exhibit the local maxima apparently at T

_{Alo+}and T

_{Alo−}. As evidenced in Figure 16, A

_{lo+}/g and A

_{lo}

^{_}/g increase linearly, and T

_{Alo+}and T

_{Alo−}decrease with an increasing H

_{0}/h

_{0}(or nonlinearity). For example, A

_{lo+}/g (= −A

_{lo}

^{_}/g = 0.0434, 0.1075, and 0.1523) for Cases C, E and I are about 6.11, 15.14, and 21.45 times that (= 0.0071) for Case A, demonstrating the obvious nonlinear effect on A

_{lo+}/g and A

_{lo}

^{_}/g. The reason is assignable to η(T)/H

_{0}and u

_{o}(T)/C

_{0}both becoming more concentrated around T = 0 with narrower symmetric bell shapes for a greater H

_{0}/h

_{0}. Namely, if H

_{0}/h

_{0}gets larger, a shorter time is taken to achieve a complete motion with a greater value of the maximum free-stream velocity (i.e., translating with relatively larger |Δu| under the same Δt), thus resulting in a larger ∂[u

_{o}(T)/C

_{0}]/∂T.

_{co}/g, for Cases A–I (H

_{0}/h

_{0}= 0.050–0.550). As illustrated in Figure 3, t < 0, t = 0, and t > 0 correspond to the times before, at, and after the wave crest intersection with the SMS. This leads to the maximum free-stream velocity, [u

_{o}(x)]

_{max}, appearing at x < 0, x = 0 or x > 0; and subsequently, ∂u

_{o}(t)/∂x < 0, ∂u

_{o}(0)/∂x = 0 or ∂u

_{o}(t)/∂x > 0 at the SMS. Accordingly, A

_{co}/g (= [u

_{o}(t) × ∂u

_{o}(0, t)/∂x]/g) takes negative, zero, or positive values for T < 0, T = 0, or T > 0, as observed in Figure 17 almost with the odd-function feature [i.e., A

_{co}(T)/g ≈ −A

_{co}(−T)/g]. For all cases, the magnitudes of negative and positive maxima in the dimensionless convective acceleration, A

_{co−}/g and A

_{co+}/g, occur at the dimensionless characteristic time T = T

_{Aco−}(< 0) and T = T

_{Aco+}(> 0), and increase with increasing H

_{0}/h

_{0}. Note that |T

_{Aco−}| and T

_{Aco+}are smaller than |T

_{Alo+}| and T

_{Alo−}. For example, for Case E, −T

_{Aco−}= T

_{Aco+}≈ 1.090 < 1.390 ≈ −T

_{Alo+}= T

_{Alo−.}The fact points out that the dimensionless times for the occurrence of A

_{co−}/g and A

_{co+}/g are closer to T = 0 than the counterparts of A

_{lo+}/g and A

_{lo}

^{_}/g. Further, A

_{co−}/g and A

_{co+}/g are about 1/22.0–1/4.3 times A

_{lo+}/g and A

_{lo}

^{_}/g, justifying much more contribution from the local acceleration than from the convection acceleration in the pressure gradient. Further, the averaged value of the dimensionless convective acceleration, [A

_{co}(T)/g]

_{mean}, is equal to almost zero for −6.00 ≤ T ≤ 6.00 because the time series of A

_{co}(T)/g is characterized by an odd-function form.

#### 4.4. Nonlinear Effect on Pressure Gradient in FSZ

_{o}(0, t)/∂x = −[∂u

_{o}(0, t)/∂t + u

_{o}(0, t) × ∂u

_{o}(0, t)/∂x]

= −[A

_{lo}(t) + A

_{co}(t)].

= P

_{o}(t)

_{o}(t) takes a minus value of [A

_{lo}(t) + A

_{co}(t)] (i.e., the force per unit mass in the horizontal direction). With the magnitude of A

_{lo}(t) being much larger than A

_{co}(t), A

_{lo}(t) dominates the contribution to P

_{o}(t).

_{o}(T)/g for all cases. The characteristic dimensionless times for the occurrence of a negative maximum [P

_{o}(T)/g]

_{max−}(= P

_{o}

^{_}/g) and a positive maximum [P

_{o}(T)/g]

_{max+}(= P

_{o+}/g) are defined as T

_{Po−}(< 0) and T

_{Po+}(> 0), respectively. Similar to the variation feature of A

_{lo}(T)/g (see Figure 15), an odd-function form of P

_{o}(T)/g is examined by P

_{o+}/g ≈ −P

_{o}

^{_}/g and T

_{Po+}= T

_{Alo−}≈ −T

_{Alo+}= −T

_{Po−}. P

_{o}(T)/g decreases from near zero to P

_{o}

^{_}/g for −6.00 ≤ T ≤ T

_{Po−}or from P

_{o+}/g to about zero for T

_{Po+}≤ T ≤ 6.00, indicating an increase in the favorable pressure gradient or a decrease in the adverse pressure gradient in the FSZ. Further, it increases from P

_{o}

^{_}/g (< 0), via 0, to P

_{o+}/g (> 0) for T

_{Po−}≤ T < 0, T = 0, and 0 < T ≤ T

_{Po+}, respectively, demonstrating that P

_{o}(T)/g varies correspondingly from favorable, via zero, to the adverse pressure gradient. Note that, for −6.00 ≤ T ≤ 6.00, the averaged value of the dimensionless pressure gradient, [P

_{o}(T)/g]

_{mean}, is equal to almost zero because the time series of P

_{o}(T)/g is featured with an odd-function form.

_{o+}/g (≈ −P

_{o}

^{_}/g) and T

_{Po+}(≈ −T

_{Po−}) versus H

_{0}/h

_{0}. It is found that P

_{o+}/g increases and T

_{Po+}, however, decreases with an increasing H

_{0}/h

_{0}. Namely, at a larger H

_{0}/h

_{0}value, P

_{o+}/g becomes greater and takes place closer to T = 0 (i.e., the “phase” with wave crest right passing through the SMS). For example, for Cases C, E, and I, P

_{o+}/g = 0.0350, 0.0887, and 0.1210 appear at T

_{Po+}= 0.1674, 0.1049, and 0.0943, respectively. The values of P

_{o+}/g are 5.74, 14.54, and 19.84 times that of Case A (= 0.0061, with T

_{Po+}= 0.2463), underlining the distinct effect of nonlinearity on P

_{o}(T)/g and P

_{o+}/g.

_{Po+}> T

_{ifr}, T

_{Po+}= T

_{ifr}, or T

_{Po+}< T

_{ifr}. In this study, T

_{ifr}is determined from the flow-visualized images. To well observe the incipient flow reversals with “particle-dotted” and “path-lined” images for all cases, the ranges of framing rate of the camera were set at 1600–2500 for the HSPIV measurements and 30–100 Hz for flow visualization tests. The former enhanced the temporal resolution (i.e., 1/2500–1/1600 s) in precisely identifying T

_{ifr}from the continuous display (with frame-by-frame operation) of the recorded images. The latter provided visual evidence related to the temporal variation of flow structure in the near-bottom zone. Together with a concise description of other cases, the observation results are here detailed only for Case E. For ease of understanding, an original motion picture displaying Case E’s flow structure in Supplementary Material, in which eight images of the instantaneous flow structure are drawn and then elucidated as follows.

_{0}≤ 0.10 and 0 ≤ y/h

_{0}≤ 0.10 for Case E. The water particles in the FSZ all move from the left to right side in each image, as evidenced by the temporal variation of free-stream velocities [i.e., u

_{o}(T) > 0] at y/h

_{0}= 0.05–0.35 (Figure 8). The longer the length of each pathline, the larger the magnitude of each particle velocity. As observed in Figure 20a, the free-stream velocity reaches its maximum and P

_{o}/g = 0 at T = 0. Immediately after the passing of wave crest through the SMS, say 0 < T ≤ 1.67, the lengths and the corresponding free-stream velocities in the FSZ decrease with an increasing T (see Figure 20b,c for T = 0.89 and 1.76). It is thus indicated that the flow in the near-bottom zone, subjected to adverse pressure gradients, does decelerate temporally.

_{0}< 0.004 at T = T

_{ifr}= 1.67 (Figure 20c). After T > T

_{ifr}, flow reversal occurs right beyond the bed. Namely, the pathlines located very near the bed start translating from right to left, indicating that the particle velocities are negative, see Figure 20d–h at T = 2.55–6.09. The counterparts observed in the FSZ move, however, from left to right. This reveals that the corresponding particle velocities are positive, but keep decreasing with T. The thickness of the flow reversal layer increases with an increasing T if T > T

_{ifr}. This feature shows that the particle velocities right on the bed and exactly at the edge of flow reversal are both zero with negative velocities in between. Interestingly note that, for Case E, the incipient flow reversal occurs at T = T

_{ifr}= 1.67 (or t = t

_{ifr}= 0.1508 s), soon after but not simultaneously with the occurrence of P

_{o+}/g at T = T

_{Po+}= T

_{Alo+}= 1.39 (or t = t

_{Po+}= t

_{Alo+}= 0.1255 s). The reason for such a slight temporal delay with (ΔT)

_{delay}= 0.28 [or (Δt)

_{delay}= 0.0253 s] is attributable to the viscous damping effect in the bottom boundary layer.

_{o}(T)]/g (Figure 18) have been demonstrated for the first time to shed light on the relationship between T

_{ifr}and T

_{Po+}(= T

_{Alo+}). For each visualized case other than Case E, the incipient flow reversal does take place immediately after the maximum adverse pressure gradient.

_{ifr}is slightly larger than T

_{Po+}. Finally, the relationship between T

_{ifr}and H

_{0}/h

_{0}is also shown in Figure 19, indicating T

_{ifr}decreasing prominently with an increase in H

_{0}/h

_{0}. This demonstrates the effect of nonlinearity on the incipient flow reversal in the near-bottom zone.

## 5. Conclusions

_{0}/h

_{0}= 0.050–0.550, a series of experimental results at the SMS are presented, systematically showing the results of the relevant flow parameters. The findings are summarized as follows.

- For all the cases exclusive of Case E, the FSZs are positioned between y/h
_{0}= (0.035–0.055) and (0.335–0.366), nearly identical to those between y/h_{0}= 0.05 and 0.350 in Case E. - If H
_{0}/h_{0}increases, the dimensionless free surface elevation, η(T)/H_{0}, and the dimensionless free stream velocity, u_{o}(T)/C_{0}, become more concentrated around T = 0 with a narrower symmetric bell-shape, exhibiting shorter time taken to generate a complete wave motion. This trend indicates that the change of ascending or descending free surface elevation per unit (dimensionless) time becomes greater in magnitude. - For −6.00 ≤ T < 0 and 0 < T ≤ 6.00, the dimensionless free-stream velocity, [u
_{o}(T)/C_{0}], increases from near zero to a maximum and decreases from the maximum to about zero, highlighting the temporal acceleration and deceleration in the FSZ. - The relationship between [u
_{o}/C_{0}]_{max}and H_{0}/h_{0}is uniquely expressed in Equation (3), stating that the former gets large with an increasing H_{0}/h_{0}. For H_{0}/h_{0}= 0.179, 0.363 and 0.550, the values of [u_{o}/C_{0}]_{max}are about 3.10, 5.32, and 6.20 times that (= 0.0473) for H_{0}/h_{0}= 0.050. This trend demonstrates the nonlinear effect on [u_{o}/C_{0}]_{max}. - The dimensionless local acceleration, A
_{lo}/g, is positive for −6.00 ≤ T < 0 and negative for 0 < T ≤ 6.00. At T = 0 with wave crest intersecting the SMS, A_{lo}/g is equal to zero and the free-stream velocity reaches its maximum. - The magnitudes of positive and negative maxima in the dimensionless local acceleration, A
_{lo+}/g and A_{lo−}/g (≈ −A_{lo+}/g), increase linearly, and the counterparts of the dimensionless characteristic time T_{Alo+}(< 0)and T_{Alo−}(> 0) decrease with an increase in H_{0}/h_{0}. For H_{0}/h_{0}= 0.179, 0.363 and 0.550, the values of A_{lo+}/g are about 6.11, 15.14 and 21.45 times that (= 0.0071) for H_{0}/h_{0}= 0.050, indicating the nonlinear effect on A_{lo+}/g and A_{lo−}/g. - The magnitudes of negative and positive maxima in the dimensionless convective acceleration, A
_{co−}/g, and A_{co+}/g, increase when H_{0}/h_{0}increases. However, their magnitudes are about 1/22.0–1/4.3 times those of A_{lo+}/g and A_{lo−}/g. With the magnitude of A_{co}(T)/g being much smaller than that of A_{lo}(T)/g, the contribution to the dimensionless pressure gradient, P_{o}(T)/g (= −[A_{lo}(T) + A_{co}(T)]/g), is thus governed mainly by A_{lo}(T)/g. - P
_{o}(T)/g decreases from near zero to P_{o}^{_}/g (< 0) for −6.00 ≤ T ≤ T_{Po−}or from P_{o+}/g (≈ −P_{o}^{_}/g > 0) to about zero for T_{Po+}≤ T ≤ 6.00, exhibiting an increase in the favorable pressure gradient or decrease in the adverse pressure gradient in the FSZ. Moreover, it increases from P_{o}^{_}/g, via 0, to P_{o+}/g for T_{Po−}≤ T < 0, T = 0, and 0 < T ≤ T_{Po+}, indicating the change from favorable, via zero, to an adverse pressure gradient. - With an increase in H
_{0}/h_{0}, P_{o+}/g increases but T_{Po+}decreases. For H_{0}/h_{0}= 0.179, 0.363 and 0.550, the values of P_{o+}/g are about 5.74, 14.54 and 19.84 times that (= 0.0061) for H_{0}/h_{0}= 0.050, showing the strong nonlinear effect on P_{o}(T)/g and P_{o+}/g. - For each case, the incipient flow reversal occurs immediately after the maximum adverse pressure gradient. Namely, T
_{Po+}is slightly less than T_{ifr}. Further, T_{ifr}decreases if H_{0}/h_{0}increases, which accentuates the nonlinear effect on the incipient flow reversal right above the bed.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic diagram showing a solitary wave propagating over a horizontal bed, two wave gauges, the (x, y) coordinate system, and (u, v) velocity components.

**Figure 2.**A comparison among five repeated runs for the original time series of free surface elevation η

_{o}(t) (Case E).

**Figure 3.**A comparison of time series between ensemble-averaged free surface elevation η(t) and predicted one (Case E).

**Figure 4.**The (ensemble-averaged) velocity fields obtained at (

**a**) t = −0.045 s; (

**b**) t = 0.045 s (Case E).

**Figure 5.**Time series comparisons of horizontal velocity between measured and predicted using Boussinesq solution (Case E) at y = (

**a**) 7.04 cm; (

**b**) 4.40 cm; and (

**c**) 1.44 cm.

**Figure 6.**Time series comparisons between calculated local accelerations (which are based on time series of measured horizontal velocities) and predicted ones using Boussinesq solution (Case E) at y = (

**a**) 7.04 cm; (

**b**) 4.40 cm; and (

**c**) 1.44 cm.

**Figure 7.**(

**a**) Time series of dimensionless free surface elevation, η(T)/H

_{0}; Dimensionless horizontal and vertical velocities profiles are shown at T = (

**b**) −2.50, (

**c**) −1.39, (

**d**) −0.50, (

**e**) 0, (

**f**) 0.50, (

**g**) 1.39, and (

**h**) 2.50 (Case E).

**Figure 8.**Temporal variation of the horizontal velocity profile obtained in the boundary layer and the FSZ at T = (

**a**) −2.50; (

**b**) −1.39; (

**c**) −0.50; (

**d**) 0; (

**e**) 0.50; (

**f**) 1.39; and (

**g**) 2.50 (Case E). Note that the FSZ is located within the region as marked by the two horizontal dashed lines, and that the internal zone or boundary layer is situated beyond the upper or beneath the lower horizontal dashed line.

**Figure 10.**A comparison of temporal variations in the non-dimensional free surface elevation for H

_{0}/h

_{0}= 0.050–0.550.

**Figure 12.**A comparison of temporal variations in the dimensionless free-stream velocity for H

_{0}/h

_{0}= 0.050–0.550.

**Figure 15.**A comparison of temporal variations in the non-dimensional local acceleration for H

_{0}/h

_{0}= 0.050–0.550.

**Figure 16.**The relationships of T

_{Alo+}and T

_{Alo−}as well as A

_{lo+}/g and A

_{lo}

^{_}/g versus H

_{0}/h

_{0}.

**Figure 17.**A comparison of temporal variations in the non-dimensional convective acceleration for H

_{0}/h

_{0}= 0.050–0.550. Note that the vertical scale used herein is distinct from that shown in Figure 15.

**Figure 18.**A comparison of temporal variations in the non-dimensional pressure gradient of the free stream for H

_{0}/h

_{0}= 0.050–0.550.

**Figure 19.**Variation trends of the non-dimensional times for occurrence of flow reversal and maximum adverse pressure gradient, and of the values of maximum adverse pressure gradient for H

_{0}/h

_{0}= 0.050–0.550.

**Figure 20.**Flow-visualized images at T = (

**a**) 0; (

**b**) 0.89; (

**c**) 1.67; (

**d**) 2.55; (

**e**) 3.43; (

**f**) 4.32; (

**g**) 5.20; (

**h**) 6.09 for −0.10 ≤ x/h

_{0}≤ 0.10 and 0 ≤ y/h

_{0}≤ 0.10.

Case | H_{0} (cm) | H_{0}/h_{0} * | C (cm/s) | C_{0}(cm/s) | C_{0}/C | Framing Rate of HSPIV (Hz) | Framing Rate of FV (Hz) | Size (cm × cm) (Length × Width) |
---|---|---|---|---|---|---|---|---|

A | 0.40 | 0.050 | 88.59 | 90.78 | 1.025 | 1600 | 30 | 2.05 × 1.15 |

B | 0.90 | 0.112 | 88.59 | 93.44 | 1.055 | 2500 | 50 | 2.05 × 1.15 |

C | 1.43 | 0.179 | 88.59 | 96.18 | 1.086 | 2500 | 50 | 2.05 × 1.15 |

D | 2.28 | 0.285 | 88.59 | 100.42 | 1.134 | 2500 | 100 | 2.05 × 1.15 |

E | 2.90 | 0.363 | 88.59 | 103.41 | 1.167 | 500 | - | 16.80 × 16.80 (HSPIV) |

1000 | - | 2.00 × 1.00 (HSPIV) | ||||||

- | 100 | 2.05 × 1.15 (FV) | ||||||

F | 3.08 | 0.385 | 88.59 | 104.26 | 1.177 | 2500 | 100 | 2.05 × 1.15 |

G | 3.52 | 0.440 | 88.59 | 106.31 | 1.200 | 2500 | 100 | 2.11 × 1.18 |

H | 4.00 | 0.500 | 88.59 | 108.50 | 1.225 | 2000 | - | 2.11 × 1.18 |

I | 4.40 | 0.550 | 88.59 | 110.29 | 1.245 | 2500 | 100 | 2.11 × 1.18 |

_{0}= 8.0 cm for all cases.

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Lin, C.; Kao, M.-J.; Yang, J.; Yuan, J.-M.; Hsieh, S.-C.
Effects of Nonlinearity on Velocity, Acceleration and Pressure Gradient in Free-Stream Zone of Solitary Wave over Horizontal Bed—An Experimental Study. *Water* **2022**, *14*, 3609.
https://doi.org/10.3390/w14223609

**AMA Style**

Lin C, Kao M-J, Yang J, Yuan J-M, Hsieh S-C.
Effects of Nonlinearity on Velocity, Acceleration and Pressure Gradient in Free-Stream Zone of Solitary Wave over Horizontal Bed—An Experimental Study. *Water*. 2022; 14(22):3609.
https://doi.org/10.3390/w14223609

**Chicago/Turabian Style**

Lin, Chang, Ming-Jer Kao, James Yang, Juan-Ming Yuan, and Shih-Chun Hsieh.
2022. "Effects of Nonlinearity on Velocity, Acceleration and Pressure Gradient in Free-Stream Zone of Solitary Wave over Horizontal Bed—An Experimental Study" *Water* 14, no. 22: 3609.
https://doi.org/10.3390/w14223609