Breaking Undular and Breaking Surges Advancing in Still Water

Zheng, Feidong; Li, Xueyi

doi:https://doi.org/10.1155/2023/6733848

Advances in Civil Engineering

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Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 6733848 | https://doi.org/10.1155/2023/6733848

Breaking Undular and Breaking Surges Advancing in Still Water

Feidong Zheng¹and Xueyi Li²

Academic Editor: Iacopo Carnacina

Received02 Feb 2023

Revised02 Apr 2023

Accepted14 Jun 2023

Published28 Jun 2023

Abstract

The propagation of secondary surges can lead to unexpected interaction events with navigation ships and downstream hydraulic structures; hence, they may cause navigation accidents and structural damage. In the present study, physical experiments were performed in a long water wave tank to study the evolution of breaking undular and breaking surges advancing in still water. Both free-surface and velocity measurements were conducted. It was observed that the transition from a breaking undular surge to a breaking surge occurred as the ratio of the mean water depth behind the surge to the still water depth equal to 1.68–1.71. Empirical expressions based on the experimental data were proposed to predict the wave amplitude and wave steepness of breaking undular surges. The unsteady flow results demonstrated that the normalized longitudinal velocity component beneath the surge front was governed mainly by the normalized free-surface elevation, independently of the surge type. Moreover, the maximum vertical velocity component during the surge front passage had a linear dependence on the vertical location. The present result provided new quantitative information on breaking undular and breaking surges advancing in still water.

1. Introduction

Three types of secondary surges can develop at the wave face of a long wave: nonbreaking undular surge, breaking undular surge, and breaking surge [1]. Such surges can be observed in shallow water conditions when a tide wave enters a converging river [2] or after the operation of lock chambers in a navigation channel [3]. Secondary surges can also occur in intermediate or deep water conditions during the nonlinear evolution process of long-crested wave groups [4]. The generation and development of secondary surges results from imbalance between wave nonlinearity and dispersion effects [5–7]. These surges have a great impact on the ecology and estuarine processes [8, 9]. The propagation of secondary surges may lead to extreme interaction events with coastal and marine structures, compromising the stability of these structures, and even causing loss of structural functionality [4, 10–12]. In navigation channels, secondary surges can also impose violent impact loads on navigation ships and downstream hydraulic structures (e.g., lock gates, sluices, and dams), causing navigation accidents and structural damage [13]. Therefore, quantitative characterization of the nonlinear evolution of secondary surges is important for evaluating potential damages to infrastructure and navigation ships.

Because of the potential impact caused by secondary surges, the mechanisms governing their evolution have been widely investigated. Most of previous investigations have focused on secondary surges propagating against an initially steady flow, such as Chanson [14], Gualtieri and Chanson [15, 16], Docherty and Chanson [17], Simon [18], and Leng and Chanson [19]. According to these studies, the following important conclusions were drawn. First, the transition from a nonbreaking undular surge to a breaking one depended on the surge Froude number and the degree of the boundary layer development at the toe of the surge. Second, the wave parameters (i.e., wave amplitude, wavelength, and wave steepness) and the wave shape of an undular surge could not be accurately predicted using existing solutions derived from linear wave theory and the Boussinesq equations, independently of the surge type. Third, the streamwise velocity component exhibited an oscillating feature with maximum and minimum velocities obtained beneath the wave troughs and crests of an undular surge, respectively. Fourth, the largest turbulent Reynolds stress components occurred beneath the wave crests of an undular surge in the absence of flow reversal induced by the surge. A different insight into the flow structures beneath both nonbreaking undular and breaking surges was obtained by considering the Kolmogorov complexity [20]. This has demonstrated a marked effect of an undular surge passage on the vertical distribution of the Kolmogorov complexity for the streamwise velocity, which was not influenced by a breaking surge passage. Recently, a new analytic solution has been developed based on the potential wave theory by Chen et al. [21] to describe the wave feature of nonbreaking undular surges propagating against a steady flow. The comparison between the proposed analytic solution and experimental results indicated that both the wave amplitude and wave celerity were well reproduced. In contrast, the study of the evolution of secondary surges in still water has received less attention except a few investigations (e.g., Peregrine [22], Hornung et al. [23], and Lin et al. [24]). In fact, the wave features of secondary surges were highly dependent upon the initial flow conditions prior to surge arrival (i.e., still water and steady flow), which has been demonstrated by Benet and Cunge [25], and Zheng et al. [26]. There are indeed fundamental differences between the two types of secondary surge flows.

Recently, the nonlinear evolution of nonbreaking undular surges advancing in still water has been systematically investigated by Zheng et al. [27]. Their results indicated that the leading wave front of a nonbreaking undular surge was close to that of a solitary wave in shallow water region during the entire wave evolution process. This finding obtained for nonbreaking undular surges encourage further quantitative investigation of breaking undular and breaking surges advancing in still water as well. The mechanisms governing the free-surface dynamics and velocity distributions of breaking undular and breaking surges are not yet well understood. However, detailed investigation on these aspects is inevitable for understanding the overall evolution process of secondary surges advancing in still water. In addition, the requirement for datasets for the validation of the corresponding numerical models developed for breaking undular and breaking undular surges advancing in still water remains imperative.

To date, limited quantitative information is available on the evolution characteristics of breaking undular and breaking surges advancing in still water. The study presents such an investigation, which was performed in a long water wave tank. This paper is organized as follows. The experimental methodology is presented in Section 2. In Section 3, the transition from a breaking undular surge to a breaking surge is first identified. The wave features of breaking undular surges are subsequently investigated. The velocity distributions beneath two types of secondary surges are further analyzed and discussed. Finally, conclusions of this study are drawn in Section 4.

2. Experimental Setup and Experiments

A schematic description of the experiment setup is shown in Figure 1. The details of the experimental arrangement can be found in Zheng et al. [27], and a brief description is provided here. The experiments were carried out in a rectangular horizontal wave tank, which was 0.3 m wide, 0.35 m deep, and 38 m long. The tank was made of plexiglass walls and a smooth polyvinyl chloride bed. A vertical wall was placed at right angle to the direction of surge propagation. The tank was connected to a reservoir (1 × 1 m in plane xy) through a pipe, which was equipped with a plate gate and the corresponding shaft. In the current study, the still water depth d₀ and the gate lifting time t_v were fixed at 0.08 m and 20 s, respectively. The only variable determining the development process of a surge was the drop height H_u. The details of the experiments are listed in Table 1. Overall, four different drop heights ranging from H_u = 0.2–0.5 m were tested. When lifting the gate at t_v = 20 s, a nonbreaking undular surge induced by water flow in the tank was first observed. During wave propagation, the nonbreaking undular surge exhibited significant wave amplification and subsequently evolved into a breaking undular surge. This process was observed in all experiments. For some experiments (i.e., H_u ≥ 0.4 m), the breaking undular surge transferred to the breaking surge regime. The present research focused on the evolution of breaking undular and breaking surges. Each experiment was stopped when the leading wave arrived at the vertical wall. The (x, z) coordinate system was chosen with the vertical upward pointing z-axis and the horizontal x-axis coinciding with the tank bed z = 0.

The free-surface features and velocity distributions of secondary surges were measured using a set of wave gauges and an acoustic Doppler velocimeter Nortek™ Vectrino+, respectively, with a sampling frequency of 200 Hz. The wave probes were located from x = 3 to 36 m with an interval of 3 m between two neighbor probes. The accuracy of the wave probes was 0.1 mm. Unsteady velocity measurements were performed at x = 15 and 30 m for one experiment with H_u = 0.4 m. The secondary surge was a breaking undular type at x = 15 m, whereas it was in the breaking surge region at x = 30 m. For each longitudinal location x, instantaneous velocity data at five vertical elevations (i.e., z/d₀ = 0.08, 0.13, 0.41, 0.71, and 0.91) were recorded with an accuracy of 1 cm/s. In the present study, tap water was used. The reservoir water was seeded with some vegetable dye to improve the quality and accuracy of the velocity data during instantaneous velocity measurements.

3. Results and Discussion

3.1. Flow Patterns

The temporal evolution of wave profiles at different longitudinal locations is shown in Figure 2, where the water depth d was normalized with d₀, and the time t was normalized with (d₀/g)^0.5 with g denoting the gravitational acceleration. Note that, the wave profile immediately prior to wave breaking at the leading wave crest (i.e., a nonbreaking undular surge at x/d₀ = 150) is also shown for comparison. It can be seen that the wave amplitude of an undular surge exhibited a sharp decrease shortly after the appearance of some wave breaking (i.e., a breaking undular surge at x/d₀ = 187.5). Over a large propagation distance (e.g., x/d₀ = 262.5), the free-surface undulations eventually disappeared and the breaking undular surge transferred to a breaking surge.

Pertaining to secondary surges propagating against an initially steady flow, the transition from a breaking undular surge to a breaking surge was usually characterized by the transition surge Froude number, e.g., Chanson [14] and Gualtieri and Chanson [15, 16]. However, a combined analysis of the results from these studies indicated that the transition surge Froude number was closely dependent on the inflow velocity. This revealed that the surge Froude number might not be a suitable index of transition flow conditions. Recently, the change in wave regime for secondary surges propagating against an initially steady flow was considered by Pelinovsky et al. [1]. Their results revealed that the surge type could be simply identified by the ratio of the mean water depth behind the surge front, d_m, to the still water depth. The same concept was applied herein to secondary surges advancing in still water. Note that the mean water depth was defined as the average of the water depths at the leading wave crest d_1c, and trough d_1t, for undular surges, whereas it referred to the water depth measured immediately after the surge front for breaking surges [28]. The present data showed that breaking undular surges occurred for 1.48 ≤ d_m/d₀ ≤ 1.71, whereas breaking surges were observed for 1.68 ≤ d_m/d₀ ≤ 1.89. Thus, criterion d_m/d₀ = 1.68–1.71 can be used as an estimate of the change in surge regime. It is worth noting that this criterion seems also valid for secondary surges propagating against an initially steady flow in field situations [1].

3.2. Wave Features of Breaking Undular Surges

3.2.1. Water Depths at the Leading Wave Crest and Trough

The variations of the normalized water depths at the leading wave crest and trough with d_m/d₀ during surge evolution are illustrated in Figure 3. It is clearly seen that the values of d_1c/d₀ and d_1t/d₀ for d_m/d₀ = 1.48 (these data were collected at x = 18 m, which was very close to the breaking location, i.e., x = 17.2 m) deviated significantly from the remaining data. This discrepancy was attributed to the water depth redistribution process at the leading wave induced by wave breaking; therefore, the wave form for d_m/d₀ = 1.48 was instable. To ensure that this different behavior is fully declared, the measured data corresponding to d_m/d₀ = 1.48 are excluded in subsequent analyses. As d_m/d₀ increased, both d_1c/d₀ and d_1t/d₀ exhibited linearly increasing trends. In the experimental range of 1.50 ≤ d_m/d₀ ≤ 1.71, the evolution of d_1c/d₀ and d_1t/d₀ could be related to the normalized mean water depth bywith a correlation coefficient r² = 0.85 and 0.97 for Equations (1) and (2), respectively.

3.2.2. Wave Amplitude and Steepness

In undular surges, the wave amplitude A_w was defined as half of the difference between the water depths at the leading wave crest and trough. The wavelength, L_w, referred to the distance between the first and second wave crests. The dominant influence of the normalized mean depth d_m/d₀ on the wave amplitude is shown in Figure 4. It was expected that the wave amplitude decayed with increasing d_m/d₀. In the range 1.50 ≤ d_m/d₀ ≤ 1.71, the variation of A_w/d₀ during surge evolution could be described bywith a correlation coefficient r² = 0.78.

The dependency of the wave steepness on the normalized water depth is presented in Figure 5. A decrease in the wave steepness with increasing d_m/d₀ was clearly observed. The experimental data suggested a relationship between A_w/L_w and d_m/d₀with a correlation coefficient of 0.68.

3.3. Velocity Field beneath Advancing Surges

3.3.1. Temporal Evolution of Velocity Components

Figures 6(a) and 6(b) illustrate the effects of a breaking undular surge and a breaking surge on velocity field. In each graph, the variations of V_x/(gd₀)^0.5, V_y/(gd₀)^0.5, V_z/(gd₀)^0.5, and d/d₀ were plotted against the normalized time relative to the first maximum elevation of the water surface, (t−t₀)(g/d₀)^0.5.The experimental data highlighted the rapid increase of the streamwise velocity component during the surge front passage. The V_x/(gd₀)^0.5 oscillated with the free-surface undulations behind the breaking undular surge front, while it was almost constant after the breaking surge front. Note that for all experiments, a short time lag was observed between the first maximum elevation of the water surface and the maximum longitudinal velocity component. During the passage of the breaking undular surge, the transverse velocity component showed some irregular fluctuations. In contrast, the V_y/(gd₀)^0.5 exhibited a larger fluctuation range associated with some low frequency pattern during the breaking surge passage. The experimental data from Koch and Chanson [28] demonstrated a similar behavior of the transverse velocity component in a breaking surge propagating against an initially steady flow. For both types of surges, the vertical velocity component reached its maximum value, V_zm, beneath the surge front.

(a)

(b)

3.3.2. Velocity Distributions beneath Surge Fronts

The numerical results of Zheng and Li [13] showed that the depth-averaged longitudinal velocity component beneath the surge front of a nonbreaking undular surge was close to that of a solitary wave in shallow water region [29]. Herein, a quantitative investigation on the longitudinal velocity component distributions beneath the surge fronts of the breaking undular and breaking surges was carried out. For each vertical location z, instantaneous velocity data at several representative moments (i.e., (t−t₀)/(g/d₀)^0.5 = 0, −0.5, −1.0, −1.5, −2.0, and −2.5) were extracted. The dependance of the normalized longitudinal velocity component beneath the surge fronts on (d−d₀)/d₀ is shown in Figure 7. The relation between V_x/(gd₀)^0.5 and (d−d₀)/d₀ for two types of surges was well described by the following equationwith a deviation of ±20% and a correlation coefficient of 0.97. This finding revealed that the normalized longitudinal velocity component distributions beneath the surge fronts exhibited a similar trend to that predicted by the second-order solitary wave solution derived by Laitone [29].

Figure 8 shows the variation of the maximum vertical velocity component along the water column. Note that the vertical velocity component was null at the impervious tank bed. A linear distribution of the maximum vertical velocity component was observed during the surge front passage for both breaking undular and breaking surges. This finding was in agreement with the earlier observation of Docherty and Chanson [17], although they studied the velocity distribution induced by a breaking surge propagating against an initially steady flow. It is noteworthy that the linear distribution of the maximum vertical velocity component was consistent with Boussinesq’s approximation for a solitary wave [30].

4. Conclusions

In this study, the evolution of breaking undular and breaking surges advancing in still water was physically investigated in a series of experiments conducted in a long water wave channel, extending a previous study by Zheng et al. [27]. The main findings are summarized as follows.

A breaking undular surge was observed for the relative mean water depth behind the surge front d_m/d₀ in the range from 1.48 to 1.71, whereas a breaking surge occurred for 1.68 ≤ d_m/d₀ ≤ 1.89. The change in surge regime could be approximated as d_m/d₀ = 1.68–1.71, which was close to the transition criterion for secondary surges propagating against an initially steady flow [1], indicating a negligible effect of the initial flow conditions on the transition of the surge types. Pertaining to breaking undular surges, the water depths at the leading wave crest and trough exhibited linear increasing trends with increasing d_m/d₀, whereas the distributions of wave amplitude and steepness followed a power law. For both breaking undular and breaking surges, the longitudinal velocity component beneath the surge front depended primarily on the free-surface elevation. Moreover, the maximum vertical velocity component presented a linear distribution along the water column, coinciding with Boussinesq’s approximation. This study provided new quantitative results on the evolution characteristics of breaking undular and breaking surges advancing in still water, which might be used to validate numerical models for secondary surge evolution.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the China Postdoctoral Science Foundation (grant no. 2021M700620); Natural Science Foundation of Chongqing, China (grant no. cstc2021jcyj-bshX0049).

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Copyright © 2023 Feidong Zheng and Xueyi Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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