Limitations of the Yang’s breaking wave force formula and its improvement under a wider range of breaker conditions

1 Introduction

Sloping sea dikes are commonly used types of coastal protection structures [1,2] to withstand wave and tide impacts in extreme weather and climate events (e.g., typhoon and tsunami). Wave breaking is a common phenomenon on seaward sea dikes [3]. Compared with a non-breaking wave, breaking wave can produce extremely high slamming pressure on the sea dike surface in a short period of time [4], which may lead to the failure of the sea dike. Therefore, breaking wave impinging induced by strong waves is one of the main causes of sea dike failure [5,6]. Experimental, numerical, and theoretical investigation of breaking waves based on wave flume experiments (including numerical wave flumes) can be found in some literature [7,8,9,10,11,12,13]. Only a few results of the field measurements are available because field measurement is very difficult [14,15,16,17].

Results from previous studies mentioned above show that plunging breaker is the most destructive of the three types (i.e., spilling breaker, plunging breaker, and surging breaker), which is characterized by acceleration of the fluid near the crest, causing the jet with air bubbles impinging and the devastating forces on the coastal structure surface [18,19]. Unfortunately, the plunging wave-breaking mechanism had not been fully understood. The maximum slamming pressure due to breaking waves acting on the sea dike slope is a stochastic variable even for regular waves because the instabilities at the breaking point are influenced by the highly turbulent water–air mixture produced by the preceding breakers. This leads to the difficulty of reliable estimation of breaking wave loads, which is necessary for the effective design of coastal structures like sea dikes.

Most predictions of breaker loads are based on empirical or semi-empirical formulas calibrated from laboratory data because wave breaking mechanism is complex and filed data available are remarkably slight [20,21]. Since the last century, some breaker load formulas had been suggested and adopted in the coastal structure design, such as improved Morison’s formula [22,23,24], Soviet CHиПII57-75 formula widely used in China, the Ikeno–Tanaka formula of Japan [25], Führböter–Sparboom formula of Germany [26,27], and Stanczak formula derived from the Führböter–Sparboom formula [28]. The comparison of these formulas with the measurements was done by Yang [29], which showed that the difference with the measurements was primarily because of some important shortcomings in these formulas.

These formulas mentioned above do not take into account the comprehensive influence of the wave steepness, slope angle, and breaker height and depth on breaker loads. To eliminate this drawback, Yang [29] reviewed some of the recent literature and gave a derivation of a formula for wave impacts, based on an analysis of the impulse in the breaking wave tongue and the kinetic and potential energy in the wave tongue. This is a rather simple method, which does not include important aspects, like air entrainment in the impacting water. Nevertheless, the resulting formula is interesting because the comparison of the formula with some measurements was performed well. But Yang [29] believes that further analysis is still required for the accuracy improvement in predicting wave slamming pressures.

Yang formula’s validity is limited according to the experimental data from Führböter [26] and Stagonas et al. [32]. In practice, the calculated result may be larger than the true value in some breaker conditions. Moreover, the formula only works for small slope angles. So, this article aims to revise the Yang formula to improve its performance and make it suitable for a range of hydraulic conditions and bottom slope conditions. The improved Yang formula will be compared with some formulas, of which Soviet CHиПII57-75 formula has been adopted in the Chinese design guidelines of coastal structures for many years. The selected breaker data used in this comparative analysis include some laboratory data and field data. The article is divided into four main sections:

1. Section 1 briefly reviews the existing breaker load formulas.

2. Section 2 is the improvement of the existing Yang formula.

3. Section 3 presents the results and discussion.

4. Section 4 details the main conclusions.

2 Breaker load formulas including the Yang formula

The comparison of the revised formula with those of Soviet CHиПII57-75, Führböter–Sparboom, Stanczak formulas, and Yang formula will be done in this study. The Ikeno–Tanaka formula is usually suitable for vertical structures. The Führböter–Sparboom formula does not take into account the influence of the wave steepness on the maximum slamming pressure, and the Stanczak formula eliminates the drawback. The Yang formula gives the influence of the wave steepness, slope angle, and breaker height and depth. These formulas suitable for slope structures are as follows:

1. In Soviet CHиПII57-75 formula, the maximum slamming pressure p _max [Pa] on the slope is described by

where ρ is the density of the water (kg/m³); g is the acceleration due to gravity (m/s²); n is the dike bottom slope ratio denoted by 1:n; H is the wave height at the dike toe (m); L is the deep water wave length (m); k ₂ is the constant coefficient as listed in Table 1; p ¯ is the relative wave pressure as listed in Table 2. The formula was suggested to be used for the range of 1.5 ≤ n ≤ 5.0 and 10 ≤ L/H ≤ 35.

1. The complexity of wave breaking makes the breaker load a highly random process, even in a regular wave train. So, in the Führböter–Sparboom formula and the Stanczak formula, the notation p _max,i [Pa] is used to indicate the maximum pressure with no more than i% probability. The p _max,99 can be used as the highest maximum slamming pressure in practice [28]. The two formulas were suggested to be used for the range of surf similarity parameter ξ < 2.6 for plunging breaker, with ξ = tan α H / L where the deep wave length L = gT ²/(2π).

1. The Führböter–Sparboom formula is described by

   (2) p max , i = const i ρ g H tan α ,

   where const _i is the coefficient depending on the characteristic maximum value considered, const₅₀ = 12, const₉₀ = 16, const₉₉ = 20, and const₂₄ = 24; α is the outer dike slope (deg).

2. The Stanczak formula is described by

   (3) p max , i = k i ρ g H tan α , k 50 = − 289 H g T 2 + 11.2 , k 90 = 1.33 k 50 , k 99 = 1.67 k 50 , k 99.9 = 2.5 k 50 ,

   where T is the wave period (s).

   1. The Yang formula is as follows:

where K _KG is the slope effect coefficient [30]; γ depends on both the wave steepness and the out dike slope angle α [31]. The Yang formula was suggested to be used for the range of 0.5 ≤ ξ ≤ 3.3 and 0.01 ≤ tan α ≤ 0.44.

3 Revised Yang formula

The limited range of the Yang formula is 0.5 ≤ ξ ≤ 3.3 and 0.01 ≤ tan α ≤ 0.44; otherwise, the formula’s prediction error will greatly increase, especially when tanα increases. In addition, the results from Yang [29] show that the calculated results using the formula are larger compared with some experimental data. The main aim of this section is to modify the formula so that it can better predict a wider range of breaker conditions. The wave-breaking process is very complex, so some assumptions and simplifications are necessary in order to be able to calculate the breaker loads. Following Yang [29], some construction derivation steps of the Yang formula under some hypotheses are listed in Figure 1.

Figure 1

Main steps in the derivation of Yang formula [29].

As shown in Figure 1, the most important assumptions implied in the derivation of the Yang formula are as follows:

1. Conservation of cross sections 1 – 1, 2 – 2, and 3 – 3 momentum in control volume.

2. A homogeneous velocity distribution across the cross Sections 1 – 1, 2 – 2 and 3 – 3.

3. Ignoring the incidence angle α ₁ and reflection angle α ₂ of the breaking wave plunging on the dike slope.

Under assumption (iii), “p _y = ρu ² A(sin α ₁ + sin α ₂) where A is section area” is modified to “p _y ≤ 2ρu ²,” where constant 2 is also a constant term in the Yang formula (equation (4)). The larger constant term should be one of the main reasons that the calculated results by the Yang formula are obviously larger in some breaker conditions. This drawback can be weakened by replacing the constant value 2 in equation (4) with an empirical function that depends on the incidence angle α ₁° and reflection angle α ₂°.

Let tanα ₁ ≈ 1.8ƞ _b /(u _b t) empirically, where u _b is the water particle velocity at the breaking wave crest; ƞ _b is the maximum free surface height from the mean water level (MWL) at the breaking location, ƞ _b ≈ 0.8H _b [29]; t is the time that the water particles on the breaking wave crest travel to MWL. The derivation of t is as follows:

Let u _b be proportional to the square root of water depth, u b = g d b with d _b = H _b /γ [29], where d _b is the water depth at the breaking location; H _b is the breaking wave height at the breaking location. Then

Let K _KG = 0.56 if tan α > 0.44 [30]. Assume that α ₁ = α ₂ in the symmetric case and replace the constant value 2 in equation (4) with equation (6). Then, the revised Yang formula is as follows:

with

4 Results and discussion

4.1 Predictive performance of revised Yang formula in experimental conditions

We will verify the revised Yang formula with the laboratory breaker data from the studies of Führböter [26] and Stagonas et al. [32]. In the experimental work of Führböter [26], eight breaking wave cases (H = 0.09–1.3 m; T = 1.3–4.6 s) with the bottom slope ratio of 1:4 were selected. In the experimental condition of Stagonas et al. [32], two breaking wave cases (H = 0.09 m, 0.13 m; T = 1.3 s, 1.43 s) with the bottom slope ratio of 1:3 were selected. The purpose of Figure 2 is to investigate the predictive performance of the revised Yang formula in the above cases. The calibration of the formula indicates that

1. The calculated result using the revised Yang formula is about 25% smaller than that of the original formula.

2. The calculated result using the revised Yang formula is generally between P _max,99 by the Führböter–Sparboom formula and P _max,99 by the Stanczak formula.

3. The relative errors between the calculated results from the revised formula and the measured values are 0.96% (case H = 0.92 m, T = 3.37 s) ∼ 32.7% (case H = 0.45 m, T = 2.38 s), and the average relative error is 12.6%, which is better than other formulas in most of the cases.

Figure 2

Calibration of revised Yang Formula in experimental conditions: (a) conditions from Führböter [26] and (b) conditions from Stagonas et al. [32].

4.2 Predictive performance of revised Yang formula in field conditions

An earlier field investigation completed in 2004 provides some breaker data. The primary goal of this investigation was to evaluate the safety of the Dongpuxintang sea dike located at Wenling City, Zhejiang Province, China. Figure 3 shows the field investigation site (28°19′52″ N, 121°30′34″ E) and the cross-section of the investigated sea dike. The crest elevation of the dike is 6.95 m (vertical datum: 1985 National Height Datum), and the crest is 5.0 m wide.

Figure 3

Schematic diagram of field investigation.

The incident wave height was measured using a capacitance-wire wave recorder at a distance of 100 m toward offshore. The sensor has a wave elevation metering range of 0–7 m. The maximum slamming pressure on the dike slope was measured using six pressure transducers at elevations ranging from 3.46 to 6.23 m. Due to the presence of comparatively large noise, field measuring equipment commonly suffers from poor signal quality [20]. Compared with the sampling of wave loads at 680 Hz [32] and 2,000 Hz [26] in laboratory, and at 200 Hz in field [20], the sampling frequency of this field investigation is only 100 Hz.

The calibration of the formula in Figure 4(a) indicates that

1. The calculated results using formulas are commonly larger than field data, which may be because that the low sampling frequency makes it difficult for us to capture the real breaker load peak.

2. The calculated results using the revised Yang formula and Soviet CHиПII57-75 are closer to field data. The relative errors between calculation and measurement are 24.9% (case H = 1.37 m, T = 4 s)–44.2% (case H = 1.18 m, T = 5 s).

Figure 4

Calibration of revised Yang formula in filed data: (a) from the author and (b) from Bird et al. [20].

Figure 4(b) describes the performance of the revised Yang formula on the vertical wall under the field condition by Bird et al. [20]. The results show that the relative error between the revised Yang formula and the field data is 20.2% and that of the Ikeno–Tanaka formula suitable for vertical structures is 38.4%. However, it is still not possible to judge which formula is better or worse, because the field data are affected by sampling frequency as well as possible noise interference.

4.3 Comparative analysis of empirical formulas for given breaker data

The computations of the formulas (i.e. revised Yang, Yang, Führböter–Sparboom, Stanczak, and CHиПII57-75) are carried out with 90 cases from nine sets of data (Figure 5), with H = 0.1–3.0 m, T = 2–5 s, and n = 1–5. For cases beyond the applicable scope of these formulas, such as ξ > 2.6 for the Stanczak formula, tan α > 0.44 for Yang formula, k ₂ is out of scope in Table 1 for CHиПII57-75 formula and so on, the calculation results are not presented in Figure 5. It is shown that

1. For all formulas, the maximum slamming pressure generally increases as the wave period increases. With the decrease in slope ratio, the calculation result of the revised Yang formula (including Yang formula) increases, but Führböter–Sparboom and Stanczak formulas show the opposite trend.

2. The calculation result of the Führböter–Sparboom formula is larger than that of other formulas, and it increases rapidly with the increase in slope ratio. In the case of slope ratio 1:1, its calculation result is about six times that of the revised Yang formula.

3. The calculation result of CHиПII57-75 formula is generally smaller than that of other formulas. The calculated result using the revised Yang formula is closer to p _max,90 using the Stanczak formula under the case condition that both are suitable.

Figure 5

Comparative analysis of formulas for given data: (a) n = 5, T = 2 s, (b) n = 5, T = 3 s, (c) n = 5, T = 5 s, (d) n = 3, T = 2 s, (e) n = 3, T = 3 s, (f) n = 3, T = 5 s, (g) n = 1, T = 2 s, (h) n = 1, T = 3 s, and (i) n = 1, T = 5 s.

5 Conclusion

Plunging breaker impinging is one of the main causes of sea dike failure. The main aim of this study is to find a reliable empirical formula to better predict the slamming pressure on the dike slope for a wider range of breaker conditions. Compared with other widely used empirical formulas, the Yang formula takes into account more comprehensive factors and shows good performance under some wave breaking conditions. However, the prediction error of the Yang formula will increase with the increase in dike slope. The unreasonable constant term also causes the calculation result of Yang formula to be larger in some cases. For this, the Yang formula was revised in this study. The major conclusions of this study are obtained as follows:

1. Yang formula does not take into account the incidence angle α ₁ and reflection angle α ₂ of the breaking wave plunging on the dike slope. The revised Yang formula eliminates this drawback. Moreover, the Yang formula is only applicable under the condition of 0.01 ≤ tan α ≤ 0.44. By modifying the slope effect coefficient K _KG , the revised formula can be applied for a range of tan α > 0.44. The revised formula may be applicable to vertical walls under some breaker conditions. By replacing the constant term in the Yang formula with a function related to α ₁° and α ₂°, the accuracy of the formula is also improved.

2. Some laboratory data and field data were employed in the verification of the revised Yang formula. The performance of the formula was also compared with that of CHиПII57-75 formula, the Ikeno–Tanaka formula, the Führböter–Sparboom formula, and the Stanczak formula. It is found that the revised formula predicts well. The average relative error between the calculated results from the revised Yang formula and the measured values from Führböter [26] and Stagonas et al. [32] was 12.6%, which was better than other formulas in most of the cases. Validation based on field data also showed that the Yang formula performed best.

3. The analysis results of empirical formulas for given breaker data showed that the Yang formula could adapt to more wave-breaking conditions compared with other empirical formulas. As Stanczak [28] pointed out that the complex characteristics of wave breaking made the breaker loads highly random. Even in a regular wave train, the maximum slamming pressure of each wave was quite different [28]. In addition, no matter whether in the laboratory or field, noise interference is inevitable in wave load measurement. All of this can undermine the authenticity of the wave load data. An important solution is to conduct more reliable laboratory tests or field observations used for validation.

This study would significantly contribute to the sea dike design against wave. However, Yang formula’s validity may be limited due to the limited breaker data for verification. The subject is important but has not been solved sufficiently yet. Addressing this subject is still relevant. In addition, some important aspects of this study are worthy of further discussion, including the probability form of the revised Yang formula since the slamming pressure is stochastic, the improvement of the proposed Yang formula by considering the effect of different revetments on wave breaking, and the further improvement of the proposed formula by using some numerical models and optimization algorithms.