Feasibility Study on Common Methods for Wave Force Estimation of Deep Water Combined Breakwaters

YU Dingyong, TANG Peng, and SONG Qingguo

College of Engineering,Ocean University of China,Qingdao266100,P. R. China

Feasibility Study on Common Methods for Wave Force Estimation of Deep Water Combined Breakwaters

YU Dingyong*, TANG Peng, and SONG Qingguo

College of Engineering,Ocean University of China,Qingdao266100,P. R. China

China’s newly enacted Breakwater Design Specifications (JTS154-2011) explicitly state that breakwaters with water depths greater than 20 m are categorized as deep-water breakwaters, and emphasize that design principles, methods and construction requirements are different from those of common shallow water breakwaters. However, the specifications do not make any mention of how to choose wave force calculation methods of deep-water breakwaters. To study the feasibility of different formulae for wave force estimation of deep water combined breakwaters, the wave force calculated by the Sainflou’s, Goda’s, modified Goda’s and specifications’ methods are compared for various water depths and wave heights in this paper. The calculated results are also compared with experimental data. The total horizontal forcing and the pattern of pressure distributions are presented. Comparisons show that the wave pressure distributions by the four methods are similar, but the total horizontal forces are different. The results obtained by the Goda’s method and the specified formulae are much closer to the experimental data. As for the wave force estimation for the deepwater mixed embankment foundation bed parapet, the Goda’s formula is applied in the case with a water depth of less than 42 m. The Specifications method is suitable for standing waves. In the wave force estimates of breastworks, Sainflou’s and the modified Gaoda’s formulae are no longer applicable for the foundation bed of mixed embankment.

combined breakwater; breast; wave force; deep water; feasibility

1 Introduction

Breakwater is an important marine/hydraulic structure in coastal engineering commonly-used breakwater structures include the slope, the vertical and the combined types. The type of breakwaters to be built is determined based on natural environment, construction facilities, materials as well as cost. In deep water the vertical type or the slope type is normally a good choice in terms of the cost, but the combined one is usually the first choice. There are different descriptions on the combined structure. Xie (1994) pointed out that although enginners began to pay attention to deep water structures, the concept of the combined deep water breakwater has not been clear until the 1990s. Goda (1985) indicated that the vertical breakwater includes both the structure directly placed on the seabed and that built on artificial rubble foundation. Hur and Mizutani (2003) developed numerical models to estimate the wave forces acting on a three-dimensional submerged breakwater. Romitiet al.(1988) that emphasized the combined breakwater can be defined as a structure with the vertical wall placed on a slope rubble bed. This kind of structure can keep the typical characteristicsof the vertical or slope breakwater. Kawasaki (1999) proposed a numerical model to simulate wave deformation and wave breaking based on the SOLA-VOF method originally developed by Hirt and Nichols (1981).

Breakwater Design and Construction Specifications (JTS154-1-2011) has classified a breakwater constructed in water deeper than 20 m as the deep-water breakwater. Having characteristics and advantages of both the slope and the vertical breakwater, the combined structure is often found in deep water areas with high waves. The Busan Port breakwater was built in 60 m deep water, with 35 m wide slope foundation and 30 m vertical caisson (Su and Qu, 2011). With the development of construction technology, deepwater breakwaters become common, such as the 38 m Ofunato Port breakwater, the 42 m Shimoda Port breakwater, the 50 m Portugal Sine Port breakwater, the 30 m Shanghai Yangshan Port dike, and the 50 m Daishan breakwater (Yu and Su, 2012).

In deep water areas, it is of great importance to calculate wave loads on structures. Wang (2010) pointed out that more attention should be given to the deepwater embankment because of the powerful wave forces and wave reflection. In Breakwater Design and Construction Specifications (JTS154-1-2011) it is stated that the design principles, methods and construction requirements of breakwaters in deep water is different from those in shallow water; however, no wave force calculation methodsare recommended for deep water breakwaters.

The Sainflou’s, Goda’s, the modified Goda’s formulas and the method in the Code of Hydrology for Sea Harbor (TJ213-98) are the commonly-used methods for wave force estimation for shallow water breakwaters. Goda’s formula has been used in many countries after it was modified in 1975 (Su and Wang, 2003). The British used to apply the Sainflou’s formula in the their standard before it was modified, while the new version of the standard, BS6439, adopted the Goda’s formula. The modified Goda’s formula is applied to open foundation bed with big width base shoulder in various situations, strong wave pressure being calculated using the method in Code of Hydrology for Sea Harbor (TJ213-98). About the wave force calculation in deep water. Zhanget al.(2010) referred to the irregular wave experiment data for deep water mixture basement, in which the embankment body was caisson, the water depth in front of the embankment was 20.6 m,H1/3=8 m, andT1/3=9-13 s. WhenH1/3/dwas larger (H1/3is the significant wave height anddis the water depth), the calculated horizontal wave force was overestimated by Goda’s formula. Unsalan and Gurhan (2005) pointed out that the Goda’s formula has been a good estimator for the maximum pressure values in all cases. Wang (2010) showed that measured data given consistent results by the Chinese standard formula and the Japanese Goda’s formula when standing waves exist and waves are breaking in front of deep water vertical breakwater. After wave breaking, the measured data are significantly greater than the calculated results by the Chinese standard formula and Japanese Goda’s formula. As for the wave force calculation in mixed deep water embankment on middle foundation bed, no definite conclusion has been drawn on selecting a better calculation method.

In order to study the feasibility of the Sainflou’s, Goda’s, the modified Goda’s formula and the method in the Code of Hydrology for Sea Harbors in estimation of wave loads for combined deep water breakwaters, the wave forces were calculated for four different water depths and three different wave heights in this study. An experimental test was carried out to obtain the wave loads on the vertical wall of the combined breakwater. The calculated results are compared with the measured data. Among the above-mentioned four methods one is suitable for combined deep water breakwater.

2 Commonly Used Breakwater Wave Force Calculation Methods

2.1 Goda’s

Goda (1974) developed an empirical formula to estimate non-breaking and breaking wave pressures on vertical walls, which has been widely used in Japan for the design of vertical caisson type breakwaters and is one of general methods for calculating breakwater wave force. It is mainly applied in standing wave and breaking wave statess. Fig.1 shows the wave pressure distribution by the Goda’s formula, wheredis the water depth in the position 5Hoff the foundation,d1is the height from the top of the foundation to the still water surface,d2is the water depth above the foundation armour,hcis the height of vertical wall above the still water surface,His the wave height,Psis the wave pressure at the still water surface, andPbis the wave pressure at the vertical wall bottom.

Vertical incidence：η* = 1.5H.

η* is the height of zero pressure point above the still water surface.

The total horizontal wave force is：

Fig.1 Wave pressure distribution by Goda’s formula.

2.2 Sainflou’s

Sainflou (1928) proposed a theoretical method for calculating the dynamic pressures due to non-breaking waves on vertical walls. Experimental observations by Rundgren (1958) indicated that the Sainflou’s method may significantly overestimate the non-breaking wave force, particularly for steep waves. Wave pressure distribution by the Sainflou’s formula can be found in Fig.2, where

h0is the ultra height over the still water surface;Hiisthe wave height;kis the wave number;Lis the wave length.

Wave pressure on the vertical wall at the foundation armour is：

Wave pressure at the still water surface is：

Wave pressure at the vertical wall bottom can be expressed as：

The total horizontal wave force is：

Fig.2 Wave pressure distribution by Sainflou’s formula.

2.3 The Modified Goda’s

The researchers of the Japan Harbor Research Institute considered that the strong wave pressure can be generated at the front of a vertical wall in the case of higher rubble bed and larger width of base shoulder of mixed embankment. Based on a large number of experiments, they proposed to use the modified Goda’s formula for calculating strong wave pressure (Xie, 1994). Fig.3 shows the wave pressure distribution by the modified Goda’s formula, wherePSis the wave pressure at the still water surface,Pbis the wave pressure at the vertical wall bottom,

η* is the height of the zero pressure point above the still water surface.

αI1can be obtained from Fig.4.

The total horizontal wave force without overtopping is：

Fig.3 Wave pressure distribution by the modified Goda’s formula.

Fig.4 Value ofαI1.

2.4 JTJ213-98’s

Application conditions for the specification’s formula are described in Code of Hydrology for Sea Harbor(JTJ213-98), according to which the wave state can be determined in front of the breakwater (Table 1). The wave pressure distribution by the JTJ213-98’s formula is shown in Fig.5, where,dis the water depth from the foundation bottom to the still water surface,d2is the water depth in front of the vertical wall.

Table 1 Wave state

Fig.5 Wave pressure distribution by the JTJ213-98’s formula.

Under the condition ofH/L≥1/30 and 0.2＜d/L＜0.5, the wave force is obtained as follows：

whereγis the specific gravity of water (kN m-3),Pis the total horizontal wave force in unit length.

From the four methods shown above, no straightforward correlation between wave force and deep wave depth is given.

3 Wave Load Comparison of the Results by Four Different Formulae

3.1 Conditions

In order to compare the four wave force formulae for deep water, a breakwater with four water depths (30.0, 36.0, 42.0 and 48.0 m) is examined (Table 2). The designed wave height is 6.5, 7.0 and 7.5 m and the wave period in all cases is 11.0 s.

Table 2 Wave state with 1/3＜d2/d1≤2/3

3.2 Comparison

Figs.6-8 show the calculated total horizontal wave force for different water depths and wave heights, and Fig.9 shows the distribution of wave pressure by the four formulae.

Based on the above figures, the calculated total horizontal wave force by the Sainflou’s, Goda’s formula and the modified Goda’s formula increases with water depth and wave height, but it is independent of wave state. By assuming a linear wave pressure distribution the fastest increasing rate of the Sainflou’s formula is calculated. The increasing rates of the Goda’s and the modified Goda’s formula are relatively low.

The calculated total horizontal wave force by the standard formula increases with water depth and wave height in the standing wave state, and the increasing rate by the standard formula is slightly higher than that of the Goda’s formula. While waves are breaking, the calculated total horizontal wave force increases with the decrease of water depth for the same wave height.

The calculated total horizontal wave force by theGoda’s formula is close to that by the standard formula, but the calculated results by both formulae are smaller than the results calculated by the Sainflou’s and the modified Goda’s formula.

Fig.6 The total horizontal forces by the four formulae (H=6.5 m).

Fig.7 The total horizontal forces by the four formulae (H=7.0 m).

Fig.8 The total horizontal forces by the four formulae (H=7.5 m).

Calculated by all four methods, the distribution of parapet wave pressure intensity is similar. The maximum wave pressure appears at still water level. And the wave pressure has a linear distribution from still water level to both sides and gradually decreases. Shown in Fig.9 is the calculated wave pressure distribution for a water depth of 30m and a wave height of 7.0m.

Fig.9 Pressure distribution by the four formulae.

4 Comparative Analysis Between Calculated Results and Experimental Data

The physical model tests in this paper were conducted in a wave tank which has a length of 81 m, a width of 1.4 m, and a height of 2.6 m. The wave tank was divided into two parts by a glass plate placed in the longitudinal direction, each with a width of 0.6 m and 0.8 m, respectively. The physical model tests were conducted in the part with the 0.6 m width and the reflected wave energy was reduced in the part with the 0.8 m width.

The experimental system before hybrid breakwater breastwork was set up with the four water depths 30 m, 36 m, 42 m, 48 m and three wave heights 6.5 m, 7.0 m, 7.5 m. Wave parameters of the experiments can be found in Table 3.

Table 3 Experiment wave parameters

Comparison between the calculations by the four methods and the experimental results is shown in Figs.10-12 with the percentage difference given in Table 4.

1) The total horizontal wave force calculated by the Goda’s formula for breastwork well agrees with the experimental results and the percentage difference is within10% at a water depth of less than 42 m.

Fig.10 Comparison between the calculations by the four methods and the experimental data (middle-height foundation,H=6.5 m).

Fig.11 Comparison between the calculations by the four methods and the experimental data (middle-height foundation,H=7.0 m).

Fig.12 Comparison between the calculations by the four methods and the experimental data (middle-height foundation,H=7.5 m).

2) The total horizontal wave force calculated by Code of Hydrology for Sea Harbor (TJ213-98)’s formula is less than the experimental results in the standing wave case and the percentage difference is within 10%. The calculated incrassation ratio between wave height and water depth changes from less than 1% to greater than 15% of the experimental results in the breaking wave case.

3) The percentage differnce between the calculations by the Sainflou’s formula and the experimental results is larger and varies from 22% to 40%. Because the modified Goda’s formula considers the influence of foundation shoulder width and wave impact load, the percentage differnce of the total horizontal wave force between the calculations and the experimental results is from 8% to 24%. Sainflou’s formula and the modified Goda’s formula had not been applied in our study to calculate wave force in mixed embankment breast wall on middle foundation bed.

Table 4 Wave force difference between the calculations by the four methods and the experimental results

5 Conclusions

As water depth increases, traditional formulae for the estimate of breastwork wave force cannot simply be applied.

For foundation bed parapet wave force estimates for the deepwater mixed embankment, the Goda’s modified wave pressure formula applies to the case of a water depth of less than 42 m; the method is suitable for the standing wave case.

For wave force estimates of breastworks on the foundation bed of mixed embankment, the Sainflou’s and the modified Goda’s wave pressure formulas are no longer applicable.

Acknowledgements

This work is supported by the Shandong Sci-tech Development Plan (Item No. 2008GGB01099).

Goda, Y., 1974. New wave pressure formula for composite breakwater.Coastal Engineering Proceeding, 1 (14)： 1702-1720.

Goda, Y., 1985.Random Seas and the Design of Maritime Structures. University of Tokyo Press, Tokyo, 42-56.

Hirt, C. W., and Nichols, B. D., 1981. Volume of fluid (VOF) method for the dynamics of free boundaries.Journal of Computational Physics, 39 (1)： 201-225.

Hur, D.-S., and Mizutani, N., 2003. Numerical estimation of the wave forces acting on a three-dimensional body on submerged breakwater.Coastal Engineering, 47 (3)： 329-345.

Kawasaki, K., 1999. Numerical simulation of breaking and postbreaking wave deformation process around a submerged breakwater.Coastal Enginerring Journal, 41 (3-4)： 201-223.

Ministry of Transportation, 1998.Code of Hydrology for Sea Harbor (JTJ213-98). People’s Communication Press, Beijing, 52-66.

Ministry of Transportation, 2011.Breakwater Design and Construction Specifications (JTS154-1-2011). People’s Communication Press, Beijing, 23-27.

Romiti, C., 1988. Italian hybrid breakwater experience.Port Engineering Technology, 51： 11-18.

Rundgren, L., 1958.Water Wave Forces. Bulletin No. 54, Royal Institute of Technology, Division of Hydraulics, Stockholm, Sweden, 112-125.

Sainflou, M., 1928.Treatise on Vertical Breakwaters. Annales des Ponts et Chaussees, Paris, 45-75.

Su, N., and Wang, M. Y., 2003. Application of goda formula for wave pressure in british standard.China Harbour Engineering, 1： 18-22.

Su, Y., and Qu, Y., 2011. Progress in methods for calculating wave loads on breakwater crown wall in deep water.Coastal Engineering, 3： 43-48.

Unsalan, D., and Gurhan, G., 2005. A comparative study of the first and second order theories and Goda’s formula for waveinduced pressure on a vertical breakwater with irregular waves.Ocean Engineering, 32 (17-18)： 2182-2194.

Wang M. R., 2010. Inquiry on design method for deep-water breakwater.Port Engineering Technology, 47 (3)： 1-7.

Xie, S. L., 1994. Latest advances in deepwater breakwater.Port Engineering Technology, 2： 1-10.

Yu, D. Y., and Su, Y., 2012. Study on calculation method for wave loads on deep water breakwater crown wall.Periodical of Ocean University of China, 42 (1-2)： 136-140.

Zhang, S. S., Zhang, X. W., and Li, Y. B., 2010. Progress in the research of irregular wave force on vertical wall.Port Engineering Technology, 19 (5)： 79-86.

(Edited by Xie Jun)

(Received May 23, 2013; revised July 2, 2013; accepted April 7, 2015)

? Ocean University of China, Science Press and Springer-Verlag Berlin Heidelberg 2015

* Corresponding author. Tel： 0086-532-66786009 E-mail： dyyu01@ouc.edu.cn