1   Introduction

Ports are considered to be the most important transit locations to carry out the world trade through seaways, which need to be protected from the disturbances due to the incoming waves. A seaport plays an important role in the sector of sea transportations, exports, imports, tourism, and travel, and thus, it is an important ingredient of economic growth. Water transport is a major economy input for a nation with over 82% of world trade in tons and 94% of world trade in tons-kilometers which are moved by shipping and thereby through ports.

The basic requirement of any port, harbor, or marina is a sheltered area free from the sea waves. In the coastal areas where natural protection from sea waves is not available, the development of harbor requires artificial protection for the creation of calm areas. Large structures such as rubble mound breakwaters or vertical wall breakwaters were constructed in major harbors and nearer to other marine structures where more tranquility conditions are essential for its sustainability. Breakwaters are also provided in the lagoon and at the entrance channel of ports, for maneuvering of ships and port operations. Sometimes, they are used to protect beaches from erosion due to the destructive wave forces.

Recent advances in science and technology led to the development in various types of composite type breakwater structures with different layouts and using modern construction techniques. The design of breakwater depends on the wave conditions, locality, and their appearance. As a result, variety of breakwater structures has been evolved in the past few decades. Innovative concepts for types of breakwater are in progress as a measure for calming down the destructive effect caused by sea waves. Semicircular breakwater is a composite structure first developed in Japan at the beginning of the 1990s and first adopted for the formation of the harbor in Miyazaki Port, Japan. Xie et al. (2006), developed the concept of QBW, and has conducted studies comparing the performance of semicircular and QBW. Quarter circle breakwater (QBW) has various advantages such as lightweight and cheaper to construct. It is also found to be favorable in all soil conditions, can be easily transported and placed, provide better esthetic view, sustainable, and can withstand severe wave conditions.

Recent studies in this field are related to the effect of earthquake and tsunami waves on the stability of the breakwater foundation. Chaudhary et al. (2017) investigated various effective reinforcement techniques for the foundation of the breakwater in order to provide resiliency to overcome earthquake and tsunami. It was found that the duration and magnitude of acceleration of earthquake loadings had significant impacts on the settlement and horizontal displacement of the breakwater. Chaudhary et al. (2018a, b) evaluated the effectiveness of the reinforced foundation by conducting a series of shaking table tests and hydraulic model tests, and comparisons were made between the conventional and reinforced foundations. Numerical analyses were also performed to clarify the mechanism of the soil–breakwater–reinforcement–fluid system. Chaudhary et al. (2018a, b) carried out physical model studies using geo-grid, gabions, and sheet piles for reinforcing a foundation for reducing the damage of the breakwater brought by the earthquake and tsunami. Chaudhary et al. (2019) conducted studies on the development of reinforcing countermeasures for a breakwater foundation that can produce a resilient breakwater against earthquakes and tsunamis, such as foundations reinforced with sheet piles and gabions. Physical model tests were carried out for scaled-down breakwater models to examine the performance of the reinforcing countermeasures under an earthquake and tsunami.

The present study investigates the stability of an emerged seaside perforated quarter circle breakwater with varying percentage of perforations, for different water depths and wave conditions. The major objectives include the study of the effect of incident wave steepness, S/D (spacing between perforations/diameter of perforations) ratio or percentage perforations, and water depth on the stability of quarter circle breakwater with varying seaside perforations.

2   Experimental Investigations

The experimental studies were carried out in a 2D wave flume existing in the Marine Structures Laboratory under Applied Mechanics and Hydraulics Department, National Institute of Technology Karnataka, Mangalore, India (Hughes 1993).

2.1   Wave Flume and Instrumentation

The physical model study for regular waves was conducted in a 2D wave flume having a length of 50 m long, width 0.71 m, and depth of 1.1 m (refer Figure 1). Ranges of wave height varying from 0.08 to 0.24 m and periods from 0.8 to 4.0 s can be generated with this facility (Shi et al. 2011).

Figure  1  Longitudinal section of wave flume

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2.2   Breakwater Test Model

The model for the quarter circle breakwater was constructed as two parts: the topmost part is a caisson having a quarter-circular shape and a bottom part consisting of base slab constructed with precast concrete placed on rubble mound foundation (Luwen et al. 2013).

Quarter circular caisson is made-up of the galvanized iron (GI) sheets having 2 mm thickness and is joined together. For the experimental study, physical models with geometrical similarity having a scale of 1:30 are used to simulate the present wave conditions of the Mangalore coast and suitable size of perforation in accordance with the Froude's law.

Based on the wave conditions and water depth available for the present study, QBW having a radius of 0.55 m radius was selected so that there will be no wave overtopping and the model serves as the emerged type QBW under all adverse conditions. Stiffeners are used for fixing the breakwater model and the base slab. The perforations are made in the quarter circle shaped front part of the QBW model, with the help of the drilling machine. The typical cross-section of QBW considered for the study is shown in Figure 2.

Figure  2  Typical cross-section of perforated QBW

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3   Methodology

Rubble mound foundation having a slope 1:2 and thickness 0.05 m was laid with stones having weight 50 to 100 g and the QBW model after casting was placed over it. The entire QBW model with the foundation was placed at 30 m distance from the wave flap (Binumol et al. 2017).

For measuring the wave height for the incident and reflected waves, the 3 probe method proposed by Isaacson (1991) was used. The distance between the first probes from the center of the model is kept as one wavelength L, but for other probes, space between the probes was maintained as L/3. To avoid continuous reflection, a burst of five waves was generated for each trial used. The elevation of the wave surface obtained by the wave probes was recorded by the wave recorder and the voltage signals are transformed into wave heights and wave period with the help wave recorder software developed by EMCON (Environmental Measurements and Controls), Kochi, India. The list of experimental variables and the ranges used for the present study are tabulated as shown in Table 1.

The experimental investigations carried out are based on the following assumptions:

· The sea bed is rigid and horizontal and it is assumed that the sediment movement does not interfere with the wave motion and does not affect the model performance.

· The waves are periodic and monochromatic.

· There is no interference between the incident waves and the structure since the waves are generated by the flume as bursts.

· The generation of secondary waves is not taken into account.

· Reflection caused by the waves on the flume bottom or flume side walls is avoided.

· The difference between the density of freshwater and seawater is neglected.

· The frictional effects between the base slab and structure have not been accounted

4   Results and Discussions

Studies are conducted on emerged seaside perforated QBW with 0.55 m radius and different spacing to diameter (S/D) ratios having the depth of water 0.35, 0.40, and 0.45 m under varying wave conditions. Variations of minimum weight to prevent sliding for different wave conditions and structural parameters were studied, and the variations were then presented in graphical format using dimensionless parameters obtained from a dimensional analysis by Buckingham's π theorem. The stability based on sliding performance was denoted by a dimensionless stability parameter (W/γH_i²), where W is the minimum weight of the QBW required (including the additional weight) to resist the sliding per unit length of the breakwater, γ is the specific weight of water, and H_i is the incident wave height. The sliding of QBW occurs when the wave force exceeds the frictional resistance offered by the QBW. The stability of QBW against wave force increases with an increase in the total weight of the structure. Therefore, the experiment is repeated by the addition of weight in increments until the structure stops sliding.

4.1   Influence of Incident Wave Steepness on W/γH_i²

The graphs of non-dimensional stability parameter (W/γH_i²) against the incident wave steepness, Hi/gT² (H_i = incident wave height in m, g = acceleration due to gravity in m/s², and T = wave period in sec), are plotted for different values of relative water depth, d/h_s, and a constant spacing to diameter of perforation ratio (S/D).

4.1.1   Seaside Perforated QBW with S/D = 2

The variation of W/γH_i² with H_i/gT² at different water depths for QBW radius 0.55 m and S/D ratio equal to 2 was plotted as shown in Figure 3.

Figure  3  Influence of H_i/gT² on W/γH_i² for S/D = 2

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It is clear from the graphs that W/γH_i² decreases with increase in H_i/gT² for all values of d/h_s. This may be due to the reason that waves of higher steepness (lower wave period for a given wave height) interact with the QBW surface only for short interval of time causing lesser wave force. Therefore, lesser weight has to be added to prevent sliding and hence lower values of W/γH_i².

Considering all values d/h_s, W/γH_i² varies from 2.225 to 10.532 for 6.24 × 10⁻⁴ < H_i/gT² < 6.4 × 10⁻³ and S/D = 2. The maximum value for W/γH_i² observed is 10.532 for H_i/gT² = 6.318 × 10⁻⁴ and at 0.45 m water depth (d/h_s equal to 0.732). The minimum value for W/γH_i² observed is 2.225 for H_i/gT² = 6.241 × 10⁻³ and at water depth equal to 0.35 m (d/h_s = 0.569).

4.1.2   Seaside Perforated QBW with S/D = 2.5

When S/D ratio is increased to 2.5, the observed values for W/γH_i² vary from 2.110 to 10.269 for 6.24 × 10⁻⁴ < H_i/gT² < 6.4 × 10⁻³. The maximum value for W/γH_i² observed is 10.269 for H_i/gT² = 6.318 × 10⁻⁴ and at water depth equal to 0.45 m (d/h_s equal to 0.732). The minimum W/γH_i² observed is 2.110 for H_i/gT² = 6.241 × 10⁻³ and at water depth equal to 0.35 m (d/h_s = 0.569) (refer Figure 4).

Figure  4  Influence of H_i/gT² on W/γH_i² for S/D = 2.5

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4.1.3   Seaside Perforated QBW with S/D = 3

For S/D = 3, the values obtained from the experiments for W/γH_i² are observed to be varying from 2.472 to 11.223 for 6.24 × 10⁻⁴ < H_i/gT² < 6.4 × 10⁻³ (Refer Figure 5). The maximum W/γH_i² observed is 11.223 for H_i/gT² = 6.318 × 10⁻⁴ and at water depth equal to 0.45 m. The minimum W/γH_i² observed is 2.472 for H_i/gT² = 6.241 × 10⁻³ and at 0.35 m water depth.

Figure  5  Influence of H_i/gT² on W/γH_i² for S/D = 3

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4.1.4   Seaside Perforated QBW with S/D = 4

From Figure 6 showing the variation of W/γH_i² with H_i/gT² for different water depths with radius of breakwater constant (R = 0.55 m or h_s = 0.615 m) and S/D ratio equal to 4, the values of W/γH_i² are found to be varying from 2.729 to 11.668 for 6.24 × 10⁻⁴ < H_i/gT² < 6.4 × 10⁻³. The maximum W/γH_i² observed is 11.688 for wave height of 0.03 m and a wave period of 2.2 s (H_i/gT² = 6.318 × 10⁻⁴) and at water depth equal to 0.45 m (d/h_s equal to 0.732). The minimum W/γH_i² observed is 2.729 for wave height of 0.12 m and a wave period of 1.4 s (H_i/gT² = 6.2410 × 10⁻³) and at water depth equal to 0.35 m (d/h_s equal to 0.569).

Figure  6  Influence of H_i/gT² on W/γH_i² for S/D = 4

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4.1.5   Seaside Perforated QBW with S/D = 5

From Figure 7, it is observed that for S/D equal to 5 and for all values d/h_s and QBW of 0.55 m radius, W/γH_i² varies from 3.006 to 12.850 for 6.24 × 10⁻⁴ < H_i/gT² < 6.4 × 10⁻³. The highest value for W/γH_i² observed is 12.850 for H_i/gT² = 9.439 × 10⁻⁴ and at water depth equal to 0.45 m. The lowest W/γH_i² observed is 3.006 for wave height of 0.12 m and a wave period of 1.4 s (H_i/gT² = 6.241 × 10⁻³) and at water depth equal to 0.35 m.

Figure  7  Influence of H_i/gT² on W/γH_i² for S/D = 5

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For all models, it was concluded that when H_i/gT² increases, the dimensionless stability parameter decreases. Hence, for a given wave height, considering waves of long period (lower values of H_i/gT²) impart a larger force on the caisson resulting in more minimum weight and short period (steep) waves transfer low force, hence minimum structural weight. The safety due to sliding for higher values of wave force is increased by providing more weight into the QBW caisson thereby increasing the structural weight. The variation of W/γH_i² with H_i/gT² for different water depths and S/D ratio for seaside perforated QBW was summarized in Table 2.

4.2   Influence of Water Depth on W/γH_i²

From Figure 8, it is concluded that by increasing the water depth, the values of W/γH_i² will be more and the lesser value for W/γH_i² is obtained for 0.35 m water depth. If water depth changes from 0.35 to 0.40 m, the values for W/γH_i² were found to be increased by 17.30% to 31.24%. When water depth increases from 0.35 to 0.45 m, there is an increase in W/γH_i² by 31.17% to 33.28%. Therefore, it is clear that the structure is safe against sliding with minimum weight including additional weight at lower water depths. For deep water conditions, more area of the QBW will be subjected to wave action resulting in greater force and higher values of W/γH_i². Hence, for a constant S/D ratio, heavier structure is required for sliding stability in more water depths compared with lower values for water depth.

Figure  8  Influence of d/h_s on W/γH_i²

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4.3   Influence of S/D Ratio on W/γH_i²

Figure 9 shows the variation on the values of W/γH_i² for different values of S/D and H_i/gT². From dimensionless graphs, it was concluded that the values of W/γH_i² are more for higher values of the S/D ratio. When depth of water changes to 0.40 m, there will be decrease in W/γH_i² from 5.92% to 9.20%, 10.42% to 10.55%, 16.51% to 20.09%, and 14.36% to 18.04% for S/D = 4, 3, 2.5, and 2 when compared with S/D equal to 5. Similarly, when the depth of water changes to 0.45 m, the percentage of W/γH_i² reduces for S/D = 4 from 4.3% to 9.2% compared with S/D = 5.

Figure  9  Influence of S/D values on W/γH_i² for different H_i/gT²

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The percentage reduction in W/γH_i² for S/D = 3, 2.5, and 2 is 12.3% to 12.7%, 20.08% to 21.2%, and 17.8% to 18.1% with respect to S/D = 5.When values of S/D are decreased, W/γH_i² was found to lower when compared with higher values of S/D except for the QBW model having S/D = 2.

For lower values of S/D, more perforations will be affected causing the absorption of a significant amount of energy due to incoming waves, and therefore, force exerted on the caisson will be less. Hence, lesser additional weight to be added for resisting sliding stability and hence the values for W/γH_i² will be lower.

4.4   Equations Developed for Stability Parameter

The results for the experimental studies on stability characteristics for impermeable QBW for different breakwater radius at different water depths and wave conditions are combined into suitable dimensionless terms (Jiang et al. 2008). The regression analysis is done by using the Excel statistical software—XLSTAT—and the equation for the best fit curve is obtained.

Nonlinear regression analysis was conducted using the XLSTAT software with 1254 sets of data's for W/γH_i² corresponding to different values of H_i/gT², d/h_s, and S/D. Model parameters are assigned according to the limitations while selecting dependent and independent variables. For this case, W/γH_i² is the dependent variable which depends on values of H_i/gT², d/h_s, and S/D. Summary of the statistics is shown in Table 3 and the fitness of the statistics is shown in Table 4.

From the above table, it is clear that regression coefficient R² is 0.909 which means the curve obtained is the best fit curve with minimum error. The resulting equation is obtained as the output from the XLSTAT software is noted (Misra 2001).

The equation for W/γH_i² for seaside perforated QBW was obtained as follows:

Figure 10 shows the residual of the errors obtained after the regression analysis. Residual errors are obtained due to the difference of the measured results from the actual values.

Figure  10  Residuals values of W/γH_i² for perforated QBW

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Figure 11 shows the comparison between the measured and predicted values of stability parameter W/γH_i² for perforated QBW obtained using the XLSTAT software.

Figure  11  Comparison between measured and predicted values of W/γH_i² for perforated QBW

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5   Conclusions

Based on the model studies conducted on seaside perforated QBW, following are the conclusions obtained:

1) For varying d/h_s and S/D ratio, the values of W/γH_i² will be lesser for higher values of H_i/gT².

2) For different values for S/D and H_i/gT², the values for stability parameter (W/γH_i²) are found to be more for higher values of water depth when compared with lower water depth.

3) The stability parameter W/γH_i² is found to be reduced for lower S/D ratio with different values of H_i/gT² and d/h_s, but a reverse trend was obtained for seaside perforated QBW having S/D = 2. When S/D is reduced, there will be more dissipation of wave energy and QBW is subjected to less wave force resulting in less weight of QBW to prevent sliding or lesser values of W/γH_i². Further decreasing S/D to 2 causes back propagation of wave and hence exerts more force on QBW resulting in higher values for W/γH_i².

4) Nonlinear regression analysis was carried out for the various tests conducted and the equation for the best fit curve with a good regression coefficient (R² value) was found out.

5) From the experiments conducted and the comparative analysis, it is clear that the perforated QBW of S/D = 2.5 has better performance characteristics and can be adopted in the site with the favorable condition.