Performance Characteristics of an OWC in Regular and Random Waves

Abstract

Typically, the dimensions of OWC chamber have a single resonant frequency which captures efficiently wave energy over a narrow range of frequency of the waves. The high-frequency waves are reflected. Provision of a pair of additional harbour walls in front of the device creates an added resonance enabling it to capture substantial energy from a wider range of frequency of the waves. Although, there exists a substantial loss in energy due to wave deformation as the wave from the offshore approach shallow waters, in which case, the harbour walls prevent any further dissipation till it reaches the device. This chapter reports the influence of harbour walls on the hydrodynamic characteristics of a bottom - mounted OWC. The study was carried out with different configurations of harbour walls, i.e., the projecting length was varied in terms of the ratio of harbour wall length (c) to the breadth of OWC (b), (c/b = 1, 1.5 and 2). The opening angle of harbour wall was varied in the range of [π/2, 7π/8] in an interval of π/8 with respect to the front lip wall of OWC. The results reveal that the efficiency of the OWC with the harbour walls has been significantly improved. The details of such a behaviour are herein discussed.

5.1 Introduction

The concept of OWC was introduced by Yoshio Masuda in the 1940s and since then, several OWC plants have been deployed at several locations around the globe. The concept of harbour walls at the entrance of an OWC was later introduced by [10]. Theoretical studies revealed that the harbour wall concept could improve the performance of the device and arguably may reduce the cost of electricity. This concept caught the attention later, and several researchers investigated the effect of harbour walls in enhancing the performance of OWC (e.g. [8], [17]).

The inclusion of harbour walls improved the performance of OWC based on creating an added resonance to the system, thus causing the incident wavefield to amplify between the harbour area (the area between the two parallel projecting sidewalls) which is explained by the quarter-wavelength analogy in acoustics [10]. This amplification forces the incoming wave with apparent power having both active and reactive elements in it. Such wave amplification triggers the active elements present in the wave to rush into the device chamber, producing maximum oscillation of trapped water and hence the air column. The active elements define the degree of the kinetic energy of the system, whereas the potential energy of the system is controlled by the reactive elements. When the device experiences wave frequency differing from the resonance frequency of OWC, standing waves are likely to be formed in the harbour area, which results in the absorption of more power from the incoming wavefield over a wider range of frequencies. Whereas the conventional OWC chamber has a single resonant frequency that captures wave energy over a narrow band of wave frequencies.

It is noteworthy to mention, several existing studies focussed on different geometric properties of the conventional OWC device through both analytical or numerical and physical experiments e.g. [11, 19, 21, 24]. Furthermore, to absorb wave power over a wide range of wave frequencies and to increase the competitiveness of OWC with other devices, the multi-resonant oscillating water column was proposed and studied e.g. ([18], [22, 23, 25, 27]. For more comprehensive reviews on the history of OWCs, set-up and operation and theoretical optimal control and basis, readers are referred to [9, 13, 14].

Although a few full-scale prototypes OWCs have been tested till date, e.g., the Mutriku power plant in Spain [15], the LIMPET shore connected experimental plant installed at Islay, Scotland [3], U-OWC plants in the port of Civitavecchia, Italy [1], the PICO power plant in Portugal [20], there still remains a lot to be investigated on improving on the performance of OWCs. As harbour walls could play a vital role in improving the performance, for instance, by allowing larger energy of the incident wave to penetrate into the chamber the aim is to improve the performance by optimizing the geometries of harbour walls. In addition to improving the performance of the system, harbour walls can also be useful by preventing the rubbles from blocking the path of the incoming wave to the chamber (if it were to be integrated with rubble-mound breakwater structures) and could also be competent in an environment dominated by oblique waves. To the best knowledge of authors, there is no distinct efficient geometry of harbour walls (side wall length and opening angle) has been reported in the literature. This necessitates the goal of the present work, which is to optimally convert the incident wave energy into electrical power by analysing the influence of harbour walls and optimizing the projecting length and opening angle.

5.2 Experimental Investigations

5.2.1 OWC Model

A scale ratio of 1:20 as per Froude’s law was considered for the present study considering the wave characteristics that could be achieved in the experimental test facility. The scale effects on air compressibility in scaled-down model experiments related to OWCs were emphasized by [26] as the fact that air compressibility cannot be accurately represented by the geometrical scaling of the device. In addition, the exact dynamic similarity of the air-compressibility effect in the chamber requires the atmospheric pressure (or room pressure) during the model testing to be much smaller than the atmospheric pressure at full scale as proposed by [12]. This may, however, not be practically feasible.

The geometric dimensions of the OWC model were adopted based on the various optimization studies carried out by researchers, e.g. [2, 11, 19, 21, 27]. The model was fabricated with 12 mm thick acrylic material. The interior length of the water column was 540 mm, and the width was 283 mm. The interior breadth was 180 mm with a circular orifice diameter of 21 mm modelled at the top of the device. The bottom rectangular opening was fixed as 180 mm by 283 mm with a circular curve bottom profile of radius 180 mm. A detailed plan and sectional view of the model [6] with the location of all sensors is shown in Fig. 5.1.

Fig. 5.1

Plan and sectional view of the OWC model

5.2.2 Experimental Program

Although contemporary theoretical models provide valuable information, a physical model study can also classify and isolate specific problems, often system configurations, and provide a significant contribution to the future generation of models. The experimental investigation was carried out in a 72.5 m long, 2 m wide, and 2.5 m deep wave flume at the Indian Institute of Technology Madras, India. The wave flume is well-equipped with a wavemaker at one end, which can function in the piston mode or hinged mode to generate shallow water or deep-water waves, respectively. At the other end, a perforated wave absorber is equipped with the capability of absorbing the incident waves effectively. The model rig was fixed rigidly at a distance of 36 m from the wavemaker. Throughout the experimental investigation, the water depth ‘d’ and front lip wall submergence ‘s’ were kept constant at 0.312 m and 0.120 m, respectively. The illustration of the experimental setup and the position of the model and wave gauges in the wave flume is shown in Fig. 5.2. All the instruments were calibrated at every change in the harbour wall configurations for accurate measurements.

Fig. 5.2

The experimental setup

5.2.3 Harbour Walls in OWC

For illustrative purpose, experiments were carried out in two stages, with the first stage involving the identification of efficient resonant length and the second stage focussed on the opening angle of the harbour walls. The resonant length is defined in terms of c/bi.e. the ratio of harbour wall length (c) to the breadth of OWC (b). To understand the influence of harbour walls resonant length, experimental results for various c/b (1, 1.5, 2) were compared with the conventional OWC (without harbour walls). The second stage involved the identification of effective opening angle (θ) by varying its inclination in the range of [π/2, 7π/8] at an interval of π/8 with respect to the front lip wall of OWC. The selection of the wall inclinations has been adopted based on an analytical study of [8] in which the efficiency was found to be maximum for the harbour wall inclination ranging between π/2 and 3π/4. In total, 13 sets of experiments were carried out by changing the harbour wall configurations as stated above. The results from the physical model studies with models exposed to both regular and random waves were reported by the authors [4], [6]. The plan view representation of harbour wall configurations is shown in Fig. 5.3a and b, respectively.

Fig. 5.3

a Plan view representation of resonant length formations. b Definition of the orientation of the harbour walls

5.2.4 Wave Characteristics for the Study

The waves of periods varying between 1 and 2.5 s in steps of 0.1 s were considered for experimental investigation with regular waves. For each of the wave periods, three-wave heights (0.03 m, 0.06 m, and 0.09 m) were employed. The said wave characteristics covered the relative water depth, d/L of range from 0.074 to 0.23 and wave steepness, H/L of range 0.0074 to 0.065, where ‘H’ denotes the wave height and L denotes the wavelength. A similar range of wave conditions was adopted for the case of random waves defined by the JONSWAP spectrum.

5.2.5 Hydrodynamic Factors

As there are numerous terminologies to define the performance of the device, there is no consensus or standard term to designate this quantity. In a three-dimensional study, Non-dimensional Capture Width, Relative Capture Width (RCW), and non-dimensional absorption length are the terminologies used to define the performance. In the case of a two-dimensional study, the term efficiency is used in place of capture width to assess the performance. To understand a few definite variables, mostly two-dimensional studies are carried out, and efficiency played a prominent role in laboratory comparison. In addition, developing a two-dimensional numerical model is easier than three-dimensional models as it involves a number of variables. In general, both efficiency and the relative capture width are similar non-dimensional measures; efficiency is the ratio of output power to the input power for a unit width of wave crest with respect to the width of the device. As it is evaluated for the unit width of the device and unit crest of the wave, the maximum possible value it can reach is unity i.e., the maximum percentage of 100%. On the other hand, as a three-dimensional study involves total wave crest and not limited to the width of the device, the relative capture width may possess a value greater than unity. Both the measure of performance is applicable to regular and random waves. In wave energy conversion, the definition of relative capture width is unambiguous, while the same definition does not hold well in the case of efficiency. In this work, we used RCW as a measure to compare the hydrodynamic performances. For estimating the RCW, the incident wave power flux for plane progressive waves are calculated based on the relation defined according to the linear wave theory as given by,

$$incident\,power_{Reg} = \frac{{\rho gH^{2} }}{8}\frac{C}{2}\left( {1 + ~\frac{{2kd}}{{\sinh 2kd}}} \right)w$$

(5.1)

In which, celerity, C = L/T, wave number, k = 2π/L, ρ = mass density of fluid, g = acceleration due to gravity, and w = width of OWC. However, in the case of the random wave field, the time-average input power is calculated as by the expression (5.2),

$$incident\,power_{{Ran}} = w\rho g\mathop \sum \limits_{{i = 1}}^{N} (S_{i} \left( {C_{g} } \right)_{i} \nabla F)$$

(5.2)

whereas N = number of frequency bands having a width of Δf, S_i = spectral density at ‘i’th band and (C_g)_i = group velocity at ‘i’th band, which is calculated using (5.3) and (5.4).

$$\left( {C_{g} } \right)_{i} = n_{i} C_{i}$$

(5.3)

$$n_{i} = \frac{1}{2}\left( {1 + ~\frac{{2k_{i} d}}{{\sinh 2k_{i} d}}} \right)$$

(5.4)

The pneumatic output power due to the volume flux of water inside the OWC chamber is calculated by the instantaneous integration of free surface oscillation inside the chamber fluctuating with a velocity (V_w).

$$output\,power = \frac{1}{T}\mathop \smallint \limits_{t}^{{t + T}} p_{{air}} AV_{w} dt$$

(5.5)

Here, ‘A’ is the area of the OWC chamber, V_w is discretised by the first derivative of third-order approximation as given below,

$$V_{w} = \frac{{2Z_{{i + 1}} + 3Z_{i} - 6Z_{{i - 1}} + ~Z_{{i - 2}} }}{{6\Delta t}}$$

(5.6)

where, Z_iis the free surface elevation at time t_i and Δt = sampling interval.

The energy conversion performance of OWC is evaluated by the ratio of power incident on OWC to the output power.

$$Relative\,Capture\,Width\left( {RCW} \right) = \frac{{input\,power}}{{output\,power}}$$

(5.7)

A detailed methodology of the study to evaluate the performance of OWC is represented in the form of the flow chart in Fig. 5.4.

Fig. 5.4

Flow chart for a methodology for the evaluation of the performance of the OWC

5.3 Results and Discussion

5.3.1 Regular Wave Tests

5.3.1.1 Wave Amplification Factor

The performance of the OWC device mainly depends on the rate at which the compressed/decompressed air pressure exits/enters the vent, which in turn relies on the volume flux of water entering inside the chamber. The wave amplification factor (β_out) is one of the key parameters in defining the performance of the device. For illustrative purpose, hydrodynamic factors subjected to regular and random waves are differentiated by (…)_reg (…)_ran respectively. Wave amplification in between the areas of harbour walls (β_out) unswervingly enhances the volume flux of water inside the chamber (β_in). The wave amplification factor (β_out)_reg is established based on the ratio of measured wave run-up, R_u1 to the wave height, H measured from the incident wave probe (W.P1, see Fig. 5.2). The comparison of (β_out)_reg for different configurations of harbour wall (c/b) for the different relative wave heights, H/s of 0.25, 0.50, and 0.75, is depicted in Fig. 5.5a, b and c, respectively. Herein, ‘s’ is the front lip wall submergence.

Fig. 5.5

Variation of (β_out) _reg with d/L for different c/b a H/s = 0.25 b H/s = 0.50 and c H/s = 0.75

The efficiency of the OWC increases when it functions closer to the resonant frequency, and in the case where wave frequencies are away, its performance is optimised through phase control mechanisms. In this study, it is observed that the influence of harbour walls on wave amplification (in-between the harbour area) is found to be consistent for all the wave frequencies tested, which implies that even for the out of resonance condition, the performance could be improved. The quarter-wavelength analogy in acoustics explains the large amplification of incident wave field inside the harbour area due to a frequency of disturbance having a wavelength corresponding to four times the length of a parallel-sided duct with one end closed.

In the absence of a harbour wall, the variation of (β_out)_reg is uniform and does not vary much over the entire frequency range tested. The trend in its variation is observed to be similar for the different H/s as well. For all the H/s tested, the introduction of harbour walls is found to result in an increase in (β_out)_reg to a maximum extent of thrice the incident wave height. Although (β_out)_reg is found to be high for c/b of 2.0 for the least H/s tested, as H/s increases, the models with c/b = 1.5 and 2.0 exhibit almost similar results in their peak values. The increase in (β_out)_reg further leads to an increase in the volume flux of water inside the chamber, which eventually enhances the device efficiency. The observed trend also reveals that H/s has less influence on (β_out)_reg for different harbour wall configurations over the entire frequency range tested.

The amplification in front of the device is also due to the interaction of reflected waves from the front lip wall of OWC and diffracted waves from the harbour walls with the incident wave field. The wave amplification factor (β_in)_reg inside the water column for different c/b is shown in Fig. 5.6 and compared with different H/s i.e., (a) H/s = 0.25, (b) H/s = 0.50, and (c) H/s = 0.75. Though the wave amplification inside the water column is measured at both the rear wall, and behind the lip wall of OWC, due to linear variation, both the measured data showed a similar quantity and hence the results pertaining to the rear wall measurements are plotted in the above figure.

Fig. 5.6.

Variation of (β_in) _reg with d/L for different c/b a H/s = 0.25 b H/s = 0.50, and c H/s = 0.75

When the incoming wave frequency entering the chamber is out of resonance with the natural frequency of the water column, phase variation between the rear and lip walls will be a maximum. This results in a resultant sub-harmonic component due to the sum of propagating waves in the opposite direction, which will lead to the circulation of pneumatic pressure inside the chamber, leading to stalling of the turbine. It results in an increase in the horizontal water particle motion, whereas the relative capture width (RCW) is directly proportional to the vertical motion inside the water column. Theoretical studies also specified that the dominant reflections from the walls of OWC could be partial standing waves, as reported by [11].

The above results clearly illustrate that the presence of harbour walls enhances the volume flux of water inside the chamber. On comparing the variation of (β_in)_reg as a function of d/L for different H/s, the wave amplification factor is higher for low-frequency waves and less for higher frequency waves due to reduced run-up inside the chamber similar to the variation observed for (β_out)_reg. However, as H/s increases, the (β_in)_reg for all the c/b decreases gradually, and for d/L > 0.16, it tends to be less than unity for OWC and c/b = 1.0. This shows that H/s has significant influence over (β_in)_reg. The least amplification has been observed for the short period waves incident on the OWC with the least c/b of 1.0. The wave amplification results paved the way to the understanding of the device configuration with optimum harbour wall projected length to have maximum volume flux of water.

Though the magnitude of (β_in)_reg is found to be a maximum for c/b of 2.0, the rate of change of volume flux of water plays a key role in the energy conversion process from hydrodynamic energy to pneumatic. The velocity with which the water oscillates inside the water column is established based on the relation, Eq. (5.6). Each of the frequency components contributes to the volume flux of water. However, when the wave frequency close to the natural frequency of the OWC, a larger flux of water with a maximum velocity is likely to happen.

5.3.1.2 Pneumatic Pressure

The analysis of the pneumatic pressure developed inside the chamber is another important factor in dictating the performance of the OWC. The rate at which pressure enters/exits from the air hole depends on the volume flux of water inside the water column. The top plate with an air hole of a/A = 0.68% acts as a damper which allows the oscillating water to compress air at the upper end of the chamber. This air is then forced out through a small orifice which drives the turbine to generate clean energy. Essentially, this is an air-flow exchange between the air in the pneumatic chamber and the atmosphere outside of the OWC chamber. The volume flux of water inside the chamber acts as a piston by the dynamic interaction and produces a bi-directional airflow through the orifice on the top of OWC. The issue of scale effect, particularly on-air compressibility, has been discussed briefly in Sect. 1.2, however, with careful attention, all the measurements were carried out [5].

A pressure transducer mounted on the top of the OWC model registered the pneumatic pressure developed inside the chamber. The velocity at which the water oscillates inside the column and pneumatic pressure measured is used to calculate the hydrodynamic power developed within the conversion chamber. The dimensionless air pressure (P_air)_reg due to the action of regular waves is based on the relation of measured air pressure (p_air)_reg inside the OWC chamber to the pressure head due to the incident wave as given in Eq. (5.8),

$$\left( {P_{{air}} } \right)_{{reg}} = \frac{{p_{{air}} }}{{\gamma H}}$$

(5.8)

wherein, γ is the specific weight of the fluid. The effect of c/b on the variation of pneumatic pressure inside the air chamber as a function of d/L for different H/s is presented in Fig. 5.7.

Fig. 5.7

Variation of (P_air)_reg with d/L for different c/b a H/s = 0.25 b H/s = 0.50 and c H/s = 0.75

The air pressure associated with wave amplification inside the chamber is greatly influenced by the harbour walls. In the absence of harbour walls, the (P_air)_reg is found to be the minimum for all the H/s tested. This is due to the fact that there exists dissipation of a large amount of incident wave energy in the system as the model does not extend over the entire width of the flume. The trend in the variation of normalized air pressure is influenced and varies as H/s increases. The introduction of the harbour walls creates an added resonance to the system and also confines the wave energy leak, which otherwise is expected to be more in the absence of the walls. Further, the results of the wave amplification show that the harbour wall increases the wave amplification in between them and thereby increases the volume flux of water inside the chamber, which influences the air pressure.

The (P_air)_reg increases with an increase in c/b up to 1.5, and a further increase in c/b to 2.0 does not have a significant influence on (P_air)_reg for high-frequency waves. However, (P_air)_reg is found to decrease gradually towards high-frequency waves. This trend in variation in (P_air)_reg is observed for all the H/s tested. The peak magnitude of (P_air)_reg for all H/s is found to follow a similar trend which substantiates that the device resonates closer to the natural frequency as designed. Further, for each of the configurations of harbour walls, the peak magnitude shifts towards low frequency as the natural frequency of the water column changes. This behaviour indicates the frequency dependence nature of the system and it is not influenced by the addition of harbour walls.

5.3.1.3 Pressure Outside OWC

The pressure acting on the front lip wall of OWC due to non-breaking regular waves and the influence of harbour walls on the front lip wall pressure are discussed in this section. The pressures on the lip wall induced by the incident waves can be a combination of the static pressure forces, the dynamic pressure forces, and the shock forces or impact pressures. The structure exposed to unbroken waves, a standing wave will be generated seaward of the wall, in which case only the static and dynamic forces exist, which can be determined from linear wave theory [7]. Though impact pressure forces are of short duration (of the order of tenths of a second) and consequently affect the dynamic response of the structure rather than its static equilibrium, the pressure due to non-breaking waves is equally important for the understanding of the cyclic loading experienced by the OWC which could result in fatigue failure. The results pertaining only to non-breaking waves are discussed here.

The non-dimensional pressure outside the lip wall (P_out)_reg is established based on the ratio as given in Eq. 5.9.

$$\left( {P_{{out}} } \right)_{{reg}} = \frac{{p_{{out}} }}{{\gamma H}}$$

(5.9)

The influence of harbour walls on pressure variation outside the lip wall subjected to regular waves for different H/sis plotted and depicted in Fig. 5.8.

Fig. 5.8

Variation of (P_out)_reg with d/L for different c/b a H/s = 0.25 b H/s = 0.50 and c H/s = 0.75

When a vertical impermeable wall with a bottom opening to the water column obstructs a major portion of the kinetic energy in the waves is reflected, and in addition, some spent in the wave run-up over the lip wall, and some are spent due to propagation into the water column. The upward component of the energy over the wall can result in the wave crests to rise to double the incident wave in deeper waters. However, the harbour walls create an added resonance, which results in further amplification of the waves, thereby, increases the pressure acting on the lip wall. For OWC, the non-dimensional pressure is of the order of 1.5 in the low-frequency zone and slightly reduces towards the higher frequencies for H/s of 0.25. As H/s increases (P_out)_reg reduces to the order of unity and does not fluctuate throughout the entire frequency range. Introducing harbour walls is found to significantly influence (P_out)_reg due to the additional resonance. According to the fundamental principle, the pressure would be more for low-frequency waves. Similar trends in variation are observed, as can be seen from the results in the earlier figure. Increasing the resonant length of harbour walls from c/b = 1.0, 1.5, and 2.0 further increases (P_out)_reg towards the low-frequency zone. However, as H/s increases, the variation in magnitude for low-frequency waves decreases gradually, whereas, for the high-frequency waves, the variation is insignificant.

5.3.1.4 Pressure Inside OWC

The influence of harbour walls on the pressure variation inside the OWC subjected to regular waves for different H/s are superposed in Fig. 5.9. The non-dimensional pressure inside OWC (P_in)_reg is established based on the relation given in Eq. 5.10.

$$\left( {P_{{in}} } \right)_{{reg}} = \frac{{p_{{in}} }}{{\gamma H}}$$

(5.10)

Fig. 5.9

Variation of (P_in)_reg with d/L for different c/b a H/s = 0.25 b H/s = 0.50 and c H/s = 0.75

The variation of (P_in)_reg is directly related to (P_out)_reg as its variation is a function of (P_in)_reg Similar to (P_out)_reg, the magnitude of variation due to the harbour wall’s resonant length is larger at the low-frequency zone and marginal for high-frequency waves for H/s of 0.25. Further, as H/s increases, the variation of (P_in)_reg due to harbour walls gradually reduces for the entire frequency range tested. On comparing with the results of (P_out)_reg, the overall magnitude of (P_in)_reg is slightly less than (P_out)_reg. This is due to the fact that, as most of the energy is spent in reflection and run-up of the wave along the seaside of the lip wall, less energy flows into the OWC chamber.

5.3.1.5 Effect of Wave Steepness

The relative capture width due to regular waves, RCW_reg is established based on the ratio of output power in terms of air pressure to the incident wave power as per Eq. (5.7). The incident wave power flux for plane progressive waves is calculated based on Eq. (5.1), and output power is calculated using Eq. (5.5). In order to analyse the effect of wave steepness for different configurations of harbour walls, RCW_reg is plotted against d/L for different H/L ranges of 0.007–0.016, 0.018–0.027, 0.030–0.037, and 0.041– 0.065, as shown in Fig. 5.10a–d, respectively. The steepness limits are considered in the operational limit of wave energy device, i.e., above small steepness wave (H/L > 0.0062) and below the limit of very steep waves (H/L < 0.1) in the intermediate water range.

Fig. 5.10

Relative capture width RCW_reg as a function of d/L for H/L a OWC b c/b = 1.0 c c/b = 1.5 and d c/b = 2.0

The hydrodynamic RCW_reg for OWC without the harbour walls is found to steadily increase from low frequency to high-frequency waves up to the resonance period of the device. The wave steepness (H/L) does not influence the RCW_reg significantly over the intermediate water depth range. The addition of a pair of harbour walls in front of the device greatly influences the RCW_reg. The RCW_reg of the convertor with harbour wall of c/b = 1.0 is found to increase up to d/L of about 0.14. An increase in c/b to 1.5 exhibits a significant increase in RCW_reg about twice the incident wave power near the resonance range of 0.12 < d/L > 0.14. A further increase of c/b to 2.0 shows a decrease in the value of RCW_reg compared to c/b = 1.5. For all the configurations of the harbour walls, RCW_reg is found to be less than 0.6 for d/L < 0.05. Among the three c/b tested, RCW_reg is found to be a maximum for c/b = 1.5. Towards high-frequency waves, RCW_reg is found to reach an asymptotic value of more than 1.0.

5.3.1.6 Effect of Wave Frequency

The hydrodynamic RCW_reg as a function of d/L for different c/b including the case for without the harbour walls, OWC for H/s = 0.25, 0.50, and 0.75 are superposed in Fig. 5.11a, b, and c respectively. A comparison of RCW_reg for different configuration of harbour walls for a close range of wave conditions, as stated above, enables to identify the efficient system. The effect of wave frequency on each configuration also helps in locating the peak RCW_reg, and its variation with respect to d/L can be related to the natural frequency of the oscillating water column.

Fig. 5.11

Variation of RCW_reg with d/L for different c/b a H/s = 0.25 b H/s = 0.50 and c H/s = 0.75

The trend in the variation of RCW_reg with d/L for different c/b is found to be similar to RCW_reg found to rapidly increase up to d/L = 0.142 for both c/b = 1.5 and 2.0. However, the difference between these two c/b being marginal. The said trend in variation is observed to be similar for the three H/s tested. The RCW_reg is found to be a maximum of 2.3 for the least H/s of 0.25, and the corresponding peak is found to be 2.0 and 1.6 for H/s = 0.50 and 0.75 respectively. An increase in c/b to 2.0 has shown a decrease in RCW_reg, which proves that there is an optimum c/b for maximum RCW_reg to be at its peak. For the long period waves, d/L < 0.142, although RCW_reg is found to be higher for c/b = 2.0, its rate of increase is found to be marginal for all H/s. Furthermore, for H/s = 0.25, the variation exhibits a secondary peak in RCW_reg for all c/b corresponding to d/L = 0.173 in the laboratory model scale. This secondary peak is not observed for higher H/s.

As long waves are incident on the OWC, a greater part of the energy is also likely to be reflected, and the less amount of energy entering the chamber contributes to the air pressure that leads to a reduction in RCW_reg. The chamber oscillation does not amplify from the incident wave surface profile in this case. Further, the harbour walls have facilitated the concentration of energy for higher frequency waves by focussing on the OWC. Similar to the trend observed from the results of (P_air)_reg, the peak of RCW_reg is found shifting towards low frequency for all the H/s. This peak shift indicates that as the natural frequency of the water column changes due to the resonant length of the harbour walls its corresponding resonance also varies. This again implies that the device takes in energy closer to the natural period of the water column. Furthermore, the OWC system still relies on the wave frequency, and the addition of harbour walls alters the natural frequency of the water column.

The natural frequency of the water column is established based on the relation given as,

$$T_{n} = 2\pi \sqrt {\frac{l}{g}}$$

(5.11)

wherein, T_n = natural frequency of water column in secs and l = length of flow line (outside to inside the water column). According to this relation, the length of the flow line of water particles determines the natural frequency of the water column. The theoretical natural frequency established for the models are tabulated and shown in Table 5.1. The peak RCW_reg for each of the configurations of harbour walls and OWC compares well with the theoretical natural frequency of the water column estimated.

Table 5.1 Theoretical natural frequency (T_n) for different harbour walls

5.3.1.7 Percentage of Increase

For the purpose of quantifying the amount of increase in RCW_reg for OWC with harbour walls compared that for without the harbour walls, defined as compared for the three c/b for H/s = 0.25, 0.50, and 0.75 are projected in Fig. 5.12a–c. As the said percentage increase is found to be negligible for d/L < 0.09, the results for the percentage increase is calculated for d/L > 0.09 alone are plotted. These results clearly demonstrate the effectiveness of harbour walls.

Fig. 5.12

a Percentage increase in RCW_reg as a function of d/L for H/s = 0.25. b Percentage increase in RCW_reg as a function of d/L for H/s = 0.50. c Percentage increase of RCW_reg as a function of d/L for H/s = 0.75

The above results imply an increase in RCW_reg thus, proving that harbour walls enhance the performance of the system. The harbour walls of c/b = 1.0 show a considerable increase in the RCW_reg for all the wave frequencies and for all the H/s tested. Increasing the harbour walls to c/b1.5 and 2.0 shows a tremendous increase in RCW_reg to an extent of about 70% towards low-frequency waves and about 40% for high-frequency waves. However, the increase for c/b = 1.5 is found to be a maximum and is consistent for the entire d/L range, particularly closer to the natural frequency region. This proves that there is an optimum resonant length of the harbour wall. Even though there is only a marginal variation in the RCW_reg for c/b = 1.5and 2.0, the optimization should also consider other parameters such as the cost of construction, maintenance, workability etc. In such a case, the cost of additional length with only a marginal variation of RCW_reg indicates that c/b = 1.5can be adopted as an efficient resonant length of the harbour wall.

5.3.1.8 Phase Difference

In the OWC system, generally, the phase variation exists between the incident wave and the volume flux of water as it is separated by the front lip wall with a bottom opening. It also leads to phase variation between the pneumatic pressure and the pressure at the front lip wall of the OWC. The instruments being fixed at a certain horizontal space in the direction of wave propagation, there exists a default phase between all measurements. The phase variation between outside and inside the OWC is important as it qualifies the energy conversion capacity of the system. As the quantity of power captured by an OWC due to the flow due to incident waves is controlled by the amplitude as well as the phase of the oscillation, the phase between the wave oscillations outside and inside the OWC should be as small as possible to achieve maximum performance.

The phase difference, ϕ between the wave amplification outside and inside the OWC is computed based on the relation,

$$\phi = 360 \times \frac{{\Delta T}}{T}$$

(5.12)

$${\varPsi }\left( {\Delta T} \right) = \left\langle {R_{{u1}} \left( t \right)} \right.\left. {R_{{u3}} \left( {t + \Delta T} \right)} \right\rangle$$

(5.13)

In which, ${\varPsi }(\Delta T)$ is the absolute maximum phase lag is obtained from the cross-correlation between wave amplification outside (R_u1) at the front lip wall of OWC and the wave amplification inside (R_u3) at the rear wall of OWC. The phase difference between outside and inside OWC for a typical wave condition i.e., T = 1.5 s and H = 0.06 m for all the models tested in the first stage as a function of d/Lis depicted in Fig. 5.13.

Fig. 5.13

Phase difference between outside and inside OWC

From the above result, it is explicit that the phase difference is found to vary linearly with respect to the wave frequency for the OWC models tested. Further, the natural frequency of the water column is observed to have the least influence over the phase variation even if the flow of streamline of water particles varies due to the variation in the harbour wall resonant length. On examining the phase variation between the R_u2 and R_u3, both being measured inside the water column, ϕ showed negligible variation as the volume flux varies simultaneously inside the water column.

5.3.2 Random Waves

5.3.2.1 General

The study on the behaviour of models using random waves is crucial as it increases the reliability of the system performance under realistic sea states. In the laboratory, the random wavefield with pre-defined spectral characteristics is generated through a wavemaker. Though ocean waves are multidirectional, only unidirectional waves are considered for the present study. The response of the harbour walls integrated with OWC subjected to oblique waves will be considered in the future.

5.3.2.2 Wave Amplification Factor

Similar to regular waves, the non-dimensional wave amplification factor due to random waves (β_out)_ranis established based on the ratio of significant R_u1 to the significant wave height (H_s) as given in Eq. 5.14.

$$(\beta _{{out}} )_{{ran}} = \frac{{\left( {R_{{u1}} } \right)_{{m0}} }}{{H_{s} }}$$

(5.14)

The influence of harbour walls on (β_out)_ran due to different resonant lengths of harbour walls for different H_s/s are shown in Fig. 5.14a–c correspondingly.

Fig. 5.14

Variation of (β_out)_ran with d/L_p a H_s/s = 0.25 b H_s/s = 0.50 and c H_s/s = 0.75

From the results of(β_out)_ran, it is observed that for OWC (without harbour walls), the variation of (β_out)_ran is not fluctuating throughout the entire d/L_p range examined. However, the introduction of harbour walls has amplified the incoming waves to a greater extent by increasing the resonant length which, the chamber oscillation is further amplifies the (β_out)_ran. For H_s/s = 0.25, the amplification shows a drastic change in the low-frequency region for d/L_p between0.083 and 0.109, whereas, towards high frequency, the variation is observed to be marginal. In contrast, as H_s/s increases to 0.50 and 0.75, the amplification gradually reduced, and the variation is identical towards the entire d/L_p range.

The non-dimensional wave amplification factor inside the water column subjected to random waves (β_in)_ran is established based on the ratio of significant run-up measured from R_u3 to the significant wave height (H_s) as given in Eq. 5.15.

$$(\beta _{{in}} )_{{ran}} = \frac{{\left( {R_{{u3}} } \right)_{{m0}} }}{{H_{s} }}$$

(5.15)

The influence of harbour walls on (β_in)_ran due to different resonant lengths of harbour walls for different H_s/s is brought out in Fig. 5.15a– c, respectively.

Fig. 5.15

Variation of (β_in)_ran with d/L_p a H_s/s = 0.25 b H_s/s = 0.50 and c H_s/s = 0.75

As the calculation of total power output from the OWC involves the volume flux of water and pneumatic pressure, the study of wave amplification due to real field condition is essential for the present study. On comparing the results of (β_in)_ran with that of (β_out)_ran, both the overall magnitude inside the water column is less than the wave amplification outside the water column, especially towards the high-frequency zone which is similar to the results of (β_in)_reg. Unlike the trend observed in (β_out)_ran that increasing the resonant length increases the amplification, (β_in)_ran increases up to c/b = 1.5 and beyond, which decreases the volume flux of water. This trend in variation of (β_in)_ran is found to follow for all the H_s/s. However, the influence of H_s/s on (β_in)_ran reveals that the magnitude of flux reduces as H_s/s increases, particularly for low-frequency waves which shows a drastic reduction in volume flux, whereas a considerable reduction is observed for high-frequency waves. Based on the above results, the overall wave amplification is significantly influenced due to the presence of the harbour walls.

5.3.2.3 Pneumatic Pressure

The variation of normalized pneumatic pressure (P_air)_ran due to random waves as a function of d/L_p for varying projecting length of harbour walls is shown in Fig. 5.16.

Fig. 5.16

Variation of (P_air)_ran with d/L_p. a H_s/s = 0.25 b H_s/s = 0.50 and c H_s/s = 0.75

Unlike (β_in)_ran and (β_out)_ran, (P_air)_ran resonates differently for varying significant wave height. From the results for different H_s/s, it is observed that (P_air)_ran for OWC (without harbour walls) is minimal compared to the presence of harbour walls. The trend in variation of (P_air)_ran appears to be similar for H_s/s 0.25 and 0.50, whereas, for H_s/s = 0.75, due to an increase in the nonlinearity of waves, a slight difference in its trend is observed. The rate of increase in (P_air)_ran by increasing the length of harbour walls from c/b of 1.5 and 2.0 is found marginal.

The pneumatic pressure resonates closer to the natural frequency of the water column, similar to the results of regular wave tests. As the natural frequency of the water column changes for each configuration, the peak of (P_air)_ran shifts correspondingly. In addition, the device configuration should aim for higher pneumatic pressure flux for the efficient performance of the system.

5.3.2.4 Pressure Outside OWC

The dynamic pressures due to random waves acting on an OWC are lesser than the wave-induced pressure on a vertical wall. The analysis and applicability of dynamic wave pressure are still important for estimating the cyclic pressure and the momentum, which could result in horizontal sliding and structure overturning. The influence of harbour walls on the variation in the dynamic pressures along the front lip wall due to different peak frequencies for varying configurations of harbour walls is shown in Fig. 5.17a–c, respectively.

Fig. 5.17

Variation of(P_out)_ran with d/L_p. a H_s/s = 0.25 b H_s/s = 0.50 and c H_s/s = 0.75

The measured (P_out)_ran due to varying resonant length of harbour walls show a peak of about 2.2 when d/L_p is around 0.09 for H_s/s of 0.25. For smaller H_s/s, a smooth variation of (P_out)_ran was observed, whereas, increasing the H_s/s, the variation in trend gradually reduces, which could be associated with the non-linearity of incoming waves.

5.3.2.5 Pressure Inside OWC

The influence of harbour walls on dynamic pressure variation inside the water column under different sea states due to the volume flux of water is shown in Fig. 5.18a–c for the different H_s/s.

Fig. 5.18

The variation of (P_in)_ran with d/L_p a H_s/s = 0.25 b H_s/s = 0.50 and c H_s/s = 0.75

Inside the rectangular chamber of OWC, the volume flux of water, dynamic pressure, and pneumatic pressure are inter related to each other as one influences the other. On comparing both the results, the variation in trend coincides with each other with a slight reduction in the magnitude of (P_in)_ran, as (P_out)_ran being exposed to the direct incoming waves. The reduction in magnitude is mostly due to energy loss by front lip wall reflection. Similar to (P_out)_ran, due to the non-linearity of the waves for higher H_s/s (e.g., 0.75), the variation for all configurations of OWC models exhibits a marginal variation, especially for c/b 1.5 and 2.0.

Based on the comprehensive experimental study, the results show that the non-dimensional dynamic pressure outside the front lip wall is more than that of inside the OWC. Furthermore, the pneumatic pressure is found to be minimal compared to other pressure results i.e., P_out > P_in > P_air. This is because, as waves propagate from the free surface into the water column, there exists a substantial loss of energy due to reflection from the front lip wall of OWC and the transfer of energy from hydrodynamics to aerodynamics. This variation of pressure is identical for both regular as well as random wave studies. The observed results are in agreement with the conclusions drawn by [16].

5.3.2.6 Relative Capture Width (RCW)

The Relative capture Width (RCW)_ran subjected to random sea states is also calculated based on the ratio of output power to the input power similar to regular waves. The variation of (RCW)_ran for different OWC models of varying resonant length of harbour walls H/s = 0.25, (b) H/s = 0.50, and (c) H/s = 0.75 are shown in Figs. 5.19a–c, respectively.

Fig. 5.19

Variation of RCW_ran with d/L_p for different c/b a H_s/s = 0.25 b H_s/s = 0.50 and c H_s/s = 0.75

The results from random wave studies are found similar to those obtained through regular wave tests. For OWC without harbour walls, the RCW_ran is less than unity for all peak wave frequency except during the resonant frequency. Similar to regular wave studies, random wave studies also reveal that the harbour walls with c/b of 1.5 show an increase in RCW_ran with a maximum of about 2.5 corresponding to d/L_p of 0.1312. It is also evident that the energy due to low-frequency waves are not absorbed appreciably and beyond d/L_p of about 0.16. The variation in trend seems to be almost asymptotic. This observed variation in trend specifies that wave energy conversion due to random wave field mainly is governed by the system configuration and peak wave period similar to the findings of [27]. Proving that the energy conversion due to water surface oscillation inside the chamber can be maintained optimum if b/L = 0.1, where ‘b’ is the breadth of OWC normal to the wave crest.

Though the experimental runs with random waves are carried out with a peak period interval of 0.25 s, the observed peak of RCW_ran still can be observed closer towards the natural frequency of the water column, which shifts for each of the OWC models with and without harbour walls. Unlike regular wave study, the c/b1.5 shows a considerable increase in RCW_ran for all H_s/s tested under random sea states, which implies an optimum c/b could be adopted. The study of hydrodynamic RCW_ran enables to identify the efficient resonant length of harbour wall system under random sea states.

5.3.2.7 Spectral Density Comparison

In order to have a clear understanding of the energy loss and wave amplification due to harbour wall resonance, the energy spectral density for the measured parameters is plotted and compared. A typical spectral density plot for the harbour wall configuration of c/b = 1.5 during the resonance peak frequency of T_p1.5 s is shown in Fig. 5.20.

Fig. 5.20

Spectral density of η₁, η₂, η₃, R_u1and R_u3 as a function of f(Hz)

From the above spectral density result, the variation of energy density measured at different locations is quantified and compared with the incident wave spectrum (η₁), η₂, and η₃ are the measurements from W.P2 and W.P3 (see Fig. 5.2) respectively. The energy density of R_u1 measured in the harbour area shows the extent of amplification of wave energy which could be attributed to the harbour resonance. As the wave amplification outside the OWC leads to volume flux of water inside the chamber, and it also gets reflected due to the vertical front lip wall, the spectral density of R_u3 shows a slight reduction of energy density compared to R_u1. However, the energy density of R_u3 is still more than that of incident energy density. The energy density of η₂and η₃showed a significant reduction compared to the energy density of η_1. A slight shift of peak spectral density towards the high-frequency region for all the parameters is observed compared to that of incident peak wave frequency. The variations of spectral density for pressure outside, inside, and pneumatic pressure comparing with incident wave pressure are depicted in Fig. 5.21.

Fig. 5.21

Spectral density of ρgη_1, p_out, p_in, and p_air as a function of f(Hz)

The spectral density of pressure outside the lip wall has been greatly amplified due to harbour walls which agree with the spectral density of wave amplification outside the lip wall, as shown in Fig. 5.20. Further, due to the volume flux of water inside the OWC, the spectral density of pressure inside the OWC shows a marginal reduction compared to p_out. The reduction of spectral density is explained by the front lip wall reflection; however, the energy density of the p_in is more than the incident wave pressure. All these pressure variations contribute to the pneumatic pressure, of which the spectral density is slightly higher than the incident wave pressure. For OWC, the energy density of pneumatic pressure would fall well below the incident wave pressure. This increase of pneumatic pressure is solemnly due to the resonance in the harbour area, which amplified the incident wave energy to a considerable extent. The spectral density of pressure plots also showed a slight shift of peak pressure towards the high-frequency region. This illustrates how the harbour area resonance alters the incident wave frequency.

5.3.3 OWC with Inclined Harbour Walls in Regular and Random Wave Fields

In the second stage of the experiments, we investigate the influence of the opening angle of the harbour walls. The concept of providing the harbour walls in an inclined manner instead of straight is to trap a large amount of incident wave energy in front of the bottom opening. Providing harbour walls in such a manner increases the width trapping the incoming waves and thereby increases the volume flux of water inside the column. A system with a wider capture width also competent in an environment dominated by oblique waves. As the efficient resonant length identified in the present study as c/b = 1.5 (discussed in Sect. 1.4), we first focus on the opening angle variation for c/b = 1.5. However, inclination corresponding to other resonant lengths is summarized in Sect. 1.6. The harbour wall inclination,θ is studied by changing the outward inclination of each harbour wall in the range of [π/2 to 7π/8] with respect to the front lip wall of OWC at an interval of π/8. The range of wall inclination has been adopted based on the analytical study by [8] in which the efficiency was found to be a maximum for θin the range of π/2 to 3π/4.

5.3.3.1 Wave Amplification Factor

When the harbour wall is perpendicular to the front lip wall of OWC, i.e., θ = π/2, the width traps incoming wave equal to the bottom opening width of OWC. As θincreases, the width trapping the incoming waves also increases, leading to a huge amplification of waves inside the harbour region (see Fig. 5.22 and 5.23). In particular, the presence of the harbour wall increases the wave run-up as the θ increases up to 3π/4. The results also indicate that the wave amplification does not increase with a further increase in θ. Though (β_out)_reg is found to be increasing with an increase in θup to 3π/4, a further increase in the inclination yielded only a marginal variation in (β_out)_reg. For H/s = 0.25, (β_out)_reg is found to be a maximum for low-frequency waves, whereas it increases considerably for short period waves, in particular for the lower H/s. The variation of (β_out)_reg with d/L is observed to be similar for all the H/s. Though, the inclination of the harbour wall was varied, the resonant length being constant resulted in the peak of (β_out)_reg to occur at the same wave frequency. It may also be noted that the variation of (β_out)_reg for each of the inclination shows a considerable variation for the least H/s tested, whereas for higher H/s_, the variation in magnitude for each of the inclination diminishes, and a marginal variation being noticed.

Fig. 5.22

Variation of (β_out)_reg with d/L for different θ. a H/s = 0.25 b H/s = 0.50 and c H/s = 0.75

Fig. 5.23

Variation of (β_out)_ran with d/L_p for different θ a H_s /s = 0.25 b H_s /s = 0.50 and c H_s /s = 0.75

The trend in the variation of (β_out)_ran is found to be different from that of (β_out)_reg. Though the peak of (β_out)_ran occurs at the same wave frequency for all the H_s /s, the trend in variation vanishes and shows marginal variation for all the d/L_p. On comparing both (β_out)_ran and (β_out)_reg, the maximum amplification occurs for θ = 3π/4, however, the magnitude of (β_out)_ran is comparatively less than (β_out)_reg. This can be illustrated by the presence of a number of frequency components in the random waves which give rise to a number of harmonics, and the resonance is found to decrease considerably in the harbour area. Overall, the above results confirm that the inclination of the harbour walls has a significant effect on wave amplification.

The effect of θ on the variation in the volume flux of water inside OWC subjected to regular and random waves are shown in Figs. 5.24 and 5.25, respectively.

Fig. 5.24

Variation of (β_in)_reg with d/L for different θ a H/s = 0.25 b H/s = 0.50 and c H/s = 0.75

Fig. 5.25

Variation of (β_in)_ran with d/L_p for different θ a H_s /s = 0.25 b H_s /s = 0.50 and c H_s /s = 0.75

Due to the wave amplification inside the harbour area, the active and reactive elements in the wave particles influence the volume flux of water inside the OWC. As the resonance inside the OWC and harbour resonance are two different consequences, the trend in the variation of (β_out)_reg and (β_in)_reg is different from each other. On comparing the results of (β_out)_reg with (β_in)_reg, it is observed that the wave amplification outside is more than that inside the OWC chamber. Unlike (β_out)_reg, the wave amplification inside the chamber is a maximum for low-frequency waves and a minimum for high-frequency waves. Furthermore, as H/s increases, the rate of growth of (β_in)_reg decreases gradually, and when H/s = 0.75, the harbour wall inclination has a marginal influence on (β_in)_reg.

In the case of random waves, smooth variation with higher volume flux for low-frequency waves and slightly lesser flux for higher frequency waves were observed. The observed trend for each of the θ is quite similar to the trend observed for (β_in)_reg. The presence of a harbour wall increases the wave run-up from 1.5 times to a maximum of 2.5 times for low-frequency waves, and it increases to a considerable extent for high-frequency waves. This volume flux of water further compresses the air column above and leads to an increase in the pneumatic pressure which eventually enhances the performance of the device.

5.3.3.2 Pneumatic Pressure

The change in the pneumatic pressure inside the air chamber subjected to the action of regular and random waves are shown in Figs. 5.26 and 5.27, respectively.

Fig. 5.26

Variation of for different θ a H/s = 0.25 b H/s = 0.50 and c H/s = 0.75

Fig. 5.27

Variation of (P_air)_ran with d/L_p for different θ a H_s /s = 0.25 b H_s /s = 0.50 and c H_s /s = 0.75

Similar to the variation found with (β_in)_ran and (β_in)_reg, the (P_air)_reg and (P_air)_ran. Also, follow a similar trend for the different θ. The peak of (P_air)_reg and (P_air)_ran observed to fall on the same wave frequency (d/L_p of 0.131), which again implies that the device takes in energy from the frequency components closer to its natural period. The results indicate that widening of harbour walls increased the pneumatic pressure, especially near the resonant frequency of the structure. The contributions from other frequency components remain marginal. This proves that the natural frequency of the system remains unaltered with the variation of harbour wall inclination. It is also observed that the overall peak pneumatic pressure is a maximum for greater θ and minimum for θ = $\pi$/2. Unlike (P_air)_reg, the results pertaining to (P_air)_ran show smooth variation for all the inclinations. Furthermore, similar to the results observed for varying resonant length (see Fig. 5.7 and 5.16), the inclination of harbour walls has also yielded a similar conclusion that(P_air)_reg > (P_air)_ran.

5.3.3.3 Pressure Outside OWC

The influence of harbour walls inclinations on the non-dimensional pressure variation (on the lip wall)subjected to the action of regular and random waves for different H/s are plotted in Figs. 5.28 and 5.29, respectively.

Fig. 5.28

Variation of (P_out)_reg with d/L for different θ and wave conditions a H/s = 0.25 b H/s = 0.50 and c H/s = 0.75

Fig. 5.29

Variation of (P_out)_ran with d/L_p for different θ a H_s /s = 0.25 b H_s /s = 0.50 and c H_s /s = 0.75

Similar to the wave amplification factor and pneumatic pressure, the non-dimensional pressure outside the front lip wall subjected to the action of regular waves expressed a higher magnitude compared to random waves. Besides, the observed trend follows the same for both regular and random waves.

5.3.3.4 Pressure Inside OWC

The non-dimensional pressure in the lee of the lip wall (inside OWC) due to regular and random waves are shown in Figs. 5.30 and 5.31, respectively.

Fig. 5.30

Variation of (P_in)_reg with d/L for different θ a H/s = 0.25 b H/s = 0.50 and c H/s = 0.75

Fig. 5.31

Variation of (P_in)_ran with d/L_p for different θ a H_s /s = 0.25 b H_s /s = 0.50 and c H_s /s = 0.75

In general, due to the convergence of direct incoming waves to the OWC system, the internal pressure, and the volume flux of water are expected to be increased and the results indicate a similar pattern as reported in the above figures. Furthermore, the variations of (P_in)_reg and (P_in)_ran are found to follow similar trends with the other hydrodynamic parameters discussed so far.

5.3.3.5 Relative Capture Width

In the calculation of hydrodynamic RCW, the width traps the incoming wave is taken into account for establishing incident power. It may also be noted that as the inclination of harbour walls increases, the width capturing the incoming wave energy increases as well. However, the bottom opening width of the water column remains constant thus it makes two different entities. The arguments of calculating the incident wave power either by considering the width of the bottom opening or the width capturing the wave energy appear to be reasonable, and hence it was calculated in both ways for a better comparison. The variation of RCW_reg with d/L by considering the constant bottom opening width of the water column is plotted as a function of relative water depth is shown in Fig. 5.32.

Fig. 5.32

Variation of RCW_reg with d/L for different θ a H/s = 0.25 b H/s = 0.50 and c H/s = 0.75

The RCW_reg increases up to d/L of about 0.142, beyond which it decreases. Further, RCW_reg gets increased as θ increases. The peak of RCW_reg appears to be about 4.6 for θ = 7π/8for the least H/s of 0.25, which reduces by about 22% for the highest H/s of 0.75. On the contrary, as the capture width is at its maximum, the incoming waves converge towards the OWC, leading to a large run-up on the front of the lip wall of the device, and splashing was noticed for steeper waves of H/L > 0.041 during the experimental investigation. RCW_reg was found to be less for long waves, as a major part of wave energy is liable to get reflected back, and only a minor part of the energy is driven into the system to contribute to the air pressure. Even the widest θ failed to counterpart the natural frequency of the device resulting in a less RCW_reg for low-frequency waves. However, the provision of the flared opening of the harbour wall has resulted in an increase of output power from that of the model with a straight harbour wall. On the other hand, the harbour walls have facilitated the concentration of energy due to higher frequency waves.

The foregoing discussion of the results again reveals that the wave energy conversion due to the incoming wave field is mainly governed by the system configuration and the wave period. However, the inclination of the harbour wall has no influence on the frequency dependence nature of the system. The resonance condition of the device remains unaltered if the projecting length is fixed same. The output power of the device, however, is enhanced by the inclination of harbour walls, and so does RCW_reg.

5.3.3.6 Percentage of Increase in RCW_reg

In order to have a clear understanding of the effect of θ on RCW_reg, the percentage of increase in RCW_reg from that for c/b = 1.5 is plotted as a function of d/L for different H/s in Fig. 5.33. As the change in RCW_reg for d/L, less than 0.09 is merely negligible, the percentage increase is depicted for d/L greater than 0.09.

Fig. 5.33

a Percentage of increase with d/L for H/s = 0.25. b Percentage of increase with d/L for H/s = 0.50. c Percentage of increase with d/L for H/s = 0.75

It is observed that although, the width trapping the incoming waves (capture width) is increased to about 80% of ‘w’ for θ = 7π/8, the corresponding average increase in RCW_reg is found to be about 40%. Similarly, an increase of capture width by 65% (θ = 3π/4) and 45% (θ = 5π/8) yields an average increase in RCW_reg to an extent of about 25% and 20% respectively. Hence, a trade-off to be adopted while deciding on the inclination of the harbour walls. If the cost of additional lengths of the walls could be justified for the benefit in the efficiency that is attained with higher θ, it could be considered.

However, if the capture width of the harbour wall is taken into account instead of the OWC chamber width of the OWC, the magnitude of RCW_reg for different θ follows a different pattern. The value of RCW_reg, considering the varying capture width of the harbour wall is depicted in Fig. 5.34.

Fig. 5.34

Variation of RCW_reg with d/L for varying capture width a H/s = 0.25 b H/s = 0.50 and c H/s = 0.7

In contrast to the results projected in Fig. 5.33, an increase in θ results in a reduction of RCW_reg for all the H/s tested. This is due to the fact that, as RCW_reg is the ratio of output power to the input power, and even though, the output power is a maximum for the widest θ, its capture width for trapping the incoming wave is also maximum and thereby, the corresponding ratio appears to be decreasing. The results calculated based on varying the capture width is contrary to the results on the variation in the air pressure and wave amplification which showed maximum values for the widest θ.

The influence of harbour walls on RCW due to the action of random waves was analysed for further information. Therefore, RCW_ran by considering the constant bottom opening of the water column is studied, and the results for different H_s /s is shown in Fig. 5.35.

Fig. 5.35

Variation of RCW_ran with d/L_p for uniform capture width a H_s /s = 0.25 b H_s /s = 0.50 and c H_s /s = 0.75

Unlike RCW_reg the variation of RCW_ran shows a different trend for each inclination of harbour walls. The trend of increase in RCW for each inclination of harbour walls is not observed in the case of random waves, though RCW_ran has been enhanced due to the inclination of harbour walls. Only marginal variations were observed with the increment of H_s/s, and the results are slightly contradicting the regular wave studies. We noted that RCW_ran is very minimum for the widest inclination, both H_s /s of 0.50 and 0.75, which could be associated with the non-linear dynamics inside the harbour area for higher significant wave heights. Furthermore, this could also be illustrated by the reflection of incident waves, and the interaction of a number of frequency components with the reflected waves from harbour walls disturbs the resonance in the harbour area and thereby the resonance inside the water column is greatly affected. Furthermore, RCW_ran, by considering the capture width of the harbour walls, makes it more complex as it reveals the similarity with the regular waves results of varying capture width. The influence of harbour walls on RCW_ran and calculation considering the varying capture width is depicted in Fig. 5.36.

Fig. 5.36

Variation of RCW_ran with d/L_p for varying capture width a H_s /s = 0.25 b H_s /s = 0.50 and c H_s /s = 0.75

The above results show similar kinds of variations observed in the case of regular waves. The RCW_ran decreases with an increase in θ with peaks occurring at the same wave frequency for all θ. The widest θappears to fall less than unity for all the H_s /s tested. We also carried out the analysis for all inclinations with varying resonating lengths of harbour walls e.g., c/b1 and 2. As we noted, similar variations as discussed with c/b1.5 it therefore not presented here. However, for a better comparison, all the resonant lengths and various inclinations subjected to regular and random waves are compared in Fig. 5.37a and b, respectively.

Fig. 5.37

a Comparison of RCW_reg for different configurations of harbour walls. b Comparison of RCW_ran for different configuration of harbour walls

It is evident from the results that the hydrodynamic characteristics of inclined harbour walls for all the resonant length have shown appreciable results compared to that of straight harbour walls. This proves the concept of inclined harbour walls is worth considering. The above results unveil that θ of 3π/4 shows enhanced RCW during the regular wave study and for random wave, maximum RCW is observed for θ of 7π/8 when the resonant length is at its maximum i.e., c/b = 2.0. The above statement conveys that, increasing the width trapping incoming wave increases the performance. The percentage increase of width for the widest inclination is 85% compared to straight harbour walls, and the corresponding increase of RCW is not appreciable. The trend in variation of regular and random waves shows minimum RCW towards the low-frequency region, and maximum RCW occurs closer towards the natural frequency of water column region and moderate RCW towards the  high-frequency region.

5.4 Summary and Conclusions

The hydrodynamic characteristics of the terminator category, OWC type wave energy device, was studied by integrating the harbour walls of different configurations. As the sustainable generating potential from ocean waves is of the order of 1700 TWh per year which is about 10% of the global energy needs, researchers from the multi-disciplinary field have been claiming that ocean energy is promising and fulfilling the deficiency at least partially. Due to the nascent scientific development in ocean science and technology, the capability to improvise the performance of wave energy devices are promising. Numerous works on the conversion of wave energy into clean electricity have evinced intense interest amongst researchers worldwide over the last few decades.

As the working and performance of the OWC system rely on the geometric parameters (which influences the energy conversion based on the natural period of the system), the influence of harbour walls on the performance of OWC is addressed in this work.

The intent of the harbour walls is to create an added resonance such that it enables the device to resonate for a wider range of frequencies. The extended study on the energy output from the OWC system for different configurations of harbour walls indicates that there exists an optimal geometry of harbour walls for the device to extract energy from the incident waves. Moreover, no major disagreement has been observed on the study of the natural frequency of water column till this point between theory and experiment. The addition of harbour walls increases the natural period, and thereby, energy from longer period waves can also be trapped.

The well-controlled experimental program paved the way to understand the hydrodynamic characteristics due to the influence of the harbour walls when subjected to the action of regular as well as random waves. The energy conversion capacity of conventional OWC is compared with the OWC integrated with harbour walls which revealed that the concept of harbour walls is earnest of going over. The quantitative analysis of the phase difference between inside and outside the water column and the influence of harbour walls are reported. The pneumatic compressibility and hydrodynamics necessitate different model scales for the complete understanding, and the influence of vortex shedding, flow separation, and viscous effects which are difficult to infer from small-scale experiments. The hydrodynamic RCW due to the harbour wall’s resonant length and its opening angle is studied distinctly.

Even though the magnitude of air pressure and wave amplification is found to be a maximum for harbour walls of length c/b = 2.0, the overall RCW seems to be maximum and consistent for c/b = 1.5. The results are also in agreement with the analytical studies of [10] indicate that the presence of the harbour halls enhances the performance of OWC in terms of its efficiency.

The influence of the harbour wall’s inclination on air pressure and volume flux of water inside OWC indicate that for θ of 7π/8theRCWis found to be a maximum. As RCW increases with an increment of θ, it makes it more difficult to report an optimum inclination. Furthermore, an optimum θ cannot be adopted based on the RCW alone as it involves various other factors such as the overall output power (if deployed in arrays), the location where space is a constraint, and the cost investment, including the maintenance cost over the design life of the OWC and grid connection, etc. In addition, as the behaviour of the device is not known for OWC providing with harbour walls in a field dominated by oblique waves, where the principal axis of the OWC is not exactly parallel to the wavefronts, the results could be different for different θ.

Considering a model scale of 1:20, the experiments tested here may be applied to a shoreline of 10 m exposed to a uniform wave climate, which can accommodate 32 devices with a harbour wall angle θ = π/2 without any spacing. While increasing the harbour wall angle θ = 5π/8 would reduce the number of devices to 18. Similarly, when accommodating OWCs with harbour wall angles of θ = 3π/4 and θ = 7π/8, the number of devices reduces to 12 and 6 respectively. In such a case, the overall RCW is at a maximum for θ = π/2, which also comes with a huge capital cost as it involves more number of devices. This overall RCW for a given number of devices decreases as θ increases. Practically, the cost involved for the widest opening angle would be a maximum, and hence a compromise between conversion efficiency and construction cost needs to be adopted while deciding on the opening angle of harbour walls.

However, while integrating OWC with breakwater, considering the fact of space constraint, overall RCW based on the number of devices that can be accommodated for a given length of the breakwater and the outcome based on the cost investment, the shortcomings of “θ” of 7π/8 is more compared to that of other θ This leaves to a conclusion that the harbour wall of θ between 5π/8 and 3π/4 could be consistent. The results also flagged a way to understand the natural frequency of the system, which is independent of the harbour wall inclination, whereas the RCW of the system relies on the wave frequency. Furthermore, the performance of the device in wave power absorption is appreciable when the relative water depth is 0.12 < d/L < 0.16, and it shows poor performance for d/L < 0.09.

The salient conclusions drawn from the present study are given below,

• 1. The addition of harbour walls substantially enhances the performance of the device compared to that of an OWC without harbour walls.

• 2. The Harbour wall’s resonant length, c/b of 1.5, has been identified to be optimum and efficient as the hydrodynamic characteristics and RCW appears to be consistent for both regular and random waves. The efficiency of the configuration is recognized to be in the vicinity of the resonant region of the device. In addition, the optimum c/bis close to the range suggested by [8] based on his studies with circular OWC. Hence, it is suggested to consider a system with a harbour wall configuration of c/b1.5.

• 3. The opening angle, θ of the harbour wall, can be adopted between 5π/8 and 3π/4 by maintaining the resonant length constant as c/b1.5.

• 4. The resonating length of the harbour wall determines the natural frequency of the water column, whereas the inclination of harbour walls does not influence natural frequency.

• 5. The pneumatic pressure resonates with respect to the natural frequency of the water column, and the peak of P_air shifts towards low frequency as the resonant length of the harbour wall increases. However, the inclination of the harbour has no significance towards pneumatic pressure as the resonant length was maintained constant for every increment of θ. This agrees with the fact that pneumatic pressure also resonates close to the natural frequency of the water column.

• 6. The phase difference of water level oscillation between inside and outside the OWC chamber varies linearly with the wave frequency.

• 7. The variation of dynamic pressure inside and outside the OWC chamber due to the incident wave field is mainly governed by the system configuration and peak wave period.

• 8. The P_out is greater than P_in, and P_air is found to be minimum.

• 9. The energy conversion capacity of the device can be maintained optimum if the natural frequency of the OWC device is closer to the predominant wave frequency, which is in agreement with the claims made by [27].

• 10. The results of RCW indicate that the performance of the device relies on the wave frequency, and this confirms the frequency dependence nature of the device. The performance results indicate that substantial changes occur on adding harbour walls.