1 Introduction

The exploitation of energy from ocean waves is a challenge that progressively attracts the interest of researchers thanks to the significant energy potential available in the seas. In order to take advantage of this immense potential, several devices are patented as energy converters. Among these converters are the oscillating water columns (OWCs) which have shown significant efficiency against the turbulent states of the oceans. In most cases, the OWC consists of a partially emerged chamber with an opening that includes an air turbine. Its operating principle is to generate pressure due to wave action, which alternately compresses and expands the trapped air, which—in return—is forced to flow through the air chamber into a turbine coupled to a generator acting as a PTO (power take-off). General reviews on the OWC cited in references [1,2,3,4,5,6,7,8] for turbines specific references can be found in [9,10,11,12,13], a review by [14] cited an important summary on the hydrodynamic performance of wave energy converters, including fundamental understandings on numerical modeling, physics and techniques for these wave energy converters.

OWC’s hydrodynamic performance is mainly influenced by the converter’s ability to extract hydraulic energy and convert it into pneumatic energy. While this performance factor is linked to the geometric parameters of the different components of the converter, climatic conditions remain an important issue in the optimization of energy extraction. In fact, to characterize this connection, several researchers have proven the existence of an input between the dimensions and the location of the air chamber with the power absorbed at the PTO level. We can conclude that the change in a parameter related to the geometry of the OWC device attaches considerable optimization of energy standpoint [15].

The geometric parameters of the air chamber; namely the chamber walls, width, thickness of the frontal wall, slope at the bottom of the chamber, and size of the opening; were the subject of numerous investigations. Morris-Thomas et al. [16] examined experimentally the effect of frontal wall geometry on the efficiency of an OWC; they concluded that a curved wall was more effective than the rectangular one. Sheng [17] showed that the rearward curved duct has the best hydrodynamic performance as the forward curved duct especially for wave periods between 5 and 10. In the same direction, Dizadji and Sajadian [18] proved a dependence between the geometry of the chamber and the properties of the incident waves. The numerical study developed by [19] showed the effect of geometric parameters on the hydrodynamic performance of a U-shaped OWC. The results found by this search initially indicate an increase of ηmax and Pmax (denote the amplitudes of the free-surface elevation and the air pressure inside the chamber, respectively) then a decrease depending on the width of the vertical duct for the high frequency region. Deng et al. [20] also address the effects of wall-front dimensions. Çelik and Altunkaynak [21] have also studied the effect of the various opening heights on the fluctuation of the free surface of OWC, this research [21] shows a linear relationship between the captured wave ratio and the transmitted wave heights.

In this paper, we have produced a prototype of an oscillating water column (OWC) which consists of a caisson (air chamber) in the form of parallelepipeds. Our novelty in this work will be a combination of experimental study between the geometric parameters of the air chamber and the OWC installation conditions to determine the different hydrodynamic performances.

As a first step, we will present the equations that govern the calculation of the powers absorbed by the OWC or the average incident power during a test period, this calculation of power will subsequently allow evaluating the performance of this device.

In a second step, we will present all the necessary materials for such an experiment specifying the method chosen for the experimental approach.

The exploitation of the results will be summarized by means of tables and graphs in terms of power for each variation of the geometric quantities.

In order to reach our main objective, we will evaluate the efficiency of our converter taking advantage of the values of the optimal powers for all the variations of the previous quantities.

1.1 OWC's primary output

The efficiency of an OWC is determined by the capacity of converting from wave energy to the so-called pneumatic energy. In general, the efficiency of such a device is defined by the power available to the turbines in relation to the power supplied to the system by the incident waves. This efficiency is illustrated below by the relationship:

$$\eta =\frac{ \mathrm{total}\, \mathrm{energy}\, \mathrm{absorbed}}{\mathrm{total} \,\mathrm{incident}\, \mathrm{energy} }=\frac{{P}_{\mathrm{OWC}}}{{P}_{\mathrm{W}}}$$
(1)

where POWC the average power absorbed by the OWC given by the relation:

$${P}_{\mathrm{OWC}}=\frac{1}{T}{\int }_{0}^{T}PQ\,\mathrm{d}t$$
(2)

With P the pressure of the area inside OWC measured by a manometer during a test, time T and Q the volume flow rate through the turbine.

The total power PW of the incident wave averaged over a wave period can be solved using the equation:

$${P}_{\mathrm{W}}={E}_{\mathrm{W}}{C}_{\mathrm{g}}$$
(3)

With Ew, the wave energy density is the amount of energy that is transported in a horizontal wave front region, perpendicular to the direction of the wave, which is given by the equation:

$${E}_{\mathrm{W}}=\frac{1}{2}\rho gb{A}^{2}$$
(4)
$${C}_{g}=\frac{1}{2}\frac{\varOmega }{k}\left(1+\frac{2kh}{\mathrm{sinh}\left(2kh\right)}\right)$$
(5)

With ρ the density of the water, g the gravitational acceleration, b the width of the part perpendicular to the direction of the incident waves of an OWC, A the amplitude of the waves and Ω the angular frequency which can be related with K the wave number by the following dispersion expressions:

$${\varOmega }^{2}=gk\,\mathrm{tan}h\left(kh\right)$$
(6)

2 Method and materials

The operating principle of OWC is to generate pressure due to the action of the incidental waves, which alternately causes a compression, then an expansion of the trapped air. The latter is forced to circulate via the air chamber in a turbine coupled to a generator playing the role of a power take-off. In this sense, the design of the air chamber is necessary for the measurement of the pneumatic energy supplied by the OWC (Fig. 1).

Fig. 1
figure 1

The air chamber

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In order to carry out the various experimental tests, a hydraulic flow channel is available, (Fig. 2). The channel is made up of five segments of Plexiglas (width: 10 m, height: 0.30 m and length: 0.30 m), a tank: (width: 0.705 m, height: 0.905 m. and length: 2 m), a centrifugal pump with a maximum flow rate of 3500 l/min and an analogue volumetric meter.

Fig. 2
figure 2

The hydraulic channel

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To avoid disturbances due to wave reflections an inclined plane was fixed upstream of the hydraulic channel (Fig. 3):

Fig. 3
figure 3

Inclined plane

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The wave generating system is powered by a 105 W motor with a maximum speed of 2000 tr/min and a speed variator. This mechanism is coupled to a square beater attached to the channel's bottom by an axis that allows back and forth movement (Fig. 4).

Fig. 4
figure 4

The wave generating system

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To obtain the corresponding values for each characteristic of the wave we use:

  • A chronometer: to determine the period of the generated wave (the time it takes for a wave takes to leave a peak point and return to the same point) We take time of ten successive waves for good accuracy.

  • A camera: to take pictures to measure the length and height of the wave (Fig. 5).

Fig. 5
figure 5

Measure the length and height of the wave

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The design of the chamber is necessary for the measurement of the pneumatic energy supplied by this device. It is a box in the form of a parallelepiped with an opening through which water enters and an opening at the top through which an airflow exits to a turbine, from which the passage of energy mechanical to electrical energy is necessary (Fig. 6).

Fig. 6
figure 6

The geometry of the OWC in the hydraulic channel

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The anemometer is used to measure the air speed of the air chamber each time the water depth in the hydraulic column is changed as well as when the speed of the wave generator motor is varied. The following results are obtained (Table 1):

Table 1 The characteristics of waves as a function of motor speed
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With: w and v the motor speed, λ wavelength, T period, H wave height, A = H/2 the amplitude, and D the water depth. Based on the results found and the equations quoted in the literature section, we can calculate the different wave characteristics for each water depth (Table 2):

Table 2 Calculation of wave power
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In the following tests, several parameters are played with:

  • The distance between the wave generator and the device for two positions (see Fig. 7).

  • The depth of the water in the hydraulic channel noted D.

  • The immersion depth of the front wall of the chamber noted d.

  • The opening at the bottom of the prototype.

Fig. 7
figure 7

Positions of the device

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3 Result and discussion

We set ourselves in a water depth of D = 18 cm, where the operating conditions for the chamber are: Air chamber opening = 5 cm and Orifice diameter = 1 cm (Table 3), Fig. 8 describe all the experimental parameters under discussion.

Table 3 The operating conditions: water depth of D = 18 cm, air chamber opening = 5 cm and orifice diameter = 1 cm
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Fig. 8
figure 8

All the experimental parameters under discussion

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The results obtained shown in the following Fig. 9 for the two positions at D = 18 cm, indicate that the immersion depth of the front wall of the chamber of the prototype influences the power available in the OWC. When the prototype is placed at a depth of 12 cm, it can be seen that higher powers are obtained at depths of 0 cm and 6 cm. This is due to the energy losses by friction of the particles with the soil of the hydraulic channel for deep media. When the prototype is placed in a shallow medium closer to the free surface where the friction is approximately low and the oscillations are high, more pneumatic energy is generated.

Fig. 9
figure 9

The results obtained for the two positions at D = 18 cm

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In order to determine the best operating environment of the device, we compare between the speeds where the powers are maximum for all the test (see Fig. 10). The maximum power generated during the tests in position 2 is higher than in position 1 (obtained for the immersion depth of the front wall of the chamber = 12 cm).

Fig. 10
figure 10

Comparison between the 2 positions at D = 18 cm

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During the tests when the prototype is fixed in position 1, which is the closest to the wave generator, we notice that there is a reflection of the waves arriving at the prototype through its wall. This has a direct influence on the characteristics of the wave and its energy potential; therefore; an impact on the pneumatic energy. This confirms the description of the wave energy distribution in an OWC in Fig. 10. We proceed in the same way for the 18 cm depth. Now we fix the water depth D = 20 cm with the same operating conditions (Opening of the air chamber = 5 cm and the diameter of the orifice = 1 cm) (Table 4).

Table 4 The operating conditions: water depth of D = 20 cm, air chamber opening = 5 cm and orifice diameter = 1 cm
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Using the data from the tests in a water depth D = 20 cm, Fig. 11 is plotted for the 2 positions. It can be seen that for average water depths, the available power is better when the device is placed at the bottom of these depths (prototype depth = 0 cm), which is completely different from the results obtained in D = 18 cm. For the pattern representing the depth of the prototype = 12 cm. It can be observed that the different speeds of the wave generator (between 62.5 and 245 rpm) lead to considerable powers (see Table 5) which is not true for the other two patterns. This high speed of the particles leads to friction and therefore loss of kinetic energy in the air chamber (pneumatic energy).

Fig. 11
figure 11

The results obtained for the two positions at D = 20 cm

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Table 5 The operating conditions: water depth of D = 23 cm, air chamber opening = 5 cm and orifice diameter = 1 cm
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The two figures in Fig. 12 represent a prototype depth = 0 cm. If we neglect the friction losses in this case, the explanation of the maximum values of the powers can be linked to the oscillations of the water inside the air chamber, which reach a maximum height of water, leading to a very important air compression, thus a high airflow. This time, the maximum power corresponding to position 1 is higher than that of position 2, but it is important to note that for low speeds of the wave generator in this position one can have a significant power.

Fig. 12
figure 12

Comparison between the 2 positions at D =  20 cm

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The following table presents the results corresponding to a water depth is set to D = 23 cm.

The graphs show the variation of power in relation to W (rmp) in tow position for position 1 and position 2. The results for the different depths are confirmed, and the observations mentioned above are valid. We notice that in a depth of the prototype equal to 0 cm where the prototype is sunk to the bottom of the hydraulic channel the power is maximum, and zero for the last speeds. It is important to note that almost all the depths d achieve maximum power values at speeds between 80 and 157.5 rpm (Fig. 13).

Fig. 13
figure 13

The results obtained for the two positions at D = 23 cm

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Figure 14 represents the powers available in the OWC device for a depth of the prototype equal to 0 cm. Position 2 gives much better powers than position 1. This is due to reflection losses from the incident waves as position 1 is closer to the wave generator. It can also be seen that for the last speeds the powers are zero; this could be due to the energy losses by particle friction, and the low amplitudes of the waves.

Fig. 14
figure 14

Comparison between the 2 positions at D = 23 cm

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Referring to the results acquired previously, we can see that in order to be able to exploit the OWC device and produce significant power, the OWC must be fixed deeper in a medium or deeper environment (its position in relation to the sea or water level in our case). Because of the oscillations that can move inside the air chamber in a large volume that would subsequently allow for significant compression, and therefore a very large airflow. If we return to the case where the depth is low (D = 18 cm), we fall into a paradox from which we obtain better powers only this time the prototype is not fixed deep (depth of the prototype = 12 cm); whereas for a depth of the prototype equal to 0 cm the powers generated are not very important because of the losses by friction of the particles by the walls of the channel.

It is, therefore, necessary to avoid the areas where the phenomenon of wave reflection exists. From the above, we conclude that most of the results are given by the position 2 of the prototype so we will continue our studies only on this position.

Additionally, we will change another parameter of device. We will make an opening of 11 cm following the same procedure:

It can be seen that when the prototype is placed in a depth of 12 cm (D = 18 cm) (Table 6 and Fig. 15), there is no air flow, which results in a glued pattern on the horizontal axis of the engine speed (zero power). This is due to the water level not reaching the air chamber because of the large opening (11 cm). In this case, the maximum powers are generated for a prototype depth of 12 cm, which is compatible with the above results.

Table 6 The operating conditions: water depth of D = 18 cm, air chamber opening = 11 cm
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Fig. 15
figure 15

The results obtained for D = 18 cm and air chamber opening = 11 cm

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The maximum powers are given by a prototype depth of 12 cm (Table 7 and Fig. 16), which is completely different to the results of other tests seen previously. During the tests, and at the highest level of the device, it was noticed that the water does not cover the entire opening until the arrival of the wave crests, which subsequently causes a quantity of atmospheric air to enter with the water, and therefore a significant compression, thus explaining of the results obtained.

Table 7 The operating conditions: water depth of D = 18 cm and Air chamber opening = 11 cm
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Fig. 16
figure 16

The results obtained for D = 20 cm and air chamber opening = 11 cm

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The graphs (Table 8 and Fig. 17) represent powers that are larger than the results for other water depths, whether for a 5 cm or 11 cm opening. The explanations in the graph for the 20 cm water depth (11 cm opening) are also valid in this case.

Table 8 The operating conditions: water depth of D = 18 cm and air chamber opening = 11 cm
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Fig. 17
figure 17

The results obtained for D = 23 cm and air chamber opening = 11 cm

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If we compare the three maximum powers available in the OWC device of the three water depths (D = 18, 20, 23 cm) (Fig. 18), we notice that the one of the water depth 23 cm and the prototype 12 cm is the most important. As already explained, when the air chamber opening has been changed and when it is fixed at the highest level, there is an entrainment of atmospheric air with the incoming water in the device, which—in return—gives an additive amount of energy (more compression than the air in the air chamber alone).

Fig. 18
figure 18

Comparison between the three maximum powers available in the OWC device of the three water depths (D = 18, 20, 23 cm)

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3.1 Efficiency of the first opening

The effectiveness of the device is a challenge that must always be raised. In order to evaluate any kind of optimization, the dependence between geometric quantities and performance has been approved by several authors such as: Bouali and Larbi [22] who reported that the maximum efficiency of OWC at the frontal wall is related to the depth of submersion with 0.45 h (where h is the depth of water). The same authors in [23] also showed a performance gain about 7% obtained for three optimal parameters. This study also shows that there is a unique triplet (T, h, H) that makes the device more efficient. In the same sense [18] have conceptualized several chamber geometries in order to examine their efficiency. On the one hand, the results have shown a dependence between the chamber geometry and the properties of the incident waves. On the other hand, the optimal geometry examined with this study is 32% in efficiency term.

Figure 19 represents the efficiency for a water depth D = 18 cm for the two positions and for the different depths of the prototypes. The maximum efficiency can be generated by the prototype at this depth is 37.5%. It is given by a wave power Pw = 0.244 W knowing that this power is the smallest among all the other wave powers. This translates to the fact that the efficiency of OWC depends strongly on the conditions of the sites chosen for the installation. In fact, to generate more energy in an area with a low wave power, it is recommended to install the chamber with a depth very close to the free water surface.

Fig. 19
figure 19

The efficiency for a water depth D = 18 cm of the first opening

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Although there is not much drop in power for the depth of D = 20 cm (Fig. 20), it has a very low efficiency which does not exceed 0.4% due to the high power generated by the waves during the tests. Even with this very low efficiency result, it is not possible to judge the benefit of OWC since it can generate more power compared to the depth of D = 18 cm.

For the water depth D = 23 cm (Fig. 21), the maximum efficiency does not exceed 8% which is also low for the depth of D = 18 cm. But the system response in terms of power for this depth is still relevant among all other depths and this choice of depth may be the optimal choice despite the low efficiency of OWC.

Fig. 20
figure 20

The efficiency for a water depth D = 20 cm of the first opening

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Fig. 21
figure 21

The efficiency for a water depth D = 23 cm of the first opening

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3.2 Efficiency of the second opening

In the previous paragraph, we have seen the effect of variation of depth D on the efficiency of the converter for a constant opening. Then we will present the effect of variation of opening on the efficiency of the device.

For the second opening, it can be seen that the maximum efficiency generated is 38.12% with an increase of 1% compared to the first opening for the same depth of D = 18 cm (Fig. 22). But again, it can be seen that this efficiency will be cancelled out for a prototype depth of d = 0 cm. This is due to the level of the water not reaching the air chamber because of the opening becoming large (11 cm). Therefore, it can be proposed that for shallow water depths it is recommended to work with such small openings in order to take advantage of the maximum pneumatic energy.

Fig. 22
figure 22

The efficiency for a water depth D = 18 cm of the second opening

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At a depth of D = 20 cm, the efficiency is still very low which agrees with the other results found (Fig. 23). In fact, the maximum efficiency generated at this depth does not exceed 0.6% with an increase of 0.2% compared to the other opening. This decrease in efficiency is due to the high power generated with the wave generator.

Fig. 23
figure 23

The efficiency for a water depth D = 20 cm of the second opening

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The maximum efficiency generated at a depth of D = 23 cm is about 12% with an increase of 4% (Fig. 24. This is also low compared to D = 18 cm to the same aperture. However, this decrease in efficiency is still due to the high power of the waves generated during the tests.

Fig. 24
figure 24

The efficiency for a water depth D = 23 cm of the second opening

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4 Use of goodness-of-fit statistics in this study

We examine the goodness-of-fit statistics in order to visualize the quality of fit. The Curve Fitting tool of the MATLAB software supports these goodness-of-fit statistics for parametric models:

4.1 Sum of squares due to error SSE

This statistic calculates the overall deviation of the response values from the response values' fit. It is frequently abbreviated as SSE and is known as the summed square of residuals:

$$\mathrm{SSE}=\sum_{{\varvec{i}}={\mathbf{1}}}^{{\varvec{n}}}{{\varvec{w}}}_{{\varvec{i}}}{({{\varvec{y}}}_{{\varvec{i}}}-\widehat{{{\varvec{y}}}_{{\varvec{\imath}}})}}^{\mathbf{2}}$$

The fit will be more accurate for prediction if the value is closer to 0, which means that the model has a smaller random error component.

4.2 The mean square error MSE

The mean square error or the residual mean square MSE is defined as:

$$\mathrm{MSE}=\frac{\mathrm{SSE}}{v}$$

\({\varvec{v}}\) denotes the number of independent pieces of information concerning the n data points needed to compute the sum of squares.

Identical to SSE, an MSE value closer to 0 denotes a fit that is more useful for prediction.

5 Conclusion

OWC's power response proved the existence of an optimal relationship between geometric quantities and converter efficiency. The experimental results found by this study showed the different optimum of:

  • The distance between the wave generator and the device (2 positions).

  • The depth of water in the hydraulic channel.

  • The immersion depth of the front wall of the chamber.

  • The opening at the bottom of the prototype.

Moreover, make it possible to size the OWC.

For the depth of water in the hydraulic channel noted D, we can confirm a rather direct correlation between the absorbed power and the depth D. This correlation is no longer valid when we approach the free surface of water (depth d = 12 cm).

These results can be explained by the losses due to the friction of the particles with the channel walls as well as the reflection of the waves.

We have also concluded that the installation of the OWC at a level very near to the free surface causes a quantity of atmospheric air to enter the chamber, which can contribute to an additional quantity of energy.

The comparison between the two positions (1 and 2) shows that most of the maximum power is given by the second position.

For the response of OWC in terms of efficiency, it was shown that the efficiency increases with the decrease of the water depth D and increases with the increase of the prototype depth d. This is due to the decrease of the power arriving to the prototype, which can subsequently decrease the friction between the water particles and the oscillations, thus generating more pneumatic energy.

For the effect of the opening of the prototype, we can observe a significant increase either in terms of power or in terms of efficiency. Provided that we avoid areas where the depth of water and prototype is small, in our case the power was zero for the couple (D = 18 cm, d = 0 cm), an increase of 4% of efficiency was generated for a depth of D = 23 cm with the second opening = 11 cm. Therefore, we can conclude that the opening is an important factor that should be taken into account when designing OWC in order to take maximum advantage of it.

It is very important to note that the power response is the most compatible to describe the efficiency of OWC because of the low values calculated by the efficiency. This difference in compatibility between the two responses is due to the high power generated by the waves during their movement and also the effect of the different random characteristics of the waves such as the reflection phenomenon of the waves.