Effects of Structure Geometry on Energy Harvesting Efficiency of Multi-Stage Overtopping Wave Energy Converters

2.1 Numerical domain

The numerical wave tank domain used in the present study is shown in Fig. 3. The domain is 300 m long and has water depth of 20m approximately corresponding to 5.4 λ and 0.36 λ respectively, where λ is the wavelength calculated by dispersion relation. This condition falls in the intermediate water depth. The length of absorber zone equals to a wavelength (1 λ). If this zone is too short, the wave energy could not be effectively absorbed. On the contrary, if this zone is too long, it affects the computational expense. Moreover, numerical wave probes are employed in front of and behind the device in order to record surface elevation. The conditions are applied differently to boundaries surrounding the domain. Wall condition combined with dynamics mesh model is applied on the left boundary to generate wave. The function of wave generation is described by a UDF (Userdefined function) code in this paper is based on the previous study (Jungrungruengtaworn and Hyun [2017]). The motion of the piston type wavemaker x(t) is defined by:

Fig. 3.

Two-dimensional numerical wave tank used to simulate wave flow behavior.

where x₀ is the maximum displacement of the wavemaker and ω is the angular frequency.

The pressure outlet and open channel flow condition combined with porous media model is used at outlet (right boundary) acting as a wave absorber which is similar to the boundary conditions of numerical wave tank used by Du and Leung[2011] and Hu and Liu[2014]. The porous media condition is defined by an additional momentum source term which is added to the Navier-Stokes equations as:

and

where S_i is the additional source term, consisting of two parts: a viscous loss term and an inertial loss term, for the i-th (x, y or z) momentum equation, |v_i| is the velocity magnitude, 1/α is the viscous resistance coefficient, and C₂ is the inertial resistance factor.

The viscous resistance coefficient (1/α) is the key parameter in order to define the rate of absorption which has been investigated and shown the accuracy of the wave profile in the previous study (Jungrungruengtaworn and Hyun[2017]). Pressure outlet and wall condition are applied on top and bottom boundary respectively.

In the present study, adaptive refinement is utilized in order to construct grid system of the computational domain. A dense or more refined mesh is generated in regions of interest: the vicinity of free surface and OWEC device. The mesh size is dependent on the wave height and length. In this work, the number of cells per wave height is 20 cells and 100 cells per wavelength is used following the previous study (Jungrungruengtaworn and Hyun [2017]) which is considerably dense compared to the grid system suggested by Connell and Cashman[2016].

2.2 Optimum design

The optimization of the OWEC design concerns particularly the ramp geometry in order to maximize the energy captured by the reservoirs. There are two strategies to achieve the goal: devices with different overlap ramp and devices with different submerged ramp.

The former investigation focuses on the influence of overlap ramp on the hydraulic performance. The base-line model is based on non-adaptive device which is schematically illustrated in Fig. 4b. The ramp height consists of two parts, the submerged depth R_s and freeboard height R_c4 that both parameters are fixed as 2 m. The slope of each ramp S is defined as the ratio between the ramp height R and horizontal ramp length L which is kept constant as S=1/2 for both freeboard and draught parts. This consequently results in the 3$$ \overline{w}$$ extended length of the ramp.

Fig. 4.

Layouts of non-overlap ramp multi-stage devices base on the relative slot width of 0.2.

These multi-stage devices have three extra slots which are defined as w₁ (the lowest), w₂ (the middle) and w₃ (the highest). The adaptive layout gives difference in size of the slots as Δw=w₂-w₁=w₃-w₂, so that the average width of each device is $$ \overline{w}$$=w₂ which is fixed as 0.4 m in the present study. Moreover, the parameter slot width, $$ \overline{w}$$ is represented in dimensionless form namely relative slot width as: $$ \overline{w}$$/R_c4 of 0.2. The adaptive device has two layouts: reservoir width varies from large to small (w₁=1.4$$ \overline{w}$$) and vice versa. The illustration of overlap distance is shown in Fig. 5 and is defined by parameter y that is varied within the range of 0.025 to 0.100 m with a 0.025 m increment. This design is applied to both adaptive and non-adaptive devices.

Fig. 5.

Example of overlap ramp layout of non-adaptive device.

The slot width of non-adaptive and adaptive devices are summarized in Table 1. The distance of slot width is measured horizontally at crest level. Fig. 4 shows the layouts of non-overlap ramp multi-stage devices based on the relative slot width of 0.2 which is corresponding to the Table 1.

Table 1.

Slot width of multi-stage devices

The second investigation concerns the influence of submerged ramp on the performance of land-based OWEC. The parameter R_s is varied within the range 2 to 8 m with a 2 m increment. The schematic illustration of OWEC device installed on breakwater is shown in Fig. 6.

Fig. 6.

Schematic of multi-stage device to study the effect of submerged depth on performance.

2.3 OWEC performance

The overall efficiency of OWEC device consists of four partial efficiencies, i.e., hydraulic efficiency, reservoir efficiency, turbines efficiency and generator efficiency. In the present study, the numerical results are represented by the hydraulic efficiency η calculated based on crest levels R_c,i and averaged discharge Q_i. The hydraulic efficiency η is defined as the ratio between the potential power P_crest in the overtopping water and the power in the incident wave P_wave which is also used by Margheritini et al.[2009] and Tanaka et al.[2015] as:

where

and

here ρ is the water density, g is the gravitational acceleration, Q_i is the average overtopping flow rate per unit length of the i^th reservoir, R_c,i is the crest height of the i^th reservoir and H is the wave height.

The wave energy flux or wave power shown in the Eq. 6 consists of three terms: wave energy density E, phase velocity c and factor n. The factor n is relates to the region of water depth, for deep water, n=1/2 and for shallow water, n=1.

The incident, reflected and transmitted wave power can be calculated using Eq. 6, when the height or amplitude of these waves is known. The transportation of energy conservation of the overtopping phenomenon can be explained by:

here P_r is the reflected wave power, P_t is the transmitted wave power and P_L is the dissipated wave power or loss.

Since the wave power is proportional to H² or a² then the dimensionless wave energy in reflection wave is proportional to $$ K_r^2$$ where K_r is the reflection coefficient which is ratio between the reflected and incident wave amplitudes: K_r=a_r/a_i. In addition, Eq. 7 can be expressed in dimensionless form by:

where $$ K_r^2$$ is the reflection rate, $$ K_t^2$$ is the transmission rate and L is the loss rate which is defined as ratio between the dissipated wave power P_L to the incident wave power: L=P_L/P_wave. This equation is similar to transportation of energy conservation used by Tanaka et al.[2015].

The reflection coefficient used in Eq. 8 can be determined using recorded wave profiles, since the free surface elevation is assumed to be the superposition of sinusoidal incident and reflected wave trains. A detailed explanation of the method is summarized by Isaacson [1991]. By using two fixed numerical wave probes, the incident and reflected wave amplitudes can be represented respectively by:

and

where A_n is the measured amplitude of the n^th probe, Δ is the dimensionless distance between two probes, and δ is the measured phase angle of second probe relative to that of the first probe and is obtained by fast Fourier transform (FFT) algorithm. Similarly, the transmission coefficient is expressed as the ratio between the transmitted and incident wave amplitudes: K_t=a_t/a_i which can be calculated using measured wave profiles.

3.1 Combination of overlap ramp and adaptive slot width

The overlap distance y, which is presented in dimensionless form as y/$$ \overline{w}$$, is shown in the Table 3. This overlap design, i.e., extended the ramp of each extra slot, is applied to both adaptive and non-adaptive layouts.

Table 3.

Parameters condition studying effect of overlap ramp

The numerical results are shown in Fig. 7~9. It can be seen from Fig. 7 that the applied overlap distance insignificantly affects the overall hydraulic efficiency whether the device is adaptive or non-adaptive. An example of energy flow for case of y/$$ \overline{w}$$=0.125 is shown in Fig. 8 for both adaptive and non-adaptive devices. A similar trend of energy flow is found for devices with other overlap distances, i.e., y/$$ \overline{w}$$=0.0625, 0.1875, 0.25 resulting in the overall hydraulic efficiency of about 35~39%.

Fig. 7.

Effect of overlap ramp applied to multi-stage devices on overall hydraulic efficiency as function of relative extended ramp. The non-overlap ramp is represented by the relative extended ramp y/w¯=0.

Fig. 8.

Comparison of overall hydraulic efficiency, reflection, transmission and loss rates of overlap ramp non-adaptive and adaptive device base on y/w¯=0.125.

However, the overlap distance has strong influence on the partial hydraulic efficiencies as shown in Fig. 9. By increasing the overlap distance, the amount of water collected by lower reservoirs is decreasing while that of the higher reservoirs is increasing. The overlap ramp reduces the gap between two consecutive ramps, as if the slot width is getting smaller and therefore the water can run-up smoothly. The increase in energy captured by lower reservoirs compensates for the decrease of the higher ones giving insignificant change in overall efficiency. This affects the strategy of power extraction process via multi-stage turbines. The water capture mechanism of both run-up and fall-down processes are shown in Fig. 10~11.

Fig. 9.

Partial hydraulic efficiencies of each reservoir versus the relative extended ramp.

Fig. 10.

The water flow run-up processes of overlap ramp multi-stage device based on y/w¯=0.25 at different time.

Fig. 11.

The water flow fall-down processes of overlap ramp multi-stage device based on y/w¯=0.25 at different time.

3.2 Multi-stage OWEC installed on breakwater

The investigation concerns the influence of submerged ramp on the performance of land-based OWEC. Table 4 summarizes the specification of submerged ramp.

Table 4.

Parameters condition studying effect of submerged ramp

The overall performance of land-based device is considerably superior compared to that of stationary devices with equivalent submerged ramp as shown in Fig. 12. For small depth, extending the submerged part of ramp results in relatively great performance enhancement. Further extending the ramp gives gradual increase in efficiency and tends to converge to a certain value. Fig. 13a and 13b show partial efficiencies of different submerged depth. The overall performance enhancement is achieved by increasing the energy captured by the highest reservoir, while the lower ones are unaffected.

Fig. 12.

Overall hydraulic efficiencies of single- and multi-stage device as function of relative submerged depth.

Fig. 13.

Effect of submerged depth on overall hydraulic efficiencies of OWEC devices.

In case of stationary buoyant devices, a portion of un-captured energy transports with transmission wave. However, a portion of this wave and its energy transforms and runs-up, then overtop into reservoirs in case of land-based devices, resulting in greater efficiency as seen in Fig 14a~14b. Increasing the submerged depth is similar to the effect of wave behavior in shoaling water. Wave shoaling is the effect by which surface waves entering shallower water, resulting in change in wave height, and therefore the amount of water in the higher reservoirs is increasing.

Fig. 14.

Water flow into reservoirs of multi-stage device with and without breakwater.