Numerical and experimental investigation of wave dynamics on a land-fixed OWC device

Energy 115 (2016) 326e337

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Numerical and experimental investigation of wave dynamics on a land-fixed OWC device De-Zhi Ning a, *, Rong-Quan Wang a, Ying Gou a, Ming Zhao b, Bin Teng a a b

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024, China School of Computing, Engineering and Mathematics, Western Sydney University, Locked Bag 1797, Penrith, NSW, 2751, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 March 2016 Received in revised form 23 June 2016 Accepted 1 September 2016

An Oscillating Water Column (OWC) Wave Energy Converter (WEC) is a device that converts the energy of ocean waves to electrical energy. When an OWC is designed, both its energy efficiency and the wave loads on it should be considered. Most attentions have been paid to the energy efficiency of an OWC device in the past several decades. In the present study, the fully nonlinear numerical wave model developed by Ning et al. (2015) [1] is extended to simulate the dynamic wave forces on the land-fixed OWC device by using the acceleration potential method, and the experimental tests are also carried out. The comparisons between numerical results and experimental data are performed. Then the effects of wave conditions and chamber geometry on the wave force on the front wall of the chamber are investigated. The results indicate that the total wave force decreases with the increase of the wavelength and increases with the increase of the incident wave height. The wave force is also strongly influenced by the opening ratio, i.e., in the low-frequency region, the larger the opening ratio, the smaller the wave force and it shows an opposite tendency in the high-frequency region. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Oscillating water column (OWC) Wave loads Air chamber HOBEM Model testing

1. Introduction Due to its non-polluting nature and environment friendliness, renewable energy has attracted increasing attention and renewable energy harvesting technique has been the topics of many theoretical and empirical studies [2]. Wave energy is considered to be one of the most promising forms of clean renewable energy because of its high energy flux density and low negative environmental impact. To harvest the wave energy, various types of Wave Energy Converters (WECs) have been proposed. Due to their high efficiency and structural simplicity, OWC devices are believed to be feasible devices to harvest wave energy and the performance of OWC devices has been studied extensively in the past decades [3]. An OWC device generates energy from the cyclic rising and falling of water in an air chamber caused by waves in the ocean as shown in Fig. 1. The incoming waves make the water surface inside the air chamber oscillate. The air is driven in and out of the chamber through a power-take-off system, which is represented by a small hole on the roof of the chamber as shown in Fig. 1. The chamber design and the turbine characteristics are

* Corresponding author. E-mail address: [email protected] (D.-Z. Ning). http://dx.doi.org/10.1016/j.energy.2016.09.001 0360-5442/© 2016 Elsevier Ltd. All rights reserved.

important factors that affect the efficiency of an OWC device. To achieve the optimum hydrodynamic efficiency, numerous studies have been carried out either experimentally [2,4e6] or numerically [1,7e9] to optimize the chamber design of OWC devices. Because the turbine plays a very important role in the energy conversion, the effect of turbine characteristics on the efficiency are studied by many researchers [8,10e13]. Studies have also been carried out to investigate the effects of wave conditions [14e17] and sea bottom profiles [18e21] on the hydrodynamic efficiency of OWC devices. However, most of the previous studies about OWC devices were focused on the overall hydrodynamic efficiency, the dynamic wave loads on the OWC devices have been paid little attention. The large difference between the internal and external water surface levels of the chamber can cause a dynamic wave load on the front wall of the chamber, which may be a threat to the safety operation of OWC devices [22]. The largest bottom standing wave energy plant (OSPREY plant in Scotland) was destroyed by the sea shortly after installment [7] and the concrete subsurface structure of the Pico plant in Portugal was significantly damaged by waves [22]. As the working in the ocean is both dangerous and expensive, it could be more cost effective to sacrifice some overall wave energy conversion efficiency to ensure the safety of the system [23]. An experimental study was carried out by Ashlin et al. [24] to investigate the

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Air orifice Air chamber

Incident wave

Oscillating water Column

Fig. 1. Sketch of an OWC device.

horizontal and vertical wave forces on the OWC devices. Their study was mainly focused on the effects of the wave steepness and the relative water depth on the wave loads. To provide better understanding of the wave forces acting on the OWC device and provide a guidance for the device design and safe operation, it is important to find out the factors that affect the wave dynamics through a systematic study. In the present study, the effects of wave conditions and chamber geometry on the wave loads on the front wall are studied both numerically and experimentally over a wide range of wave conditions and OWC chamber dimensions. The rest of the present paper is organized as follows. The numerical model is introduced and the experimental procedure is described in Section 2 and Section 3, respectively. Then, the comparison between numerical results and experimental data, the effects of wave conditions and chamber geometry on the wave loads on the front wall are discussed in detail in Section 4. Finally, the conclusions of this study are summarized in Section 5.

2. Numerical model To investigate the hydrodynamic performance of a land-fixed OWC, the two-dimensional (2-D) fully nonlinear numerical wave flume based on the potential theory and time-domain HOBEM developed by Ning et al. [1] was used to simulate the wave interaction with the fixed OWC device. The numerical model is extended to simulate the nonlinear wave forces on the structure in the present study. The sketch of the numerical wave flume is shown in Fig. 2. A Cartesian coordinate system Oxz is chosen with its origin on the still water level, and the z-axis pointing upward. In Fig. 2, h denotes the static water depth, B the chamber width, C the thickness of the front wall, d the immersed depth of the front wall, Ld the length of the damping zone, Lo the width of the orifice and hc the

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height of the air chamber above the still water level. The potential flow theory assumes that the fluid is incompressible, inviscid, and the fluid motion is irrotational. The fluid motion can therefore be described by the velocity potential f related to the fluid velocity. In the time-domain HOBEM model by Ning et al. [1], the incident waves are generated using the inner sources in the computational domain, the governing equation is described with Poisson equation and a damping layer with a damping coefficient m1(x) at the inlet of the numerical flume is applied to absorb the reflected waves from the OWC device. To model the viscous effect due to the water viscosity and the flow separation in the potential flow model, linear damping terms can be introduced into free surface boundary conditions. This technique has been used to study water sloshing in a container [25] and wave surface elevation in the narrow gap between twin floating objects [26,27]. In this study, an artificial viscous damping term with a damping coefficient m2 is applied to the dynamic free surface boundary condition inside the OWC chamber in the numerical model. Thus, the fully nonlinear free surface boundary conditions are modified as

8 !  > !  > > d X ðx; zÞ ¼ Vf  m1 ðxÞ ! X  X0 < dt ; > > df 1 pa vf > : ¼ gh þ jVfj2   m1 ðxÞf  m2 dt 2 vn r

! where X ðx; zÞ denotes the position vector of a fluid particle on the ! free surface and X 0 ðx0 ; 0Þ the initial static position of the fluid particle, h the vertical elevation of the free surface, g the gravity acceleration, r the water density and t the time. The material derivative is defined as d/dt ¼ v/vt þ Vf∙V. The damping coefficient m1(x) is defined by

8  2 > < u x  x1 ; Ld m1 ðxÞ ¼ > : 0; x  x1

x1  Ld < x < x1

z Wave

hc x

O

d

W1

W2

Ld Sources

h

;

(2)

where x1 is the starting position of damping zone, Ld is the length of the damping zone given to be 1.5 times the incident wavelength (i.e., 1.5L, where L is the wavelength) in the present study. The artificial viscous damping coefficient m2 is determined by trial and error method by comparison with the experimental data and is only implemented inside the chamber. Outside of the chamber, the air pressure pa on the free surface is set to be zero (i.e., atmospheric pressure), while inside the chamber, the pneumatic pressure is specified on the free-surface:

o

Damping zone

(1)

C

Fig. 2. Sketch of the numerical wave flume.

B

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pa ðtÞ ¼ Cdm Ud ðtÞ;

(3)

where Cdm is linear pneumatic damping coefficient and Ud(t) the air flow velocity through the orifice. Based on the assumption of negligible spring-like effect of air compressibility inside the chamber, the air flow velocity Ud(t) can be expressed as follows:

Ud ðtÞ ¼

DV ; S0 Dt

(4)

where DV ¼ VtþDt-Vt represents the change of air volume in the chamber within each time step Dt, which can be calculated with the variation of the free surface. S0 is the cross-sectional area of the air orifice as shown in Figs. 1 and 2. The boundaries at the bottom, backward wall and front wall of the air chamber are considered impermeable. Therefore, the zero normal velocity condition is imposed as follows:

vf ! ¼ 0; vn

(5)

! where n is the unit normal vector with a positive direction point out of the fluid domain. To solve the above boundary value problem in the time domain, the initial conditions are given as

fjt¼0 ¼ hjt¼0 ¼ 0:

(6)

By applying Green's second identity to the fluid domain U, the above boundary value problem can be converted into the following boundary integral equation:

að pÞfð pÞ ¼

Z  G

vGðp; qÞ vfðqÞ  Gðp; qÞ dG fð qÞ vn vn

Z

þ

(7)

¼ r

Z 

Gb

where q*t is the time derivative of q*, related to the incident wave acceleration and can be obtained analytically. On the free surface Gf, ft satisfies the Bernoulli equation as

vf 1 ¼ g h  jVfj2 : vt 2

 vf 1 pa vf þ g h þ jVfj2 þ nxðzÞ dG;  m2 vt 2 vt r

(10)

As the OWC device is fixed, the boundary condition for ft on the surface of the OWC walls boundaries can be written as

vft ! ¼ 0: vn

(11)

By replacing f by ft, the equation to be solved (Eq. (7)) becomes

Z 

ft ðqÞ

Z

þ

vGðp; qÞ vf ðqÞ  Gðp; qÞ t dG vn vn

q*t Gðp; qÞdU:

(12)

U

! p w n dG

Gb

(9)

G

q* Gðp; qÞdU;

where G represents the entire computational boundary, p and q represent the source point (x0, z0) and the field point (x, z), respectively, and a is the solid angle coefficient determined by the surface geometry at a source point position. q* is the pulsating volume flux density of the internal sources related to the incident velocity. G is the simple Green function, and can be written as G (p, q) ¼ lnr/2p, where r ¼ [(x-x0)2þ(z-z0)2]1/2. More details regarding the numerical model can be found in Ning et al. [1]. Once Eq. (7) is solved, the velocity potential on the body surface is known. Then the pressure can be obtained from the Bernoulli equation, and the hydrodynamic force F ¼ {fx, fz} on the front wall of the OWC device can be calculated from the following integration of the transient wave pressure over the wetted surface of the wall (Gb) as

Z

V2 ft ¼ q*t ;

aðpÞft ð pÞ ¼

U

fxðzÞ ¼

the outer side of the wall. Due to the fact that the viscous effect is considered in the free surface boundary conditions in the chamber, the viscous term can not be considered in the calculation of the force on the inner side of the wall. The main difficulty to calculate the forces is to accurately evaluate the time derivative of the potential, because the common backward difference scheme for estimating the time derivative term is inaccurate and prone to instability. The method used in the present study to avoid the numerical instability and inaccuracy is the so-called acceleration-potential method, an effective method first mathematically formulated by Tanizawa [28] and successfully used by Koo and Kim [29]. In the acceleration-potential method, the governing equation and boundary conditions are rewritten to calculate the time derivative of the potential. The temporal derivative of the velocity potential satisfies the Poisson equation as

(8)

where pw is the hydrodynamic pressure on the wetted surface. The fourth term (i.e., pa/r) on the right side of equation (8) represents the pressure from the air pressure in the chamber and it is only implemented in the calculation of the wave force on the inner side of the wall. The last term (i.e., m2vf/vt) in the equation is the viscous term and it is only applied in the calculation of the wave force on

To solve Eq. (12), the same method can be used as Eq. (7). Once ft is obtained, the hydrodynamic forces can be calculated from Eq. (8). The acceleration-potential method is more accurate and more stable than the simple backward difference scheme, especially for the structures piercing through the free surface. 3. Experimental setup The physical model tests were carried out in the wave-current flume at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, China. The wave-current flume is 69 m long, 2 m wide and 1.8 m high and is equipped with a piston-type unidirectional wave maker, which can generate regular and irregular waves with periods from 0.5 s to 5.0 s. The test section of the wave flume was divided into two parts along the longitudinal direction, which were 1.2 m and 0.8 m wide, respectively. The sketch of the experimental setup is shown in Fig. 3 (a). The fixed OWC model was placed 50 m (about 9e30 times of the wavelength) from the wave maker in the 0.8 m wide part of the flume. This long distance enables us to obtain enough data before the reflected waves from the OWC model in the opposite direction of the incident waves reaches the wave maker. To avoid wave energy transfer through the device, the model was designed to span across the width and depth of the flume. The main body of the model was made of 8 mm thick Perspex sheets. The circular orifice situated on the roof of the chamber that represents the power take-off system is 0.2 m from the front wall.

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329

Wave Maker

S7

S8 WG

WG

(a)

h

S3

S6

S2

S5

S1

S4

hc

d

C

B

2m

50 m

(b)

Pressure Sensors

Fig. 3. (a) Sketch of the experimental setup and (b) A photo of the model OWC.

In Fig. 3 (a), D denotes the diameter of the orifice and other notations are the same with it in the numerical wave flume. In the experiment, two pressure sensors (S7 and S8) were placed 2 cm from the edge of the orifice to measure the air pressure in the chamber. Six pressure sensors were symmetrically fixed on the both sides of the front wall to record the wave pressure as shown in Fig. 3 (a) and (b). One pair of them (S1 and S4) were situated at 1.5 cm above the bottom edge of the wall, one pair (S2 and S5) were fixed at the middle point of the submerged part of the front wall and the third pair (S3 and S6) were fixed at the positions of the still water surface. Two wave gauges (WG) were used to record the instantaneous water surface elevation. One was placed 2 m upstream the model and the other was placed at the mid-point of the

Table 1 Test groups in the experiments.

Test Test Test Test Test Test Test

group group group group group group group

1 2 3 4 5 6 7

B (m)

d (m)

D (m)

Hi (m)

T (s)

0.55 0.70 0.85 0.55 0.55 0.55 0.55

0.14 0.14 0.14 0.17 0.20 0.14 0.14

0.06 0.06 0.06 0.06 0.06 0.04 0.08

0.04e0.14 0.06 0.06 0.06 0.06 0.06 0.06

1.037e2.350 1.037e2.350 1.037e2.350 1.037e2.350 1.037e2.350 1.037e2.350 1.037e2.350

chamber. The sample frequency of the pressure and the water surface elevation are 50 Hz. The test for each case was repeated twice and it was found that the differences between the two repetitions are less than 5%. The average measured data of the tests was used as the final data. Seven groups of experiments were carried out to investigate the effects of the incident wave height, the chamber width, the front wall draft and the orifice diameter on the wave pressure on the front wall. The test cases are listed in Table 1. The front wall thickness C ¼ 0.04 m and the chamber height hc ¼ 0.20 m were remained constant in the experiments. By keeping the still water depth constant at h ¼ 0.8 m, different wave conditions were carried out in each test group, including 6 wave heights Hi in the range of 0.04e0.14 m and 13 wave periods T in the range of 1.037e2.350 s. Note that for the cases of testing the influence of the chamber geometry, only wave height Hi ¼ 0.06 m was used. For each wave condition, the incident wave parameters were recorded by conducting a test of wave propagation in the wave flume without the OWC model, and the wave parameters at the position where the OWC model was to be fixed were measured. Then the OWC model was placed and the test was conducted. Each wave condition has been run for 4096 time steps (time step Dt ¼ 0.02 s, totally about 35e79 waves were sampled), to ensure the stability of the experimental data.

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Fig. 4. Time series of the simulated (solid line) and measured (dashed line) surface elevation at the chamber center for different wave periods T ¼ 1.423 s and 1.735 s.

Fig. 5. Time series of the simulated and measured air pressure in the chamber for different wave periods T ¼ 1.423 s and 1.735 s.

4. Results and discussions 4.1. Comparison of numerical and experimental results In the experiments, the geometry parameters are: chamber width B ¼ 0.55 m, front wall draft d ¼ 0.14 m and orifice diameter D ¼ 0.06 m. In the 2-D numerical simulations, the air orifice width is Lo ¼ 0.0036 m, which is equivalent to the area of the circularshaped air orifice in the physical model. The other parameters in the numerical model are kept the same as in the experiments. By calibrating against the experimental data, the viscous coefficient and the linear pneumatic damping coefficient are chosen as m2 ¼ 0.2 and Cdm ¼ 9.5, respectively. The length of the numerical wave flume is 5L, in which 1.5L at the left side is used as the damping layer. In all cases, there are 30 mesh segments per wave length on the free surface, 15 mesh segments are distributed on the front wall surface, and 10 mesh segments are used across the depth of the numerical wave flume. For each case, 30 wave periods are simulated with a time step of Dt ¼ T/80. Figs. 4 and 5 show the time histories of the free surface elevation at the chamber center and the air pressure in the chamber for wave periods of T ¼ 1.423 s and T ¼ 1.735 s, respectively. The incident wave height is Hi ¼ 0.06 m. The predicted amplitudes of the wave surface elevation and the air pressure agree well with the experimental data. The higher harmonics with very small amplitudes in the surface elevations was not captured by the numerical model. This is due to the fact that the linear pneumatic model is unable to predict the higher harmonics generated by the interaction between the high frequency wave and the inhaled air flow [6]. Because the high frequency component is very weak in the wave surface elevation, the simplified pneumatic model can well simulate the interaction of water and air in the chamber.

The time series of wave pressures at T ¼ 1.423 s and T ¼ 1.735 s are shown in Fig. 6. To prove that the viscous term introduced in Eq. (8) to calculate the hydrodynamic force on the outer side of the front wall is rational, the comparison between the measured and simulated (including both conditions with and without the viscous term) pressures at the different measuring points are also given in the figure. Good agreements between the simulations and the experiments of the wave pressure on inner side of the front wall are achieved as shown in Fig. 6 (b) and (d). From Fig. 6 (a) and (c), it can be seen that while the simulated amplitude without the viscous term is larger than that of the laboratory measurements, the results with viscous term agree well the measured data. This indicates that though the vortex and flow separation can be observed clearly near the front wall according to the flow field studies [8,17,30], their effects on the pressure on the front wall can be simulated well by introducing an artificial viscous term to the pressure expression. Thus, the simulations in the following text are presented with viscous term considered. What's more, though both the measuring points are at still water surface, the pressures at S3 are zero when the free surface is below the still water surface, and zero pressure is not observed at S6, because of the air pressure fluctuation in the air chamber.

4.2. Hydrodynamic pressure In this subsection, the hydrodynamic pressure distribution on the front wall is investigated. The following parameters are fixed: the water depth h ¼ 0.8 m, the wave height Hi ¼ 0.06 m, the chamber width B ¼ 0.55 m, the front wall draft d ¼ 0.14 m, the air orifice diameter D ¼ 0.06 m (air orifice width Lo ¼ 0.0036 m), the front wall thickness C ¼ 0.04 m and the chamber height hc ¼ 0.2 m. To better understand the hydrodynamic pressure on the front

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331

Fig. 6. Comparisons of the simulated and measured wave pressures at different measuring points.

Fig. 7. Time series of the hydrodynamic pressure obtained from the experiments.

wall. Firstly, the hydrodynamic pressure time series is shown in Fig. 7, from which we can observe that the largest crests of the pressure on the outer and inner sides of the front wall are at S3 and S6, respectively. The smallest trough on the outer sides of the front

wall is at S1. The pressure curves are distorted at the measuring points on the inner side as shown in Fig. 7 (b), which may be due to the combined effects of multi-reflections and air pressure fluctuation in the air chamber. Then the maximum hydrodynamic

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Fig. 8. Crest of hydrodynamic pressure at different positions on the front wall versus kh.

Fig. 9. Pressure distributions on the front wall due to the wave crest and wave trough actions (T ¼ 1.735 s).

pressures at different positions on the front wall versus kh are plotted in Fig. 8. Although some differences exist in the simulations and laboratory measurements, the overall good agreements are achieved between them. Both simulations and the experiments show that the maximum pressure on the outer side of the front wall (i.e., S1, S2 and S3) firstly decreases with increasing kh in a very small kh range and then increases with further increasing kh, while the maximum pressure on the inner side of the front wall (i.e., S4, S5 and S6) decreases with increasing kh continuously. Note that, the pressure on the outer side of the wall is usually larger than that on the inner side except several cases in the low-frequency region. To provide a direct description for the pressure on the front wall, the pressure distribution on the front wall under the action of wave crest and wave trough are shown in Fig. 9 (a) and (b), respectively. It can be seen that the positive maximum pressure on the outside of the front wall occurs on the free surface under the action of the

Fig. 10. Maximum horizontal wave forces versus relative wavelength (Hi ¼ 0.06 m).

D.-Z. Ning et al. / Energy 115 (2016) 326e337

wave crest. While under the action of wave trough, the pressure is negative and the absolute value of the pressure on the inside surface is greater than that on the outside surface of the front wall in this case. That is to say, the dynamic wave pressure directing the landward side is produced under the action of wave trough, which is different from the wave action on a bottom mounted vertical wall. As for the latter, the pressure directs seaward. 4.3. Effect of wave condition In this subsection, the influence of the wave condition on the wave force on the front wall of the OWC device is investigated by the validated numerical model. The experimental wave force is not measured and presented. The following parameters are kept constant: the water depth h ¼ 0.8 m, the chamber width B ¼ 0.55 m, the front wall draft d ¼ 0.14 m, the air orifice width Lo ¼ 0.0036 m, the front wall thickness C ¼ 0.04 m and the chamber height hc ¼ 0.2 m. 4.3.1. Effect of incident wavelength Fig. 10 shows the variations of the dimensionless maximum horizontal wave forces on the front wall with the relative wavelength L/h. The maximum total horizontal wave force Fhm, the maximum horizontal wave forces on the outer side of the front wall Fom and inner side of the front wall Fim are discussed, where the subscript “m” stands for the maximum force. It can be seen that Fhm decreases with the increase of the relative wavelength L/h. In addition, an opposite tendency is observed for Fim. Fom initially decrease to its minimum and then increases with the increase of L/ h. The variation of the wave forces with L/h can be explained using the relationship between the wave transmission ability and

Fig. 11. Snapshot of wave elevation along the wave flume at t ¼ T/4 and t ¼ 3T/4 (Hi ¼ 0.06 m).

333

wavelength as following texts. Fig. 11 (a), (b) and (c) show the snapshots of wave elevation along the wave flume for different wave periods of T ¼ 1.037 s, 1.545 s and 2.350 s, respectively. The corresponding wavelength to the water depth ratios are L/h ¼ 2.1, 4.2 and 7.4, respectively. From Fig. 11 (a) we can see that, the fluctuation of the water free surface on the inner surface of the front wall is very weak, and the free surface on the inner wall surface almost stays at the still water surface level. But the amplitude of the wave elevation outside the chamber is nearly twice the incident wave amplitude. Almost full reflection in front of the front wall happens because of the weak transmission ability of the short wave. Thus, the maximum wave force on the outer side of the wall Fom is large and the maximum wave force on the inner side of the wall Fim is small. The time history and the spectrum analysis of the wave force on the front wall for the incident wave period T ¼ 1.037 s shown in Fig. 12 (a) and Fig. 13(a) can further explain this. From the figures, we can see that the total horizontal wave force Fhm is mainly contributed by Fom because of the large wave reflection from the front wall. That is to say, under the action of short waves, the horizontal wave force on the front wall is mainly contributed by the incident and reflected waves outside the chamber. As the wave period increases to T ¼ 1.545 s, its corresponding wavelength is increased to L ¼ 3.367 m, the transmission ability of the wave is enhanced. Then from Fig. 11 (b) we can see that the fluctuation of the inner surface is much greater than that for T ¼ 1.037 s and the outside wave elevation is decreased. Consequently, the wave force on the outer surface of the front wall Fom is decreased and the wave force on the inner surface of the front wall Fim is increased. With the further increase in the wave period to T ¼ 2.350 s, the transmission ability of the wave is further enhanced. From Fig. 11 (b) and (c) we can see that, the surface elevation in the wave flume is greater than that for T ¼ 1.545 s. This is because that the long waves are mostly transmitted into the chamber through the front wall, and then is totally reflected by the rear wall of the chamber. The reflected waves from the rear wall also pass through the front wall with very little reflection. Further, notice that the wave surface motions on the inner and the outer surfaces of the front wall are almost inphase with each other and have similar motion amplitudes. As the result, both Fi and Fo increases and there is a T/2 phase difference between them. The crest of Fi and the trough of Fo with similar values, and it's vice versa with the trough of Fi and the crest of Fo. Therefore, the value of Fh is relatively small as shown in Fig. 12 (b). The total horizontal wave force Fh has mainly two frequency components with comparable energy as shown in Fig. 13 (b). 4.3.2. Effect of wave nonlinearity Fig. 14 shows the variations of the maximum total horizontal wave forces on the front wall with the dimensionless wave number kh for wave heights Hi ¼ 0.04 m, 0.06 m and 0.08 m. It can be clearly observed that the force increases with the increase of wave height. A small reduction of the force near kh ¼ 1.6 can also be noted. This is because of the high hydrodynamic conversion efficiency at resonant frequency. The reduction of the force near the resonant frequency is also observed in the experimental results presented by Ashlin et al. [24]. They found that at the natural frequency of system, the force on the structure is small due to high-energy absorption by the OWC. Fig. 15 shows the variations of the maximum horizontal wave forces on the front wall with wave steepness Hi/L for different wave periods T ¼ 1.423 s and 1.735 s. We can see that the wave forces increase with the increase of Hi/L. It is because that the wave nonlinearity becomes strong with the increasing wave steepness.

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Fig. 12. Time history of the horizontal wave force on the front wall for different wave periods T ¼ 1.037 s and 2.350 s (Hi ¼ 0.06 m).

Fig. 13. Spectrum analysis of the horizontal wave force on the front wall for different wave periods T ¼ 1.037 s and 2.350 s (Hi ¼ 0.06 m).

Fig. 14. Maximum horizontal wave forces versus dimensionless wave number kh for different wave heights Hi ¼ 0.04 m, 0.06 m and 0.08 m.

At the same time, due to the shield of the front wall, the increment ratio of the wave force on the inner side is lower than the others. 4.4. Effect of geometry parameters In this subsection, the influence of three geometry parameters: the chamber width B, the front wall draft d and the air orifice width Lo on the total horizontal wave force on the front wall of the OWC device is investigated. When the influence of one of the three geometry parameters is investigated, the rest two parameters are kept constant. Simulations are performed for a constant wave height of Hi ¼ 0.06 m, and the incident wave periods T varying from 1.037 s to 2.350 s.

4.4.1. Effect of chamber width The influence of the chamber width on the total horizontal wave force is analyzed by varying the chamber width (B ¼ 0.55, 0.70 and 0.85 m) and keeping the other geometry parameters constant as d ¼ 0.14 m and Lo ¼ 0.0036 m. Fig. 16 shows the dimensionless maximum total horizontal wave force for the three simulated chamber width. Due to the strong transmission ability of long waves, the chamber width has little influence on the total horizontal wave force in the low-frequency region. While the wave force decreases with the increase of chamber width B in the high-frequency region. To explain this phenomenon, the time series of the surface elevation at both the outer and inner sides of the front wall (i.e., W1 and W2) at wave period T ¼ 1.037 s is shown in Fig. 17, which are directly related to the wave force. We can see that the surface elevation at the outer side of the wall (i.e., W1) is insensitive to the chamber width. However, the surface elevation at the inner side of the wall (i.e., W2) increases with the increase of chamber width to some extent. That is to say, the wave force on the outer side of the wall changes little, while the force on the inner side of the wall increases with the increasing of the chamber width. Thus, the total force decreases with the increasing chamber width in the high-frequency region.

4.4.2. Effect of front wall draft Fig. 18 shows the dimensionless maximum total horizontal wave force Fhm obtained from front wall draft d ¼ 0.14, 0.17 and 0.20 m and by keeping the other parameters B ¼ 0.55 m and Lo ¼ 0.0036 m constant. It can be observed that the effect of the draft on the dimensionless maximum total horizontal wave force is very limited. This may be due to the fact that the variation of the draft itself is not obviously. But it should be noted that the total wave

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335

Fig. 15. Maximum horizontal wave forces versus wave steepness Hi/L for different wave periods T ¼ 1.423 s and 1.735 s.

Fig. 16. Maximum total horizontal wave force on the front wall versus kh for B ¼ 0.55 m, 0.70 m and 0.85 m.

Fig. 17. Time series of surface elevation at W1 and W2 for B ¼ 0.55 m, 0.70 m and 0.85 m at T ¼ 1.037 s.

force actually increases with the increasing of draft.

4.4.3. Effect of air orifice scale The size of an opening can be described by the opening area ratio ε ¼ S0/S, where S0 and S are the cross-sectional areas of the orifice and the air chamber, respectively. The analysis of the influence of the air duct width on the dimensionless maximum total horizontal wave force Fhm is carried out by keeping the chamber width B ¼ 0.55 m and front wall draft d ¼ 0.14 m constant. Three air orifice width Lo ¼ 0.0016, 0.0036 and 0.0064 m (with the same area

Fig. 18. Maximum total horizontal wave force on the front wall versus kh for d ¼ 0.14 m, 0.17 m and 0.20 m.

Fig. 19. Maximum total horizontal wave force on the front wall versus kh for Lo ¼ 0.0016 m (ε ¼ 0.29%), 0.0036 m (ε ¼ 0.66%) and 0.0064 m (ε ¼ 1.17%).

of the circular-shaped orifice diameter of D ¼ 0.04, 0.06 and 0.08 m in the physical test) correspond to the opening ratios of 0.29%, 0.66% and 1.17%, respectively. Fig. 19 shows the dimensionless maximum total horizontal wave force curve for each opening ratio. It can be noticed that the dimensionless maximum total horizontal wave force Fhm decreases with increasing opening ratio ε in the low-frequency region. This is because the surface-elevation difference between the outer side and inner side of the front wall decreases gradually with the increasing opening ratio in the low-frequency region. And the surface elevation at the inner side of the front wall even exceeds

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Fig. 20. Variation of the (a) surface elevation at W1 and W2 and (b) air pressure in the chamber for different orifice widths.

that at the outer side when the opening width is large enough as shown in Fig. 20 (a). Thus, the wave forces on the both sides of the wall are getting closer to each other with increasing opening ratio, which results in the total wave force decreasing. However, an opposite trend is observed in the high-frequency region, namely the wave force Fhm increases with increasing the opening ratio. Though the surface elevations are very similar under different opening ratios in the high-frequency region (i.e., the hydrodynamic pressures are similar), the air pressures in the chamber are much different. The air pressure in the chamber transmits to the water surface, and then forwards to the inner side of the front wall in the form of dynamic pressure. The smaller the opening ratio, the larger the air pressure as shown in Fig. 20 (b). As a result of that, the dynamic pressure on the inner side is increased. 5. Conclusions Both numerical simulations and laboratory tests were carried out to investigate the wave forces on the front wall of a fixed OWC device. The aim of the work is to study the effects of wave parameters and the dimensions of the OWC chamber on wave forces acting on the device and provide guidance for the geometry design. Though some small differences exist in the simulations and experiments, such as the higher harmonic of the surface elevation not predicted well. The overall good agreements of the surface elevation and air pressure in the chamber between simulations and experiments suggest that the hydrodynamic performance of the fixed OWC device can be simulated successfully by the timedomain HOBEM model with the properly calibrated pneumatic damping coefficient and artificial viscous damping coefficient. It is particularly worth mentioning that the effect of vortex and flow separation, which occur nearly the front wall, on the pressure on the front wall can be simulated well by introducing another artificial viscous term to the dynamic pressure expression. The largest wave pressure occurs on the outside of the front wall on the free surface under the action of the wave crest. It was found that the wave number affects the pressure significantly. While the dimensionless wave pressure on inner surface of the front wall decreases with the increase of dimensionless wave number kh, the pressure on the outer surface shows an opposite trend. The total dimensionless horizontal wave force decreases with increasing wavelength and increases with increasing wave height. For a given chamber geometry, under the action of short waves, the total horizontal force on the front wall is mainly due to the large amplitude water surface oscillation on the outer surface of the front wall, while the very small amplitude on the inner surface makes little contribution to the total force. And under the action of long waves, the total wave force is small because the crest (trough) value of force on outer side of the front wall and the trough (crest) value

of force on inner side of the front wall are of similar values and there is a T/2 phase difference between them, although their magnitudes are large. Although the chamber width has little influence on the total horizontal wave force in the low-frequency region, the wave force decreases with the increasing chamber with in the high-frequency region. In general, the front wall draft has very limited effect on the total dimensionless horizontal wave force. The opening ratio strongly influences the wave force, i.e., in the low-frequency region, the larger the opening ratio, the smaller the wave force and it shows an opposite tendency in the highfrequency region. Note that, the present study focuses on revealing the general characteristics of the wave force on the front wall under the action of non-breaking waves with different chamber geometries. Although numerous researches have been carried out to study the hydrodynamic performance of the OWC device [1e9], most of them are carried out under the action of non-breaking waves. The extreme loading on WECs is an extremely challenging area and is likely to be the hardest area to be simulated numerically [31]. And the present potential numerical model is difficult to simulate the broken phenomenon induced by the extreme wave action. The relating research will be carried out in the future work. Acknowledgements The authors also would like to gratefully acknowledge financial support from the National Natural Science Foundation of China (Grant Nos. 51490672, 51679036), the Program for New Century Excellent Talents in University (Grant No. NCET-13-0076) and HighTech Ship Research Projects Sponsored by the Ministry of Industry and Information Technology (MIIT) of China. References [1] Ning D-Z, Shi J, Zou Q-P, Teng B. Investigation of hydrodynamic performance of an OWC (oscillating water column) wave energy device using a fully nonlinear HOBEM (higher-order boundary element method). Energy 2015;83: 177e88. [2] Dizadji N, Sajadian SE. Modeling and optimization of the chamber of OWC system. Energy 2011;36(5):2360e6.  Y, Lewis A. 3D hydrodynamic modelling of fixed oscillating water [3] Delaure column wave power plant by a boundary element methods. Ocean Eng 2003;30(3):309e30. [4] Morris-Thomas MT, Irvin RJ, Thiagarajan KP. An investigation into the hydrodynamic efficiency of an oscillating water column. J Offshore Mech Arct Eng 2007;129(4):273e8. [5] He F, Huang Z, Law AW-K. An experimental study of a floating breakwater with asymmetric pneumatic chambers for wave energy extraction. Appl Energy 2013;106:222e31. [6] Ning D-Z, Wang R-Q, Zou Q-P, Teng B. An experimental investigation of hydrodynamics of a fixed OWC wave energy converter. Appl Energy 2016;168: 636e48. [7] Zhang Y, Zou Q-P, Greaves D. Airewater two-phase flow modelling of hydrodynamic performance of an oscillating water column device. Renew

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