Numerical prediction of the natural frequency of an Oscillating Water Column operating under resonant conditions

Abstract

Among the different technologies developed in order to harness wave energy, the Oscillating Water Column devices are the most accredited for an actual diffusion. Recently, Boccotti has patented the REWEC1 (REsonant sea Wave Energy Converter solution 1), a submerged breakwater that performs an active coast protection, embedding an Oscillating Water Column device, which is capable of operating under resonant conditions with that sea state, which gives the highest yearly energy contribution. The REWEC1 dynamic behavior can be approximated by means of a mass-spring-damper system. According to this approximation, a criterion for evaluating the oscillating natural frequency of the REWEC1 has been derived. This criterion has been validated against both experimental results and computational fluid dynamics simulations, performed on a REWEC1 laboratory-scale model. The numerical simulations have shown a good agreement between measurements and predictions.

Keywords

Renewable energy wave energy converter oscillating water column computational fluid dynamics damped harmonic oscillations

Introduction

The first examples of patents on wave energy converters (WECs) were filed more than two centuries ago. Until today, more than 500 devices have been proposed, such as “rocking ducks” (Salter, 1974), “articulated rafts” (Cockerell, 1980); McCabe, 1992), “oscillating buoys” (Falnes and Budal, 1978), and “oscillating water columns” (Wells, 1993). Despite the inventive fervor, only few of these WECs were actually manufactured and tested. The largest obstacle to the commercial development of WECs is the high total cost. For instance, in the case of Oscillating Water Column (OWC) devices, the construction of a reinforced concrete structure is required and this can constitute up to 66% of total costs. A possible solution to reduce these costs is the insertion of the OWC device inside a breakwater. An example of this type of installation is the one operating in the port of Sakata (Takahashi et al., 1992) since 1989. In this way, the incidence of the cost due to the structure for the OWC is drastically reduced, and the plant fulfills a double purpose: to defend the stretch of water behind it and to produce electrical energy by means of generators connected to one or more turbines (for instance, high solidity Wells turbines). The Wells turbine is actually a self-rectifying axial flow turbine (Raghunathan, 1995) suitable for the conversion of the kinetic energy of the oscillating air flow into unidirectional mechanical energy of the turbine shaft. Another possibility is the use of impulse turbines (Thakker and Hourigan, 2004). Both Wells and impulse turbines have been examined in various configurations for wave energy conversion (e.g. Setoguchi and Takao, 2006). In order to improve the energy conversion process, both high and low solidity Wells turbines need to be characterized by good efficiencies over a wide range of flow rates (Curran et al., 1997; Raghunathan, 1995). Furthermore, turbine performance has to be optimized not only considering the turbine as a standalone machine but also as a whole together with the OWC plant. Several experimental analyses have been carried out in order to investigate the performance of Wells turbines. For instance, wind tunnel tests have been considered under both constant or reciprocating flow conditions (Torresi et al., 2007), using scaled devices (Filianoti and Camporeale, 2009), or in operative plants, such as the LIMPET (Heath, 2009) and the PICO ones (Falcao, 2001). More detailed investigations of the fluid dynamics of Wells turbines have been carried out by means of computational fluid dynamics (CFD) analysis, regarding hydraulic low solidity Wells turbines (Camporeale et al., 2003), or high solidity Wells turbines (Dhanasekaran and Govardhan, 2005; Torresi et al., 2008). More recently, Boccotti (1998, 2001) invented a family of WECs named REWEC1 (REsonant sea Wave Energy Converter solution 1), which consists of several caissons placed close to each other, in order to form a submerged breakwater (Figure 1(b)). Each caisson encloses a pressurized air volume, which is connected to a completely submerged vertical duct open to the sea, where a hydraulic low solidity Wells turbine can be placed for the direct conversion of the kinetic energy of the OWC into mechanical energy (Torresi et al., 2004).

Figure 1.

(a) Conventional breakwater and (b) REWEC1 schematics.

Actually, the incident waves cause fluctuations of the water free surface inside the pressurized caisson, and the alternate exhalation and inhalation of the water in the vertical duct drive the turbine. The main characteristic of the REWEC1 is that it can be designed in order to operate under resonance conditions with the characteristic frequency of the sea state, which gives the greatest yearly energy contribution. Under such conditions, the amplitude of the fluctuations of the water free surface inside the pressurized caisson increases, increasing the amount of kinetic energy of the OWC. Actually, the REWEC1 dynamic behavior can be approximated by means of a mass-spring-damper system. According to this approximation, a criterion for calculating the oscillating frequencies of the REWEC1 can be derived. This criterion has been validated against both the experimental results and CFD simulations, performed on a REWEC1 laboratory-scale model. Previously, CFD simulations were also carried out in order to study the dynamic behavior of a scaled model placed in a wave flume (Scarpetta et al., 2015).

Experimental set-up

The REWEC1 (Figure 1(b)) is essentially a submerged caisson with a U-tube inside. The first branch of the U-tube is a vertical duct connecting the free water surface to the second branch, a chamber in which an air pocket is generated by means of compressed air. The value of the REWEC1 undamped natural frequency, $f_{n}$ , depends on the value of the height of the air pocket inside the chamber, $ζ_{0}$ , under resting conditions (Figure 2(a)). In order to reach the resonance conditions with the frequencies of the fundamental seas, the value of $ζ_{0}$ can be modified varying the value of the pressure inside the chamber. In order to replicate the REWEC1 dynamic behavior in laboratory, an experimental test rig (Filianoti, 2004) was specifically designed (Figure 2(b)), in order to generate free damped harmonic oscillations. A lower caisson replicates the REWEC1, whereas an upper caisson is used in order to simulate the REWEC1 submergence. The use of the upper caisson is needed in order to be able to generate the non-equilibrium initial condition, as will be explained in the test procedure description. As a result of the free damped harmonic oscillations, the value of the damped natural frequency, $f_{d}$ , was obtained. Then, taking into account the damping ratio, $ξ$ , the value of the undamped natural frequency, $f_{n}$ , was derived

f_{n} = \frac{f_{d}}{\sqrt{1 - ξ^{2}}}

(1)

All the geometrical parameters (Figure 2(a)) of the experimental device are summarized in Table 1.

Figure 2.

(a) Section of the experimental device and (b) the experimental device.

Table 1.

Geometrical dimension of the experimental device.

Geometrical items	Dimension (m)
$L_{c}$	0.85
c	0.797
w	0.1
t	0.153
b	0.366
d	0.797
s	0.032

The test procedure is as follows:

The valves of both upper and lower caissons are opened, connecting each air pocket with the atmosphere.

A specified amount of water is introduced in the upper caisson and part of it overflows in the lower one. During this operation, the valve in the lower caisson is closed, in order to generate an air pocket inside it with a $ζ$ lower than $ζ_{0}$ .

Air is pumped inside the lower caisson until the desired resting condition is achieved.

In order to generate the initial non-equilibrium condition ( $ζ_{i n} \neq ζ_{0}$ and $p_{2, i n} \neq p_{2, 0}$ ), the valve of the upper caisson is closed and air is pumped inside it, reaching $p_{1} \neq p_{a t m}$ .

When the valve of the upper caisson is opened, the value of the pressure in the upper caisson drops to the atmospheric value and the free damped harmonic oscillations occur.

In order to evaluate $f_{d}$ and $ξ$ , the values of the pressure inside the air pocket of both the upper, $p_{1}$ , and the lower, $p_{2}$ , caissons were measured by means of pressure transducers. Table 2 summarizes the initial and resting conditions during the test. The results of this test have been compared with those computed by the analytical model and the CFD simulations.

Table 2.

Initial and resting conditions of the CFD simulations.

Initial conditions			Resting conditions
$ζ_{i n}$ (m)	$p_{1, i n}$ (mbar)	$p_{2, i n}$ (mbar)	$ζ_{0}$ (m)	$p_{1, 0}$ (mbar)	$p_{2, 0}$ (mbar)
0.452	1204	1282	0.527	1013	1097
0.60	800	911	0.513	1013	1112

CFD: computational fluid dynamics.

Analytical model

An analytical model, describing the behavior of the water flow inside a REWEC1, has been developed under the simplifying hypothesis of one-dimensional and incompressible flow. Actually, the REWEC1 is essentially a U-tube (Figure 3) and its fluid-dynamic behavior can be correctly described along a streamline, which starts from the air–water interface inside the lower caisson (point A) and ends at the air–water interface inside the upper caisson (point B).

Figure 3.

Streamline describing the fluid dynamics of water inside the REWEC1.

Applying the generalized Bernoulli equation along that streamline, one obtains

\frac{1}{g} \int_{A}^{B} \frac{\partial u}{\partial t} d s = H_{A} - H_{B} - \sum H_{W}

(2)

where $H_{A}$ and $H_{B}$ are the total heads in the points A and B, respectively, while $\sum H_{W}$ is the sum of the head losses along the streamline. The left hand side of Equation 2 is actually the integral of the Euler equation and, for the sake of simplicity, can be computed considering only the two vertical paths along the streamline, neglecting the contribution of the local inertia of the water below the bottom end of the duct. The first path goes from the air–water interface inside the lower caisson to the bottom opening of the vertical duct (C); the second goes from the aforesaid aperture to the air–water interface inside the upper caisson. Hence, the integral can be rewritten as follows

\frac{1}{g} \int_{A}^{B} \frac{\partial u}{\partial t} d s = \frac{1}{g} \int_{A}^{C} \frac{\partial u}{\partial t} d s + \frac{1}{g} \int_{C}^{B} \frac{\partial u_{w}}{\partial t} d s

(3)

where, in the first integral, $u$ can be expressed as the derivate of the height, $ζ$ , of the air pocket inside the lower caisson, while $u_{w}$ is the velocity in the vertical duct. According to the continuity equation, the velocity $u_{w}$ can be written as a function of $u$ as follows

u_{w} = \frac{b}{s} \frac{d ζ}{d t}

(4)

where $b$ and $s$ represent the cross sections of the lower caisson and the vertical duct, respectively (Figure 2(a)). In this model, we assume small values of the oscillations of $ζ$ and $a$ , with respect to the height of the air pocket in resting condition, $ζ_{0}$ . According to this assumption, the values of $ζ$ and $a$ are always assumed to be equal to $ζ_{0}$ and $a_{0}$ in the geodetic heads. Therefore, the integration along the first and the second path gives

\frac{1}{g} \int_{A}^{B} \frac{\partial u}{\partial t} d s = \frac{1}{g} (c - w - ζ_{0}) \frac{d^{2} ζ}{d t^{2}} + \frac{1}{g} (\frac{b}{s}) L_{c} \frac{d^{2} ζ}{d t^{2}}

(5)

The total head, $H_{A}$ , is related to the absolute pressure, $p_{A}$ of the air pocket inside the lower caisson, as follows

H_{A} = - (a_{0} + t + ζ_{0}) + \frac{p_{A}}{ρ g} + \frac{1}{2 g} {(\frac{d ζ}{d t})}^{2}

(6)

Assuming a polytropic law for the thermodynamic behavior of the air inside the chamber, $p_{A}$ can be evaluated as a function of the air volume, which is directly proportional to the height, $ζ$ , of the air pocket

p_{A} = [p_{1, 0} + ρ g (a_{0} + t + ζ_{0})] {(\frac{ζ_{0}}{ζ})}^{m}

(7)

where the subscript “0” refers to the resting condition and m is the polytropic index. Actually, an isentropic law for the thermodynamic behavior of air has been assumed; then the value of the polytropic index has been considered to be equal to 1.4. The total head, $H_{B}$ , is related to the absolute pressure, $p_{B}$ , at the duct mouth, which is evaluated as a function of the pressure in the upper caisson, $p_{1, 0}$ as follows

H_{B} = \frac{p_{1, 0}}{ρ g} + {(\frac{b}{s})}^{2} \frac{1}{2 g} {(\frac{d ζ}{d t})}^{2}

(8)

The head losses, $\sum H_{W}$ , can be estimated as the sum of the continuous friction losses, $H_{c}$ , and the minor head losses in the vertical duct, $H_{d}$ , whose general form is as follows

H_{W, i} = \frac{1}{2 g} k_{i} | u | u

(9)

where $k_{i}$ can be assumed as the value of the coefficient of either the friction losses, $k_{c}$ , or minor head losses, $k_{d}$ . Therefore, the head losses are written as

\sum H_{W} = \frac{1}{2 g} {(\frac{b}{s})}^{2} (k_{c} + k_{d}) | \frac{d ζ}{d t} | \frac{d ζ}{d t}

(10)

Substituting Equations 6, 8, 10 in Equation 2 leads to a non-linear second order differential equation

[c - w - ζ_{0} + (\frac{b}{s}) L_{C}] \frac{d^{2} ζ}{d t^{2}} - \frac{1}{2} {(\frac{d ζ}{d t})}^{2} + \frac{1}{2} {(\frac{b}{s})}^{2} {(\frac{d ζ}{d t})}^{2} + \frac{1}{2} {(\frac{b}{s})}^{2} (k_{c} + k_{d}) | \frac{d ζ}{d t} | \frac{d ζ}{d t} + [\frac{p_{1, 0}}{ρ} + g (ζ_{0} + t + a_{0})] {(\frac{ζ_{0}}{ζ})}^{m} + \frac{p_{1, 0}}{ρ} + g (ζ_{0} + t + a_{0}) = 0

(11)

In order to obtain a simple solution for this problem, Equation 11 needs to be linearized, in particular the terms associated to polytropic law inside the air pocket and the head losses. The linearized form of Equation 11 becomes

A \frac{d^{2} ζ}{d t^{2}} + B \frac{d ζ}{d t} + C ζ = 0

(12)

where

A = [(c - w - ζ_{0}) + (\frac{b}{s}) L_{C}]

(13)

B = \frac{π}{8 g} (k_{c} + k_{d}) [{(\frac{b}{s})}^{2} ω ζ^{'}] + \frac{π}{8} [{(\frac{b}{s})}^{2} - 1] ω ζ^{'}

(14)

C = \frac{m}{ζ_{0}} [\frac{p_{1, 0}}{ρ} + g (ζ_{0} + t + a_{0})]

(15)

where $ζ^{'}$ is related to the amplitude of the $ζ$ oscillations in the time domain. Equation 12 expresses the equation of the damped harmonic oscillator, associated to the device, as a function of $ζ$ . Therefore, the dynamic behavior of the plant can be approximated by means of a mass-spring-damper system. According to vibration theory, the undamped natural frequency of the system is then

f_{n} = \frac{1}{2 π} \sqrt{\frac{C}{A}}

(16)

which, actually, depends on the geometry of the device and the value of the pressure inside the air pocket. In order to validate the analytical model, the value of $f_{n}$ calculated from Equation 16 will be compared to these resulting from both the experimental and CFD analyses.

CFD analysis

The CFD analysis has been carried out using a computational domain, which reproduces the 2D schematic reported in Figure 2(a)). This domain has been discretized by means of the grid generator Pointwise adopting a multi-block structured mesh. The total mesh size has approximately 60,000 cells. The numerical domain has been imported in the ANSYS Fluent solver. The physical model under investigation is described by means of the Navier–Stokes equations written for an unsteady two-phase flow. These equations are discretized according to a Finite Volume approach, adopting a pressure-based algorithm in its implicit formulation. The two-phase interaction is taken into account by means of the Eulerian Volume of Fluid (VOF) Model. Air is assumed to be compressible, in order to take into account the compressibility of the air pocket inside the chamber. The SIMPLE algorithm has been used for the pressure-velocity coupling. Regarding the spatial discretization of the equations, the Green-Gauss cell-based scheme has been used for gradients, the PRESTO! scheme for pressure, whereas first-order upwind schemes for momentum and the energy. In order to evaluate the surface of separation of the different phases, the Geo-Reconstruct model has been used. Regarding the temporal discretization, a time step t = 0.001 s has been chosen and the first-order implicit scheme has been used for the transient formulation. Every two time steps, the values of the pressure inside both the lower and the upper caissons have been registered, allowing the analysis of the dynamic behavior of the water inside the device during post-processing.

Results

In order to make a comparison with the experimental test, a numerical simulation has been carried out, setting the values of the initial and the resting conditions reported in Table 2. The graph in Figure 4 shows a comparison between the values of $p_{2}$ registered during the simulation and the experimental test (Filianoti, 2004). From the comparison, we can see that the values of the pressure registered in both cases are in quite good agreement in terms of amplitude, while the values registered in the experimental test seems to have a time delay with respect to the CFD ones. The value of the time period between the second and the third peak has been assumed to be equal to the damped natural period, $T_{d}$ , from which the values of the damped natural frequencies, $f_{d}$ , are equal to 0.893 Hz in the experimental test and 0.896 Hz in the CFD simulation.

Figure 4.

Values of $p_{2}$ registered during the simulation and during the experimental test.

From both the experimental and the CFD analyses, the value of the undamped natural frequency, $f_{n}$ , has been calculated from the value of $f_{d}$ (Equation 1), in order to make a comparison with the analytical result (Equation 16). The value of the damping ratio, $ξ$ , has been evaluated computing the logarithmic decrement, $δ$ , between the second and the third peak of the oscillations. From the value of $δ$ , the following equation has been used for the calculation of the damping ratio, $ξ$

δ = 2 π \frac{ξ}{\sqrt{1 - ξ^{2}}}

(17)

Table 3 summarizes the results derived from the analytical model, the experimental test, and the CFD simulation. Other CFD simulations have been carried out considering the same resting conditions and varying the initial ones (Table 4). From the registered value of $p_{2}$ , the values of $ξ$ and $f_{d}$ have been calculated for each simulation according to the same methods described before.

Table 3.

Comparison between the analytical, experimental, and CFD results.

	Analytical model	Experimental test	CFD analysis
$f_{n}$	0.863 (Hz)	0.898 (Hz)	0.900 (Hz)
$f_{d}$	−	0.893 (Hz)	0.896 (Hz)
$ξ$	−	0.107 (–)	0.096 (–)

CFD: computational fluid dynamics.

Table 4.

Initial and resting conditions of the CFD simulations.

Initial conditions			Resting conditions
$ζ_{i n}$ (m)	$p_{1, i n}$ (mbar)	$p_{2, i n}$ (mbar)	$ζ_{0}$ (m)	$p_{1, 0}$ (mbar)	$p_{2, 0}$ (mbar)
0.40	1442	1525	0.513	1013	1112
0.45	1221	1311	0.513	1013	1112
0.50	1052	1149	0.513	1013	1112
0.55	914	1018	0.513	1013	1112

CFD: computational fluid dynamics.

The graph reported in Figure 5(a)) shows the values of $ξ$ for the different values of $ζ_{i} n$ . The trend shown in the graph is due to the fact that the value of the damping ratio grows with the absolute value of the difference between $ζ_{i n}$ and $ζ_{0}$ due to the increase of dissipations in the water flow. The graph reported in Figure 5(b)) shows the values of $f_{d}$ for the different values of $ζ_{i n}$ . Considering that the value of $f_{n}$ remains constant, from Equation 1, we expect that also the value of $f_{d}$ grows with the absolute value of the difference between $ζ_{i n}$ and $ζ_{0}$ . As shown in Figure 5(b), the calculated values of $f_{d}$ are in agreement with this prediction only if the value of $ζ_{i n}$ is smaller than the value of $ζ_{0}$ . Finally, the graph reported in Figure 6 shows the comparison between the values of $f_{n}$ calculated from the CFD simulation and the ones calculated by means of the analytical model (Equation 16). As expected from the previous considerations, the values are in quite good agreement when the value of $ζ_{i n}$ is smaller than the value of $ζ_{0}$ .

Figure 5.

Values of (a) $ξ$ and (b) $f_{d}$ for the different values of $ζ_{i n}$ .

Figure 6.

Comparison between the values of $f_{n}$ .

Conclusion

The results derived from the CFD simulations are in good agreement with the experimental test ones, showing that the proposed 2D numerical model can be successfully used for the evaluation of both the damped, $f_{d}$ , and undamped, $f_{n}$ , natural frequencies of a REWEC1. The proposed analytical model, based on a simple analogy with a mass-spring-damper system, has been also validated, allowing a quite good prediction of the undamped natural frequency, $f_{n}$ , with respect to both numerical and experimental calculations. Finally, by means of the CFD analysis, the effect of the initial conditions on both damped, $f_{d}$ , and undamped, $f_{n}$ , natural frequencies has been considered under the hypothesis of maintaining constant the resting condition.

Footnotes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Author biographies

Torresi Marco is research assistant in Fluid Machinery since 01/03/2005 at Politecnico di Bari University. He’s got his Ph.D. in Mechanical Engineering in 2007. His research activity is mainly focused on renewable energy applications. Specifically, he works on Wells turbines for wave energy conversion devices and Vertical Axis Wind Turbines.

Filippo Scarpetta graduated in Mechanical Engineering in 2014 from Politecnico di Bari and he is currently working on his PhD degree in Mechanical Engineering at Politecnico di Bari. His research activity is mainly focused on the CFD analysis of an innovative OWC device, named REWEC (Resonant Wave Energy Converter), for the conversion of sea wave energy into electricy.

Giuseppina Martina graduated in Mechanical Engineering in 2014 at “Politecnico di Bari” University, Italy. Her expertise is the CFD simulation of wave motion. In particular, during her post-doc internship, she carried out (ANSYS Fluent) the simulations of the interaction of waves with a submerged breakwater inside a wave tank.

Pasquale Fabio Filianoti is Associate Professor in Hydraulic and Marine Constructions and Hydrology since April 2001 at the Mediterranean University of Reggio Calabria, Italy. His research activity is mainly focused on: shoreline deformations; wave-structure interactions; stability of harbor defense structures; sea state statistical properties; wave energy conversion devices; wave climate analysis.

Sergio M Camporeale is associate professor in Fluid Machinery since 2001 at “Politecnico di Bari” University, Italy. His main fields of expertise are: a) performance, dynamics, cycle innovation and diagnostics in gas turbine power plants, and, recently, modelling of thermo-acoustic combustion instabilities; b) renewable energy sources, mainly wind energy and tidal and ocean wave energy.

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