Evaluation of air compressibility effects on the performance of fixed OWC wave energy converters using CFD modelling

Accepted Manuscript Evaluation of air compressibility effects on the performance of fixed OWC wave energy converters using CFD modelling

I. Simonetti, L. Cappietti, H. Elsafti, H. Oumeraci PII:

S0960-1481(17)31223-5

DOI:

10.1016/j.renene.2017.12.027

Reference:

RENE 9524

To appear in:

Renewable Energy

Received Date:

25 January 2017

Revised Date:

20 October 2017

Accepted Date:

05 December 2017

Please cite this article as: I. Simonetti, L. Cappietti, H. Elsafti, H. Oumeraci, Evaluation of air compressibility effects on the performance of fixed OWC wave energy converters using CFD modelling, Renewable Energy (2017), doi: 10.1016/j.renene.2017.12.027

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ACCEPTED MANUSCRIPT Highlights 

The errors induced by neglecting the effect of air compressibility in modelling OWC devices are systematically evaluated and quantified.



The study is conducted by using multiphase, compressible CFD modelling of the OWC device at scale 1:50, 1:25, 1:10, 1:5 and 1 the prototype scale.



Neglecting air compressibility results in an overestimation up to about 15% for the air pressure in the OWC chamber and the subsequent air volume flux, but less than 10% for the capture width ratio.



Results are analysed in terms of dimensionless parameters; a newly defined parameter, strongly related to the compressible effects, is proposed and used to provide generalized equations delivering correction factors.



These equations might be beneficial for the correction of results obtained from incompressible numerical models as well as from small-scale laboratory tests of OWC devices.

ACCEPTED MANUSCRIPT Evaluation of air compressibility effects on the performance of fixed OWC wave energy converters using CFD modelling I. Simonetti1*, L. Cappietti1, H. Elsafti2, H. Oumeraci2 1 Dept.

of Civil and Environmental Engineering, University of Florence, Florence, Italy Leichtweiß-Instritut für Wasserbau, Division of Hydromechanics and Coastal Engineering, TU Braunschweig, Brunswick, Germany *corresponding author, e-mail address: [email protected] 2

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Abstract

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This study presents the evaluation of the error induced by neglecting the effect of air compressibility in modelling Oscillating Water Column (OWC) wave energy converters. A compressible two phases CFD model in the open-source software package OpenFOAM® is validated and then used to simulate a fixed OWC device, detached from the sea bottom. A comparative analysis of the results obtained by simulating the device at full-scale (1:1) and four smaller scales (1:50, 1:25, 1:10 and 1:5) is performed in order to assess the scale effects associated with air compressibility. Indeed, for the air pressure levels considered in the simulations (up to 350 Pa at model scale 1:50), the effect of neglecting the air compressibility results in an overestimation up to about 15% for the air pressure in the OWC chamber and the subsequent air volume flux, but less than 10% for the capture width ratio. This overestimation increases with increasing pressure level. Results are analysed in terms of dimensionless parameters and a new parameter, strongly related to the compressible effects, is proposed and used to provide generalized equations delivering correction factors.

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Keywords: Wave Energy Converters, Oscillating Water Column, Air compressibility, Scale effects, Numerical

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1. Introduction Due to the running out of fossil energy sources and the environmental impacts implied by their use, the interest in the harvesting of renewable energy sources is increasing. The yearly average value of the total energy transferred from the wind to the ocean is quantified in about 0.17 W/m2 [1]. The wave energy globally reaching the coastline is estimated at 32000 TWh/yr [2].

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Though a large variety of Wave Energy Converter (WEC) concepts have been proposed [3], at the current status the exploitation of wave energy is quite limited. The Oscillating Water Column (OWC) is one of the few WECs that have reached the development stage of full size prototype, e.g. the LIMPET experimental plant in Islay island, Scotland [4], the Mutriku power plant in Spain [5], the Pico power plant, in Portugal [6]. Moreover a prototype of harbour breakwater embodying several OWCs has also been built, at Civitavecchia harbour (Italy), with the main objective of absorbing the wave energy thus reducing wave reflection plaguing a navigational channel [7]. It consists, basically, in a hollow chamber partially submerged where the inner water column is excited by the external incident waves, pressurizing and depressurizing the inner air that flows through a turbine.

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As for any WEC concepts, maximizing the wave energy conversion is one of the essential steps in order to consolidate the development of the OWC technology towards the commercial stage. In this framework, reliable modelling tools (experimental or numerical), able to accurately simulate the OWC dynamics and to predict its power output are needed. Laboratory tests, although fundamental for reproducing the complex non-linearity and the multi-scale interactions as well as for the calibration and validation of numerical models, are expensive and highly time consuming. Such limitations do not allow to comparatively test a wide range of device geometries and wave conditions. Moreover, small-scale laboratory models are unavoidably affected by scale effects such as, for the case at hand, those related to the thermodynamics effects of air compressibility [8]. On the other hand, models at relatively large scale or field testing highly increases the

study, Correction factors

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costs, making it difficult to use such an approach for fundamental studies to support the development and the optimization of the WEC concept. Therefore, numerical modelling is necessarily required.

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Within the framework of potential flow theory, the interaction between incident waves and OWCs has mainly been modelled using two approaches: the rigid piston model [9-19] and the uniform pressure distribution model [20-23]. The so-called Lattice Boltzmann Method [24], has been applied for the simulation of the hydrodynamic only [25] showing that also this numerical technique performs well in this specific field.

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Recently, advanced Computational Fluid Dynamic (CFD) techniques have been used to fully capture the non-linear interaction between waves and the OWC device (i.e. viscous flow separation, turbulence and wave breaking) and to properly take into account the most relevant hydrodynamic and thermodynamic processes involved in the interaction between waves, OWC and air in the chamber. The suitability of Numerical Wave Tanks (NWT) based on CFD modelling to accurately analyse the OWC, including both the hydrodynamics and aerodynamics effects and simulating the air phase as incompressible, was demonstrated in several studies [26-35]. Senturk & Ozdamar [26] experienced up to 30% relative error (overestimation) for the airflow velocity with respect to the theoretical prediction of Hiramoto [36] and the authors concluded that further CFD analysis should include compressibility effects. Moreover, a slight over-prediction of the device hydrodynamic efficiency compared with the small-scale experiments by Morris-Thomas et al. [37], has been pointed out by Zhang et al. [27]. Iturrioz et al. [28-30] and Vyzikas et al. [34] highlighted the differences between the inhalation phase and the exhalation phase. Kamath et al. [31] and López et al. [32-33] revealed a significant effect of the turbine damping.

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One of the peculiar aspects of the CFD approach is its capability to also account for compressible flows, and to be applied to simulate the OWC device at different model scales, up to the prototype scale. This aspect is particularly important since it is recognized that at prototype scale, the thermodynamic effects related to air compressibility may be crucial for the OWC system dynamics [8, 38-39]. In small-scale laboratory tests, an appropriate simulation of the thermodynamic effects associated with air compressibility in the OWC chamber is difficult. In fact, to fully fulfil dynamic similarity between model and prototype, the scale ratio for the volume of the air chamber should not necessarily correspond to the scale ratio for the submerged part of the converter [8]. If the same scale ratio is adopted, the exact dynamic similarity of air compressibility springlike effects would require the atmospheric pressure during the experimental testing to be much smaller than the atmospheric pressure at prototype scale. This would be hardly feasible in most of the available experimental testing facilities. If the atmospheric pressure were the same at both scales, it would be necessary to use a volume of the air chamber larger than that which would result from Froude scaling [40-43]. One possible way to achieve the required volume scale factor for the OWC air chamber in the laboratory would be to connect the air chamber to a rigid reservoir [8]. This method might, however, induce additional forces and moments on the system, which may severely affect its dynamics, particularly for the case of floating OWCs.

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The effect of air compressibility on the performance of an OWC device was theoretically analysed by Thakker et al. [44], under the hypothesis of isentropic compression/decompression processes. A 5-8% reduction in the device conversion efficiency was found because of the compressibility of the flow. Sheng et al. [39], conducted an experimental study considering the airflow through an orifice connected to a chamber pressurized and depressurized by the motion of a piston. In their work, a power loss due to air compressibility of about 2% was found for a relative pressure in the chamber of about 2.2 kPa. Very recently, Elhanafi et al. [45] applied a compressible CFD model to evaluate air compressibility effects for a fixed, detached OWC, finding a reduction up to 12% of the maximum device efficiency due to air compressibility near resonance conditions. Furthermore, the air compressibility effects, in terms of discrepancies between results at different model scales, showed a dependence on the turbine damping and the incident wave height.

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Worth to note that, at present, the effect of air compressibility was only assessed on a restricted set of reference cases, and a systematic CFD study of this phenomenon has not yet been performed.

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The focus of our study is to enhance the current knowledge by providing a systematic evaluation of the error induced by neglecting air compressibility in modelling OWC devices with the provision of correction factors for different pressure levels inside the OWC chamber and scale ratios.

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The study is based on a multiphase and compressible CFD modelling of a OWC device. The compressible solver is used for simulating the OWC device at five scales: 1:50, 1:25: 1:10, 1:5 and 1:1, in order to quantify the magnitude of the scale effects due to the incorrect scaling of air compressibility. These results are analysed in terms of dimensionless quantities in order to provide generalized correction factors. The latter and further results might be beneficial for the assessment of scale effects and the correction of results obtained from incompressible numerical models as well as from small-scale laboratory tests of OWC devices.

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This paper is organized as follows. In section 2, first the features of the numerical model are briefly summarized, then the model validation in simulating the compressibility effects arising from the air-water dynamic interaction is assessed versus schematic benchmarks and laboratory measurements on an OWC model at small scale. In section 3, the results arising from numerical simulations of the OWC device at different scale (1:50, 1:25: 1:10, 1:5 and 1:1) are presented and discussed in terms of compressibility correction factors are provided. Concluding remarks are finally provided in section 4.

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2. Materials and Methods

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The study is based on multiphase and compressible CFD modelling performed by using the open-source software package OpenFOAM®. The performance of the compressible solver is, at first, evaluated on the basis of a set of benchmarks proposed in the open literature in order to test the capability of CFD solver to effectively reproduce compressible effects in idealized air-water systems (section 2.1). Then, the solver is used to simulate the OWC device in a NWT (section 2.2), and the OWC model is validated against laboratory tests at small scale, designed and conducted by the authors (section 2.3).

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The compressible flow is solved with the multiphase solver compressibleInterFoam, a pressure-based solver that can describe low Mach number flows of two compressible, non-isothermal immiscible fluids. In the pressure-based approach, finite variations of pressure take place, regardless the value of Mach number Ma [46], hence the convergence problems of density-based methods in the low Mach number regime are avoided. Density-based solver, in fact, may fail to appropriately reproduce the incompressible flow for Mach number Ma << 0.3 [47]. On the other hand, in the case of high speed compressible flows, pressure based solver may have the disadvantage of low temporal order of accuracy or require several inner loop iterations to converge to an accurate solution [48]. In this work, a pressure-based solver was preferred since the flow conditions are expected to be at maximum weakly compressible, as resulting from previous studies [45].

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The governing equations for fluid dynamics are mass conservation, momentum conservation and energy conservation, together with an equation of state, relating the fluid density ρ to its pressure p and temperature Te. The Eulerian fluid mixture approach is used, i.e. momentum, density and other fluid properties are of the air-water mixture, and properties vary according to the volume fraction of each phase (phase fraction, α), as in Eq. 1:

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  w  a 1   

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where ϕ generically denotes one of the fluid mixture properties, and the subscripts w and a refer to the water phase and the air phase respectively. The air-water system is treated with the Volume of Fluid (VOF) approach [49].The perfect gas equation of state is used for air, ρa=p/RaTe (with Ra=287 J/kgK). The water phase is treated as a compressible perfect fluid, with equation of state ρw=p/RwTe+ρw,0 (with ρw,0=1000 kg/m3, Rw=3000 J/kgK). The use of this equation of state for the water phase in NWTs was previously assessed and validated [50].

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The pressure-velocity coupling is performed using the so-called PIMPLE algorithm, which is hybrid in the sense that it consists of two classical algorithms, PISO (Pressure Implicit with Splitting of Operators, [51]) and SIMPLE (Semi-Implicit Method for Pressure-Linked Equations, [52]). Both PISO and SIMPLE algorithm are extensively explained and applied in the VOF method framework by Jasak [53]. The main

(1)

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structure of the PIMPLE algorithm is inherited from the PISO algorithm, but it additionally allows underrelaxation to ensure the equation convergence when bigger time steps are used.

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2.1. Benchmark tests for the compressible multiphase solver

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In order to evaluate the capability of the numerical solver to deal with air compressibility, the following benchmark tests are performed. The selected benchmarks are used in the literature as reference cases to assess the performance of numerical methods for air-water systems in which air compressibility is relevant, and were particularly used in the Sloshel Joint Industry Project [54-56]. The Sloshel project extensively investigated air compressibility bias in small-scale model testing by comparison with data from full scale sloshing experiments.

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2.1.1 Gravity-induced liquid piston in a tube (1D case)

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The first benchmark consists in a 1D closed tube, with a total length of 15 m, filled with an 8 m water column surrounded by two air pockets (Fig. 1). The case is very close to the classical Bagnold problem [57] and its wide application for benchmarking purposes is well-documented [55, 58-60]. Friction effects are ignored, and the density of air is set to ρa=1 kg/m3, while the density of water is ρw=1000 kg/m3. At the beginning, the velocity of both air and water is zero and the air pressure is equal to 1 bar in the computational domain. Then, the water column starts falling due to gravity effects, compressing the air in the lower pocket. The water decelerates while the pressure in the lower pocket increases, until the water velocity reduces to zero. At that point, the air pocket starts to expand pushing the water column upwards.

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Figure 1: 1D benchmark case (left) and time series of pressure at the bottom centre of the piston (right), comparison between values numerically simulated by compressibleInterFoam in this study.

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In order to reproduce the case with the numerical code, 300 cells equally spaced were distributed along the vertical tube, and the time series of pressure was recorded at the bottom of the tube. The adopted discretization schemes are second order accurate schemes in both time and space. A time step Δt equal to 10-3 s is used, and the computational cells have a vertical dimension of 0.05m. The results (Fig.1), in terms of both peak pressure inside the lower air pocket and its time of occurrence, are in good agreement with data from the literature [59].

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2.1.2. Gravity-induced liquid piston in a tank (2D case)

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The second test case considered is an extension to 2D of the aforementioned gravity induced liquid piston case: it consists in the gravity fall of a rectangular water column in a closed tank (Fig. 3). This test case has been widely used as a benchmark case for numerical codes [54,55,59,60]. In the 2D tank, the water (ρw =1000 4

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kg/m3) is surrounded by air (ρa=1 kg/m3), the velocity field is initially set to zero, and the initial pressure in the air phase is equal to 1 bar. Under gravity, the water column drops and impacts the bottom after about 0.65 s. The pressure peak at the time of impact has a maximum value at the centre of the tank [60], where a small amount of air is trapped between the water column and the tank bottom, and undergoes compression and expansion. Therefore, a pressure probe is set at this location to record the pressure time history. The adopted discretization schemes are second order accurate in both time and space. A time step Δt equal to 10-3 s is used. Mesh independent results are obtained by adopting a mesh having 1000 cells in the horizontal direction 750 cells in the vertical direction. A satisfactory agreement between calculated results and data from the literature is achieved in terms of both peak pressure at the time of the first impact of the water column on the bottom and its time of occurrence t=0.65 s (Fig. 2). In particular, the value of the estimated pressure peak (ca. 50 bar) is coherent with the most recent reference in the literature [50].

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Figure 2: 2D benchmark case (left) and time series of absolute pressure at the wave gauge located at the tank bottom centre (right), comparison between values numerically simulated by compressibleInterFoam in this study and data available in literature.

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2.2. Set-up of the numerical model of the OWC

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In the numerical model of the OWC, wave generation and absorption are simulated in the NWT with the waves2Foam toolbox [61] of OpenFOAM®. A relaxation zone approach is used, that is a zone of the NWT where the computational solutions are replaced with values calculated by weighing predefined analytical and solved values according to a relaxation function either to generate and/or absorb waves at each time step. A Large Eddy Simulation (LES) approach to turbulence modelling is used. The LES applied in this study uses a k-equation eddy-viscosity closure, which solves a transport equation for the sub-grid scale (SGS) turbulent kinetic energy kSGS [62,63]. This sub-grid model was developed to have better performance on relatively coarse grid with respect to the Smagorinsky model, since a transport equation kSGS is solved, instead of considering only production and dissipation terms under the hypothesis is of local equilibrium [63,64]. The use of such a model for LES of compressible flows is well documented [65,66]. The OWC considered in this work is a fixed and detached from the bottom device having a rectangular shape. It is equipped with a circular vent on the top cover, to mimic a pressure drop characterized by a quadratic air flow-pressure relation, which is typical of an impulse turbine. A laboratory test campaign at the model scale 1:50 was presented in Crema et al. [67] and Simonetti et al. [68] (Fig. 3-a). The computational domain of the three-dimensional NWT adopted to simulate the OWC device has a total length in wave propagation direction of 4.3λ (where λ is the length of the incident wave), a height of approximately 1 m and a width of 0.4 m. Since a symmetry condition is applied, the effective width of the computational domain is reduced to 0.4 m. The OWC model is placed at a distance equal to 2.5λ from the end of the inlet relaxation zone (Fig. 3-b). A hexahedral dominant mesh is used to discretize the geometry. The mesh is refined around the free surface zone (Fig. 4), based on the characteristics of the incident wave 5

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(wave height H and wave length λ), in order to achieve, in the far field of the OWC, the resolution determined in the grid independence tests previously performed [69], i.e.: (i) number of cells per wave height, H/cells=8 and cells aspect ratio AR=2; (ii) ratio between the length of the computational cells in the unrefined zone and that in the refined free surface zone equal to 4; (iii) length of the inlet and outlet relaxation zone equal to λ and 0.5λ, respectively (Fig. 3-b). In the near field of the OWC, the mesh is also refined around the OWC structure (OWC chamber mesh region) and around the OWC top cover vent and pipe (PTO mesh region). The length of the cells inside the OWC walls is ca. 1.5 mm at model scale 1:50 (corresponding to ca. W/130, being W the OWC chamber width). In the PTO mesh region, the mesh is further refined to properly discretize the circular geometry, in order to have a resolution of about V/cells=20, where V is the diameter of the top cover pipe. Furthermore, 10 prism layers, with an expansion factor 1.3, are placed around the OWC walls, to obtain a fist cell height equivalent y+≈1 in the submerged region near the OWC walls. The mesh has a size of approximately 3.200.000 cells (slightly varying depending on the considered λ and on the dimensions of the OWC chamber). The mesh resolution described above (called hereafter mesh M2) was chosen based on a mesh independence study. Three mesh resolutions were comparatively considered in the mesh independence study performed: starting from mesh M2, mesh M1 was obtained by doubling the mesh size in the OWC chamber mesh region, and mesh M3 was obtained by halving the mesh size in this region. As the mesh was refined, numerical results converged to experimental results. Reasonably good agreement was achieved with mesh M2 and M3 (Normalized Root Mean Square Error, NRMSE < 10% and determination coefficient R2 > 0.98 on the surface elevation inside the OWC, ηowc, the relative air pressure in the chamber, powc and the vertical component of the velocity in the top cover pipe, Uy). The relative difference between the results obtained with the finest mesh M3 and the intermediate mesh M2 is lower that 3% (Tab. 1), therefore the intermediate mesh size M2 was selected to perform the numerical simulations of this study. A detailed validation of the numerical model with experimental results is provided in Section 2.3.

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Table 1: Mesh convergence tests for the OWC. Comparison of determination coefficient R2 and Normalized Root Mean Square Error NRMSE between numerical results and laboratory data for the relative pressure powc in the OWC air chamber, the vertical component of the air velocity in the top cover pipe, Uy, and the water level ηowc in the OWC chamber.

Number of cells

powc R2

Uy

NRMSE [%]

R2

ηowc

NRMSE [%]

R2

NRMSE [%]

M1

1200000

0.96

12.5

0.95

12.0

0.96

12.4

M2

3200000

0.98

9.2

0.97

8.2

0.98

8.1

M3

6500000

0.98

9.0

0.97

8.0

0.98

8.0

224

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Figure 3: The small scale model of the OWC tested in the laboratory (a); definition of the NWT dimensions and position of the OWC model in the NWT (b).

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Figure 4: Overview of the mesh in the computational domain, refinement mesh region and close up of the mesh near the OWC structure; definition of the boundaries of the Numerical Wave Tank (NWT) with the OWC device (bottom, rigth).

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Considering the designation of the boundaries of the NWT with the OWC device defined in Fig.4-b, no slip conditions are set at the bottom and at the OWC walls. The atmosphere boundary of the NWT, occupied by the air, is set as a constant pressure boundary (with relative pressure equal to zero). The velocity u and the volume phase fraction α at the inlet boundary are strongly imposed by a Dirichlet boundary condition, with values given according to Stokes second order wave theory. For the fluid temperature, a value of 293 K is imposed for at the inlet with a Dirichlet boundary condition. A symmetry boundary condition is specified on the front boundary of the NWT to reduce the computational domain. It is worth to note that even if symmetry boundary conditions are not strictly applicable for LES, in this case the use of the symmetry constrain was accepted once the errors arising from this approximation were found to be negligibly small by means of the model validation with laboratory results (see Section 2.3). This supports the hypothesis of a fundamental two-dimensional nature of the turbulence in the studied physical system, which is governed by a twodimensional incident wave, propagating in a wave tank and interacting with the three-dimensional OWC structure. This hypothesis seems to be also confirmed by different literature references in which twodimensional CFD models of OWCs are successfully validated with laboratory tests [31,33,70,71].

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A second order discretization scheme is used for time derivatives, blended with a first order Euler scheme to improve stability. It is worth to note that the blended scheme of OpenFOAM® yields a first-order temporal accuracy, as discussed in Lee et al. [72]. Convection terms in momentum equation are discretized with a total variation diminishing (TVD) scheme with a central differencing interpolation scheme bounded by a Sweby limiter [73]. For the convection term in the transport equation of the phase fraction α, Monotone Upwind Schemes for Scalar Conservation Laws (MUSCL, [74]) interpolation scheme is used, which is a TVD scheme with a high accuracy level even in the case of shocks or elevate gradient of the solution. For generic gradient operators, second order accurate discretization schemes are applied. Regarding the linear solvers, a preconditioned conjugate gradient solver (PCG) is used for the pressure equation. The Krylov subspace solver bi-conjugate gradient solver for axisimmetric matrix (PBiCG) is used for u and for α. An adaptive time step is used, determined based on the maximum imposed Courant number Co=0.6. The solution 7

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is considered converged when the residuals has reached the tolerance of 10-7 (for p and α) and 10-9 (for u). Two iterations of the PIMPLE loop are used.

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2.3. Comparison between laboratory tests, compressible and incompressible CFD simulations

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The compressibleInterFoam code is, at first, used to reproduce the flow processes observed in the OWC device at laboratory scale 1:50 as well as to determine any systematic differences between the results of a compressible and an incompressible CFD model at such a small scale. The numerical set-up of the incompressible CFD model used for this comparative analysis was previously presented [35]. A set of 16 simulations with different OWC geometrical configurations (draught of the front wall D, length of the chamber in wave propagation direction W, diameter of the top cover pipe V) are tested (Tab. 2).

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Table 2: Geometrical parameters of the OWC device simulated and water depth.

parameter

[unit]

scale 1:50

scale 1:1

Chamber length B

[m]

0.2

10

Front wall draught D

[m]

0.07-0.3

3.5-6

Chamber width W

[m]

0.1-0.2

5-10

Back wall length G

[m]

0.45

22.5

Top cover vent diameter V

[m]

0.022-0.038(*)

1.1-1.9

water depth h

[m]

0.5

25

Nine values of the top cover vent diameter, ranging from 0.022 to 0.038 m (corresponding to 1.1 and 1.9 m at prototype scale) with a step of 2 mm (corresponding to 10 cm at prototype scale) (*)

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The regular wave trains tested are characterized by a wave height H=0.04 m and periods T ranging from 0.8 s to 1.4 s. The simulation time is set to around 20T (in order to observe at least 10 completely developed oscillation periods inside the OWC chamber, Fig. 5). The inner OWC surface elevation (ηowc) and air pressure (powc) are sampled at the chamber centre, the vertical component of the velocity of the incoming/outgoing air flux Uy is sampled at the centre of the top cover pipe. The time series of powc (Fig. 5-a), Uy (Fig. 5-b) and ηowc (Fig. 5-c) are compared with experimental data by means of the determination coefficient R2 and the Normalized Root Mean Square error (NRMSE). The value of the determination coefficient R2 with laboratory data over the complete validation dataset is higher than 0.93 for all considered parameters (ηowc, powc and Uy), for both the compressible and the incompressible model, with an average value of about 0.98 for powc and ηowc, and 0.97 for Uy (Tab. 3). The NRMSE over the validation dataset is lower than 18% for all parameters, with an average value of about 9% (Tab. 3). A slightly higher maximum value of the NRMSE (around 17.5%) is found on powc in the compressible CFD simulations than in the incompressible CFD simulations (where the NRMSE on powc is ca. 15%). However, the difference in the accuracy of the of the compressible model over the validation data set is judged to be negligibly small (the determination coefficient R2 is almost identical and the difference in the NRMSE is lower than 3%).

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Figure 5: Time series of relative pressure powc in the OWC air chamber (a), of vertical component of the air velocity, Uy, in the OWC top cover pipe (b) and of water level ηowc in the OWC chamber (c). Comparison between experimental data and CFD simulations (compressible and incompressible) at model scale 1:50 for the OWC geometry with W=0.1 m, D=0.09 m, V=0.036 m and incident wave H=0.04 m and T=1 s, water depth h=0.5 m.

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Considering values referred to the average OWC inner oscillation period, called hereafter period average values, the maximum value of the air velocity Uymax, the amplitude of the inner water level oscillation Δηowc and the amplitude of the air chamber pressure oscillations Δpowc have a determination coefficient R2 with the experimental data of about 0.98, for both compressible and incompressible CFD models (Fig. 6). No relevant or systematic differences are found between the performance of the compressible and incompressible model in simulating the OWC device at scale 1:50. This is valid when considering both the instantaneous values and period averaged values, and has the following implications: (i) an incompressible CFD code can properly reproduce small-scale OWC model tests, therefore confirming that compressibility effects are not reproduced in such small-scale OWC model tests; (ii) the compressible model is also appropriate for reproducing the flow in the incompressible regime (Ma << 0.3).

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Table 3: Determination coefficient R2 and NRMSE for comparison of results of the incompressible and compressible CFD simulations with laboratory data at scale 1:50 for relative air chamber pressure, powc, vertical component of the air velocity, Uy and water levels ηowc.

R2

Incompressible model

Compressible model

NRMSE [%] average maximum

average

minimum

Uy

0.98 0.97

0.94 0.93

9.2 8.2

15.1 15.9

ηowc

0.98

0.94

8.1

15.6

powc Uy

0.98 0.97

0.93 0.93

9.6 8.5

17.5 16.3

ηowc

0.98

0.94

8.4

15.3

powc

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Figure 6: Scatter plot between laboratory and numerical data (obtained with the compressible and the incompressible CFD models) of Δpowc, Uymax and Δηowc at laboratory scale 1:50.

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3. Results and discussion of air compressibility effects

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A comparative analysis of the results from the compressible CFD model applied at scale 1:1 (prototype) and at four smaller model scales N (1:50, 1:25: 1:10 and 1:5) is performed in order to assess the errors induced by neglecting the effect of air compressibility for each scale and to derive correction factors. Starting from the simulations at the laboratory scale 1:50, performed in accordance with Froude similarity, the OWC device geometry and the characteristics of the incident waves are up-scaled by keeping the same mesh resolution (in terms of number of cells per wave height H, per wave length λ, number of cells per OWC chamber width W), in order to ensure a consistent comparison between the results at different scales. Achieving the nearwall resolution needed to perform a wall-resolved LES in the full-scale model would be much more computationally expensive. For this reason, for the 1:1 scale (and for the intermediate scales) a spacing of the first cell from the OWC walls such that a y+≈30 would result is used, adopting wall functions. The use of wall-modelled LES is, indeed, well documented in the literature, as a way to perform high-Reynolds number engineering applications of LES at an affordable computational cost [62,75,76]. In order to verify the absence of relevant differences arising from the use of different wall treatment, a comparison of the numerical results obtained with the incompressible model at scale 1:50 and the incompressible model at scale 1:1 is made. For the relative pressure in the chamber powc, the average NRMSE and the R2 between the incompressible CFD results at the two scales are respectively lower than 3% and 0.99, once the regime state is achieved (after about 7T from the simulation beginning, Fig. 7,a). Similar values are obtained for ηowc (Fig. 7,b) and for the volumetric airflow rate, qowc (Fig. 7,c). The very good agreement between results at scale 1:1 and 1:50 (once compressibility effects are excluded) seems to also indicate that scale effects arising from the inability to keep other relevant force ratio constant when using Froude scaling (e.g. different Reynolds number, Re) are negligible. This results is supported by considering that, for the OWC system, characteristic values of Re>104 are found also for the smaller scale 1:50 (when Re is estimated by considering a reference 10

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flow velocity of the order of 10-1 m/s and a characteristic length given by the OWC width, i.e. of the order of 10-1 m). This imply a fully turbulent regime, where losses are independent of Re. Negligible differences between CFD simulation results at different model scales when using an incompressible model were also observed by Elhanafi et al. [70].

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Figure 7: Comparison between powc (a), ηowc (b) and qowc (c) at scales 1:50 and 1:1, obtained by using the incompressible CFD model for the OWC geometry with W=10m, D=5m, V=1.5m and incident wave H=2m and T=7s, water depth h=25m.

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The simulations are performed by accessing the computational resources of CINECA, the largest High Performance Computing centre in Italy (http://www.cineca.it/en). The used system architecture is an IBM neXtScale model, with 516 nodes, 2 processors/node (2.40 GHz Intell Haswell 8 cores). The needed computational time for a single run (i.e., 20 wave periods T) on the aforementioned architecture is about 45 hours for a parallel run on 16 cores (i.e. a single node). In order to consider a wider range of pressure oscillation amplitude Δpowc inside the OWC than the range tested in the laboratory, different top cover vent diameters, chamber widths W and incident wave heights (H= 1.2-2.8 m with T= 7.07 s and 8.48 s at prototype scale, Tab. 4) are tested.

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Table 4: Characteristics of tested incident wave heights H and periods T in a constant water depth (h=25 m at scale 1:1 and h=0.5 m at scale 1:50) selected for the scale effects study.

wave code H01 H02 H03 H04 H05 H06

scale 1:50 H [m] T [s] 1.00 0.024 1.00 0.030 1.00 0.034 1.00 0.043 1.20 0.050 1.20 0.056 11

scale 1:1 H [m] T[s] 7.07 1.20 7.07 1.50 7.07 1.70 7.07 2.15 8.48 2.50 8.48 2.80

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When using the compressible CFD model, differences are observed between the simulation results at scale 1:50 and those at scale 1:1 (Fig. 8). As a consequence of air compressibility, both the oscillation amplitude of the relative pressure powc and that of the air flow rate qowc are reduced, with a stronger effect during the inhalation phase (as discussed more detailed in quantitative terms, section 3.1). Moreover, due to the springlike effect of air compressibility, a phase difference arises between the peak powc and qowc in the 1:50 and the 1:1 OWC models, with a delay (ca. 0.1·T) in the time of occurrence of the peak values at model scale 1:1. No appreciable differences are observed in the time series of the water surface elevation ηowc inside the OWC.

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Figure 8: Comparison between powc (a), ηowc (b) and qowc (c) obtained at model scale 1:50 and at model scale 1:1 by using the compressible CFD model for the OWC geometry with W=10 m, D=5 m, V=1.8 m and incident wave H=2 m and T=7s, water depth h=25 m.

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3.1. Proposed correction factors for the air compressibility effect

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In this section, the results of the compressible CFD model at model scale 1:50, 1:25, 1:10, 1:5 and 1:1 are analyzed in terms of dimensionless quantities in order to assess air compressibility effects and to provide corrections factors. The effect of air compressibility is assessed on those quantities that are relevant in the evaluation of the OWC device performance, i.e. pressure inside the chamber powc, air flow rate qowc, OWC capture width ratio εowc, water surface elevation ηowc inside the OWC. εowc is computed as the ratio between the period averaged pneumatic power absorbed by the OWC and the period averaged incident wave power. Considering incident waves with characteristics corresponding to Stokes second order wave theory, the period averaged incident wave power per unit width [W/m], for generic a water depth h, is expressed as

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 wave 

1  2kh   9 H2   gH 2 1    1  16 k  sinh(2kh)   64 k 4 h6 

(2)

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where ρ is the water density, ω is wave frequency and k is the wave number, g is gravitational acceleration. The mean pneumatic power Πowc [W] absorbed by the OWC, is estimated by integrating over the total length of the analysis windows, Ttest, the product of air pressure powc in the chamber and air volume flux qowc [40]:

 owc

1  Ttest

Ttest



powc  qowc dt

(3)

1

εowc is then computed as:

 owc 

 owc  wave B

(4)

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where B is the chamber length perpendicular to the wave direction.

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For each of the aforementioned physical quantities and for each of the tested model scales, a correction factor CF is defined as in Eq. 5, where χ indicates the generic parameter, 1:N the model scale and the numerator value is Froude upscale to the 1:1 scale. (5)

CF 1:N  1:N 1:1

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To take into account the variation of the air compressibility effect on the system dynamics as a function of both the pressure variation inside the OWC air chamber and the characteristic of the incident wave (T and H), correction factor CF is expressed as a function of the non-dimensional pressure parameter Γ, defined as:

  powc1:N

 cosh k  h  D    H  cosh  kh   

(6)

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where Δpowc1:N is the period averaged value of the pressure oscillation amplitude in the OWC, γ=ρg is the water specific weight, k is the wave number of the incident wave, h is the water depth and D is the OWC front wall draught. The denominator in Eq. 6 is proportional to the maximum wave-induced pressure at a depth y=-D (according to Airy wave theory). For a given wave number k, a given OWC geometry and turbine applied damping, the air volume variation in the OWC chamber during a wave cycle is related to the incident wave height H. Since Γ is defined as a function of H, it allows to also implicitly account for the air volume variation in the OWC chamber during a compression/decompression cycle.

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3.1.1. Effect of air compressibility on pressure in the air chamber powc

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The period average value of pressure oscillation amplitude, Δpowc, is found to be relevantly affected by air compressibility. The magnitude of the scale effect is well correlated to Γ within all the investigated range, 0.4<Γ< 1.0. The maximum difference between Δpowc in the simulations at scale 1:50 and those at scale 1:1, expressed in terms of correction factor CFΔp1:50, is about 1.15 (for Γ=1). For decreasing values of Γ, lower CFΔp1:50-values would result, and CFΔp1:50 tends to 1 (i.e., the results from scale 1:50 tend to converge to that at prototype scale) when Γ tends to zero. For Γ~0.4, CFΔp1:50-values close to 1.02 would result, meaning that air compressibility has a negligible effect on the OWC system dynamics when for low values of the dimensionless pressure parameter Γ. The relation between CFΔp1:50 and Γ is approximately quadratic (Fig. 9), with a relatively high value of the determination coefficient for the fitting (R2=0.96).

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For model scale 1:25, no relevant differences are found in the values of CFΔp, as compared to those at model scale 1:50. Therefore, for model scale 1:25, the quadratic relation to relate CFΔp1:25 and Γ is approximately the same proposed for model scale 1:50 (Fig. 9 and Tab. 5). This indicates that, in the considered range of Γ=0.45-1.0, the air phase behaves as incompressible in the 1:25 scale as well as in the 1:50 scale. This results suggest that, since the air behaves as incompressible at scale 1:25 and at smaller scales, the correction factors based on the 1:25 experiments can be adopted also for smaller scales. The period average value of pressure oscillation amplitude at model scale 1:10 shows a maximum difference (expressed in terms of CFΔp1:10) from 13

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that at scale 1:1 of about 1.11 (for Γ=1, Fig. 9). Considering model scale 1:5, a maximum value of CFΔp1:5=1.07 is found in the considered range of the pressure parameter Γ (Fig. 9).

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Considering the values of Mach number (Ma) in the OWC air chamber, computed as the ratio between the maximum value of the speed of the airflow in the simulations and the speed of sound, we can observe that for both model scales 1:50 and 1:25, Ma is about 0.05, i.e. the flow behaves as approximately as incompressible. Further increasing the model scale (1:10 and 1:5), Ma increases to 0.1 and 0.15 respectively, i.e. the flow regime moves towards the weakly compressible flow regime, and a moderate decrease of the scale effect related to air compressibility is observed.

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Overall, the proposed quadratic relations between CFΔp1:N and Γ are summarized in Table 5.

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Figure 9: Correction factor CFΔp for the period average air chamber pressure oscillation amplitude Δpowc (obtained from OWC model scales 1:50, 1:25, 1:10, 1:5 and scaled up to 1:1 according to Froude similarity) versus non-dimensional pressure parameter Γ.

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Differences in the system response to air compressibility effects are observed between the exhalation (outflow) and the inhalation (inflow) phases. Considering, exemplarily, the comparison between results at scale 1:50 and 1:1, the average value of the maximum pressure during the exhalation phase (p+owc), are less affected by air compressibility, with a maximum CFp+1:50 in the investigated range of Γ lower than 1.05 (Fig. 10). In the inhalation phase, the effect of air compressibility is remarkably higher (Fig. 10). The average value of the minimum chamber pressure during depressurization, denoted as p-owc, shows a correction factor CFp-1:50 that reaches a maximum value of about 1.3 in the investigated range of Γ.

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In the exhalation phase, the thermodynamic conditions in the chamber are approximately isentropic (as previously observed by Falcão & Justino [23]). The air, which is progressively compressed by the upward motion of the inner water surface of the OWC, is driven out of the top cover orifice, where it rapidly accelerates due to the sudden decrease of the cross sectional area for the flow at the orifice. Therefore, in the exhalation phase, the turbulent mixing process between the air at atmospheric conditions and the pressurized high-speed airflow takes place outside the OWC air chamber.

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In the inhalation phase, the inflow accelerates through the orifice before entering the air chamber. The mixing process, therefore, takes place inside the OWC chamber. In other words, for a given value of relative pressure oscillation amplitude Δpowc, and orifice size V, even if the airflow qowc and the velocity in the top cover pipe are approximately the same in the inhalation and the exhalation phases (with slightly higher values during exhalation, as also observed in [77]), the air velocity inside the chamber is remarkably higher during inhalation. The higher air velocity inside the chamber during inhalation results in a higher value of Mach number Ma, possibly explaining the observed difference in the air compressibility effect.

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The difference during an exemplarily wave cycle can be visualized in Fig. 11 showing that during inhalation, time 7.5T, the air velocities inside the OWC chamber are much higher than during the exhalation phase, time 7.9T.

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Figure 10: Comparison of scale effects due to air compressibility between inhalation and exhalation phase for model scale 1:50. Results for the air chamber relative pressure during the exhalation phase, powc: CFp1:50 for versus nondimensional pressure parameter Γ.

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Figure 11: 2D cross-section of vertical component of the air velocity Uy in the OWC chamber at different times during the wave period T (model scale 1:50).

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3.1.2. Effect of air compressibility on airflow rate qowc

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Air compressibility also influences the period averaged value of volume airflow oscillation amplitude Δqowc in/out the OWC air chamber (Fig. 12). The maximum difference observed between Δqowc at scale 1:50 and at scale 1:1, expressed in terms of correction factor CFΔq1:50, is about 1.16. When decreasing the value of the non-dimensional pressure parameter Γ, air compressibility has a lower effect on qowc, with a value of CFΔq1:50 lower than 1.03 for Γ<0.4.

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As observed for the relative air chamber pressure powc, the relation between CFΔq1:50 and Γ can be fitted with a second order polynomial relation (Fig. 12), with a value of the determination coefficient R2 equal to about 0.97. Also for the parameter Δqowc, the quadratic relations to relate CFΔq1:25 and Γ can be approximated with the same relations proposed for model scale 1:50 (Fig. 12 and Tab. 5), while a maximum difference with the prototype scale value of CFΔq1:10=1.10 and CFΔq1:5=1.05 is found for model scales 1:10 and 1:5 respectively.

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Also for qowc, differences in the effect of air compressibility between the exhalation phase (Fig. 13) and the inhalation phase (Fig 13) are observed. This effect may be related to the asymmetry in the thermodynamic conditions between inhalation and exhalation phases, as aforementioned. In the inhalation phase, as observed 15

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for powc, the effect of air compressibility on qowc is higher. In the considered range of Γ, the period averaged value of the maximum airflow directed towards the OWC chamber, q-owc, has a maximum CFq-1:50=1.25 (Fig. 13) when considering model scale 1:10.

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Figure 12: Correction factor CFΔq for the period averaged air volume air flux oscillation amplitude Δqowc (obtained from OWC model scale 1:50, 1:25, 1:10, 1:5 and scaled up to 1:1 according to Froude similarity) versus non-dimensional pressure parameter Γ.

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Figure 13: Comparison of scale effects due to air compressibility between inhalation and exhalation phase for model scale 1:50. Results for the relative pressure in the air chamber during the exhalation phase: CFq1:50 versus non-dimensional pressure parameter Γ.

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3.1.3. Effect of air compressibility on capture width ratio εowc

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The air compressibility effect results in a power loss as compared to the incompressible case. This loss can be evaluated in terms of the correction factor CFε1:N (Fig. 14) that applies to the period average OWC capture width ratio εowc. In the investigated range of Γ, the maximum value of CFε1:50 is about 1.09. Assessing the capture width ratio εowc by using an incompressible CFD model or laboratory tests at a small-scale is conservative, i.e. it might result in an overestimation of εowc by up to 10% for pressure conditions in the air chamber up to Γ=1. The relation between CFε1:50 and Γ can be fitted with a parabolic function with a value of R2=0.87 (Fig. 14). The effect of air compressibility on the period average value of the capture with ratio 16

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εowc is smaller than that on the period average pressure amplitude Δpowc and air volume flux Δqowc. This might be explained by the fact that part of the input power is stored in the compressible air when the air in the chamber is pressurized or depressurised. When the relative pressure in the air chamber diminishes from its minimum/maximum values, the stored power might be released, hence reducing the observed air compressibility effect on εowc. A similar interpretation was previously given by Sheng et al. [39].

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The magnitude of the maximum overestimation of εowc (up to 10%) with respect to the incompressible case, is confirmed in the theoretical studies of Thakker et al. [44] (where a reduction of εowc up to 8% was found) and the numerical study of Elhanafi et al. [45] , where a 12% decrease in εowc was obtained. Our study extends these previous findings as it reveals that the compressibility effect is strongly correlated to the newly defined dimensionless parameter Γ that is a relative measure of the prevailing pressure level in the air chamber. This finding permits to propose a set of empirical equations giving correction factors, therefore providing an operative tool to account for air compressibility effects.

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Actually, while the overestimation of εowc has approximately the same magnitude when simulating the OWC device at model scale 1:50 and 1:25 (Fig. 14 and Tab. 5), that overestimation is progressively reduced when testing the device at model scale 1:10 (CFε1:10=1.07) and 1:5 (CFε1:5=1.05). The quadratic relations between CFε1:N and Γ, proposed to account for the error committed by neglecting air compressibility at different model scales, are summarized in Table 5.

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Figure 14: Correction factor CFεowc for the period averaged capture width ratio εowc (obtained from OWC model scales 1:50, 1:25, 1:10, 1:5 and scaled up to 1:1 according to Froude similarity) versus non-dimensional pressure parameter Γ.

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3.1.4. Effect of the air compressibility on water surface elevation ηowc inside the OWC

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Only small differences in the period averaged amplitude of the water surface elevation inside the OWC, Δηowc (NRMSE<0.05 and determination coefficient R2=0.98) are found between the results of the simulations at scale 1:50 and scale 1:1 (Fig. 15). This implies that air compressibility only affects the thermodynamics of the air chamber with much less effects on the hydrodynamics inside the OWC. It is worth to note that higher discrepancies due to air compressibility might rise if higher pressure levels inside the OWC than those examined in this study were considered.

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Figure 15: Scatter plot between the period averaged amplitude of water surface elevation Δηowc simulated with the compressible CFD model at scale 1:50 (scaled up to 1:1 according to Froude similarity) and at scale 1:1.

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Table 5: Summary of the proposed correction factors for air compressibility for Δpowc, Δqowc and εowc.

parameter

CF1:50 & CF1:25

CF1:10

CF1:5

Δpowc

0.147 Γ 2  1

0.123 Γ 2  1

0.103 Γ 2  1

Δqowc

0.13 Γ 2  0.11Γ  1

0.12 Γ 2  1

0.1Γ 2  1

εowc

0.083 Γ 2  1

0.043 Γ 2  1

0.036 Γ 2  1

 cosh k  h  D     powc1:N   H  cosh  kh   

CF 1:50  1:N 1:1 with   powc , q owc and  owc

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4. Concluding remarks

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A compressible CFD model was applied to simulate the hydrodynamics and aerodynamics inside the OWC device at scale 1:1 (prototype) and at four different scales (1:50, 1:25, 1:10 and 1:5) in order to quantify scale effects related to air compressibility in the OWC chamber. For this purpose, the standard compressible VOF solver of OpenFOAM (compressibleInterFoam) is used, together with the wave model waves2Foam. Benchmark tests are performed for the new solver to ensure its capability to reproduce properly compressible effects. In this respect, a relatively good performance of the code is achieved. Using both compressible and incompressible CFD solvers to reproduce the OWC device at laboratory scale (1:50), it was shown that the results of both solvers are in good agreement with the laboratory data, thus confirming that compressibility effects are not reproduced in the small-scale model tests. This underlines the urgent need of providing correction factors for laboratory results obtained at small scale.

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A comparative analysis of the numerical results obtained by using the incompressible model at scale 1:50 and those obtained at scale 1:1 scale was performed in order to exclude any relevant further effect due to factors different from the air compressibility, which might be misleading in the interpretation of the results. For all the considered quantities (pressure inside the chamber powc, air flow rate qowc, water surface elevation ηowc), the NRMSE between incompressible model results scale 1:1 and 1:50 is lower than 3%; i.e. except scale effects due to air compressibility.

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The air compressibility effects are analysed in terms of the non-dimensional parameter Γ, which accounts for both the level of air compression in the OWC chamber and the incident wave conditions (wave height, wave 18

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number and water depth). Correction factors (CF) for the pressure inside the OWC chamber powc, the air flow rate qowc and the OWC capture width ratio εowc obtained experimentally or numerically from small scale OWC models (1:50-1:5), are proposed. The effect of neglecting the air compressibility might result in an overestimation up to about 15% for the relative air pressure powc and the air volume flux qowc, but less than 10% for capture width ratio εowc (in the tested range of Γ=0.4-1.0). This overestimation increases with increasing Γ, i.e. with increasing level of the pressure in the air chamber, while air compressibility is progressively less relevant for lower pressure levels.

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The results show that the scale effect due to air compressibility are reduced when simulating the OWC device at relatively large model scale (i.e., 1:5), resulting in a maximum overestimation of the OWC capture width ratio εowc of about 5% for Γ=1.0. For values of the non-dimensional pressure Γ lower than ca. 0.6, the air compressibility affects the results from model scales 1:50, 1:25, 1:10 and 1:5 in a similar way (the variation of CF with the scale is lower than 3%, and values of CF lower than 1.05).

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The magnitude of scale effects related to air compressibility in model scales 1:50 and 1:25 is nearly the same for the simulated range of Γ. Therefore, at both scales, the airflow in the OWC chamber behaves approximately as incompressible, and the correction factors proposed for these scales can be adopted also for smaller scales. However, at smaller scales, other scale effects may rise and grow in importance, i.e. effects related to increasing importance of viscous force (i.e. Reynolds effects) or surface tension force (i.e. Weber effects) with respect to inertial force [78]. These additional effects for smaller scales could not be accounted for by the correction factors we proposed in this work.

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Based on the work of Morris-Thomas et al. [37], the error on the capture with ratio arising when using Airy wave theory in modelling the OWC is of the order of 10%-30%. This relative comparison highlights the relevance of the correction of air compressibility effects.

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Though benchmarks on the compressible solver proved its ability to accurately reproduce compressible effects, a validation of the compressible model of the OWC at scale 1:1, on a completely equivalent system from both the dynamic and thermodynamic point of view, is required to confirm the results obtained at scale 1:1. It should also be remarked that, in this work, the air turbine effect is only introduced as an induced damping, an assumption widely used in the literature for modelling OWC devices. It should be noted that when an air turbine is used, at prototype scale, significant change in the specific entropy may occur in the flow through the turbine, due to viscous losses [8]. The possible impact of this effect on the proposed correction factors should be further investigated in future research. A CFD compressible model which includes the effect of the air turbine in a more accurate way could be applied for this purpose, i.e. substituting the orifice with an actuator disk [79]. Future research might increase the number of model scales N to be comparatively analysed and the number of tests to be performed at each model scale N, and thus possibly derive a more generic formula to evaluate air compressibility scale effects, valid for any model scale.

Acknowledgments We acknowledge the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support granted through the SCALEOWC project. The support of Civil and Environmental Engineering Department of Florence University under the NEMO project and the MARINET2 EU H2020 project, coordinator L. Cappietti, are also gratefully acknowledged.

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