# Ocean Energy Systems Wave Energy Modeling Task 10.4: Numerical Modeling of a Fixed Oscillating Water Column

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## Abstract

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## 1. Introduction

## 2. Experimental Measurements

#### 2.1. Analysis of the Experimental Data

- $Q={A}_{c}\phantom{\rule{0.166667em}{0ex}}\overline{u}$, where $\overline{u}={\partial}_{t}\overline{\eta}$, with $\overline{\eta}$ the mean of the nine internal wave gauge measurements and ${\partial}_{t}$ the time derivative (here taken by a fourth-order finite-difference scheme).
- $Q={A}_{o}{u}_{a}$, where ${u}_{a}$ is the measured air flow velocity through the orifice. Here, it was determined that the most accurate result was found by taking the value from the upper gauge for the outflow and from the lower gauge for the inflow.
- From Equation (6), we can write $Q=\sqrt{2{\left({C}_{d}{A}_{o}\right)}^{2}/\rho \phantom{\rule{0.277778em}{0ex}}\left|\overline{p}\right|}\phantom{\rule{0.166667em}{0ex}}\mathrm{sign}\left(\overline{p}\right)$, where $\overline{p}$ is the mean of the three pressure difference measurements.

## 3. Numerical Modeling of OWC Chambers

#### 3.1. Weakly-Nonlinear Potential Flow Theory in the Frequency Domain

#### 3.2. Weakly-Nonlinear Potential Flow Theory in the Time Domain

#### 3.3. Potential Flow Modeling of the Orifice Plate Damping—Incompressible Flow

#### 3.4. Potential Flow Modeling of the Orifice Plate Damping—Compressible Air Volume with Incompressible Orifice Flow

#### 3.5. Potential Flow Modeling of the Orifice Plate Damping—Compressible Air Volume with Compressible Orifice Flow

#### 3.6. Numerical Potential Flow Solutions

#### 3.7. CFD Solutions

## 4. Description of the Participating Teams Solution Strategies

#### 4.1. The Technical University of Denmark Team

**DTU_FD**: The Technical University of Denmark (DTU) weakly-nonlinear frequency-domain contributions solve Equation (7) using coefficients computed by the high-order BEM solver, WAMIT [26]. The computed coefficients are read into a post-processing program, and combined with an initial guess for ${B}_{77}^{0}$ at each value of $\omega $ and $\u03f5$, to solve Equation (7) iteratively until the maximum change in ${\xi}_{j}$ is below a tolerance of ${10}^{-6}$. This gives the weakly-nonlinear, frequency-domain solution.

**DTU_TD**: The DTU weakly-nonlinear time-domain contributions solve Equation (11) using the open-source package, DTUMotionSimulator [36], which is described in the associated documentation. This code is based on the solution originally described by Bingham [37]. In brief, the Fast Fourier Transform (FFT) is used to compute the coefficients in Equation (12), and the explicit fourth-order Runge-Kutta integration scheme is applied to evolve the equations in time, with the fourth-order Simpson rule used to evaluate the convolution integral over the nonzero range of ${K}_{jk}$. Equation (14) is used to compute the incompressible-flow model of the nonlinear power-take-off damping force at each stage of the time integration.

#### 4.2. The National Renewable Energy Laboratory Team

**NREL_WECSim**: The National Renewable Energy Laboratory Team (NREL) WEC-Sim model was developed in MATLAB/SIMULINK and solves Equation (7) in the time domain [38]. In WEC-Sim, the rigid-body dynamics are solved using the multibody dynamics solver, Simscape Multibody, but the additional degrees of freedom from the generalized modes are formulated in a state-space form [39]. Similar to

**DTU_TD**, the FFT is applied to compute the coefficients in Equation (12), and Equation (7) is solved in the time domain using the fourth-order Runge-Kutta integration scheme.

**NREL_CFD**: The high-fidelity CFD code, STAR-CCM+,is utilized to verify and validate the wave tank testing data, with the medium wave height case (WaveID = Med06) with TestID = 106 used as an example. An implicit, three-dimensional, incompressible, and unsteady RANS model is applied for all simulations. For the turbulence closure model, the Shear Stress Transport (SST) k-$\omega $ model, with “all y+ wall” treatment, is utilized. Herein, the “all y+ wall” treatment is a hybrid wall solution that attempts to combine high y+ wall treatment (y+ > 30) and low y+ wall treatment (y+ < 1). The free surface is modeled using the Eulerian multiphase volume of fluid method, utilizing typical fluid properties ($\rho $ = 1000 kg/m${}^{3}$, ${\rho}_{a}$ = 1.225 kg/m${}^{3}$).

#### 4.3. The Maynooth University/Dundalk IT Team

**1-DOF state-space model**: For the single-degree-of-freedom (1-DOF) case, two models were produced by the Maynooth University/Dundalk IT (MU/DkIT) Team. The first assumes incompressible flow through the orifice and uses Equation (17) to determine the mass flow rate of air between the chamber and atmosphere, while the second model assumes compressible flow through the orifice and uses Equation (18) to the same end. Both models use Equation (16) to model the pressure within the chamber, allowing a comparison to be made between the relative importance of the different compressible elements of the two models. Both 1-DOF models operate under the assumption that the water column acts in pumping mode only, referred to here as mode 7. The first sloshing mode, referred to as mode 8, results in an antisymmetric displacement of the free surface about a line parallel to the wave front running through the centroid of the free surface. Hence, there is no overall change in the air volume above the water column due to the sloshing mode, and it does not contribute to the power absorbed by the OWC; see Figure 13. The pumping mode is, therefore, the main ‘power’ mode of the OWC. The equation of motion of the water column in the KRISO device, for a single degree of freedom in the time domain, can be represented by a variation of Equation (11), also known as the modified Cummins equation [28]. The coupled equations Equations (11), (16), and (17) (or, in the case of the incompressible flow model, Equation (18)) are recast in state-space form. Such a form facilitates the replacement of the convolution term with a state-space approximation. The state variables are:

**2-DOF state-space model**: Following a similar procedure as for the 1-DOF models, two 2-DOF models were produced. The first 2-DOF model assumes the air flow across the orifice to be incompressible and uses Equation (17) to model the mass flow rate of air between the OWC chamber and atmosphere, while the second model assumes the air flow across the orifice to be compressible, but uses Equation (18) in place of Equation (17). Both 2-DOF models represent the motion of the water column in the KRISO using a modified version of Equation (11) for both mode 7 and mode 8, Equation (16) to model the pressure variation within the chamber, and either Equation (17) or Equation (18) to model the mass flow rate through the orifice, as appropriate. The systems of equations are then cast in state-space form. Based on the behavior of the WAMIT coefficients for the chamber, the following approximations are found to be justified:

- The sloshing mode (mode 8) and the coupling modes (mode 78 and mode 87) are waveless (i.e., the convolution integrals for these modes can be set to zero).
- The coupling added masses, ${a}_{78}$ and ${a}_{87}$, are independent of frequency and equal to their infinite frequency values, ${a}_{78}^{\infty}$ and ${a}_{87}^{\infty}$, respectively.

#### 4.4. The University of Plymouth Team

**Incompressible CFD model**: The open-source CFD code, OpenFOAM (version 4.1), is utilized in preliminary simulations of four of the physical wave tank test cases (Tests 107, 207, and 407). The code solves the unsteady, incompressible, RANS equations for two isothermal, immiscible fluids using a volume of fluid interface capturing scheme based on the multi-dimensional limiter for explicit solution (MULES) algorithm [42] and typical fluid properties (${\rho}_{water}$ = 1000 kg/m${}^{3}$, ${\rho}_{air}$ = 1.225 kg/m${}^{3}$). A variable time-stepping approach is used based on a maximum Courant number, $Co$, of 1.0. Pressure-velocity coupling is achieved via the Pressure Implicit with Splitting of Operators (PISO) algorithm [43] using one outer and three inner correctors. The k-$\omega $ SST (Shear Stress Transport) turbulence closure model is used with standard wall functions, for k, $\omega $, and ${\nu}_{t}$, on all walls (including the surface of the OWC and the tank floor).

`waves2Foam`toolbox [44], via the second-order Stokes theory, expression-based boundary conditions for the phase fraction and velocity on the upstream and downstream boundaries and 5-m-long relaxation zones adjacent to both (Figure 18). A symmetry boundary condition is applied to the boundary coincident with the symmetry plane of the problem and a pressure inlet/outlet boundary condition is applied to the top boundary. All other boundaries are modeled using a no-slip condition.

`snappyHexMesh`utility; the mesh in the free-surface region and on the OWC surface is refined to level 2 (≈0.06 m), whereas the duct and orifice regions are refined to level 4 (≈0.015 m). The surfaces of the duct and orifice are also refined with four prismatic cell layers, with a first layer thickness of 1.6 mm and an expansion ratio of 2. The mesh has 1.03 million cells in total. The simulations were run on the University of Plymouth high-performance computing cluster. Each simulation took approximately 44 h to complete and used 700 cpu.hr.

## 5. Comparison of Results

#### 5.1. Potential Flow Solutions

#### 5.2. CFD Simulations

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BEM | Boundary element method |

CFD | Computational fluid dynamics |

DkIT | Dundalk IT |

DOF | Degrees of freedom |

DTU | Technical University of Denmark |

FFT | Fast Fourier transform |

FSP | Free-surface pressure modes |

HWA | Hot wire anemometer |

KRISO | Korean Research Institute of Ships and Ocean Engineering |

MU | Maynooth University |

NREL | National Renewable Energy Laboratory |

OES | Ocean Energy Systems |

OWC | Oscillating water column |

RANS | Reynolds-averaged Navier–Stokes |

URANS | Unsteady Reynolds-averaged Navier–Stokes |

WEC | Wave energy converter |

## References

- Dallman, A.; Jenne, D.S.; Neary, V.; Driscoll, F.; Thresher, R.; Gunawan, G. Evaluation of performance metrics for the Wave Energy Prize converters tested at 1/20th scale. Renew. Sustain. Energy Rev.
**2018**, 98, 79–91. [Google Scholar] [CrossRef] - Wendt, F.; Nielsen, K.; Yu, Y.H.; Bingham, H.B.; Eskilsson, C.; Kramer, M.; Babarit, A.; Bunnik, T. Ocean energy systems wave energy modelling task: Modelling, verification and validation of wave energy converters. J. Mar. Sci. Eng.
**2019**, 7, 379. [Google Scholar] [CrossRef] [Green Version] - Heath, T. A review of oscillating water columns. Philos. Trans. Roy. Soc. Lond. A
**2012**, 370, 235–245. [Google Scholar] [CrossRef] [Green Version] - Falcão, A.F.O.; Henriques, J.C.C. Model-prototype similarity of oscillating-water-column wave energy converters. Int. J. Mar. Energy
**2014**, 6, 18–34. [Google Scholar] [CrossRef] - Vicinanza, D.; Di Lauro, E.; Contestabile, P.; Gisonni, C.; Lara, J.L.; Losada, I.J. Review of innovative harbor breakwaters for wave-energy conversion. J. Waterway Port Coastal Ocean Eng.
**2019**, 145, 1–18. [Google Scholar] [CrossRef] - Babarit, A. Wave Energy Conversion. Resource, Technology and Performance; ISTE Press Ltd.: London, UK, 2018. [Google Scholar] [CrossRef]
- Cruz, J. Ocean Wave Energy. Current Status and Future Perspectives; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Yu, Y.H.; Lawson, M.; Ruehl, K.; Michelen, C. Development and demonstration of the WEC-Sim wave energy converter simulation tool. In Proceedings of the 2nd Marine Energy Technology Symposium, U.S. Department of Energy, Seattle, WA, USA, 15–18 April 2014. [Google Scholar]
- Papillon, L.; Costello, R.; Ringwood, J.V. Boundary element and integral methods in potential flow theory: A review with a focus on wave energy applications. J. Ocean. Eng. Mar. Energy
**2020**, 6, 303–337. [Google Scholar] [CrossRef] - Windt, C.; Davidson, J.; Ringwood, J.V. High-fidelity numerical modelling of ocean wave energy systems: A review of computational fluid dynamics-based numerical wave tanks. Renew. Sustain. Energy Rev.
**2018**, 93, 610–630. [Google Scholar] [CrossRef] [Green Version] - Bingham, H.B.; Ducasse, D.; Nielsen, K.; Read, R. Hydrodynamic Analysis of Oscillating Water Column, Wave Energy Devices. J. Ocean Eng. Mar. Energy
**2015**, 1, 405–419. [Google Scholar] [CrossRef] [Green Version] - WAMIT. WAMIT; User Manual, Version 7.0; WAMIT, Inc.: Chestnut Hill, MA, USA, 2008–2012; Available online: http://www.wamit.com (accessed on 17 March 2021).
- Sheng, W.; Alcom, R.; Lewis, A. Assessment of primary energy conversions of oscillating water columns. I. Hydrodynamic analysis. J. Renew. Sustain. Energy
**2014**, 6, 053113. [Google Scholar] [CrossRef] [Green Version] - Luczko, E.; Robertson, B.; Bailey, H.; Hiles, C.; Buckham, B. Representing non-linear wave energy converters in coastal wave models. Renew. Energy
**2018**, 118, 376–385. [Google Scholar] [CrossRef] - Henriques, J.C.C.; Sheng, W.; Falcão, A.F.O.; Gato, L.M.C. A comparison of biradial and Wells air turbines on the Mutriku breakwater OWC wave power plant. In Proceedings of the 36th International Conference on Ocean, Offshore and Arctic Engineering, Trondheim, Norway, 25–30 June 2017. [Google Scholar]
- Henriques, J.C.C.; Gomes, R.P.F.; Gato, L.M.C.; Robles, E.; Ceballos, S. Testing and control of a power take-off system for an oscillating-water-column wave energy converter. Renew. Energy
**2016**, 85, 714–724. [Google Scholar] [CrossRef] - Park, S.; Kim, K.H.; Nam, B.W.; Kim, J.S.; Hong, K. Experimental and numerical analysis of performance of oscillating water column wave energy converter applicable to breakwaters. In Proceedings of the 38th International Conference on Ocean, Offshore and Arctic Engineering, Glasgow, Scotland, 9–14 June 2019. [Google Scholar]
- Fenton, J.D. Nonlinear wave theories. In The Sea; Le Mehaute, B., Hanes, D.M., Eds.; John Wiley & Sons.: Hoboken, NJ, USA, 1990; pp. 3–25. [Google Scholar]
- Bullen, P.R.; Cheeseman, D.J.; Hussain, L.A.; Ruffell, A.E. The determination of pipe contraction pressure loss coeffients for incompressible turbulent flow. Heat Fluid Flow
**1987**, 8, 111–118. [Google Scholar] [CrossRef] - Newman, J.N. Marine Hydrodynamics; The MIT Press: Cambridge, MA, USA, 1977. [Google Scholar]
- Falnes, J. Ocean Waves and Oscillating Systems; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Faltinsen, O.M. Sea Loads on Ships and Offshore Structures; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Evans, D.V. Wave-power absoption by systems of oscillating surface pressure distributions. J. Fluid Mech.
**1982**, 114, 481–499. [Google Scholar] [CrossRef] - Lee, C.H.; Nielsen, F.G. Analysis of Oscillating Water Column Device Using a Panel Method. 1996. Available online: http://www.iwwwfb.org (accessed on 17 March 2021).
- Lee, C.H.; Newman, J.N.; Nielsen, F.G. Wave Interactions with an Oscillating Water Column. In Proceedings of the 6th International Offshore and Polar Engineering Conference, Los Angeles, CA, USA, 26–31 May 1996. [Google Scholar]
- Newman, J.N.; Lee, C.H. WAMIT: A Radiation-Diffraction Panel Program for Wave-Body Interactions. 2020. Available online: http://www.wamit.com (accessed on 17 March 2021).
- Newman, J.N. Wave effects on deformable bodies. Appl. Ocean. Res.
**1994**, 16, 47–59. [Google Scholar] [CrossRef] - Cummins, W.E. The impulse response function and ship motions. Schiffstecknik
**1962**, 9, 101–109. [Google Scholar] - Kelly, T.; Campbell, J.; Dooley, T.; Ringwood, J. Modelling and results for an array of 32 oscillating water columns. In Proceedings of the EWTEC 2013, Aalborg, DK, 2–5 September 2013. [Google Scholar]
- Sheng, W.; Alcorn, R.; Lewis, A. On thermodynamics in the primary power conversion of oscillating water column wave energy converters. J. Renew. Sustain. Energy
**2013**, 5, 023105. [Google Scholar] [CrossRef] [Green Version] - Kelly, T. Experimental and Numerical Modelling of a Multiple Oscillating Water Column Structure. Ph.D. Thesis, Maynooth University, Maynooth, Ireland, 2018. [Google Scholar]
- Cunningham, R. Orifice Meters with Supercritical Compressible Flow. Trans. ASME
**1951**, 73, 625–638. [Google Scholar] - Newman, J.N. (Woods Hole, MA, USA). Personal communication. 2018.
- Davidson, J.; Costello, R. Efficient Nonlinear Hydrodynamic Models for Wave Energy Converter Design—A Scoping Study. J. Mar. Sci. Eng.
**2020**, 8, 35. [Google Scholar] [CrossRef] [Green Version] - Roenby, J.; Bredmose, H.; Jasak, H. A computational method for sharp interface advection. R. Soc. Open Sci.
**2016**, 3, 160405. [Google Scholar] [CrossRef] [Green Version] - Bingham, H.B.; Read, R. DTUMotionSimulator: A Matlab Package for Simulating Linear or Weakly Nonlinear Response of a Floating Structure to Ocean Waves. 2020. Available online: https://gitlab.gbar.dtu.dk/oceanwave3d/DTUMotionSimulator (accessed on 17 March 2021).
- Bingham, H.B. A hybrid Boussinesq-panel method for predicting the motions of a moored ship. Coast. Eng.
**2000**, 40, 21–38. [Google Scholar] [CrossRef] - WEC-Sim (Wave Energy Converter SIMulator). Available online: https://wec-sim.github.io/WEC-Sim/ (accessed on 17 March 2021).
- Guo, Y.; Yu, Y.H.; van Rij, J.; Tom, N. Inclusion of Structural Flexibility in Design Load Analysis for Wave Energy Converters. In Proceedings of the 12th European Wave and Tidal Energy Conference, EWTEC, Cork, Ireland, 27 August–1 September 2017. [Google Scholar]
- Faedo, N.; Pena-Sanchez, Y.; Ringwood, J.V. Finite-Order Hydrodynamic Model Determination Using Moment-Matching. Ocean. Eng.
**2018**, 163, 251–263. [Google Scholar] [CrossRef] [Green Version] - Guo, B.; Patton, R.J.; Jin, S.; Lan, J. Numerical and experimental studies of excitation force approximation for wave energy conversion. Renew. Energy
**2018**, 125, 877–889. [Google Scholar] [CrossRef] - Rusche, H. Computational Fluid Dynamics of Dispersed Two-Phase Flows at High Phase Fractions. Ph.D. Thesis, Imperial College of Science, Technology & Medicine, London, UK, 2002. [Google Scholar]
- Issa, R.I. Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys.
**1986**, 62, 40–65. [Google Scholar] [CrossRef] - Jacobsen, N.G.; Fuhrman, D.R.; Fredsøe, J. A wave generation toolbox for the open-source CFD library: OpenFOAM
^{®}. Int. J. Numer. Methods Fluids**2012**, 70, 1073–1088. [Google Scholar] [CrossRef]

**Figure 1.**The Korea Research Institute of Ships and Ocean Engineering (KRISO) wave basin where the tests were performed.

**Figure 3.**The test wave conditions in terms of nonlinearity, $H/\lambda $, relative water depth, $kh$, and the maximum possible wave steepness, ${H}_{max}/\lambda $, as given by Reference [18].

**Figure 4.**The particulars of the chamber model used in the experiment: (

**a**) chamber geometry and dimensions; (

**b**) duct dimensions; and (

**c**) the installed model in the basin.

**Figure 5.**The locations of the internal chamber wave gauges (indicated by WG), along with the pressure sensors (indicated by DPT) and hot wire anemometer probes (indicated by HWA) inside the duct.

**Figure 6.**Examples of the incident wave elevations at the chamber, showing the target wave amplitude and the chosen section used for analysis. (

**a**) The time for the group velocity to travel from the chamber to the beach and back again (74 m); (

**b**) Test 106; (

**c**) Test 107; and (

**d**) Test 108.

**Figure 8.**The variation in air density inside the chamber for Test 118, based on Equation (2).

**Figure 9.**Control volumes for estimating the pressure drop associated with flow from the chamber to the duct and through the orifice plate.

**Figure 10.**Estimates of the measured volume flux for two cases based on measurements of the mean chamber surface elevation (from $\overline{\eta}$), the pressure and Equation (6) (from $\overline{p}$) and the air flow velocity through the orifice (from ${\overline{u}}_{a}$). A positive flow rate represents air flowing from the chamber to the atmosphere.

**Figure 11.**Estimates of the measured power absorption computed from the product of the measured pressure and each of the flux estimates shown in Figure 10, i.e., based on each of the three measured quantities: $\overline{\eta}$, $\overline{p}$, and ${\overline{u}}_{a}$. The horizontal lines indicate the mean power absorption per wave cycle.

**Figure 12.**The WAMIT geometries used to compute the coefficients. In (

**a**), the z = 0 plane coincides with the green internal chamber surface. In (

**b**), the z = 0 plane lies at the top of the reflected geometry and the green internal chamber surface becomes a dipole patch at the level z = −h in a fluid of depth 2h.

**Figure 13.**Illustration of the two-mode decomposition of the chamber surface motion using linear theory.

**Figure 14.**Convergence of the WAMIT piston-mode added-mass and damping coefficients for three grid resolutions.

**Figure 15.**Convergence of the WAMIT piston-mode excitation forces and motion response for three grid resolutions.

**Figure 17.**STAR-CCM+ computational domain and grid refinement regions for the wave regimes, the duct, and the orifice.

**Figure 18.**OpenFOAM computational domain and grid refinement regions for the wave regimes, the duct, and the orifice.

**Figure 19.**Comparison of the pressure difference: (

**a**–

**c**) cases with low, med and high wave heights and an orifice plate size of 0.4D; (

**d**) tests with orifice plate sizes corresponding to 0.3D, 0.4D, and 0.5D at med wave height; and (

**e**,

**f**) pressure difference time history for Tests 106 and 207. Superscript * indicates the 1-DOF (Degrees of freedom) model, and all the other potential flow models are 2-DOF, using both modes to approximate the internal surface motion. Subscript I and C indicate the incompressible and compressible flow models, respectively.

**Figure 20.**Comparison of the flow rate: (

**a**–

**c**) cases with low, med and high wave heights and an orifice plate size of 0.4D; (

**d**) tests with orifice plate sizes corresponding to 0.3D, 0.4D, and 0.5D in med wave height; and (

**e**,

**f**) flow rate time history for Tests 106 and 207.

**Figure 21.**Comparison of the internal chamber surface: (

**a**–

**c**) cases with low, med and high wave heights and an orifice plate size of 0.4D; (

**d**) tests with orifice plate sizes corresponding to 0.3D, 0.4D, and 0.5D in med wave height; and (

**e**,

**f**) chamber surface elevation time history for Tests 106 and 207.

**Figure 22.**Comparison of the averaged power output: (

**a**–

**c**) cases with low, med and high wave heights and an orifice plate size of 0.4D; (

**d**) tests with orifice plate sizes corresponding to 0.3D, 0.4D, and 0.5D in med wave height; and (

**e**,

**f**) power output time history for Tests 106 and 207.

**Figure 23.**Comparison of the measured, Computational Fluid Dynamics (CFD), and potential flow pressure difference with a zoom at the peak and near the zero crossing for Test 106.

**Figure 24.**Comparison of the measured, CFD, and potential flow flux and zooms at the peak and near the zero crossing for Test 106, comparing the use of the upper/lower gauge approach and the average of the two gauges for the experimental data.

**Figure 25.**(

**a**) Comparison of the flow rate magnitude using different methods for Test 106; (

**b**) flow rate at different surface planes from the incompressible CFD simulation; and (

**c**) the orientation of those surface planes inside the duct.

**Figure 26.**(

**a**) Comparison of the measured and CFD simulated flow velocity at the upper (HWA1) and lower (HWA2) gauges; (

**b**,

**c**) velocity (in z direction) contour plots near the orifice; and (

**d**) point measurements between HWA1 and HWA2 from the incompressible CFD simulation for Test 106.

**Figure 27.**The measured, CFD simulated, and potential flow method predicted surface elevation at the center of the chamber for Test 106.

**Figure 28.**Estimated power output from the KRISO experiment, CFD simulations, and potential flow solutions for Test 106.

WaveID | T [s] | H [m] | WaveID | T [s] | H [m] | WaveID | T [s] | H [m] |
---|---|---|---|---|---|---|---|---|

Low02 | 2.25 | 0.0450 | Med02 | 2.25 | 0.0718 | High02 | 2.25 | 0.1794 |

Low03 | 2.50 | 0.0467 | Med03 | 2.50 | 0.0925 | High03 | 2.50 | 0.2198 |

Low04 | 2.75 | 0.0459 | Med04 | 2.75 | 0.0799 | High04 | 2.75 | 0.1918 |

Low05 | 3.00 | 0.0464 | Med05 | 3.00 | 0.0845 | High05 | 3.00 | 0.1961 |

Low06 | 3.25 | 0.0454 | Med06 | 3.25 | 0.0881 | High06 | 3.25 | 0.1959 |

Low07 | 3.50 | 0.0440 | Med07 | 3.50 | 0.0890 | High07 | 3.50 | 0.2020 |

Low08 | 3.75 | 0.0480 | Med08 | 3.75 | 0.0983 | High08 | 3.75 | 0.2090 |

**Table 2.**The experimental wave conditions. The cases marked by * indicate that the chamber elevations were too small to allow for synchronization of the signals.

TestID | Orifice | WaveID | TestID | Orifice | WaveID | TestID | Orifice | WaveID |
---|---|---|---|---|---|---|---|---|

402 * | $0.3D$ | Med02 | 202 * | $0.5D$ | Med02 | 302 * | $1.0D$ | Med02 |

403 | $0.3D$ | Med03 | 203 | $0.5D$ | Med03 | 303 | $1.0D$ | Med03 |

404 | $0.3D$ | Med04 | 204 | $0.5D$ | Med04 | 304 | $1.0D$ | Med04 |

405 | $0.3D$ | Med05 | 205 | $0.5D$ | Med05 | 305 | $1.0D$ | Med05 |

406 | $0.3D$ | Med06 | 206 | $0.5D$ | Med06 | 306 | $1.0D$ | Med06 |

407 | $0.3D$ | Med07 | 207 | $0.5D$ | Med07 | 307 | $1.0D$ | Med07 |

408 | $0.3D$ | Med08 | 208 | $0.5D$ | Med08 | 308 | $1.0D$ | Med08 |

TestID | Orifice | WaveID | TestID | Orifice | WaveID | TestID | Orifice | WaveID |

122 * | $0.4D$ | Low02 | 102 * | $0.4D$ | Med02 | 112 | $0.4D$ | High02 |

123 | $0.4D$ | Low03 | 103 | $0.4D$ | Med03 | 113 | $0.4D$ | High03 |

124 | $0.4D$ | Low04 | 104 | $0.4D$ | Med04 | 114 | $0.4D$ | High04 |

125 | $0.4D$ | Low05 | 105 | $0.4D$ | Med05 | 115 | $0.4D$ | High05 |

126 | $0.4D$ | Low06 | 106 | $0.4D$ | Med06 | 116 | $0.4D$ | High06 |

127 | $0.4D$ | Low07 | 107 | $0.4D$ | Med07 | 117 | $0.4D$ | High07 |

128 | $0.4D$ | Low08 | 108 | $0.4D$ | Med08 | 118 | $0.4D$ | High08 |

Maximum Percentage Difference from the KRISO Exp | ||||
---|---|---|---|---|

Model | Pressure Difference | Chamber Surface Elevation | Flow Rate | Power |

TD${}_{\mathrm{I}}$ | 14% | 16% | 5% | 15% |

FD${}_{\mathrm{I}}$ | 13% | 18% | 6% | 10% |

TD${}_{\mathrm{C}}$ | 21% | 18% | 15% | 17% |

TD${}_{\mathrm{C}}$ | 37% | 14% | 11% | 17% |

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**MDPI and ACS Style**

Bingham, H.B.; Yu, Y.-H.; Nielsen, K.; Tran, T.T.; Kim, K.-H.; Park, S.; Hong, K.; Said, H.A.; Kelly, T.; Ringwood, J.V.;
et al. Ocean Energy Systems Wave Energy Modeling Task 10.4: Numerical Modeling of a Fixed Oscillating Water Column. *Energies* **2021**, *14*, 1718.
https://doi.org/10.3390/en14061718

**AMA Style**

Bingham HB, Yu Y-H, Nielsen K, Tran TT, Kim K-H, Park S, Hong K, Said HA, Kelly T, Ringwood JV,
et al. Ocean Energy Systems Wave Energy Modeling Task 10.4: Numerical Modeling of a Fixed Oscillating Water Column. *Energies*. 2021; 14(6):1718.
https://doi.org/10.3390/en14061718

**Chicago/Turabian Style**

Bingham, Harry B., Yi-Hsiang Yu, Kim Nielsen, Thanh Toan Tran, Kyong-Hwan Kim, Sewan Park, Keyyong Hong, Hafiz Ahsan Said, Thomas Kelly, John V. Ringwood,
and et al. 2021. "Ocean Energy Systems Wave Energy Modeling Task 10.4: Numerical Modeling of a Fixed Oscillating Water Column" *Energies* 14, no. 6: 1718.
https://doi.org/10.3390/en14061718