arXiv:2005.06108v1 [physics.flu-dyn] 13 May 2020 · The inertial sea wave energy converter (ISWEC) technology: device-physics, multiphase modeling and simulations Kaustubh Khedkara,

The inertial sea wave energy converter (ISWEC) technology: device-physics,multiphase modeling and simulations

Kaustubh Khedkara, Nishant Nangiaa, Ramakrishnan Thirumalaisamya, Amneet Pal Singh Bhallaa,∗

aDepartment of Mechanical Engineering, San Diego State University, San Diego, CA

Abstract

In this paper we investigate the dynamics of the inertial wave energy converter (ISWEC) device using fully-resolved computational fluid dynamics (CFD) simulations. Originally prototyped by Polytechnic Universityof Turin, the device consists of a floating, boat-shaped hull that is slack-moored to the sea bed. Internally,a gyroscopic power take off (PTO) unit converts the wave-induced pitch motion of the hull into electricalenergy. The CFD model is based on the incompressible Navier-Stokes equations and utilizes the fictitiousdomain Brinkman penalization (FD/BP) technique to couple the device physics and water wave dynamics. Anumerical wave tank is used to generate both regular waves based on fifth-order Stokes theory and irregularwaves based on the JONSWAP spectrum to emulate realistic sea operating conditions. A Froude scalinganalysis is performed to enable two- and three-dimensional simulations for a scaled-down (1:20) ISWECmodel. It is demonstrated that the scaled-down 2D model is sufficient to accurately simulate the hull’spitching motion and to predict the power generation capability of the converter. A systematic parameterstudy of the ISWEC is conducted, and its optimal performance in terms of power generation is determinedbased on the hull and gyroscope control parameters. It is demonstrated that the device achieves peakperformance when the gyroscope specifications are chosen based on the reactive control theory. It is shownthat a proportional control of the PTO control torque is required to generate continuous gyroscope precessioneffects, without which the device generates no power. In an inertial reference frame, it is demonstrated thatthe yaw and pitch torques acting on the hull are of the same order of magnitude, informing future designinvestigations of the ISWEC technology. Further, an energy transfer pathway from the water waves to thehull, the hull to the gyroscope, and the gyroscope to the PTO unit is analytically described and numericallyverified. Additional parametric analysis demonstrates that a hull length to wavelength ratio between one-halfand one-third yields high conversion efficiency (ratio of power absorbed by the PTO unit to wave power perunit crest width). Finally, device protection during inclement weather conditions is emulated by graduallyreducing the gyroscope flywheel speed to zero, and the resulting dynamics are investigated.

Keywords: renewable energy , wave-structure interaction, Brinkman penalization method , numerical wavetank , level set method , adaptive mesh refinement

1. Introduction

Ocean waves are a substantial source of renewable energy, with an estimated 2.11 ± 0.05 TW availableglobally [1]. For perspective, the United States generated 3.7 TWy (terawatt years 1) worth of energy in2013, making up about 20% of the world’s total energy production. Of this amount, only about 9% or0.33 TWy was generated from renewable sources. It is estimated that the US will produce approximately8.65 TWy by 2050 [2]. There is an ever-increasing need to invest in renewable energy harvesting techniquesin order to accelerate economic growth while maintaining a safe and healthy planet Earth. Wave energyconversion is one of the crucial strategies towards realizing future energy sustainability. It is estimated thatabout 230 TWh/year of wave energy can be extracted from the East Coast and about 590 TWh/year from

∗Corresponding authorEmail address: [email protected] (Amneet Pal Singh Bhalla)

11 TWy = 8.76× 1012 kWh.

Preprint submitted to Elsevier May 14, 2020

arX

iv:2

005.

0610

8v1

[ph

ysic

s.fl

u-dy

n] 1

3 M

ay 2

020

the West Coast of the United States alone. In spite of this abundantly available energy source, there iscurrently no commercial-scale wave power operation that exists today.

There are several unique challenges specific to wave energy extraction processes, including hostile oceanenvironments, saltwater corrosion, stochasticity of ocean and sea waves, and costly offshore wave farmsetup. Nevertheless steady progress is being made both in the design and engineering analyses of waveenergy extraction devices, which are known as wave energy converters (WECs). Consequently, several WECdesigns have been proposed over the years after gaining popularity following the 1970s oil crisis. Howeverunlike wind turbines, an ultimate WEC architecture has not yet been identified by researchers.

(a) (b)

(c) (d)

Figure 1: The inertial sea wave energy converter (ISWEC) device developed by the Mattiazzo Group at Polytechnic Universityof Turin. (a) ISWEC device freely floating in relatively calm sea conditions. (b) Gyroscope casing mounted on the power takeoff (PTO) axis. The PTO system is housed inside the hull. (c) Front and (d) side views of the ISWEC exhibiting pitchingmotion during operation. Image courtesy of the Mattiazzo Group and Wave for Energy S.R.L., Turin.

One WEC design that addresses some of the critical wave energy extraction challenges is the inertialsea wave energy converter (ISWEC) device prototyped by Polytechnic University of Turin [3, 4, 5]. Thisdevice consists of a floating, boat-shaped hull that is slack-moored to the seabed, which internally housesa gyroscopic power take off unit (PTO); see Fig. 1. The ISWEC can be classified as a pitching point-absorber whose dimensions are shorter than the length of the water waves. The device utilizes precessioneffects produced from the spinning gyroscope and pitching hull to drive a sealed electric generator/PTO. Therotational velocity of the spinning gyroscope and the PTO control torque act as sea-state tuning parametersthat can be optimized/controlled (in real-time or via remote human-machine interfaces) to enhance theconversion efficiency of the device. Since all crucial electro-mechanical parts are sealed within the hull, theISWEC is a robust and cost-effective wave energy conversion technology. Due to its simple design, devicescan be produced by retrofitting abandoned ships, which can potentially reduce manufacturing costs and leadto easy adoption of the technology. Moreover, such devices could be lined up end-to-end just offshore, whichwould not only ensure maximal wave energy absorption but also protection of the coastline.

Although ISWEC devices have only recently been prototyped since their inception in 2011 by Bracco et

2

al. [3, 6, 7, 8, 9], their design and performance has been of much interest to the greater research communityin the past few years. Medeiros and Brizzolara [10] used the boundary element method (BEM) based onlinear potential flow equations to simulate the ISWEC and evaluate its power generation capabilities asa function of flywheel speed and derivative control of the PTO torque. They also demonstrated that thespinning gyroscopes can induce yaw torque on the hull. Faedo et al. used an alternative moment-matching-based approach to model the radiation force convolution integral, thereby overcoming the computationaland representational drawbacks of simulating ISWEC devices using the BEM-based Cummins equation [11].Although these lower fidelity methods are able to simulate ISWEC dynamics at low computational costs, theyare unable to resolve highly nonlinear phenomena often seen during practical operation such as wave-breakingand wave-overtopping. Unsurprisingly, the Turin group has extensively used carefully calibrated (with respectto wave tank experiments) BEM models to refine and optimize their preliminary designs [12, 13, 14, 15]. Incontrast, simulations based on the incompressible Navier-Stokes (INS) equations are able to resolve the wave-structure interaction (WSI) quite accurately and without making small motion approximations employed bylow-fidelity BEM models [16, 17]. However, fully-resolved INS simulations are computationally expensiveand typically require high performance computing (HPC) frameworks. In a preliminary study, Bergmann etal. enabled fully-resolved simulation of the ISWEC’s wave-structure interaction by making use of an INS-based flow solver coupled to an immersed boundary method [18]. The wave propagation in their channelfollowed the canonical “dam-break” problem setup [19] — a column of water is released from one end of thechannel, which is then reflected from the opposite end, and so-forth. Although such simple wave propagationmodels are not suitable to study the device performance at a real site of operation, Bergmann et al. werenevertheless able to capture key device dynamics in their simulations. In addition to these research efforts,industry has become interested in piloting and manufacturing these devices. Recently, the multinational oiland gas corporation Eni installed an ISWEC device off the coast of Ravenna 2 near their offshore assets. Itis clear that there is a need to further investigate ISWEC dynamics and explore the design space to enablerapid adoption of this technology, possibly through an industry-academic partnership.

In this work, we perform a comprehensive study of the ISWEC device using high fidelity simulations from apreviously developed fictitious domain Brinkman penalization (FD/BP) method based on the incompressibleNavier-Stokes equations [20]. Although the methodology is similar to the work of Bergmann et al., weconsider more realistic operating conditions by using a numerical wave tank (NWT) to generate both regularand irregular water waves. We conduct a systematic variation of control parameters (i.e. PTO control torque,flywheel moment of inertia and speed, hull length) to determine the optimal performance of the device (interm of power generation) and study its dynamics as a function of these parameters. We also provide atheoretical basis to obtain the optimal control parameters for the device’s design at a specific installationsite. Moreover, we analytically describe an energy transfer pathway from water waves to the hull, the hullto the gyroscope, and the gyroscope to the power take off (PTO) unit, and verify that it is numericallysatisfied by our simulations. A Froude scaling analysis is performed to reduce the computational cost ofsimulating a full-scale ISWEC device, which is used to define the geometry and flow conditions for both two-and three-dimensional simulations of a scaled down 1:20 ISWEC device. Additionally, we verify that the 2DISWEC model produced similar dynamics to the 3D model, thereby allowing us to obtain accurate resultsat reduced simulation cycle times. We also simulate a possible device protection strategy during inclementweather conditions and study the resulting dynamics.

The rest of the paper is organized as follows. We first describe the dynamics, power generation, geometricproperties, and scaling analysis of the ISWEC device in Sec. 2. Next, we describe the numerical wave tankapproach used to generate both regular and irregular waves for our simulations in Sec. 3. In Sec. 4, wedescribe the continuous and discrete equations for the multiphase wave-structure interaction system, andoutline/validate the solution methodology for the FD/BP technique. In Sec. 5, we briefly describe thesoftware implementation and computing hardware utilized in this study. In Sec. 6, we perform spatial andtemporal resolution tests to select a grid spacing and time step size that ensures adequate resolution ofISWEC dynamics. Finally in Sec. 7, we conduct a systematic parameter study on the various hull andgyroscope parameters and evaluate the device performance in terms of generated power.

2https://www.eni.com/en-IT/operations/iswec-eni.html

3

https://www.eni.com/en-IT/operations/iswec-eni.html

φ<latexit sha1_base64="F6Es+RKsztJTCiUrDTNzxU4+C9o=">AAAB8XicbVDLSgMxFL1TX7W+qi7dBIvgqsxUQZdFNy4r2Ae2Q8mkmTY0kxmSO0IZ+hduXCji1r9x59+YtrPQ1gOBwzn3kHtPkEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6wEnC/YgOlQgFo2ilx94gxqyXjMS0X664VXcOskq8nFQgR6Nf/rJhlkZcIZPUmK7nJuhnVKNgkk9LvdTwhLIxHfKupYpG3PjZfOMpObPKgISxtk8hmau/ExmNjJlEgZ2MKI7MsjcT//O6KYbXfiZUkiJXbPFRmEqCMZmdTwZCc4ZyYgllWthdCRtRTRnakkq2BG/55FXSqlW9i2rt/rJSv8nrKMIJnMI5eHAFdbiDBjSBgYJneIU3xzgvzrvzsRgtOHnmGP7A+fwB9bmRGg==</latexit>

δ<latexit sha1_base64="oP0M6R7sDig14VhHnbN7WtufT5g=">AAAB83icbVDLSsNAFJ34rPVVdelmsAiuSlIFXRbduKxgH9CEMpnctkMnkzBzI5TQ33DjQhG3/ow7/8Zpm4W2Hhg4nHMP984JUykMuu63s7a+sbm1Xdop7+7tHxxWjo7bJsk0hxZPZKK7ITMghYIWCpTQTTWwOJTQCcd3M7/zBNqIRD3iJIUgZkMlBoIztJLvRwnmfgQS2bRfqbo1dw66SryCVEmBZr/yZeM8i0Ehl8yYnuemGORMo+ASpmU/M5AyPmZD6FmqWAwmyOc3T+m5VSI6SLR9Culc/Z3IWWzMJA7tZMxwZJa9mfif18twcBPkQqUZguKLRYNMUkzorAAaCQ0c5cQSxrWwt1I+YppxtDWVbQne8pdXSbte8y5r9YerauO2qKNETskZuSAeuSYNck+apEU4SckzeSVvTua8OO/Ox2J0zSkyJ+QPnM8feOmR9w==</latexit>

ε<latexit sha1_base64="zx98ygV+aySoJyYTjbVMB3jlhHo=">AAAB+nicbVDLSgMxFM34rPU11aWbYBFclZkq6LLoxmUF+4B2KJk004ZmkiG5UyljP8WNC0Xc+iXu/BvTdhbaeuDC4Zx7k3tPmAhuwPO+nbX1jc2t7cJOcXdv/+DQLR01jUo1ZQ2qhNLtkBgmuGQN4CBYO9GMxKFgrXB0O/NbY6YNV/IBJgkLYjKQPOKUgJV6bqnbV5B1x0SzxHCh5LTnlr2KNwdeJX5OyihHved+2TdoGjMJVBBjOr6XQJARDZwKNi12U8MSQkdkwDqWShIzE2Tz1af4zCp9HCltSwKeq78nMhIbM4lD2xkTGJplbyb+53VSiK6DjMskBSbp4qMoFRgUnuWA+1wzCmJiCaGa210xHRJNKNi0ijYEf/nkVdKsVvyLSvX+sly7yeMooBN0is6Rj65QDd2hOmogih7RM3pFb86T8+K8Ox+L1jUnnzlGf+B8/gAmOZST</latexit>

Wavedirection

<latexit sha1_base64="p0VXxj9+mJh83plm/pQXLB5gT2Y=">AAACDHicbVC7TsMwFHV4lvAqMLJEVEgMqErKAGMFC2OR6ENqqspxb1qrjhPZTkUVpTsLv8LCAEKsfAAbf4PTZICWI1k6Pvdc2ed4EaNS2fa3sbK6tr6xWdoyt3d29/bLB4ctGcaCQJOELBQdD0tglENTUcWgEwnAgceg7Y1vsnl7AkLSkN+raQS9AA859SnBSkv9csVV8KCSNp5A6rpmfpvN3PMBFUAyT2pql12157CWiVOQCirQ6Je/3EFI4gC4IgxL2XXsSPUSLBQlDFLTjSVEmIzxELqachyA7CXzMKl1qpWB5YdCH66sufp7I8GBlNPA084Aq5FcnGXif7NurPyrXkJ5FCvgJH/Ij5mlQitrxsoTs6kmmAiq/2qRERaYKN1fVoKzGHmZtGpV56Jau6tV6tdFHSV0jE7QGXLQJaqjW9RATUTQI3pGr+jNeDJejHfjI7euGMXOEfoD4/MHf4Wb4Q==</latexit>

PTO unit<latexit sha1_base64="uWQen07cjFo2jAxWzB6iYVCzYso=">AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU0mqoMeiF29W6Be0oWy2m3bpZhN2J8US6i/x4kERr/4Tb/4bN20O2vpg4PHeDDPz/FhwDY7zbRXW1jc2t4rbpZ3dvf0D+/CopaNEUdakkYhUxyeaCS5ZEzgI1okVI6EvWNsf32Z+e8KU5pFswDRmXkiGkgecEjBS37Z7wB4hrTfunxLJYVbq22Wn4syBV4mbkzLKUe/bX71BRJOQSaCCaN11nRi8lCjgVLBZqZdoFhM6JkPWNVSSkGkvnV8+w2dGGeAgUqYk4Ln6eyIlodbT0DedIYGRXvYy8T+vm0Bw7aVcxgkwSReLgkRgiHAWAx5wxSiIqSGEKm5uxXREFKFgwspCcJdfXiWtasW9qFQfLsu1mzyOIjpBp+gcuegK1dAdqqMmomiCntErerNS68V6tz4WrQUrnzlGf2B9/gCdIpOj</latexit>

Gyroscope<latexit sha1_base64="OZiMuiQ4BPwNhKrS5PzhFK/RsNw=">AAAB+XicbVBNS8NAEN3Ur1q/oh69BIvgqSRV0GPRgx4r2A9oQ9lsJ+3STTbsTool9J948aCIV/+JN/+N2zYHbX0w8Hhvhpl5QSK4Rtf9tgpr6xubW8Xt0s7u3v6BfXjU1DJVDBpMCqnaAdUgeAwN5CignSigUSCgFYxuZ35rDEpzGT/iJAE/ooOYh5xRNFLPtrsIT5jdTZTUTCYw7dllt+LO4awSLydlkqPes7+6fcnSCGJkgmrd8dwE/Ywq5EzAtNRNNSSUjegAOobGNALtZ/PLp86ZUfpOKJWpGJ25+nsio5HWkygwnRHFoV72ZuJ/XifF8NrPeJykCDFbLApT4aB0ZjE4fa6AoZgYQpni5laHDamiDE1YJROCt/zyKmlWK95FpfpwWa7d5HEUyQk5JefEI1ekRu5JnTQII2PyTF7Jm5VZL9a79bFoLVj5zDH5A+vzB2BflCM=</latexit>

Ballast<latexit sha1_base64="FtW7RsUXGFMdzo7gvfeDQ7Mbf2Y=">AAAB9XicbVBNTwIxEO36ifiFevTSSEw8kV000SPBi0dM5COBlXTLAA3d7qadVcmG/+HFg8Z49b94899YYA8KvmSSl/dm2pkXxFIYdN1vZ2V1bX1jM7eV397Z3dsvHBw2TJRoDnUeyUi3AmZACgV1FCihFWtgYSChGYyup37zAbQRkbrDcQx+yAZK9AVnaKX7DsITplUmJTM46RaKbsmdgS4TLyNFkqHWLXx1ehFPQlDI7Qum7bkx+inTKLiESb6TGIgZH7EBtC1VLATjp7OtJ/TUKj3aj7QthXSm/p5IWWjMOAxsZ8hwaBa9qfif106wf+WnQsUJguLzj/qJpBjRaQS0JzRwlGNLGNfC7kr5kGnG0QaVtyF4iycvk0a55J2XyrcXxUo1iyNHjskJOSMeuSQVckNqpE440eSZvJI359F5cd6dj3nripPNHJE/cD5/ABlYkuY=</latexit>

(a)

x<latexit sha1_base64="hL+FaLtOT9luwfLW3Ut08xl3Pcw=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx4hkUcCGzI79MLI7OxmZtZICF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOeHjQA=</latexit>

y<latexit sha1_base64="mEcz1FLhuG1BpP6c5hi50qAIJ0g=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOl5qRfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJu1b1Lqq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8f6QuNAQ==</latexit>

z<latexit sha1_base64="VLEo6VgUnu2TnOxoOkqsMPXvyTo=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx4hkUcCGzI79MLI7OxmZtYECV/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOqPjQI=</latexit>

x1<latexit sha1_base64="cBBp6MuDbFt1W7tLqvIgQKnsIZQ=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48VTFtoQ9lsp+3SzSbsbsQS+hu8eFDEqz/Im//GbZuDtj4YeLw3w8y8MBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoqeNUMfRZLGLVDqlGwSX6hhuB7UQhjUKBrXB8O/Nbj6g0j+WDmSQYRHQo+YAzaqzkP/Uyb9orV9yqOwdZJV5OKpCj0St/dfsxSyOUhgmqdcdzExNkVBnOBE5L3VRjQtmYDrFjqaQR6iCbHzslZ1bpk0GsbElD5urviYxGWk+i0HZG1Iz0sjcT//M6qRlcBxmXSWpQssWiQSqIicnsc9LnCpkRE0soU9zeStiIKsqMzadkQ/CWX14lzVrVu6jW7i8r9Zs8jiKcwCmcgwdXUIc7aIAPDDg8wyu8OdJ5cd6dj0VrwclnjuEPnM8f0ZSOsA==</latexit>

y1<latexit sha1_base64="z2Qy524JPOlQ3GLEDP+KaRi6Mu0=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YA2lM120i7dbMLuRgihv8GLB0W8+oO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ219Y3Nru7RT3t3bPzisHB23dZwqhi0Wi1h1A6pRcIktw43AbqKQRoHATjC5m/mdJ1Sax/LRZAn6ER1JHnJGjZVa2SD3poNK1a25c5BV4hWkCgWag8pXfxizNEJpmKBa9zw3MX5OleFM4LTcTzUmlE3oCHuWShqh9vP5sVNybpUhCWNlSxoyV39P5DTSOosC2xlRM9bL3kz8z+ulJrzxcy6T1KBki0VhKoiJyexzMuQKmRGZJZQpbm8lbEwVZcbmU7YheMsvr5J2veZd1uoPV9XGbRFHCU7hDC7Ag2towD00oQUMODzDK7w50nlx3p2PReuaU8ycwB84nz/THI6x</latexit>

z1<latexit sha1_base64="QURhUBGyYxg37mjwVGgVV9e18/w=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48VTFtoQ9lsp+3SzSbsboQa+hu8eFDEqz/Im//GbZuDtj4YeLw3w8y8MBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoqeNUMfRZLGLVDqlGwSX6hhuB7UQhjUKBrXB8O/Nbj6g0j+WDmSQYRHQo+YAzaqzkP/Uyb9orV9yqOwdZJV5OKpCj0St/dfsxSyOUhgmqdcdzExNkVBnOBE5L3VRjQtmYDrFjqaQR6iCbHzslZ1bpk0GsbElD5urviYxGWk+i0HZG1Iz0sjcT//M6qRlcBxmXSWpQssWiQSqIicnsc9LnCpkRE0soU9zeStiIKsqMzadkQ/CWX14lzVrVu6jW7i8r9Zs8jiKcwCmcgwdXUIc7aIAPDDg8wyu8OdJ5cd6dj0VrwclnjuEPnM8f1KSOsg==</latexit>

ε<latexit sha1_base64="J/UA4ylXdYra8KAL+Neg17Y6Uh8=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69BIvgqSRV0GPRi8cK9gPSUDbbTbt0sxt2J4VS+jO8eFDEq7/Gm//GbZuDtj4YeLw3w8y8KBXcoOd9O4WNza3tneJuaW//4PCofHzSMirTlDWpEkp3ImKY4JI1kaNgnVQzkkSCtaPR/dxvj5k2XMknnKQsTMhA8phTglYKumOiWWq4ULJXrnhVbwF3nfg5qUCORq/81e0rmiVMIhXEmMD3UgynRCOngs1K3cywlNARGbDAUkkSZsLp4uSZe2GVvhsrbUuiu1B/T0xJYswkiWxnQnBoVr25+J8XZBjfhlMu0wyZpMtFcSZcVO78f7fPNaMoJpYQqrm91aVDoglFm1LJhuCvvrxOWrWqf1WtPV5X6nd5HEU4g3O4BB9uoA4P0IAmUFDwDK/w5qDz4rw7H8vWgpPPnMIfOJ8/ux2Riw==</latexit>

δ<latexit sha1_base64="JQcAzWaAuoYAXQfxggm9zL/KhUk=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSRV0GPRi8cK9gPaUDabTbt2sxt2J0Ip/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemApu0PO+ncLa+sbmVnG7tLO7t39QPjxqGZVpyppUCaU7ITFMcMmayFGwTqoZSULB2uHodua3n5g2XMkHHKcsSMhA8phTglZq9SImkPTLFa/qzeGuEj8nFcjR6Je/epGiWcIkUkGM6fpeisGEaORUsGmplxmWEjoiA9a1VJKEmWAyv3bqnlklcmOlbUl05+rviQlJjBknoe1MCA7NsjcT//O6GcbXwYTLNEMm6WJRnAkXlTt73Y24ZhTF2BJCNbe3unRINKFoAyrZEPzll1dJq1b1L6q1+8tK/SaPowgncArn4MMV1OEOGtAECo/wDK/w5ijnxXl3PhatBSefOYY/cD5/AJLajyA=</latexit>

δ<latexit sha1_base64="JQcAzWaAuoYAXQfxggm9zL/KhUk=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSRV0GPRi8cK9gPaUDabTbt2sxt2J0Ip/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemApu0PO+ncLa+sbmVnG7tLO7t39QPjxqGZVpyppUCaU7ITFMcMmayFGwTqoZSULB2uHodua3n5g2XMkHHKcsSMhA8phTglZq9SImkPTLFa/qzeGuEj8nFcjR6Je/epGiWcIkUkGM6fpeisGEaORUsGmplxmWEjoiA9a1VJKEmWAyv3bqnlklcmOlbUl05+rviQlJjBknoe1MCA7NsjcT//O6GcbXwYTLNEMm6WJRnAkXlTt73Y24ZhTF2BJCNbe3unRINKFoAyrZEPzll1dJq1b1L6q1+8tK/SaPowgncArn4MMV1OEOGtAECo/wDK/w5ijnxXl3PhatBSefOYY/cD5/AJLajyA=</latexit>

ε<latexit sha1_base64="J/UA4ylXdYra8KAL+Neg17Y6Uh8=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69BIvgqSRV0GPRi8cK9gPSUDbbTbt0sxt2J4VS+jO8eFDEq7/Gm//GbZuDtj4YeLw3w8y8KBXcoOd9O4WNza3tneJuaW//4PCofHzSMirTlDWpEkp3ImKY4JI1kaNgnVQzkkSCtaPR/dxvj5k2XMknnKQsTMhA8phTglYKumOiWWq4ULJXrnhVbwF3nfg5qUCORq/81e0rmiVMIhXEmMD3UgynRCOngs1K3cywlNARGbDAUkkSZsLp4uSZe2GVvhsrbUuiu1B/T0xJYswkiWxnQnBoVr25+J8XZBjfhlMu0wyZpMtFcSZcVO78f7fPNaMoJpYQqrm91aVDoglFm1LJhuCvvrxOWrWqf1WtPV5X6nd5HEU4g3O4BB9uoA4P0IAmUFDwDK/w5qDz4rw7H8vWgpPPnMIfOJ8/ux2Riw==</latexit>

(b)

Figure 2: (a) ISWEC device schematic and the main rotational velocities of the system: hull’s pitch velocity δ, gyroscope’sangular velocity φ, and the induced precession velocity of the PTO shaft ε. (b) Hull and gyroscope reference frames.

2. ISWEC dynamics

In this section, we mathematically describe the dynamics, power generation, and geometric properties ofthe ISWEC device.

2.1. ISWEC dynamics

Externally, the ISWEC device appears as a monolithic hull that is slack-moored to the seabed. Internally,the device houses a spinning gyroscopic system that drives a sealed electric generator. The pitching velocityof the hull is mainly responsible for converting the wave motion into electrical output. To simplify the modeland discussion, the other remaining degrees of freedom of the hull are not considered in this study; seeAppendix A for a comparison of one and two degrees of freedom ISWEC models. As the device operates,the combination of wave induced pitching velocity δ and spinning gyroscope/flywheel velocity φ induces aprecession torque in the ε coordinate direction. The wave energy conversion is made possible by dampingthe motion along the ε-direction by the electric generator, which is commonly referred to as the power take-off (PTO) unit. Fig. 2(a) shows the schematic of the ISWEC device, including the external hull, ballast,gyroscope, and PTO unit.

To derive the three-way coupling between the waves, hull, and gyroscopic system we consider an inertialreference frame xyz attached to the hull and a rotating non-inertial reference frame x1y1z1 attached to thegyroscope as shown in Fig. 2(b). The gyroscope reference frame is obtained from the hull reference frameby two subsequent finite rotations δ and ε. The origin of both reference frames is taken to be the center ofgravity of the device.

In the absence of waves, δ = 0 and ε = 0, and the flywheel rotates with a constant angular velocity φalong the vertical z1-axis. This configuration is taken to be the initial position of the device, in which thetwo reference frames also coincide. When the first wave reaches the hull location, it tilts the device by anangle δ and the hull attains a pitching velocity δ along the x-axis. The gyroscope structure rotates by thesame angle δ about the x- (or the x1-) axis. The rotated configuration of the x1y1z1 reference frame is shownby dashed lines in Fig. 2(b). As the hull begins to pitch, the gyroscope is subject to two angular velocities:δ along x1-axis and φ along z1-axis. This velocity combination produces a precession torque in the thirdorthogonal direction y1. This induced torque precesses the gyroscope by an angle ε about the y1-axis. As aresult of the two subsequent rotations, the gyroscope frame attains an orientation shown by bold red linesin Fig. 2(b).

The evolution of the gyroscope’s dynamics results in a gyroscopic torque MG = (Mx1 ,My1 ,Mz1),which can be related to the rotational kinematic variables using conservation of angular momentum. Theangular velocity Ω1 of the gyroscope reference frame and the angular velocity ΩG of the gyroscope are both

4

written in the x1y1z1 coordinate system and their evolution can be expressed in terms of δ, ε, and φ as

Ω1 = δ cos ε i1 + ε j1 + δ sin ε k1, (1)

ΩG = δ cos ε i1 + ε j1 + (δ sin ε+ φ) k1, (2)

in which i1, j1, and k1 are the unit vectors along x1-, y1-, and z1-directions, respectively. The rate of changeof the gyroscope’s angular momentum with respect to time is related to the gyroscopic torque MG by

MG =dHG

dt, (3)

in which HG = IGΩG is the angular momentum of the gyroscope and IG is the inertia matrix of thegyroscope. In the x1y1z1 reference frame, IG reads as

IG =

Ix1x10 0

0 Iy1y1 00 0 Iz1z1

≈I 0 0

0 I 00 0 J

. (4)

The flywheel structure, including its support brackets etc., is typically designed such that Ix1x1≈ Iy1y1 = I

and Iz1z1 = J ' I. Using Eqs. (2) and (4), the angular momentum of the flywheel is given by

HG = Iδ cos ε i1 + Iε j1 + J(δ sin ε+ φ) k1. (5)

Differentiating Eq. (5) with respect to time in the inertial reference frame involves computing time derivatives

of the unit vectors i1, j1, and k1:

di1dt

= Ω1 × i1 = −ε k1 + δ sin ε j1, (6)

dj1dt

= Ω1 × j1 = δ cos ε k1 − δ sin ε i1, (7)

dk1

dt= Ω1 × k1 = −δ cos ε j1 + ε i1. (8)

Finally after some algebraic simplification, a component-wise expression for the gyroscopic torque MG isobtained

MG =

Mx1

My1

Mz1

=

Iδ cos ε+ (J − 2I) δε sin ε+ Jφε

Iε+ (I − J)δ2 sin ε cos ε− Jφδ cos ε

Jδ sin ε+ Jδε cos ε+ Jφ

. (9)

The precession velocity ε of the generator shaft is damped using a proportional derivative (PD) controllaw implemented in the PTO unit. The PD control torque can be modeled as a spring-damper system withthe following form

Mε = MG · j1 = −kε− cε. (10)

Here, k is a spring-like stiffness parameter and c is a damper-like dissipation parameter that can be adjustedin real-time (usually through feedback) to enhance the conversion efficiency of the device when the incomingwaves change their characteristics. The wave power absorbed by the PTO unit (as a function of time) is

PPTO = cε2. (11)

Therefore, the precession component of the gyroscopic torque is balanced by the PD control torque, My1 =Mε, which is also responsible for extracting the wave energy. The other components Mx1 and Mz1 of thegyroscopic torque are balanced/sustained by the hydrodynamic torques acting on the hull and the subsequenthull-gyroscope interactions. To understand this balance, we consider the hydrodynamic torque and motion

5

of the hull about the pitch (x-direction) as observed from the inertial reference frame xyz

Mhydro = IHdδ

dt+Mδ, (12)

in whichMhydro is the hydrodynamic torque acting on the hull, IH is the moment of inertia of the hull, andMδ is the projection of the gyroscopic torque on the x-axis:

Mδ = MG · i= MG · (i1 cos ε+ k1 sin ε)

= (J sin 2ε+ I cos 2ε)δ + Jφε cos ε+ 2(J − I)δε sin ε cos ε+ Jφ sin ε. (13)

From Eq. (12) it can be seen that the gyroscopic reaction Mδ acting on the hull opposes the wave inducedpitching motion. Similarly, a second reaction torque Mφ acts on the hull along the z-direction and opposesits wave induced yaw motion:

Mφ = MG · k= MG ·

[(k1 cos ε− i1 sin ε) cos δ + j1 sin δ

]=[(J − I)δ sin ε cos ε+ δε[J(cos2 ε− sin2 ε) + 2I sin2 ε]− Jφε sin ε+ Jφ cos ε

]cos δ

+[Iε+ (I − J)δ2 sin ε cos ε− Jφδ cos ε

]sin δ (14)

In Sec. 7.3.1, we show that this yaw torque is the same order of magnitude as the pitch torque Mδ. Inpractice, however, its contribution is partially cancelled out by the mooring system of the device. Moreover,using an even number of gyroscopic units will cancel the yaw component of the gyroscopic torque acting onthe hull if each flywheel pair spins with equal and opposite velocity, as described by Raffero [21]. Therefore,we do not consider the effect of Mφ on the ISWEC dynamics in our (3D) model.

2.2. Power transfer from waves to PTO

To understand the power transfer from waves to the hull and from the hull to the PTO unit, we derivethe time-averaged kinetic energy equations of the ISWEC system. These equations highlight the coupledterms that are responsible for wave energy conversion through the ISWEC device.

First, we consider the rotation of the gyroscope around the PTO axis. The equation of motion in the εcoordinate direction, as derived in the previous section is re-written below

Iε+ (I − J)δ2 sin ε cos ε− Jφδ cos ε =Mε = −kε− cε. (15)

Rearranging Eq. (15) with the approximation I ≈ J simplifies the equation to

Iε+ cε+ kε = Jφδ cos ε. (16)

From the above equation, it can be seen that the product of the hull’s pitch velocity δ and the gyroscope’sangular velocity φ yields a forcing term that drives the PTO motion. Multiplying Eq. (16) by the precessionvelocity ε and rearranging some terms, we obtain the kinetic energy equation for the PTO dynamics

Id

dt

(ε2

2

)+ cε2 + k

d

dt

(ε2

2

)= Jφδε cos ε. (17)

Taking the time-average of Eq. (17) over one wave period, the first and third terms on the left hand side ofthe equation evaluate to zero. The remaining terms describe the transfer of power from the hull to the PTOunit:

〈cε2〉 = 〈Jφδε cos ε〉, (18)

6

in which 〈·〉 = 1T∫ t+Tt

(.) dt represents the time-average of a quantity over one wave period T 3. Here, 〈cε2〉is the average power absorbed by the PTO, denoted PPTO, and 〈Jφδε cos ε〉 is the average power generateddue to the gyroscopic motion through its interaction with the hull, denoted by Pgyro. Similarly, the kinetic

energy equation of the hull can be derived by multiplying hull dynamics Eq. (12) by the pitch velocity δ

Mhydroδ = IHδδ +Mδ δ. (19)

Under the assumptions I ≈ J and a constant gyroscope spinning speed, Mδ in Eq. (13) simplifies to

Mδ = Jδ + Jφε cos ε. (20)

Using Eqs. (19) and (20), and rearranging some terms, we obtain

Mhydroδ = IHd

dt

(δ2

2

)+ J

d

dt

(δ2

2

)+ Jφδε cos ε. (21)

Taking the time-average of Eq. (21) over one wave period, the first and second terms on the right hand sideevaluate to zero, and the expression reads

〈Mhydroδ〉 = 〈Jφδε cos ε〉. (22)

Here, 〈Mhydroδ〉 is the power transferred from the waves to the hull, denoted Phull, and 〈Jφδε cos ε〉 is thesame expression on the right side of Eq. 18. Hence, combining Eqs. (18) and (22), we obtain an equationdescribing the pathway of energy transfer from waves to the PTO:

〈Mhydroδ〉︸︷︷︸waves→hull

= 〈Jφδε cos ε〉︸︷︷︸hull→gyroscope

= 〈cε2〉︸︷︷︸gyroscope→PTO

, (23)

which is written succinctly as Phull = Pgyro = PPTO. Eq. (23) is quantitatively verified for the ISWEC modelunder both regular and irregular wave environments in Sec. 7.

2.3. PTO and gyroscope parameters

The energy transfer equation can be used to select the PTO and gyroscope parameters that achieve arated power of the installed device PR. From Eq. (23)

c =PR

〈ε2〉 =2PR

ε20

, (24)

in which ε0 is the amplitude of the precession velocity, expressed in terms of the amplitude of the precessionangle ε0 as ε0 = ε0ω. Here, we assume that all of the ISWEC components are excited at the externalwave frequency ω = 2π/T to achieve optimal performance. Based on experimental data of real ISWECdevices [5, 4], we prescribe ε0 in the range 40 ≤ ε0 ≤ 80 to obtain the damping parameter c from Eq. (24).To prescribe the rest of the gyroscope parameters, we make use of Eq. 18. Since this expression is nonlinear,we linearize it about ε = 0 (a reasonable approximation for relatively calm conditions) to estimate thegyroscope angular momentum as

Jφ =cε0

δ0, (25)

in which the amplitude of the hull pitch velocity δ0 = δ0ω, expressed in terms of the amplitude of the hullpitch angle δ0, is used. Again based on the experimental data, we prescribe δ0 in the range 2 ≤ δ0 ≤ 20,and φ in the range 250 ≤ φ ≤ 1000 RPM 4 to obtain the J value of the gyroscope from Eq. (25). The I

3For irregular waves the time-average could be defined over one significant wave period.4This range of φ is for the full-scale ISWEC device, which can be scaled by an appropriate factor for the scaled-down model.

See Sec. 2.5 for scaling analyses.

7

value of the gyroscope is set as a scaled value of J , i.e. I = γJ where γ ≤ 1. We study the effect of varyingγ in Sec. 7.3.3.

The only remaining free parameter is the PTO stiffness k used in the control torque. We make use ofreactive control theory [2] and prescribe k as

k = ω2I, (26)

ensuring that the gyroscopic system oscillates at the wave frequency around the PTO axis. Using thelinearized version of Eq. (16), it can be shown that if the gyroscope oscillates with the external wave forcingfrequency, a resonance condition is achieved along the PTO axis and the output power is maximized [2]. Inthis state, both the hull and gyroscopic systems oscillate at the external wave frequency and their couplingis maximized.

2.4. Hull shape

The ISWEC’s external hull is a boat-shaped vessel, which we idealize by a half-cylinder of length L,height H, and width W . For the actual device, a part of the outer periphery is flattened out to ease theinstallation of mechanical and electrical parts near the bottom-center location (see Fig. 3(b)). We neglectthese geometric details in our model shown in Fig. 3(a). The inside of the device is mostly hollow and thebuoyancy-countering ballast is placed around the outer periphery.

The hull length L is a function of λ, the wavelength of the “design” wave at device installation site. Asanalyzed by Cagninei et al. [4], the optimal hull length is between λ/3 ≤ L ≤ λ/2 for an ISWEC devicethat is mainly excited in the pitch direction. The hull width W is decided based on the rated power of theinstalled device PR, relative capture width (RCW) of the device (or the device conversion efficiency) η, andwave power per unit crest width Pwave. These quantities are related through the expressions

W =PR

η · Pwaveand η = PPTO/Pwave, (27)

in which P denotes time-averaged power. Sec. 3 provides closed-form expressions of Pwave for both regularand irregular waves. For the 2D ISWEC model we use W = 1, which corresponds to a unit crest width ofthe wave.

(a) Idealized ISWEC hull geometry (b) Experimental ISWEC hull geometry

Figure 3: ISWEC hull shapes.

2.5. Scaled ISWEC model

In order to reduce the computational cost of simulating a full-scale ISWEC device operating in highReynolds number (Re) regimes, we use Froude scaling [22] to scale the model problem down by a 1:20 ratio.The Froude number (Fr) measures the resistance of a partially submerged object moving through water andis defined as

Fr =characteristic velocity

gravitational wave velocity=

Uc√gLc

, (28)

in which Uc is the characteristic velocity, Lc is the characteristic length, and g is the gravitational accelerationconstant. In offshore marine hydromechanics, Froude scaling allows us to compare the dynamics of two vesselseven if their sizes are different (since they produce a similar wake). Two vessels having the same Froude

8

number may not be operating in the same Reynolds number regime. In the present study, the scaled-downmodel operates in lower Re conditions and it does not capture fine scale details such as extreme wave breakingand spray dynamics that occur at higher Reynolds numbers. These small scale features are mostly dictatedby viscous and surface tension effects, and a very fine computational mesh is needed to adequately resolvethem. However, the main quantities of interest such as power generated for the full-scale device can beinferred from scaled-down simulations by using appropriate scaling factors, some of which we derive next.

• Length scaling: The geometric parameters such as length, width or height are simply scaled by a factorof α. In the present study, we use α = 20. An exception to this length scaling is hull width in 2D,which is taken to be unity in the scaled model. Therefore in 2D, the scaling factor for hull width is Wrather than α.

• Acceleration scaling: The full-scale and scaled-down models operate under the same gravitational forcefield. Therefore, the gravitational constant g (or any other acceleration) remains unchanged.

• Density scaling: Density is an intrinsic material property, and thus it remains the same for both thefull-scale and scaled-down models.

• Volume scaling: Since volume is proportional to the length cubed, it is scaled by α3.

• Mass scaling: Mass can be expressed as a product of density ρ and volume, and its scaling for 2D and3D ISWEC models are obtained as

Mmodel

Mfull-scale=

ρ (L×H ×W )∣∣model

ρ (L×H ×W )∣∣full-scale

=Wmodel

α2 ·Wfull-scale. (29)

• Velocity scaling: Velocity scaling is obtained by equating the Froude numbers

Uc√gLc

∣∣∣∣∣model

=Uc√gLc

∣∣∣∣∣full-scale

(30)

⇒ Uc,model

Uc,full-scale=

√Lc,model

Lc,full-scale= 1/α

12 . (31)

• Time scaling: Letting tc represent a characteristic time, time scaling can be obtained from the lengthand velocity scalings as

Uc,model

Uc,full-scale=

Lc/tc

∣∣∣model

Lc/tc

∣∣∣full-scale

(32)

⇒ tc,model

tc,full-scale= 1/α

12 . (33)

Similarly, scaling factors of other quantities of interest such as force and power can be obtained in terms of α,and are enumerted in Table 1 for both two and three spatial dimensions. Full-scale (scaled-down) quantitiesshould be divided (multiplied) by factors in the third and fourth columns to obtain the scaled-down (full-scale) quantities, in three and two spatial dimensions, respectively.

2.6. Scaled hull parameters

In this section, we use the Froude scaling derived in the previous section to derive the scaled-down hullparameters required for our simulations. The scaled-down parameters of the gyroscope will be presented inSec. 7, where they are systematically varied to study their effect on device performance. The hull propertiesof the full-scale ISWEC device are taken from an experimental campaign [4, 5] conducted at the Pantelleriatest site in the Mediterranean Sea.

9

Table 1: Froude scaling of various quantities for the 3D and 2D ISWEC models. Dimensional units for the quantities used inthis work are shown in column 2.

Quantity Units Scaled 3D model Scaled 2D model

Length m α αArea m2 α2 α2

Volume m3 α3 −Time s α

12 α

12

Velocity m/s α12 α

12

Acceleration m/s2 1 1

Frequency s−1 α−12 α−

12

Angular velocity s−1 α−12 α−

12

Mass kg α3 α2 ·WDensity kg/m3 1 1Force kg · m/s2 α3 α2 ·W

Moment of inertia kg · m2 α5 α4 ·WTorque kg · m2/s2 α4 α3 ·WPower kg · m2/s3 α

72 α

52 ·W

Table 2: ISWEC hull full-scale and scaled-down parameters. Freeboard (FB) is the distance between the hull top surface andthe still waterline, which is found experimentally.

Hull property Notation Units Full-scale Scaled-down 3D model Scaled-down 2D model

Length L m 15.33 0.7665 0.7665Height H m 4.5 0.225 0.225Width W m 8 0.4 1

Freeboard FB m 1.52 0.076 0.076Center of gravity ZCG m 0.57 0.0285 0.0285

Mass MH kg 288000 36 90Pitch moment of inertia IH kg · m2 7.712× 106 2.41 6.025

10



ξ<latexit sha1_base64="Y9AsMRvmdxTQqcE8JuUntKKm/5U=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120i7dbMLuRiyhP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6mfqtR1Sax/LBjBP0IzqQPOSMGivdd594r1R2K+4MZJl4OSlDjnqv9NXtxyyNUBomqNYdz02Mn1FlOBM4KXZTjQllIzrAjqWSRqj9bHbqhJxapU/CWNmShszU3xMZjbQeR4HtjKgZ6kVvKv7ndVITXvkZl0lqULL5ojAVxMRk+jfpc4XMiLEllClubyVsSBVlxqZTtCF4iy8vk2a14p1XqncX5dp1HkcBjuEEzsCDS6jBLdShAQwG8Ayv8OYI58V5dz7mrStOPnMEf+B8/gBdM43Z</latexit>

R<latexit sha1_base64="cVRUNBy/RTcU6LUbsjbBwonoaeo=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx7ByCOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaHSPJYPZpygH9GB5CFn1Fipft8rltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AK3vjNo=</latexit>

L<latexit sha1_base64="QE9oDXOSeRcavWmZZHNQHB2S4vk=">AAAB6HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioGXQxsIiAfMByRH2NnPJmr29Y3dPCCG/wMZCEVt/kp3/xk1yhSY+GHi8N8PMvCARXBvX/XZya+sbm1v57cLO7t7+QfHwqKnjVDFssFjEqh1QjYJLbBhuBLYThTQKBLaC0e3Mbz2h0jyWD2acoB/RgeQhZ9RYqX7fK5bcsjsHWSVeRkqQodYrfnX7MUsjlIYJqnXHcxPjT6gynAmcFrqpxoSyER1gx1JJI9T+ZH7olJxZpU/CWNmShszV3xMTGmk9jgLbGVEz1MveTPzP66QmvPYnXCapQckWi8JUEBOT2dekzxUyI8aWUKa4vZWwIVWUGZtNwYbgLb+8SpqVsndRrtQvS9WbLI48nMApnIMHV1CFO6hBAxggPMMrvDmPzovz7nwsWnNONnMMf+B8/gCk14zU</latexit>

H<latexit sha1_base64="OmnsBIcI7uL9xT1nnclacUzB0Hk=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELxwhkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj+7nffkKleSwfzCRBP6JDyUPOqLFSo9YvltyyuwBZJ15GSpCh3i9+9QYxSyOUhgmqdddzE+NPqTKcCZwVeqnGhLIxHWLXUkkj1P50ceiMXFhlQMJY2ZKGLNTfE1MaaT2JAtsZUTPSq95c/M/rpia89adcJqlByZaLwlQQE5P512TAFTIjJpZQpri9lbARVZQZm03BhuCtvrxOWpWyd1WuNK5L1bssjjycwTlcggc3UIUa1KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJ7HjNA=</latexit>C

<latexit sha1_base64="ip3qj/q69+gJ0kxhZz5TBeGPSaU=">AAAB73icbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELh4xkUcCGzI7DDBhdnad6TWSDT/hxYPGePV3vPk3DrAHBSvppFLVne6uIJbCoOt+O7m19Y3Nrfx2YWd3b/+geHjUNFGiGW+wSEa6HVDDpVC8gQIlb8ea0zCQvBWMazO/9ci1EZG6x0nM/ZAOlRgIRtFK7S7yJ0xr016x5JbdOcgq8TJSggz1XvGr249YEnKFTFJjOp4bo59SjYJJPi10E8NjysZ0yDuWKhpy46fze6fkzCp9Moi0LYVkrv6eSGlozCQMbGdIcWSWvZn4n9dJcHDtp0LFCXLFFosGiSQYkdnzpC80ZygnllCmhb2VsBHVlKGNqGBD8JZfXiXNStm7KFfuLkvVmyyOPJzAKZyDB1dQhVuoQwMYSHiGV3hzHpwX5935WLTmnGzmGP7A+fwBV4GQKg==</latexit>

CG<latexit sha1_base64="/BRSAlqLJxDe+jnt7yvOQr4uyzQ=">AAAB8HicbVDLSgNBEJz1GeMr6tHLYBA8hd0o6DGYgx4jmIckS5iddJIhs7PLTK8YlnyFFw+KePVzvPk3TpI9aGJBQ1HVTXdXEEth0HW/nZXVtfWNzdxWfntnd2+/cHDYMFGiOdR5JCPdCpgBKRTUUaCEVqyBhYGEZjCqTv3mI2gjInWP4xj8kA2U6AvO0EoPHYQnTKs3k26h6JbcGegy8TJSJBlq3cJXpxfxJASFXDJj2p4bo58yjYJLmOQ7iYGY8REbQNtSxUIwfjo7eEJPrdKj/UjbUkhn6u+JlIXGjMPAdoYMh2bRm4r/ee0E+1d+KlScICg+X9RPJMWITr+nPaGBoxxbwrgW9lbKh0wzjjajvA3BW3x5mTTKJe+8VL67KFauszhy5JickDPikUtSIbekRuqEk5A8k1fy5mjnxXl3PuatK042c0T+wPn8Aes1kHs=</latexit>

CB<latexit sha1_base64="l4cj6p6gJxBYCvnC6cSFvl6ZusU=">AAAB8HicbVDLTgJBEJz1ifhCPXqZSEw8kV000SOBi0dM5GFgQ2aHBibM7G5meo1kw1d48aAxXv0cb/6NA+xBwUo6qVR1p7sriKUw6Lrfztr6xubWdm4nv7u3f3BYODpumijRHBo8kpFuB8yAFCE0UKCEdqyBqUBCKxjXZn7rEbQRUXiPkxh8xYahGAjO0EoPXYQnTGvVaa9QdEvuHHSVeBkpkgz1XuGr2494oiBELpkxHc+N0U+ZRsElTPPdxEDM+JgNoWNpyBQYP50fPKXnVunTQaRthUjn6u+JlCljJiqwnYrhyCx7M/E/r5Pg4MZPRRgnCCFfLBokkmJEZ9/TvtDAUU4sYVwLeyvlI6YZR5tR3obgLb+8SprlkndZKt9dFSvVLI4cOSVn5IJ45JpUyC2pkwbhRJFn8kreHO28OO/Ox6J1zclmTsgfOJ8/45yQdg==</latexit>

GC<latexit sha1_base64="8qSD4ghGMJ2LpHc5+ikaw8Yby6o=">AAAB8HicbVDLSgNBEJz1GeMr6tHLYBA8hd0o6DGYgx4jmIckS5iddJIhs7PLTK8YlnyFFw+KePVzvPk3TpI9aGJBQ1HVTXdXEEth0HW/nZXVtfWNzdxWfntnd2+/cHDYMFGiOdR5JCPdCpgBKRTUUaCEVqyBhYGEZjCqTv3mI2gjInWP4xj8kA2U6AvO0EoPHYQnTG+qk26h6JbcGegy8TJSJBlq3cJXpxfxJASFXDJj2p4bo58yjYJLmOQ7iYGY8REbQNtSxUIwfjo7eEJPrdKj/UjbUkhn6u+JlIXGjMPAdoYMh2bRm4r/ee0E+1d+KlScICg+X9RPJMWITr+nPaGBoxxbwrgW9lbKh0wzjjajvA3BW3x5mTTKJe+8VL67KFauszhy5JickDPikUtSIbekRuqEk5A8k1fy5mjnxXl3PuatK042c0T+wPn8Aes5kHs=</latexit>

Waterline<latexit sha1_base64="WHDv/3LVwPqVq+fXKUPPQeiMyQg=">AAAB+XicbVDLSgNBEJz1GeNr1aOXwSB4CrtR0GPQi8cI5gHJEmYnvcmQ2QczvcGw5E+8eFDEq3/izb9xNtmDJhY0FFXdM93lJ1JodJxva219Y3Nru7RT3t3bPzi0j45bOk4VhyaPZaw6PtMgRQRNFCihkyhgoS+h7Y/vcr89AaVFHD3iNAEvZMNIBIIzNFLftnsIT5i1GYLK35j17YpTdeagq8QtSIUUaPTtr94g5mkIEXLJtO66ToJexhQKLmFW7qUaEsbHbAhdQyMWgvay+eYzem6UAQ1iZSpCOld/T2Qs1Hoa+qYzZDjSy14u/ud1UwxuvExESYoQ8cVHQSopxjSPgQ6EAo5yagjjSphdKR8xxbjJQZdNCO7yyaukVau6l9Xaw1WlflvEUSKn5IxcEJdckzq5Jw3SJJxMyDN5JW9WZr1Y79bHonXNKmZOyB9Ynz9HypQT</latexit>

ZGC<latexit sha1_base64="FbEaslYpHLc4rlSE7Yqmenxe0Q4=">AAAB8nicbVBNSwMxEM36WetX1aOXYBE8ld0q6LHYgx4r2A/cLiWbZtvQbLIks2JZ+jO8eFDEq7/Gm//GtN2Dtj4YeLw3w8y8MBHcgOt+Oyura+sbm4Wt4vbO7t5+6eCwZVSqKWtSJZTuhMQwwSVrAgfBOolmJA4Fa4ej+tRvPzJtuJL3ME5YEJOB5BGnBKzkP/S6wJ4gu6lPeqWyW3FnwMvEy0kZ5Wj0Sl/dvqJpzCRQQYzxPTeBICMaOBVsUuymhiWEjsiA+ZZKEjMTZLOTJ/jUKn0cKW1LAp6pvycyEhszjkPbGRMYmkVvKv7n+SlEV0HGZZICk3S+KEoFBoWn/+M+14yCGFtCqOb2VkyHRBMKNqWiDcFbfHmZtKoV77xSvbso167zOAroGJ2gM+ShS1RDt6iBmogihZ7RK3pzwHlx3p2PeeuKk88coT9wPn8AVICRSA==</latexit>

ZCG<latexit sha1_base64="0jW28AM+6mdYFM3PVKrvGVAh+CU=">AAAB8nicbVBNSwMxEM36WetX1aOXYBE8ld0q6LHYgx4r2A/cLiWbZtvQbLIks2JZ+jO8eFDEq7/Gm//GtN2Dtj4YeLw3w8y8MBHcgOt+Oyura+sbm4Wt4vbO7t5+6eCwZVSqKWtSJZTuhMQwwSVrAgfBOolmJA4Fa4ej+tRvPzJtuJL3ME5YEJOB5BGnBKzkP/S6wJ4gq99MeqWyW3FnwMvEy0kZ5Wj0Sl/dvqJpzCRQQYzxPTeBICMaOBVsUuymhiWEjsiA+ZZKEjMTZLOTJ/jUKn0cKW1LAp6pvycyEhszjkPbGRMYmkVvKv7n+SlEV0HGZZICk3S+KEoFBoWn/+M+14yCGFtCqOb2VkyHRBMKNqWiDcFbfHmZtKoV77xSvbso167zOAroGJ2gM+ShS1RDt6iBmogihZ7RK3pzwHlx3p2PeeuKk88coT9wPn8AVHyRSA==</latexit>

ZCB<latexit sha1_base64="bS1H184nBV3ytMrdcO2N5Akdau4=">AAAB8nicbVBNSwMxEM36WetX1aOXYBE8ld0q6LG0F48V7Adul5JNs21oNlmSWbEs/RlePCji1V/jzX9j2u5BWx8MPN6bYWZemAhuwHW/nbX1jc2t7cJOcXdv/+CwdHTcNirVlLWoEkp3Q2KY4JK1gINg3UQzEoeCdcJxY+Z3Hpk2XMl7mCQsiMlQ8ohTAlbyH/o9YE+QNerTfqnsVtw58CrxclJGOZr90ldvoGgaMwlUEGN8z00gyIgGTgWbFnupYQmhYzJkvqWSxMwE2fzkKT63ygBHStuSgOfq74mMxMZM4tB2xgRGZtmbif95fgrRTZBxmaTAJF0silKBQeHZ/3jANaMgJpYQqrm9FdMR0YSCTaloQ/CWX14l7WrFu6xU767KtXoeRwGdojN0gTx0jWroFjVRC1Gk0DN6RW8OOC/Ou/OxaF1z8pkT9AfO5w9M45FD</latexit>

FB<latexit sha1_base64="IwWFMQPmKybT4tYngazPCnX7uEM=">AAAB8HicbVDLSgNBEJz1GeMr6tHLYBA8hd0o6DFEEI8RzEOSJcxOOsmQ2dllplcMS77CiwdFvPo53vwbJ8keNLGgoajqprsriKUw6Lrfzsrq2vrGZm4rv72zu7dfODhsmCjRHOo8kpFuBcyAFArqKFBCK9bAwkBCMxhdT/3mI2gjInWP4xj8kA2U6AvO0EoPHYQnTG+qk26h6JbcGegy8TJSJBlq3cJXpxfxJASFXDJj2p4bo58yjYJLmOQ7iYGY8REbQNtSxUIwfjo7eEJPrdKj/UjbUkhn6u+JlIXGjMPAdoYMh2bRm4r/ee0E+1d+KlScICg+X9RPJMWITr+nPaGBoxxbwrgW9lbKh0wzjjajvA3BW3x5mTTKJe+8VL67KFaqWRw5ckxOyBnxyCWpkFtSI3XCSUieySt5c7Tz4rw7H/PWFSebOSJ/4Hz+AOgukHk=</latexit>

Figure 4: Hull geometry with properties: L = 0.7665 m, H = 0.225 m, R = 0.4389 m and ξ = 60.83.

The scaled-down (1:20) values of the hull properties are tabulated in Table 2. To verify that the scaled-down values correlate well to the model geometry, we perform geometric estimation of the hull propertiesby assuming the hull to be a filled sector of a circle in two spatial dimensions. The geometric center (GC)and the center of buoyancy (CB) of the submerged sector can be calculated geometrically, and are found tobe located at a distance ZGC = 0.0163 m and ZCB = 0.0605 m below the still waterline, respectively (seeFig. 4). From Table 2, the scaled distance between the center of gravity of the device and waterline is ZCG

= 0.0285 m. It can be seen that the CB lies below the CG and GC, satisfying the stability condition forfloating bodies. Additionally CG lies below GC because in the real device, more mass is distributed towardsthe lower half portion.

Similarly, the scaled-down moment of inertia of the hull IH can be argued geometrically. We first estimatethe density of the hull from the scaled mass (90 kg) and the area of the sector (0.1225 m2) to be ρestimate =734.69 kg/m3. Then we use ρestimate to calculate the moment of inertia of the filled sector about its geometriccenter as IGC = 3.1768 kg ·m2. In the real device, most of the mass is concentrated along the outer periphery,resembling a ring rather than a filled sector. Since, the moment of inertia of a ring is twice as that of a filledcircle, Iestimate ≈ 2IGC = 6.3536 kg · m2, which is close to what we obtain from Table 2.

3. Wave dynamics

This section describes the types of waves, both regular and irregular, and the numerical tank approachused to simulate the ISWEC dynamics.

3.1. Regular waves

We use Fenton’s fifth-order wave theory [23] to generate regular waves of height H, time period T , andwavelength λ. According to fifth-order Stokes theory and assuming that the waves propagate in the positivey-direction, the wave elevation η(y, t) from a still water surface at depth d above the sea floor is

η(y, t) = s η1(y, t) + s2 η2(y, t) + s3 η3(y, t) + s4 η4(y, t) + s5 η5(y, t), (34)

in which, s = κH/2 is the wave steepness, η1 = κ−1 cos(ωt−κy) is the basic harmonic component, κ = 2π/λis the wavenumber, and ω = 2π/T is the wave frequency. The remaining terms in Eq. (34) are higher-ordercorrections to linear wave theory, whose details are given in [23]. The velocity and pressure solutions to thefifth-order Stokes wave can also be found in Fenton [23].

The (fifth-order) Stokes waves satisfy the dispersion relationship given by

ω2 = gκ tanh (κd), (35)

11

which relates the wavenumber κ to the wave frequency ω. Eq. (35) is an implicit equation requiring aniterative process to compute κ given ω, or vice versa. Instead, an explicit equation can be used withsufficient accuracy in all water depth regimes [24]:

κd ≈ Γ + β2 (coshβ)−2

tanhβ + β (coshβ)−2 , (36)

in which, β = Γ (tanh Γ)− 1

2 , and Γ = ω2d/g.A converter’s efficiency η is measured relative to the available wave energy at the installation site. The

traveling water waves transport (kinetic and potential) energy as they move along the sea or ocean surface,which is partially absorbed by the converter. The time-averaged wave power per unit crest width carried byregular waves in the propagation direction is given by [22]

Pwave =1

8ρwgH2cg, (37)

in which ρw is the density of water and cg is the group velocity of waves (the velocity with which wave energyis transported) given by

cg =1

2

λ

T

(1 +

2κd

sinh(2κd)

). (38)

In the deep water limit, where d > λ/2 and κd→∞, Eqs. (35) and (38) become

ω2 = gκ or λ =gT 2

2πand cg =

λ

2T . (deep water limit) (39)

Using Eq. (39) in Eq. (37), the wave power in the deep water limit can be expressed as

Pwave =ρwg

2H2T32π

≈ H2T kW/m, (deep water limit) (40)

in which the constant numerical factor ρwg2/32π ≈ 103 when evaluated with SI units.

When simulating a scaled-down model, both H and λ are scaled-down by the length scale α to generatewaves similar to the full-scale model. The scaled time period is obtained from the dispersion relationshipbetween λ and T .

3.2. Irregular waves

Irregular waves depict a more realistic sea state and are modeled as a superposition of a large number offirst-order regular wave components. Using the superposition principle, the sea surface elevation is expressedas

η(y, t) =

N∑i=1

ai cos(κiy − ωit+ θi), (41)

in which N is the number of wave components, each having its own amplitude ai, angular frequency ωi,wavenumber κi, and a random phase θi. The wavenumber κi is related to the angular frequency ωi bythe dispersion relationship given by Eq. (35). The phases θi of each wave component are random variablesfollowing a uniform distribution in the range [0, 2π].

The linear superposition of wave components also implies that the energy carried by an irregular wave isthe sum of the energy transported by individual wave components. When the number of wave components Ntends to infinity, a continuous wave spectral density function S(ω) is used to describe the energy content ofthe wave components in an infinitesimal frequency bandwidth dω. The area under the curve gives the totalenergy of an irregular wave, modulo the factor ρwg. Discretely, the component wave frequencies are typicallychosen at an equal interval of ∆ω between a narrow bandwidth of frequencies. The wave spectrum S(ω)approaches zero for frequencies outside the narrow bandwidth and peaks at a particular value of frequency

12

ωp5. The amplitude of each wave component is related to the spectral density function by

ai =√

2 · S(ωi) ·∆ω . (42)

We use the JONSWAP spectrum [22] to generate irregular waves, which reads as

S(ω) =320 · H2

s

T 4p

· ω−5 · exp

(−1950

T 4p

· ω−4

)· γA, (43)

in which Hs is the significant wave height, and Tp is the peak time period, i.e., the time period with thehighest spectral peak (see Fig. 5). The remaining parameters in Eq. 43 are:

γ = 3.3 (peakedness factor) (44)

A = exp

−( ωωp− 1

σ√

2

)2

ωp =2π

Tp(angular frequency at spectral peak) (45)

σ =

0.07 if ω ≤ ωp

0.09 if ω > ωp(46)

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.510

-4

Figure 5: The JONSWAP wave spectrum obtained using Hs = 0.1 m, and Tp = 1 s (ωp = 2π rad/s).

The mean wave power per unit crest width carried by an irregular wave is obtained from S(ω) as

Pwave = ρwg

(∫ ∞0

S(ω) dω

)cg ≈ ρwg

(N∑i=1

1

2a2i

)cg, (47)

in which the group velocity cg is calculated from Eq. (38) using the significant wavelength and peak time

5Here we consider only singly-peaked wave spectra.

13

λ<latexit sha1_base64="KR7Mt/phY1Kx38Up6egdPk98AOk=">AAAB7nicbVDLSsNAFL3xWeur6tLNYBFclaQKuiy6cVnBPqAN5WYyaYdOJmFmIpTQj3DjQhG3fo87/8Zpm4W2Hhg4nHMuc+8JUsG1cd1vZ219Y3Nru7RT3t3bPzisHB23dZIpylo0EYnqBqiZ4JK1DDeCdVPFMA4E6wTju5nfeWJK80Q+mknK/BiHkkecorFSpy9sNMRBperW3DnIKvEKUoUCzUHlqx8mNIuZNFSg1j3PTY2fozKcCjYt9zPNUqRjHLKepRJjpv18vu6UnFslJFGi7JOGzNXfEznGWk/iwCZjNCO97M3E/7xeZqIbP+cyzQyTdPFRlAliEjK7nYRcMWrExBKkittdCR2hQmpsQ2Vbgrd88ipp12veZa3+cFVt3BZ1lOAUzuACPLiGBtxDE1pAYQzP8ApvTuq8OO/OxyK65hQzJ/AHzucPPhOPgQ==</latexit>

λ<latexit sha1_base64="KR7Mt/phY1Kx38Up6egdPk98AOk=">AAAB7nicbVDLSsNAFL3xWeur6tLNYBFclaQKuiy6cVnBPqAN5WYyaYdOJmFmIpTQj3DjQhG3fo87/8Zpm4W2Hhg4nHMuc+8JUsG1cd1vZ219Y3Nru7RT3t3bPzisHB23dZIpylo0EYnqBqiZ4JK1DDeCdVPFMA4E6wTju5nfeWJK80Q+mknK/BiHkkecorFSpy9sNMRBperW3DnIKvEKUoUCzUHlqx8mNIuZNFSg1j3PTY2fozKcCjYt9zPNUqRjHLKepRJjpv18vu6UnFslJFGi7JOGzNXfEznGWk/iwCZjNCO97M3E/7xeZqIbP+cyzQyTdPFRlAliEjK7nYRcMWrExBKkittdCR2hQmpsQ2Vbgrd88ipp12veZa3+cFVt3BZ1lOAUzuACPLiGBtxDE1pAYQzP8ApvTuq8OO/OxyK65hQzJ/AHzucPPhOPgQ==</latexit>

H<latexit sha1_base64="FjVpgZ9sYnF4JX/Xxf2CItINeeg=">AAAB8nicbVDLSgMxFM3UV62vqks3wSK4KjNV0GXRTZcV7AOmQ8mkmTY0kwzJHaEM/Qw3LhRx69e482/MtLPQ1gOBwzn3knNPmAhuwHW/ndLG5tb2Tnm3srd/cHhUPT7pGpVqyjpUCaX7ITFMcMk6wEGwfqIZiUPBeuH0Pvd7T0wbruQjzBIWxGQsecQpASv5g5jAhBKRtebDas2tuwvgdeIVpIYKtIfVr8FI0TRmEqggxviem0CQEQ2cCjavDFLDEkKnZMx8SyWJmQmyReQ5vrDKCEdK2ycBL9TfGxmJjZnFoZ3MI5pVLxf/8/wUotsg4zJJgUm6/ChKBQaF8/vxiGtGQcwsIVRzmxXTCdGEgm2pYkvwVk9eJ91G3buqNx6ua827oo4yOkPn6BJ56AY1UQu1UQdRpNAzekVvDjgvzrvzsRwtOcXOKfoD5/MHfD+RYg==</latexit>



O<latexit sha1_base64="GaKRjcxSBA8vmj115Ee3A9y0aWc=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BL95MwDwgWcLspDcZMzu7zMwKIeQLvHhQxKuf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mHGCfkQHkoecUWOl+n2vWHLL7hxklXgZKUGGWq/41e3HLI1QGiao1h3PTYw/ocpwJnBa6KYaE8pGdIAdSyWNUPuT+aFTcmaVPgljZUsaMld/T0xopPU4CmxnRM1QL3sz8T+vk5rw2p9wmaQGJVssClNBTExmX5M+V8iMGFtCmeL2VsKGVFFmbDYFG4K3/PIqaVbK3kW5Ur8sVW+yOPJwAqdwDh5cQRXuoAYNYIDwDK/w5jw6L86787FozTnZzDH8gfP5A6ljjNc=</latexit>

∆d<latexit sha1_base64="BZRjcqelGtYmj/V8orNqknEvqx8=">AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9FjUg8cK9gPaUDabTbt0s4m7E6GU/gkvHhTx6t/x5r9x2+agrQ8GHu/NMDMvSKUw6Lrfzsrq2vrGZmGruL2zu7dfOjhsmiTTjDdYIhPdDqjhUijeQIGSt1PNaRxI3gqGN1O/9cS1EYl6wFHK/Zj2lYgEo2ildveWS6S9sFcquxV3BrJMvJyUIUe9V/rqhgnLYq6QSWpMx3NT9MdUo2CST4rdzPCUsiHt846lisbc+OPZvRNyapWQRIm2pZDM1N8TYxobM4oD2xlTHJhFbyr+53UyjK78sVBphlyx+aIokwQTMn2ehEJzhnJkCWVa2FsJG1BNGdqIijYEb/HlZdKsVrzzSvX+oly7zuMowDGcwBl4cAk1uIM6NICBhGd4hTfn0Xlx3p2PeeuKk88cwR84nz/YPo/X</latexit>

Vorticity damping zone<latexit sha1_base64="sO9G3nsrrpTdxJwAAMR0JrKL36o=">AAACFHicbVA9SwNBEN2L3/EraqnIoQiCEO5ioaVoYxnBxEASwt5mEhf3do/dOTEesfMH2PhXTGGhiK2Fnb/BP+EmF0GNDxbevDfD7LwgEtyg5304mbHxicmp6Zns7Nz8wmJuablsVKwZlJgSSlcCakBwCSXkKKASaaBhIOAsuDjq+2eXoA1X8hQ7EdRD2pa8xRlFKzVyOzWEK0zKSiNnHDvdm1Ro0jDisv1dXisJ3UZu08t7A7ijxB+SzYO13snn7Xqv2Mi915qKxSFIZIIaU/W9COsJ7e8S0M3WYgMRZRe0DVVLJQ3B1JPBUV13yypNt6W0fRLdgfpzIqGhMZ0wsJ0hxXPz1+uL/3nVGFv79YTLKEaQLF3UioWLyu0n5Da5BoaiYwllmtu/uuycasrQ5pi1Ifh/Tx4l5ULe380XTmwahyTFNFklG2Sb+GSPHJBjUiQlwsgdeSBP5Nm5dx6dF+c1bc04w5kV8gvO2xeH0qS0</latexit> Wave

dampingzone

<latexit sha1_base64="M9/YQaQgQo20Ik4NUf32c6fjUmM=">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</latexit>

Wavegeneration

zone<latexit sha1_base64="P7r78tuXnyFY0V81/C31nYI2j10=">AAACIHicbVC7SgNBFJ31GeMraqnIoAhWYVcLLUUbSwPmAdkQZic3cXB2dpm5K8Yl6fwMG7/C3sZCEe30F/wJJ9kUGj0wcDjnXO7cE8RSGHTdD2dicmp6ZjY3l59fWFxaLqysVkyUaA5lHslI1wJmQAoFZRQooRZrYGEgoRpcngz86hVoIyJ1jt0YGiHrKNEWnKGVmoUDH+Ea0yq7gh71/Xy/nwkdUKCHmZE8QGbdRAp6zcK2W3SHoH+JNyLbRxsPpa/bzYezZuHdb0U8CUEhl8yYuufG2EiZRsEl9PJ+YiBm/JJ1oG6pYiGYRjo8sEd3rNKi7Ujbp5AO1Z8TKQuN6YaBTYYML8y4NxD/8+oJtg8bqVBxgqB4tqidSIoRHbRFW0IDR9m1hHEt7F8pv2CacbSd5m0J3vjJf0llr+jtF/dKto1jkiFH1skW2SUeOSBH5JSckTLh5I48kmfy4tw7T86r85ZFJ5zRzBr5BefzG9N5qPY=</latexit>

2λ<latexit sha1_base64="Y8ZNJQh05frUBA+Ddty+FbkwHsE=">AAAB73icbVDLSgMxFL1TX7W+qi7dBIvgqsxUQZdFNy4r2Ae0Q8lkMm1oJjMmd4RS+hNuXCji1t9x59+YtrPQ1gOBwznnkntPkEph0HW/ncLa+sbmVnG7tLO7t39QPjxqmSTTjDdZIhPdCajhUijeRIGSd1LNaRxI3g5GtzO//cS1EYl6wHHK/ZgOlIgEo2ilTq0nbTak/XLFrbpzkFXi5aQCORr98lcvTFgWc4VMUmO6npuiP6EaBZN8WuplhqeUjeiAdy1VNObGn8z3nZIzq4QkSrR9Cslc/T0xobEx4ziwyZji0Cx7M/E/r5thdO1PhEoz5IotPooySTAhs+NJKDRnKMeWUKaF3ZWwIdWUoa2oZEvwlk9eJa1a1buo1u4vK/WbvI4inMApnIMHV1CHO2hAExhIeIZXeHMenRfn3flYRAtOPnMMf+B8/gCwKY+9</latexit>4λ

<latexit sha1_base64="K6Epy0em7+5tJZd0gGFw4CzNp/E=">AAAB73icbVDLSgMxFL3js9ZX1aWbYBFclZla0GXRjcsK9gHtUDKZO21oJjMmGaGU/oQbF4q49Xfc+Tem7Sy09UDgcM655N4TpIJr47rfztr6xubWdmGnuLu3f3BYOjpu6SRTDJssEYnqBFSj4BKbhhuBnVQhjQOB7WB0O/PbT6g0T+SDGafox3QgecQZNVbq1HrCZkPaL5XdijsHWSVeTsqQo9EvffXChGUxSsME1brruanxJ1QZzgROi71MY0rZiA6wa6mkMWp/Mt93Ss6tEpIoUfZJQ+bq74kJjbUex4FNxtQM9bI3E//zupmJrv0Jl2lmULLFR1EmiEnI7HgScoXMiLEllCludyVsSBVlxlZUtCV4yyevkla14l1Wqve1cv0mr6MAp3AGF+DBFdThDhrQBAYCnuEV3pxH58V5dz4W0TUnnzmBP3A+fwCzP4+/</latexit>

1.75d<latexit sha1_base64="VZR0q1/EYnb+WCHXSee3mDXudrw=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbB07JblXosevFYwX5Au5RsNtvGZpMlyQql9D948aCIV/+PN/+NabsHbX0w8Hhvhpl5YcqZNp737RTW1jc2t4rbpZ3dvf2D8uFRS8tMEdokkkvVCbGmnAnaNMxw2kkVxUnIaTsc3c789hNVmknxYMYpDRI8ECxmBBsrtXy3doWifrniud4caJX4OalAjka//NWLJMkSKgzhWOuu76UmmGBlGOF0WuplmqaYjPCAdi0VOKE6mMyvnaIzq0QolsqWMGiu/p6Y4ETrcRLazgSboV72ZuJ/Xjcz8XUwYSLNDBVksSjOODISzV5HEVOUGD62BBPF7K2IDLHCxNiASjYEf/nlVdKquv6FW72/rNRv8jiKcAKncA4+1KAOd9CAJhB4hGd4hTdHOi/Ou/OxaC04+cwx/IHz+QPo5Y4J</latexit>

d<latexit sha1_base64="i4T7bfCHibYNJNngwp/p1+TXADk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJu3azSbsboRS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBVcG9f9dgpr6xubW8Xt0s7u3v5B+fCorZNMMWyxRCSqE1CNgktsGW4EdlKFNA4EPgSj25n/8IRK80Tem3GKfkwHkkecUWOlZtgvV9yqOwdZJV5OKpCj0S9/9cKEZTFKwwTVuuu5qfEnVBnOBE5LvUxjStmIDrBrqaQxan8yP3RKzqwSkihRtqQhc/X3xITGWo/jwHbG1Az1sjcT//O6mYmu/QmXaWZQssWiKBPEJGT2NQm5QmbE2BLKFLe3EjakijJjsynZELzll1dJu1b1Lqq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8fyTeM7A==</latexit>

10λ<latexit sha1_base64="uIxsNxirubdBwIWVkxTOU/AfMUI=">AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqsxUQZdFNy4r2Ie0Q8lkMm1okhmSjFCGfoUbF4q49XPc+Tem01lo64HA4Zxzyb0nSDjTxnW/ndLa+sbmVnm7srO7t39QPTzq6DhVhLZJzGPVC7CmnEnaNsxw2ksUxSLgtBtMbud+94kqzWL5YKYJ9QUeSRYxgo2VHj13wG04xMNqza27OdAq8QpSgwKtYfVrEMYkFVQawrHWfc9NjJ9hZRjhdFYZpJommEzwiPYtlVhQ7Wf5wjN0ZpUQRbGyTxqUq78nMiy0norAJgU2Y73szcX/vH5qoms/YzJJDZVk8VGUcmRiNL8ehUxRYvjUEkwUs7siMsYKE2M7qtgSvOWTV0mnUfcu6o37y1rzpqijDCdwCufgwRU04Q5a0AYCAp7hFd4c5bw4787HIlpyiplj+APn8wcd4I/2</latexit>

Figure 6: Numerical wave tank (NWT) schematic showing wave generation, wave damping, and vorticity damping zones. TheISWEC device is placed in the working zone of length 7λ.

period of the spectrum. In the deep water limit, Eq. (47) simplifies to

Pwave ≈ 0.49H2sTp kW/m. (deep water limit). (48)

3.3. Wave steepness

As discussed in Sec. 2.3, if the oscillation frequencies of the hull and gyroscope system are synchronizedwith that of the wave, the coupling between the hull and the gyroscope system (and therefore the outputpower) can be increased. Along with frequency synchronization, the oscillation amplitude of the hull canalso be increased to enhance the device performance. This will result in more power transfer from the hullto the gyroscope system. The wave steepness (s) defined in Eq. 34, which gives a relation between the waveheight H and wavelength λ, plays an important role in deciding the PTO and gyroscope system parameterssuch that the hull exhibits larger pitching motion. This is achieved by adjusting the gyroscope and PTOparameters such that the maximum hull pitch angle (amplitude) δ0 is expected to reach the maximum wavesteepness angle δs of the incoming wave. An expression for δs can be obtained by approximating the incomingwave as a regular (simple harmonic) wave with elevation η(y, t) given by

η(y, t) = a · cos(κy − ωt), (49)

where a = H/2, is the wave amplitude. Differentiating the above equation with respect to y, we obtain

dη(y, t)

dy= a · (−κ sin(κy − ωt)). (50)

The maximum wave steepness (i.e. the slope) is obtained when sin(κy − ωt) = −1,(dη(y, t)

dy

)max

= κ · H2

= s. (51)

Finally, the maximum wave steepness angle is then given by

δs = tan−1

(κH2

)= tan−1

(πHλ

)(52)

When the condition δ0 = δs is used to calculate the gyroscope and PTO parameters, the ISWEC device isobserved to have maximum efficiency. A study on the variation of δ0 relative to δs for different wave heightsis conducted in Sec. 7.2.

3.4. Numerical wave tank

The wave-structure interaction of the scaled-down ISWEC device is simulated in a numerical wave tank(NWT) as shown in Fig. 6. Water waves are generated at the left boundary of the domain using Dirichlet

14

boundary conditions for the velocity components. The waves traveling in the positive y-direction are reflectedback towards the inlet side from the device surface and also from the right boundary of the domain. Thisresults in wave distortion and wave interference phenomena, which reduces the “quality” of waves reachingthe device to study its performance. Several techniques have been proposed in the literature [25, 26, 27] tomitigate these effects, including the relaxation zone method [28], the active wave absorption method [29, 30,31], the momentum damping method [32, 33], the viscous beach method [34], the porous media method [35,36], and the mass-balance PDE method [37]. In this work, we use the relaxation zone method at inlet andoutlet boundaries. The purpose of the relaxation zone near the channel inlet (the wave generation zone)is to smoothly extend the Dirichlet velocity boundary conditions into the wave tank up to a distance ofone wavelength, so that the reflected waves coming from the ISWEC device do not interfere with the leftboundary. In contrast, the relaxation zone near the right boundary (the wave damping zone) smoothlydamps out the waves reaching the domain outlet near the right end. The wave damping zone is taken to betwo wavelengths wide in our simulations. More details on the implementation of the relaxation zone methodand level set based NWT can be found in our prior work [19].

We impose zero-pressure boundary condition at the channel top boundary, zmax = 2.75d. To mitigate theinteraction between shed vortices (due to the device motion) and the top boundary of the channel, we use avorticity damping zone to dissipate the vortex structures reaching the boundary; see Fig. 6. The vorticitydamping zone is implemented in terms of a damping force fd in the momentum equation

fd = −g(z)u, (53)

in which, g(z) = ρair (cos(πz) + 1)/(4∆t) is the smoothed damping coefficient, ρair is the density of the airphase, ∆t is the time step size, z = (z − zmax)/∆d is the normalized z coordinate, and ∆d is the vorticitydamping zone width, which is taken to be six cells wide in our simulations.

4. Numerical model based on the incompressible Navier-Stokes equations

We use a fully-Eulerian fictitious domain Brinkman penalization (FD/BP) method [20] to perform fully-resolved wave-structure interaction simulations. In FD/BP methods, the governing equations for the fluid areextended into the region occupied by the solid structure, yielding a single set of PDEs for the entire domain.Additional constraints are imposed in the structure domain to ensure that the velocity field within acts likea rigid body. This is in contrast to body-conforming grid methods, in which the fluid equations are solvedonly on a domain surrounding the immersed body. For applications involving moving body fluid-structureinteraction (FSI), fictitious domain methods are less computationally expensive than body-conforming gridtechniques due to the absence of expensive re-meshing operations.

In this section, we first describe the continuous governing equations for the FD/BP formulation and theinterface tracking approach for the multiphase FSI system. Next, we briefly describe the spatiotemporaldiscretization, overall solution methodology, and time-stepping scheme. Finally, we describe the couplingused to simulate the dynamics of an inertial sea wave energy converter device, which involves modeling theeffect of a rigid body pitch torque. A validation case for vortex induced vibration of a rectangular plateexhibiting galloping motion is presented at the end of this section. We refer readers to the references byNangia et al. [19, 38], Bhalla et al. [20] and Dafnakis et al. [39] for more details on the Cartesian grid fluidsolver, FD/BP formulation, and simulating wave energy converters within this framework, respectively.

4.1. Continuous equations of motion

Let Ω ⊂ Rd with d = 3 denote a fixed three-dimensional region in space. The dynamics of a coupledmultiphase fluid-structure system occupying this domain are governed by the incompressible Navier-Stokes(INS) equations

∂ρu(x, t)

∂t+∇ · ρu(x, t)u(x, t) = −∇p(x, t) +∇ ·

[µ(∇u(x, t) +∇u(x, t)T

)]+ ρg + f c(x, t), (54)

∇ · u(x, t) = 0, (55)

15

which describe the momentum and incompressibility of a fluid with velocity u(x, t) and pressure p(x, t) usingan Eulerian coordinate system x = (x, y, z) ∈ Ω. Note that Eqs. (54) and (55) are written for the entirefixed region Ω, which can be further decomposed into two regions occupied by the fluid Ωf(t) ⊂ Ω and theimmersed body Ωb(t) ⊂ Ω. These regions are non-overlapping, i.e. Ω = Ωf(t)∪Ωb(t), and f c(x, t) representsa rigidity-enforcing constraint force density that vanishes outside Ωb(t); this ensures a rigid body velocityub(x, t) is attained within the solid region. The spatiotemporally varying density and viscosity fields aredenoted by ρ(x, t) and µ(x, t), respectively. An indicator function χ(x, t) is further used to track the locationof the solid body, which is nonzero only within Ωb(t). The acceleration due to gravity is directed towardsthe negative z-direction: g = (0, 0,−g).

The immersed structure is treated as a porous region with vanishing permeability κp 1, yielding thefollowing formula for the Brinkman penalized constraint force

f c(x, t) =χ(x, t)

κp(ub(x, t)− u(x, t)) . (56)

Sec. 4.4.5 describes the fluid-structure coupling algorithm, and Sec. 4.4.6 describes the external ISWECtorque specification, which together are used to determine the rigid body velocity ub(x, t) applied to Ωb(t).

4.2. Interface tracking

All of the cases described in the present work involve a single rigid structure interacting with an air-water interface. We briefly describe the interface tracking methodology here, and refer readers to Nangia etal. [38, 19] for further details on its implementation. A scalar level set function σ(x, t) is used to demarcateliquid (water) and gas (air) regions, Ωl ⊂ Ω and Ωg ⊂ Ω, respectively, in the computational domain. Theair-water interface Γ(t) = Ωl ∩ Ωg is implicitly defined by the zero-contour of σ. The same methodology isemployed to track the surface of the immersed body Sb(t) = ∂Vb(t) using the zero-contour of a level setfunction ψ(x, t); the aforementioned indicator function for the solid domain is computed based on σ. Theevolution of these level set fields is governed by linear advection via the local fluid velocity field

∂σ

∂t+∇ · σu = 0, (57)

∂ψ

∂t+∇ · ψu = 0. (58)

Making use of the signed distance property, the density and viscosity across the entire computational domaincan be conveniently expressed as a functions of σ(x, t) and ψ(x, t)

ρ(x, t) = ρ(σ(x, t), ψ(x, t)), (59)

µ(x, t) = µ(σ(x, t), ψ(x, t)). (60)

After every time step, both level set functions are reinitialized to maintain signed distance functions to theirrespective interfaces. Standard approaches for computing a steady-state solution to the Hamilton-Jacobiequation is used to reinitialize σ, whereas an analytical distance computation to the immersed body is usedto reinitialize ψ.

4.3. Spatial discretization

The continuous equations of motion Eqs. (54)-(55) are discretized on a staggered Cartesian grid made upof Nx×Ny×Nz rectangular cells covering the domain Ω. The mesh spacings in the three spatial directions aredenoted by ∆x, ∆y, and ∆z respectively. Without loss of generality, let the lower left corner of the rectangulardomain be the origin (0, 0, 0) of the coordinate system. Using this reference point, each cell center of the gridhas position xi,j,k =

((i+ 1

2 )∆x, (j + 12 )∆y, (k + 1

2 )∆z)

for i = 0, . . . , Nx − 1, j = 0, . . . , Ny − 1, and k =0, . . . , Nz − 1. The physical location of the cell face that is half a grid cell away from xi,j,k in the x-directionis given by xi− 1

2 ,j,k=(i∆x, (j + 1

2 )∆y, (k + 12 )∆z

). Similarly, xi,j− 1

2 ,k=((i+ 1

2 )∆x, j∆y, (k + 12 )∆z

)and

xi,j,k− 12

=((i+ 1

2 )∆x, (j + 12 )∆y, k∆z

)are the physical locations of the cell faces that are half a grid cell

away from xi,j,k in the y- and z-directions, respectively. The level set fields, pressure degrees of freedom,

16

Ωg(t)<latexit sha1_base64="HgIROftLhlWciDa+VReeo6CbM+w=">AAAB8nicbVBNSwMxEM3Wr1q/qh69BItQL2W3CnosevFmBfsB26Vk0+w2NMkuyaxQSn+GFw+KePXXePPfmLZ70NYHA4/3ZpiZF6aCG3Ddb6ewtr6xuVXcLu3s7u0flA+P2ibJNGUtmohEd0NimOCKtYCDYN1UMyJDwTrh6Hbmd56YNjxRjzBOWSBJrHjEKQEr+b17yWLSj6tw3i9X3Jo7B14lXk4qKEezX/7qDRKaSaaACmKM77kpBBOigVPBpqVeZlhK6IjEzLdUEclMMJmfPMVnVhngKNG2FOC5+ntiQqQxYxnaTklgaJa9mfif52cQXQcTrtIMmKKLRVEmMCR49j8ecM0oiLElhGpub8V0SDShYFMq2RC85ZdXSbte8y5q9YfLSuMmj6OITtApqiIPXaEGukNN1EIUJegZvaI3B5wX5935WLQWnHzmGP2B8/kDfuCQvA==</latexit>

Ωl(t)<latexit sha1_base64="b11IfULOUKM6GLuwGRWbZC4gbLE=">AAAB8nicbVBNSwMxEM3Wr1q/qh69BItQL2W3CnosevFmBfsB26Vk02wbmmSXZFYoS3+GFw+KePXXePPfmLZ70NYHA4/3ZpiZFyaCG3Ddb6ewtr6xuVXcLu3s7u0flA+P2iZONWUtGotYd0NimOCKtYCDYN1EMyJDwTrh+Hbmd56YNjxWjzBJWCDJUPGIUwJW8nv3kg1JX1ThvF+uuDV3DrxKvJxUUI5mv/zVG8Q0lUwBFcQY33MTCDKigVPBpqVealhC6JgMmW+pIpKZIJufPMVnVhngKNa2FOC5+nsiI9KYiQxtpyQwMsveTPzP81OIroOMqyQFpuhiUZQKDDGe/Y8HXDMKYmIJoZrbWzEdEU0o2JRKNgRv+eVV0q7XvIta/eGy0rjJ4yiiE3SKqshDV6iB7lATtRBFMXpGr+jNAefFeXc+Fq0FJ585Rn/gfP4AhoOQwQ==</latexit>

Vb(t)<latexit sha1_base64="WqOTPHthUdwjOmgpyVPfV0lwihU=">AAAB7XicbVBNTwIxEO3iF+IX6tFLIzHBC9lFEz0SvXjERBYS2JBu6UKl227aWROy4T948aAxXv0/3vw3FtiDgi+Z5OW9mczMCxPBDbjut1NYW9/Y3Cpul3Z29/YPyodHvlGppqxFlVC6ExLDBJesBRwE6ySakTgUrB2Ob2d++4lpw5V8gEnCgpgMJY84JWAl3++HVTjvlytuzZ0DrxIvJxWUo9kvf/UGiqYxk0AFMabruQkEGdHAqWDTUi81LCF0TIasa6kkMTNBNr92is+sMsCR0rYk4Ln6eyIjsTGTOLSdMYGRWfZm4n9eN4XoOsi4TFJgki4WRanAoPDsdTzgmlEQE0sI1dzeiumIaELBBlSyIXjLL68Sv17zLmr1+8tK4yaPo4hO0CmqIg9doQa6Q03UQhQ9omf0it4c5bw4787HorXg5DPH6A+czx/A4I6W</latexit>

Sb(t)<latexit sha1_base64="JFR9vH4Xh014syl7jrdeZd0b4dk=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoMQL2FXBT0GvXiMaB6QLGF2MpuMmZ1dZnqFEPIPXjwo4tX/8ebfOEn2oIkFDUVVN91dQSKFQdf9dnIrq2vrG/nNwtb2zu5ecf+gYeJUM15nsYx1K6CGS6F4HQVK3ko0p1EgeTMY3kz95hPXRsTqAUcJ9yPaVyIUjKKVGvfdoIyn3WLJrbgzkGXiZaQEGWrd4lenF7M04gqZpMa0PTdBf0w1Cib5pNBJDU8oG9I+b1uqaMSNP55dOyEnVumRMNa2FJKZ+ntiTCNjRlFgOyOKA7PoTcX/vHaK4ZU/FipJkSs2XxSmkmBMpq+TntCcoRxZQpkW9lbCBlRThjaggg3BW3x5mTTOKt555ezuolS9zuLIwxEcQxk8uIQq3EIN6sDgEZ7hFd6c2Hlx3p2PeWvOyWYO4Q+czx+8RY6T</latexit>

Γ(t)<latexit sha1_base64="Yv00y/NBXggma0kp9JDMs14vmOE=">AAAB8HicbVDLSgNBEJz1GeMr6tHLYhDiJexGQY9BD3qMYB6SLKF3MpsMmZldZnqFEPIVXjwo4tXP8ebfOEn2oIkFDUVVN91dYSK4Qc/7dlZW19Y3NnNb+e2d3b39wsFhw8SppqxOYxHrVgiGCa5YHTkK1ko0AxkK1gyHN1O/+cS04bF6wFHCAgl9xSNOAa302LkFKaGEZ91C0St7M7jLxM9IkWSodQtfnV5MU8kUUgHGtH0vwWAMGjkVbJLvpIYlQIfQZ21LFUhmgvHs4Il7apWeG8XalkJ3pv6eGIM0ZiRD2ykBB2bRm4r/ee0Uo6tgzFWSIlN0vihKhYuxO/3e7XHNKIqRJUA1t7e6dAAaKNqM8jYEf/HlZdKolP3zcuX+oli9zuLIkWNyQkrEJ5ekSu5IjdQJJZI8k1fy5mjnxXl3PuatK042c0T+wPn8Afgtj9w=</latexit>

(a) Continuous domain

χ = 0<latexit sha1_base64="yF3Y+AmhHlWKMswcJh5SoVA+9Mg=">AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9KIUvHisYD+gDWWznbRLN5u4uxFK6J/w4kERr/4db/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWMbqd+6wmV5rF8MOME/YgOJA85o8ZK7S4bcnJN3F6p7FbcGcgy8XJShhz1Xumr249ZGqE0TFCtO56bGD+jynAmcFLsphoTykZ0gB1LJY1Q+9ns3gk5tUqfhLGyJQ2Zqb8nMhppPY4C2xlRM9SL3lT8z+ukJrzyMy6T1KBk80VhKoiJyfR50ucKmRFjSyhT3N5K2JAqyoyNqGhD8BZfXibNasU7r1TvL8q1mzyOAhzDCZyBB5dQgzuoQwMYCHiGV3hzHp0X5935mLeuOPnMEfyB8/kDoQWPBw==</latexit>

χ = 0<latexit sha1_base64="yF3Y+AmhHlWKMswcJh5SoVA+9Mg=">AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9KIUvHisYD+gDWWznbRLN5u4uxFK6J/w4kERr/4db/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWMbqd+6wmV5rF8MOME/YgOJA85o8ZK7S4bcnJN3F6p7FbcGcgy8XJShhz1Xumr249ZGqE0TFCtO56bGD+jynAmcFLsphoTykZ0gB1LJY1Q+9ns3gk5tUqfhLGyJQ2Zqb8nMhppPY4C2xlRM9SL3lT8z+ukJrzyMy6T1KBk80VhKoiJyfR50ucKmRFjSyhT3N5K2JAqyoyNqGhD8BZfXibNasU7r1TvL8q1mzyOAhzDCZyBB5dQgzuoQwMYCHiGV3hzHp0X5935mLeuOPnMEfyB8/kDoQWPBw==</latexit>

χ = 1<latexit sha1_base64="Hfqu8ME3pzUiurljLoYAlFDVupo=">AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisYD+gDWWznbRLN5u4uxFK6J/w4kERr/4db/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWMbqd+6wmV5rF8MOME/YgOJA85o8ZK7S4bcnJNvF6p7FbcGcgy8XJShhz1Xumr249ZGqE0TFCtO56bGD+jynAmcFLsphoTykZ0gB1LJY1Q+9ns3gk5tUqfhLGyJQ2Zqb8nMhppPY4C2xlRM9SL3lT8z+ukJrzyMy6T1KBk80VhKoiJyfR50ucKmRFjSyhT3N5K2JAqyoyNqGhD8BZfXibNasU7r1TvL8q1mzyOAhzDCZyBB5dQgzuoQwMYCHiGV3hzHp0X5935mLeuOPnMEfyB8/kDo72PDA==</latexit>

ψ = 0<latexit sha1_base64="tEnMHw0U5+c2twe2MHpLC75ImQ0=">AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisYD+gDWWznbRLN5u4uxFK6J/w4kERr/4db/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWMbqd+6wmV5rF8MOME/YgOJA85o8ZK7W6iObkmbq9UdivuDGSZeDkpQ456r/TV7ccsjVAaJqjWHc9NjJ9RZTgTOCl2U40JZSM6wI6lkkao/Wx274ScWqVPwljZkobM1N8TGY20HkeB7YyoGepFbyr+53VSE175GZdJalCy+aIwFcTEZPo86XOFzIixJZQpbm8lbEgVZcZGVLQheIsvL5NmteKdV6r3F+XaTR5HAY7hBM7Ag0uowR3UoQEMBDzDK7w5j86L8+58zFtXnHzmCP7A+fwBxx6PIw==</latexit>

σ = 0<latexit sha1_base64="JGhQjzLhpZh9/UVBwoiI/2o+nMQ=">AAAB8XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6EUoevFYwX5gu5Rsmm1Dk+ySZIWy9F948aCIV/+NN/+NabsHbX0w8Hhvhpl5YSK4sZ73jVZW19Y3Ngtbxe2d3b390sFh08SppqxBYxHrdkgME1yxhuVWsHaiGZGhYK1wdDv1W09MGx6rBztOWCDJQPGIU2Kd9Ng1fCAJvsZer1T2Kt4MeJn4OSlDjnqv9NXtxzSVTFkqiDEd30tskBFtORVsUuymhiWEjsiAdRxVRDITZLOLJ/jUKX0cxdqVsnim/p7IiDRmLEPXKYkdmkVvKv7ndVIbXQUZV0lqmaLzRVEqsI3x9H3c55pRK8aOEKq5uxXTIdGEWhdS0YXgL768TJrVin9eqd5flGs3eRwFOIYTOAMfLqEGd1CHBlBQ8Ayv8IYMekHv6GPeuoLymSP4A/T5A0Jxj/w=</latexit>

(b) FD/BP discretized domain

Figure 7: (a) Schematic of a two-dimensional slice through the computational domain Ω, in which an immersed body interactswith an air-water interface. (b) Cartesian mesh discretization of the domain Ω and the value of the indicator function χ(x, t) usedto differentiate between the fluid and solid regions in the FD/BP method; χ(x, t) = 1 inside the solid domain and χ(x, t) = 0 inair and water domains. The zero contour of σ(x, t) tracks the air-water interface Γ(t), while the zero contour of ψ(x, t) tracksthe solid-fluid interface Sb(t).

and the material properties are all approximated at cell centers and are denoted by σni,j,k ≈ σ (xi,j,k, tn),

ψni,j,k ≈ ψ (xi,j,k, tn), pni,j,k ≈ p (xi,j,k, t

n), ρni,j,k ≈ ρ (xi,j,k, tn) and µni,j,k ≈ µ (xi,j,k, t

n), respectively; some ofthese quantities are interpolated onto the required degrees of freedom as needed (see [38] for further details).Here, the time at time step n is denotes by tn. The velocity degrees of freedom are approximated on cell

faces, with notation uni− 1

2 ,j,k≈ u

(xi− 1

2 ,j,k, tn)

, vni,j− 1

2 ,k≈ v

(xi,j− 1

2 ,k, tn)

, and wni,j,k− 1

2

≈ w(xi,j,k− 1

2, tn)

.

The gravitational body force and constraint force density on the right-hand side of the momentum equation(54) are also approximated on the cell faces.

Second-order finite differences are used to discretize all spatial derivatives. Subsequently, the discretizedversion of these differential operators are denoted with h subscripts; i.e., ∇ ≈ ∇h. For more details on thespatial discretization, we refer readers to prior studies [38, 40, 41, 42].

4.4. Solution methodology

Next, we describe the methodology used to solve the discretized equations of motion. At a high level,this involves three major steps:

1. Specifying the material properties ρ(x, t) and µ(x, t) throughout the computational domain.

2. Calculating the Brinkman penalization rigidity constraint force density f c based on the ISWEC dy-namics

3. Computing the updated solutions to σ, ψ, u, and p.

In the present work, we briefly review the computations above in the context of ISWEC devices with a singleunlocked rotational degree of freedom. For a more general treatment of the FD/BP method, we refer readersto previous work by Bhalla et al. [20] and references therein.

4.4.1. Density and viscosity specification

At the air-water Γ and fluid-solid Sb interfaces, a smoothed Heaviside function is used to transitionbetween the three phases. In this transition region, ncells grid cells are used on either side of the interfacesto provide a smooth variation of material properties. A given material property = (such as density ρ orviscosity µ) is prescribed throughout the computational domain first by computing the flowing phase (i.e.air and water)

=flowi,j,k = =l + (=g −=l)H

flowi,j,k, (61)

17

and then correcting =flow to account for the solid region

=fulli,j,k = =s + (=flow

i,j,k −=s)Hbodyi,j,k . (62)

Here, =full is the final scalar material property field throughout Ω. Standard numerical Heaviside functionsare used to facilitate the transition specified by Eqs. (61) and (62):

Hflowi,j,k =

0, σi,j,k < −ncells∆x,12

(1 + 1

ncells∆xσi,j,k + 1

π sin(

πncells∆x

σi,j,k

)), |σi,j,k| ≤ ncells∆x,

1, otherwise,

(63)

Hbodyi,j,k =

0, ψi,j,k < −ncells∆x,12

(1 + 1

ncells∆xψi,j,k + 1

π sin(

πncells∆x

ψi,j,k

)), |ψi,j,k| ≤ ncells∆x,

1, otherwise,

(64)

We use the same number of transition cells ncells = 2 for both air-water and fluid-solid interfaces in oursimulations, although this is not an inherent limitation of our method. We refer readers to Nangia et al. [19]for more discussion.

4.4.2. Time stepping scheme

A fixed-point iteration time stepping scheme using ncycles = 2 cycles per time step is used to evolvequantities from time level tn to time level tn+1 = tn + ∆t. A k superscript is used to denote the cyclenumber of the fixed-point iteration. At the beginning of each time step, the solutions from the previous timestep are used to initialize cycle k = 0: un+1,0 = un, pn+ 1

2 ,0 = pn−12 , σn+1,0 = σn, and ψn+1,0 = ψn. At the

initial time n = 0, the physical quantities are prescribed via initial condition.

4.4.3. Level set advection

An standard explicit advection scheme is used to evolve the two level set functions

σn+1,k+1 − σn∆t

+Q(un+ 1

2 ,k, σn+ 12 ,k)

= 0, (65)

ψn+1,k+1 − ψn∆t

+Q(un+ 1

2 ,k, ψn+ 12 ,k)

= 0, (66)

in which Q(·, ·) represents an explicit piecewise parabolic method (xsPPM7-limited) approximation to thelinear advection terms on cell centers [41, 43].

4.4.4. Incompressible Navier-Stokes solution

The following spatiotemporal discretization of the incompressible Navier-Stokes Eqs. (54)-(55) (in con-servative form) is employed

ρn+1,k+1un+1,k+1 − ρnun

∆t+ Cn+1,k = −∇hpn+ 1

2 ,k+1 + (Lµu)n+ 1

2 ,k+1+ ℘n+1,k+1g + fn+1,k+1

c , (67)

∇h · un+1,k+1 = 0, (68)

in which the discretized convective derivative Cn+1,k and the density approximation ρn+1,k+1 are computedusing a consistent mass/momentum transport scheme; this is vital to ensure numerical stability for casesinvolving air-water density ratios. This scheme is described in detail in previous studies by Nangia et al. and

Bhalla et al. [38, 20]. A standard semi-implicit approximation to the viscous strain rate (Lµu)n+ 1

2 ,k+1=

12

[(Lµu)

n+1,k+1+ (Lµu)

n]

is employed, in which (Lµ)n+1

= ∇h ·[µn+1

(∇hu +∇huT

)n+1]. The two-stage

process described by Eqs. (61) and (62) is used to obtain the newest approximation to viscosity µn+1,k+1.The flow density field is used to construct the gravitational body force term ℘g = ρflowg, which avoidsspurious currents due to large density variation near the fluid-solid interface [19].

18

4.4.5. Fluid-structure coupling

Next, we describe the Brinkman penalization term that imposes the rigidity constraint in the solid region,and the overall fluid-structure coupling scheme. In this work, we simplify the treatment of the FSI couplingby only considering immersed bodies with a single unlocked rotational degree of freedom (DOF); a moregeneral approach is described in Bhalla et al. [20].

The Brinkman penalization term is treated implicitly and computed as

fn+1,k+1c =

χ

κp

(un+1,k+1

b − un+1,k+1), (69)

in which the discretized indicator function χ = 1 − Hbody is 1 only inside the body domain and definedusing the structure Heaviside function Hbody from Eq. (64). A permeability value of κp ∼ O(10−8) has beenshown to be sufficiently small enough to effectively enforce the rigidity constraint [44, 20]. In general, thesolid body velocity ub = Ur +Wr×(x−Xcom) can be expressed as a sum of translational Ur and rotationalWr velocities. In this work, Ur = 0 and we simply have

un+1,k+1b = Wn+1,k+1

r ×(x−Xn+1,k+1

com

). (70)

Moreover, two of the rotational DOFs are locked in the present study, i.e. they are constrained to zero.Hence, the expression for Wr can be simplified even further,

Wn+1,k+1r =

(δn+1,k+1, 0, 0

), (71)

in which δ is the rotational velocity of the structure about its pitch axis.The rigid body velocity can be computed by integrating Newton’s second law of motion for the pitch axis

rotational velocity:

IHδn+1,k+1 − δn

∆t=Mn+1,k

hydro −Mn+1,kδ , (72)

in which IH is the moment of inertia of the hull,Mhydro is the net hydrodynamic torque, andMn+1,kδ is the

projection of the gyroscopic torque about the x-axis. The net hydrodynamic torque is computed by summingthe contributions from pressure and viscous forces acting on the hull and taking the pitch component

Mhydro = i ·[∑

f

(x−Xn+1,k

com

)×(−pn+1,knf + µf

(∇hun+1,k +

(∇hun+1,k

)T) · nf

)∆Af

]. (73)

The hydrodynamic traction in the above equation is evaluated on Cartesian grid faces near the hull thatdefine a stair-step representation of the body on the Eulerian mesh [20], with nf and ∆Af representing theunit normal vector and the area of a given cell face, respectively. The computation of the gyroscopic actionMδ is described in the following section.

4.4.6. Coupling ISWEC dynamics

The ISWEC is allowed to freely rotate about its pitch axis and its motion depends on the hydrodynamicand external torques acting on it. The external torque Mδ generated by the gyroscopic action is unloadedon the hull and opposes the wave induced pitching motion. Therefore,Mδ appears with negative sign on theright side of Eq. 72. The analytical expression for this pitch torque is given by Eq. 13, while its discretizationis written as

Mn+1,kδ =

(J sin 2εn+1,k + I cos 2εn+1,k

)δn+1,k + Jφεn+1,k cos εn+1,k

+2 (J − I) δn+1,kεn+1,k sin εn+1,k cos εn+1,k + Jφ sin εn+1,k, (74)

19

Table 3: Comparison of maximum pitch angle ϑmax and galloping frequency fϑ with prior numerical studies.

ϑmax fϑ

Robertson et al. [46] 15 0.0191Yang & Stern [47] 15.7 0.0198Yang & Stern [48] 16.1 0.0197

Kolahdouz et al. [49] 15 0.0198Present 15.2 0.0197

in which the pitch acceleration term δn+1,k is calculated using a standard finite difference (explicit forwardEuler) of the hull’s pitch velocity:

δn+1,k =

δn+1,k−δn

∆t , k > 0,

δn−δn−1

∆t , k = 0.

(75)

We set δn+1,0 = δn, εn+1,0 = εn, δn+1,0 = δn, and εn+1,0 = εn for cycle k = 0.The precession acceleration ε is given analytically by Eq. 15, which in discretized form reads

εn+1,k =1

I

[−kεn+1,k−1 − cεn+1,k−1 − (I − J)

(δn+1,k

)2

sin εn+1,k−1 cos εn+1,k−1 + Jφδn+1,k cos εn+1,k−1

].

(76)This newest approximation to the precession acceleration εn+1,k is explicitly calculated using only the priorcycle’s values of precession velocity εn+1,k−1 and angle εn+1,k−1. New approximations to ε and ε at cycle kare computed using the Newmark-β method [45] as follows:

εn+1,k = εn +∆t

2

(εn + εn+1,k

)(77)

εn+1,k = εn + ∆tεn +∆t2

4

(εn + εn+1,k

)(78)

As described in Sec. 2, the PTO stiffness k and damping c parameters in the control torque and the gyro-scope’s angular velocity φ, acceleration φ = 0, and moments of inertia I and J are known a priori and donot represent additional unknowns in the calculation ofMn+1,k

δ . Hence the procedure outlined by Eqs. (74)to (78) enables the calculation of the external pitch torque shown on the right-hand side of Eq. 72, thuscoupling the ISWEC dynamics to the FD/BP methodology.

4.5. FSI validation

To validate our implementation of the method described in this section, we simulate the vortex inducedvibration of a rectangular plate undergoing galloping motion. This single rotational degree of freedomcase has been widely used as a benchmark problem for FSI algorithms in prior literature. It also mimicsthe ISWEC model well, which primarily oscillates in the pitch direction. The governing equation for thespring-mass-damper plate model reads as

Iϑϑ+ Cϑϑ+Kϑϑ =Mhydro, (79)

in which ϑ is the pitch angle of the plate measured from the horizontal axis, Iϑ is the pitch moment of inertiaabout the center of mass, Cϑ is the torsional damping constant, Kϑ is the torsional spring constant, andMhydro is the hydrodynamic moment acting on the plate due to the external fluid flow.

To compare of our results with prior simulations, we consider a plate with a width-to-thickness ratioof Λ∗ = Lp/Hp = 4, a non-dimensional moment of inertia of I∗ϑ = Iϑ/(ρsH

4p) = 400, a non-dimensional

damping ratio of ζ∗ϑ = Cϑ/(2√KϑIϑ

)= 0.25, and a reduced velocity of U∗ = U∞/(fϑHp) = 40. Here, U∞ is

the free stream velocity and fϑ =√Kϑ/Iϑ/2π is the natural frequency of the spring-mass-damper system.

20

0 100 200 300 400 500 600 700

-15

-10

-5

0

5

10

15

(a)

(b) (c)

(d) (e)

Figure 8: Galloping motion of a rectangular plate with Λ∗ = 4, I∗ = 400, ζ∗ϑ = 0.25 and U∗ = 40.(a) Temporal evolution ofpitch angle ϑ; (b)—(d) Vorticity (1/s) plots at three representative time instants t = 221.25 s, t = 309 s, and t = 349.5 s,respectively. (e) Dynamic mesh/patch distribution in the domain at a representative time instant t = 352.5 s.

21

The rectangular plate is centered at the origin with an initial non-zero pitch of ϑ = 1. The computationaldomain is taken to be Ω = [−32 cm, 96 cm] × [−32 cm, 32 cm], a rectangular domain of size Lx × Ly = 128cm × 64 cm. Five grid levels are used to discretize the domain, with the structure embedded on the finestgrid level. A coarse grid spacing of hcoarsest = Ly/32 is used on the coarsest level. The finest level is refinedwith a refinement ratio of nref = 2, whereas the rest of the finer levels are refined using a refinement ratioof nref = 4 from their next coarser level. A uniform inflow velocity U = (U∞ = 1 cm/s, 0 cm/s) is imposedon the left boundary (x = -32 cm), whereas zero normal traction and zero tangential velocity boundaryconditions are imposed on the right boundary (x = 96 cm). The bottom (y = -32 cm) and top (y = 32 cm)boundaries satisfy zero normal velocity and zero tangential traction boundary conditions. The Reynoldsnumber of the flow based on the inlet velocity is set to Re = ρfU∞Hp/µf = 250. A constant time stepsize of ∆t = 0.048hfinest is used for the simulation. After the initial transients, a vortex shedding state isestablished, which results in a periodic galloping of the rectangular plate. Fig. 8(a) shows the pitch angle ofthe plate as a function of time. Figs. 8(b)-8(d) show three representative snapshots of the FSI dynamics andthe vortex shedding pattern. Fig. 8(e) shows a typical AMR patch distribution in the domain due to theevolving structural and vortical dynamics. Table 3 compares the maximum pitch angle ϑmax and gallopingfrequency of the plate fϑ with values obtained from previous numerical studies; an excellent agreement withprior studies is obtained for both these rotational quantities.

5. Software implementation

The FD/BP algorithm and the numerical wave tank method described here is implemented within theIBAMR library [50], which is an open-source C++ simulation software focused on immersed boundarymethods with adaptive mesh refinement; the code is publicly hosted at https://github.com/IBAMR/IBAMR.IBAMR relies on SAMRAI [51, 52] for Cartesian grid management and the AMR framework. Linear andnonlinear solver support in IBAMR is provided by the PETSc library [53, 54, 55]. All of the example casesin the present work made use of distributed-memory parallelism using the Message Passing Interface (MPI)library. Simulations were carried out on both the XSEDE Comet cluster at the San Diego SupercomputerCenter (SDSC) and the Fermi cluster at San Diego State University (SDSU). Comet houses 1,944 IntelHaswell standard compute nodes consisting of Intel Xeon E5-2680v3 processors with a clock speed of 2.5GHz, and 24 CPU cores per node. Fermi houses 45 compute nodes with different generations of Intel Xeonprocessors.

Between 64 and 128 cores were used for the 2D computations presented here, while 128 cores were usedfor the 3D computations. The 2D ISWEC model using the medium grid resolution described in Sec. 6.1required approximately 6,129 seconds to execute 15,000 time-steps on Comet using 80 cores. The 3D ISWECmodel using the coarse grid resolution described in Sec. 7.1 required approximately 81,998 seconds to execute10,000 time-steps on Fermi using 128 cores of Intel Xeon E5-2697Av4 Broadwell processors with a clock speedof 2.6 GHz.

6. Spatial and temporal resolution tests

In this section, we perform a grid convergence study on the 2D ISWEC model in a NWT with regularwaves using three different spatial resolutions. We also conduct a temporal resolution study to determinea time step size ∆t that is able to adequately resolve the high-frequency wave components of irregularwaves. Although our implementation is capable of adaptive mesh refinement, we use static grids for all casespresented in this section. As mentioned in Sec. 4, we lock all the translational degrees of freedom of thehull and only consider its pitching motion. Appendix A compares the rotational dynamics in the presence ofheaving motion of the device, and justifies the accuracy of the 1-DOF model to calculate the main quantitiesof interest such as power output and conversion efficiency of the device.

The size of the computational domain is Ω = [0,10λ] × [0, 2.75d] with the origin located at the bottomleft corner (see Fig. 6). The hull parameters for the 2D model are given in Table 2, and the CG of the hullis located at (5λ, d−ZCG). The quiescent water depth is d = 0.65 m, acceleration due to gravity is g = 9.81m/s (directed in negative z-direction), density of water is ρw = 1025 kg/m3, density of air is ρair = 1.2kg/m3, viscosity of water µw = 10−3 Pa·s and viscosity of air is µair = 1.8 × 10−5 Pa·s. Surface tension

22

https://github.com/IBAMR/IBAMR

Table 4: Refinement parameters used for the 2D ISWEC dynamics grid convergence study.

Parameters Coarse Medium Fine

nref 2 2 4` 2 2 2Ny 300 600 600Nz 34 68 68

∆t (s) 2× 10−3 1× 10−3 5× 10−4

effects are neglected for all cases as they do not affect the wave and converter dynamics at the scale of theseproblems.

6.1. Grid convergence study

To ensure the wave-structure interaction dynamics are accurately resolved, we conduct a grid convergencestudy to determine an adequate mesh spacing. The dynamics of the ISWEC hull interacting with regularwater waves are simulated on three meshes: coarse, medium, and fine. Each mesh consists of a hierarchy of `grids; the computational domain is discretized by a coarsest grid of size Ny×Nz and then locally refined `−1times by an integer refinement ratio nref ensuring that the ISWEC device and air-water interface are coveredby the finest grid level. The grid spacing on the finest level are calculated using the following expressions:∆ymin = ∆y0/n

`−1ref and ∆zmin = ∆z0/n

`−1ref , where ∆y0 and ∆z0 are the grid spacings on the coarsest level.

The time step size ∆t is chosen to ensure the maximum Courant-Friedrichs-Levy (CFL) number = 0.12 foreach grid resolution. The mesh parameters and time step sizes considered here are shown in Table 4.

Regular water waves, generated based on the fifth-order wave theory presented in Sec. 3.1, enter the leftside of the domain and interact with the ISWEC hull. Temporal evolution of the hull’s pitch angle δ and thegyroscope’s precession angle ε are the primary outputs used to evaluate mesh convergence. The results andthe specification of the wave, ISWEC, and gyroscope parameters are shown in Fig. 9. Fig. 10(a) shows aclose-up of the medium resolution grid and the 2D ISWEC model. A minimum of 8 grid cells vertically spanthe height of the wave, indicating that the wave elevation is adequately resolved; for the coarse (fine) gridresolution, approximately 4 (15) grid cells span the wave height (results not shown). Additionally, the numberof cells covering the ISWEC hull length is approximately 30, 60, and 119 for the coarse, medium, and finegrid resolutions, respectively. Fig. 10(b) shows well-resolved vortical structures produced by the interactionof the ISWEC device and air-water interface on the medium resolution grid. From the quantitative andqualitative results shown in Fig. 9 and Fig. 10, we conclude that the medium grid resolution can capturethe WSI dynamics with reasonable accuracy. Therefore for the remaining cases studied here, we make useof the medium grid resolution.

6.2. Temporal resolution study

Next, we conduct a temporal resolution study to ensure that WSI dynamics of irregular waves and theISWEC device are adequately resolved. As described in Sec. 3.2, irregular water waves are modeled as asuperposition of N harmonic wave components. The energy carried by each wave component is related to itsfrequency ωi (see Eq. 43 and Fig. 5). Moreover as shown in Fig. 5, high frequency waves with ωi in the rangeof 10 rad/s to 20 rad/s carry considerable amounts of energy. Hence, the time step ∆t should be chosensuch that these high frequency (small wave period Ti) components are well-resolved since they contributesignificantly to the power absorbed by the device.

The dynamics of the ISWEC hull interacting with irregular water waves are simulated using three differenttime step sizes: ∆t = 10−3 s, 5× 10−4 s and 2.5× 10−4 s. For all three cases, we use a medium resolutiongrid with refinement parameters given by Table 4. Temporal evolution of the hull’s pitch angle δ and thepower absorbed by the PTO unit PPTO are the primary outputs used to evaluate temporal convergence. Theresults and the specification of the irregular wave, ISWEC, and gyroscope parameters are shown in Fig. 11.It is observed that the hull’s pitching motion is relatively insensitive to the chosen time step size ∆t. Sinceits dynamics are governed mainly by those waves carrying the largest energy, we can conclude that the higherfrequency wave components are adequately resolved. The difference between the three temporal resolutions

23

0 5 10 15 20

-3

-2

-1

0

1

2

(a) Hull pitch angle

0 5 10 15 20

-40

-20

0

20

40

(b) Gyroscope precession angle

Figure 9: Temporal evolution of (a) hull pitch angle δ and (b) gyroscope precession angle ε, for coarse (—–, black), medium(—–, red) and fine (—–, green) grid resolutions. Fifth-order regular water waves are generated with H = 0.1 m, T = 1 s andλ = 1.5456 m, satisfying the dispersion relation given by Eq. 35. A maximum ISWEC pitch angle δ0 = 5 and a maximumgyroscope precession angle of ε0 = 70 are used to calculate the rest of the parameters following the procedure described inSec. 2.3: PTO damping coefficient c = 0.3473 N·m·s/rad, gyroscope moment of inertia J = 0.0116 kg·m2 and PTO stiffnesscoefficient k = 0.4303 N·m/rad. The speed of the flywheel is φ = 4000 RPM, and I = 0.94× J = 0.0109 kg ·m2.

(a) Locally refined Cartesian grid (medium grid resolution)

(b) Vorticity (medium grid resolution)

Figure 10: Wave-structure interaction of the 2D ISWEC model at t = 27 s using the medium grid resolution: (a) locally refinedmesh with two levels (` = 2) and (b) representative vortical and air-water interfacial dynamics resulting from the WSI.

is more apparent in Fig. 11(b), in which we calculate the average power absorbed by the PTO unit PPTO

over the interval t = 10 s and t = 20 s. For ∆t = 10−3 s, 5 × 10−4 s and 2.5 × 10−4 s, the power absorbedis PPTO = 1.7656 W, 1.8859 W and 1.9484 W, respectively. It is seen that smaller time step sizes allow forthe resolution of higher-frequency wave peaks, which directly increases the absorbed power.

Based on these results, we hereafter use the medium grid spatial resolution, and time step sizes of

24

0 5 10 15 20

-5

0

5

13.6 13.8 14

-6

-4

-2

0


0 5 10 15 20

0

5

10

15

(b) Power absorbed by PTO

Figure 11: Temporal evolution of (a) hull pitching dynamics and (b) power absorbed by PTO, for three different time step sizes:∆t = 10−3 s (—–, black), ∆t = 5 × 10−4 s (—–, red) and ∆t = 2.5 × 10−4 s (—–, green). Irregular water waves (satisfyingthe dispersion relation Eq. 35) are generated with Hs = 0.1 m, Tp = 1 s and N = 50 wave components, with frequencies inthe range 3.8 rad/s to 20 rad/s distributed at equal ∆ω intervals. A maximum ISWEC pitch angle δ0 = 5 and a maximumgyroscope precession angle of ε0 = 70 are used to calculate the rest of the parameters following the procedure described inSec. 2.3: PTO damping coefficient c = 0.1724 N·m·s/rad, gyroscope moment of inertia J = 0.0057 kg·m2 and PTO stiffnesscoefficient k = 0.2138 N·m/rad. The speed of the flywheel is φ = 4000 RPM, and I = 0.94 × J = 0.0054 kg·m2. The meanwave power per unit crest width carried by the irregular waves calculated by Eq. 47 is Pwave = 5.0798 W.

∆t = 1× 10−3 s and ∆t = 5× 10−4 s for regular and irregular wave WSI cases, respectively.

7. Results and discussion

In this section, we investigate several aspects of the dynamics of the inertial sea wave energy converterdevice:

• First, we compare the PTO power predictions by the 3D and 2D ISWEC models under identical waveconditions. Utilizing the scaling factors presented in Table 1, we show that the power predicted by the3D model can be inferred from the power predicted by the 2D model reasonably well.

• Next, we study the effect of the maximum hull pitch angle parameter δ0 and make recommendationson how to select it based on the maximum wave steepness δs. We consider different “sea states”characterized by regular waves of different heights H, and consequently of different steepnesses.

• Thereafter, a parametric analysis for the 2D ISWEC model is performed using both regular andirregular water waves to study its dynamics. We vary the following parameters to recommend “design”conditions for the device: PTO damping coefficient c, flywheel speed φ, moment of inertia J and I,and PTO stiffness coefficient k.

• Afterwards, the effect of varying hull length to wavelength ratios is studied.

• Finally, we simulate a possible device protection strategy during inclement weather conditions andstudy the resulting dynamics.

All the 2D simulations are conducted in a NWT with computational domain size Ω = [0,10λ] × [0, 2.75d]as shown in Fig. 6. For 3D cases the computational domain size is same as in 2D, with the additionaldimension having length 5W ; W is the width of 3D model of the hull. The domain sizes are large enoughto ensure that the ISWEC dynamics are undisturbed by boundary effects. The origin of the NWT is takento be the bottom left corner of the domain and shown by the point O in Fig. 6. The CG of the ISWEChull is located at (2.5W, 5λ, d− ZCG) for 3D cases and (5λ, d− ZCG) for the 2D cases. The rest of the hullparameters are presented in Table 2. The water and air material properties are the same as those describedin Sec. 6.

25

Table 5: The PTO and gyroscope parameters for various regular wave heights H and δ0 values, as calculated by the proceduredescribed in Sec. 2.3. The rated power of the device PR is taken to be the available wave power Pwave for these calculations.The prescribed gyroscope parameters are ε0 = 70, φ = 4000 RPM, and I = 0.94×J . The parameter units for c are N·m·s/rad,J and I are kg·m2, and k are N·m/rad.

Regular wave properties ParametersPrescribed hull pitch angle δ0

2 5 10 15 20 δs

H = 0.025 m and T = 1 sc 0.0217 0.0217 0.0217 0.0217 0.0217 0.0217J 0.0018 0.00072 0.00036 0.00024 0.00018 0.0012k 0.0673 0.0269 0.0134 0.0089 0.0067 0.0464

H = 0.05 m and T = 1 sc 0.0868 0.0868 0.0868 0.0868 0.0868 0.0868J 0.0072 0.0029 0.0014 0.00090 0.00072 0.0025k 0.2692 0.1076 0.0538 0.0358 0.0269 0.0928

H = 0.1 m and T = 1 sc 0.3473 0.3473 0.3473 0.3473 0.3473 0.3473J 0.0290 0.0116 0.0058 0.0039 0.0029 0.0050k 1.0777 0.4303 0.2171 0.1421 0.1065 0.1876

H = 0.125 m and T = 1 sc 0.5427 0.5427 0.5427 0.5427 0.5427 0.5427J 0.0453 0.0181 0.0090 0.0060 0.0045 0.0063k 1.6827 0.6731 0.3365 0.2243 0.1682 0.2361

7.1. 3D and 2D ISWEC models

In this section, we investigate the dynamics of the 2D and 3D ISWEC models interacting with regularand irregular water waves. We compare the motion of the hull and the power absorption capabilities ofeach model. The 2D model is simulated on a medium grid resolution and the 3D model on a coarse gridresolution using the refinement parameters specified in Table 4. The third dimension is discretized with Nx= 38 grid cells for 3D cases. Fig. 12(a) shows the configuration of the locally refined mesh (` = 2), alongwith visualizations of regular and irregular waves for the three-dimensional NWT.

First, we consider two different prescribed maximum pitch angles δ0 = 5 and 20 for each model.Regular waves are generated with properties H = 0.1 m and T = 1 s. The values for the gyroscope andPTO parameters for this choice of δ0 are given in Table 5. The rated power of the device PR is taken tobe the available wave power Pwave for calculating the parameters reported in Table 5. The hull undergoespitching motion as the regular waves impact the device, as shown in Fig. 12(b). The temporal evolutionof the hull pitch angle δ for the 2D and the 3D ISWEC models are shown in Figs. 13(a) and 13(b). Fromthese results, it is observed that the dynamics for the 2D case match well with the 3D case after the initialtransients. The power transferred to the hull from the waves Phull, the power generated through the hull-gyroscope interaction Pgyro, and the power absorbed by the PTO unit PPTO at δ0 = 5 (δ0 = 20) for the 2Dand 3D models are shown in Figs. 13(c) and 13(d) (Figs. 13(e)and 13(f)), respectively. The time-averagedpowers PPTO, Pgyro and Phull over the time interval t = 10 s and t = 20 s (after the hull’s motion achieveda periodic steady state) are also shown in Figs. 13(c)-13(f). From these results, it is seen that the energytransfer pathway Eq. (23) is numerically verified. Furthermore, the power absorbed by the PTO unit forthe full-scale device can be calculated by multiplying the power absorbed by the 2D model by the Froudescaling given in Table 1

Pfull-scale = α52 · W · P2D, (PTO unit). (80)

Similarly for the 3D model,Pfull-scale = α

72 · P3D, (PTO unit). (81)

Finally, combining the two expressions above yields

P3D =Wfull-scale

α· P2D = 0.4× P2D, (PTO unit), (82)

in which Wfull-scale = 8 m is the width of the full-scale model and α = 20 is the length scaling factor. For the2D cases, the average power absorbed by the PTO unit is P2D = 1.6972 W for δ0 = 5, and P2D = 1.1694

26

(a) Locally refined Cartesian mesh

(b) Regular waves

(c) Irregular waves

Figure 12: (a) Locally refined Cartesian mesh with two levels of mesh refinement used for the 3D NWT. Representative WSIof the 3D ISWEC model at t = 28.8 s: (b) regular waves, and (c) irregular waves.

27

W for δ0 = 20. For the 3D cases, the average powers absorbed by the PTO unit are 0.8535 W and 0.5155W for δ0 = 5 and δ0 = 20, respectively, which are close to the expected values of 0.6788 W and 0.4677 Wpredicted by Eq. 82. Note that better agreement between the simulated and expected average powers in 3Dcan be obtained by increasing the spatial and temporal resolutions. Nevertheless, we are confident that thedynamics are reasonably resolved for the chosen grid spacing and time step size.

Next, we perform a similar scaling analysis for 2D and 3D ISWEC models in irregular wave conditions.Irregular water waves are generated with properties Hs = 0.1 m, Tp = 1 s and 50 wave components withfrequencies ωi in the range 3.8 rad/s to 20 rad/s (see Fig. 12(c)). Through empirical testing, fifty wavecomponents were found to be sufficient to represent the energy of the JONSWAP spectrum. We considera maximum pitch angle of δ0 = 5 for the device. The evolution of δ for the two models are comparedin Fig. 14(a). Similar to the regular wave case presented above, the dynamics of the 2D and 3D modelsnumerically agree and the energy transfer pathway Eq. (23) is nearly satisfied. Moreover, the average powerabsorbed by the PTO unit for the 2D model is P2D = 1.8859 W, yielding an expected 3D power of 0.7543W according to Eq. (82); this is close to the power value of 0.5810 W obtained by the 3D simulation.

Based on these results, we ultimately conclude that the 2D model is sufficient to accurately simulateISWEC dynamics and to predict the power generation/absorption capability of the converter. Hereafter, wefocus on further investigating dynamics and parameter choices for the 2D model.

7.2. Selection of prescribed hull pitch angle δ0

In this section, we investigate the relationship between the prescribed hull pitch angle parameter δ0, themaximum pitch angle actually attained by the hull δmax through WSI, and the maximum wave steepness ofthe incoming waves δs. Recall that the maximum wave steepness was calculated in Sec. 3.3 by approximatingthe fifth-order wave as a linear harmonic wave. We consider the ISWEC dynamics on four regular waterwaves with same time period T = 1 s (i.e. λ = 1.5456 m) but varying wave heights: H = 0.025 m, 0.05 m,0.1 m and 0.125 m, each having maximum wave steepness δs = 2.9, 5.8, 11.48 and 14.25, respectively(see Eq 52). The prescribed PTO and gyroscope system parameters for each sea state and six maximumpitch angle values δ0 = 2, 5, 10, 15, 20 and δs are shown in Table 5. Additionally, δ0 = 1 and 30

cases are also simulated, but the parameter values are not tabulated for brevity.The results of this parameter study are shown in Fig. 15. It is observed that when δ0 < δs, δmax increases

linearly with δ0 (Fig. 15(a)), illustrating that the hull’s maximum oscillation amplitude correlates well withδ0. When the prescribed δ0 is greater than δs, it is seen that δmax no longer increases; rather it maintainsa constant value with respect to δ0. This indicates that further increasing δ0 will not lead to larger pitchoscillations, i.e. the δmax attained by the hull is the largest value permitted by the slopes of the wave. InFigs. 15(b) and 15(c), we show trends in the maximum precession angle attained by the gyroscope εmax andthe relative capture width (RCW) η, which measures the device efficiency as a ratio of the average powerabsorbed by the PTO unit to the average wave power per unit crest width (see Eq.(27)). Maximization ofboth these quantities is achieved when δ0 is set close to δs. As the hull achieves the maximum pitch anglephysically permitted by the slopes of the wave, further increasing δ0 amounts to reducing Jφ (Eq. (25)) orthe hull-gyroscope coupling, which explains the reduction in both maximum precession and device efficiency.Hereafter, we prescribe δ0 based on the value maximizing η as we conduct further parametric analyses ofthe 2D ISWEC model.

7.3. Parametric analyses of gyroscope parameters

In this section, we conduct a parameter sweep around the energy-maximizing PTO and gyroscope param-eters estimated by the theory presented in Sec. 2.3. We test the theory’s predictive capability and describe theeffect of these parameters on the converter’s performance and dynamics. In each of the following subsections,only a single parameter is varied at a time.

Simulations are conducted using both regular water waves with H = 0.1 m and T = 1 s, and irregularwaves with Hs = 0.1 m, Tp = 1 s and 50 wave components with frequencies ωi in the range 3.8 rad/s to 20rad/s. These wave conditions serve as device “design” conditions at its installation site. For regular waves,the prescribed pitch angle is taken to be δ0 = 10, and the PTO and gyroscope parameters are given inTable 5. For irregular waves, the prescribed pitch angle δ0 = 5 is used. The PTO and gyroscope parametersremain the same as those used in the temporal resolution study (see Sec.6.2). These particular values of

28

0 5 10 15 20

-3

-2

-1

0

1

2

(a) Hull motion for δ0 = 5

0 5 10 15 20

-10

-5

0

5

10

(b) Hull motion for δ0 = 20

0 5 10 15 20

0

1

2

3

4

5

6

(c) Powers for δ0 = 5 (2D model)

0 5 10 15 20

0

0.5

1

1.5

2

2.5

3

(d) Powers for δ0 = 5 (3D model)

0 5 10 15 20

-15

-10

-5

0

5

10

15

(e) Powers for δ0 = 20 (2D model)

0 5 10 15 20

-10

-5

0

5

10

(f) Powers for δ0 = 20 (3D model)

Figure 13: WSI of the 2D and 3D ISWEC models in regular water wave conditions (H = 0.1 m and T = 1 s). Temporalevolution of the hull pitch angle (δ) for the 2D and 3D ISWEC models for (a) δ0 = 5 and (b) δ0 = 20. Power absorbed by thePTO unit PPTO (—–, black), power generated through the hull-gyroscope interaction Pgyro (—–, red) and power transferredto the hull from the regular water waves Phull (—–, green) for the (c) 2D model with δ0 = 5, (d) 3D model with δ0 = 5,(e) 2D model with δ0 = 20 and (f) 3D model with δ0 = 20. Averaged power over the time period t = 10 s and t = 20 s areshown in the legends.

29

0 5 10 15 20

-6

-4

-2

0

2

4

6

(a) Hull motion for δ0 = 5

0 5 10 15 20

-5

0

5

10

15

(b) Powers for δ0 = 5 (2D model)

0 5 10 15 20

-1

0

1

2

3

(c) Powers for δ0 = 5 (3D model)

Figure 14: WSI of the 2D and 3D ISWEC models with δ0 = 5 in irregular water wave conditions (Hs = 0.1 m and Tp = 1s, N = 50, and ωi in the range 3.8 rad/s to 20 rad/s). (a) Temporal evolution of the hull pitch angle (δ) for the 2D and 3DISWEC models. Power absorbed by the PTO unit PPTO (—–, black), power generated through the hull-gyroscope interactionPgyro (—–, red) and power transferred to the hull from the irregular water waves Phull (—–, green) for the (b) 2D and (c) 3Dmodels.

30

0 5 10 15 20 25 300

2

4

6

8

10

12

(a) Maximum attained hull pitch angle

0 5 10 15 20 25 300

10

20

30

40

50

(b) Maximum attained gyroscope precession angle

0 5 10 15 20 25 300

5

10

15

20

25

30

(c) Relative capture width (RCW)

Figure 15: (a) Maximum hull pitch angle δmax, (b) maximum gyroscope precession angle εmax, and (c) relative capture widthη of the ISWEC device for various regular wave sea states and prescribed pitch angles δ0: H = 0.025 m (—–, black), H = 0.05m (—–, red), H = 0.1 m (—–, green), and H = 0.125 m (—–, blue). RCW is calculated from time-averaged powers over theinterval t = 10 s to t = 20 s

31

δ0 were found to maximize the RCW of the converter at design conditions; for an example, see Fig. 15 forregular waves with H = 0.1 m and T = 1 s.

7.3.1. PTO damping coefficient c

We first consider the PTO unit damping coefficient c, which directly impacts the power absorptioncapability of the device. We prescribe four different values, c = 0.05, 0.3473, 1.0 and 2.0 N·m·s/rad, toevaluate its impact on ISWEC dynamics. The optimal damping coefficient value of c = 0.3473 is predictedby the theory. Results for the hull interacting with regular waves are shown in Fig. 16. As expected forsmaller damping coefficients, the gyroscope is able to attain larger precession angles ε and velocities ε, asseen in Fig. 16(b). Higher precession velocities yield larger pitch torque Mδ values (see Eq. (20)), whichopposes the motion of the hull and restrict its maximum pitch oscillation; this is consistent with the dynamicsshown in Figs. 16(a) and 16(c). Moreover the hull’s pitch velocity δ is reduced with decreasing c, leadingto a smaller (in magnitude) precession torque Mε acting on the PTO shaft (see Eq. (15)); our simulationsshow this behavior as observed in Fig. 16(d).

In Fig. 16(e), we compare the time-averaged powers Phull, Pgyro, and PPTO as a function of varying PTOdamping coefficient. It can be seen that these three powers are in reasonable agreement with each other,indicating that the energy transfer pathway Eq. (23) is approximately satisfied. In terms of power generation,it is observed that the device achieves peak performance when a PTO damping coefficient c = 0.3473 isprescribed, which validates the theoretical procedure. The reason for an optimum value of c is as follows: asthe damping coefficient increases, the precession velocity decreases. The power absorbed by the PTO unitis the product of c and ε2 (Eq. (11)), and therefore these competing factors must be balanced in order toachieve maximum power generation.

Finally in Fig. 16(f), we show the evolution of the the yaw torque Mφ acting on the hull for c =0.3473, noting that its magnitude is approximately one-fifth of the pitch torque Mδ. Although this is notinsignificant, we do not consider the effect ofMφ for the 3D ISWEC model (see Sec. 2.1) since its contributionwill be cancelled out 1) by using an even number of gyroscopic units (if each flywheel pair spins with equaland opposite velocities) [21], and 2) partially by the mooring system. Discounting Mφ during the ISWECdesign phase would misalign the converter with respect to the main wave direction, which will reduce itsperformance. It is also interesting to note that the yaw torque in the gyroscopic frame of reference Mz1 isat least two orders of magnitude lower than the yaw torque in the inertial reference frame, as evidenced bythe inset of Fig. 16(f).

Similar dynamics are observed when the ISWEC model is simulated in irregular wave conditions for fourdifferent values, c = 0.05, 0.1724, 1.0 and 2.0 N·m·s/rad. The optimal damping coefficient value of c = 0.1724is obtained from the theory. The results are compared in Fig. 17 and the theoretically predicted optimum cis verified. The response of the hull and gyroscope to irregular waves can be seen in Figs. 17(a) and 17(b),respectively. The pitch torque and the precession torque are shown in Figs. 17(c) and 17(d), respectively.From Fig. 17(e), it is verified that the energy transfer pathway given by Eq. 23 is satisfied. We note thatthe device efficiency is higher in irregular wave conditions as compared to regular wave conditions. This canbe seen by comparing the maximum value of relative capture width for H = 0.1 m in Figs. 15(c) and 17(f):ηmax = 24.36% vs. ηmax = 37.61%, respectively. The power carried by irregular waves is approximatelyhalf that of regular waves when they have the same significant height and time period. Therefore for theprescribed device dimensions, the converter is more efficient in less energetic wave conditions.

7.3.2. Flywheel speed φ

Next, we conduct a parameter sweep of the flywheel speed φ and investigate its effects on ISWECdynamics. The speed of the flywheel affects not only the amount of angular momentum Jφ generated in thegyroscope, but also the magnitude of the gyroscopic torques produced as seen in Eqs. (15) and (20). Weconsider four different flywheel speeds: φ = 100 RPM, 1000 RPM, 4000 RPM, and 8000 RPM, with δ0 =10 and the remaining gyroscope parameter are prescribed based on Table 5. Recall that these values wereobtained for φ = 4000 RPM in Table 5.

The results for a hull interacting with regular waves are shown in Fig. 18. It is seen that the maximumpitch angle decreases with increasing φ (Fig. 18(a)), while a non-monotonic relationship is seen betweenthe maximum precession angle and φ (Fig. 18(b)). Time-averaged powers are shown in Fig. 18(c), whichagain shows that Eq. (23) is approximately satisfied. Power absorption is maximized at a flywheel speed of

32

(a) Hull pitch angle (b) Gyroscope precession angle

(c) Pitch torque unloaded on the hull (d) Precession torque on the PTO axis

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

(e) Time-averaged powers in the system

0 5 10 15 20-4

-2

0

2

4

10 11 12-0.02

0

0.02

(f) Yaw torque

Figure 16: Dynamics of the 2D ISWEC model for four different values of PTO damping coefficient c, with regular wave propertiesH = 0.1 m and T = 1 s. Temporal evolution of (a) hull pitch angle δ, (b) gyroscope precession angle ε, (c) pitch torque Mδ,and (d) precession torque Mε for c = 0.05 N·m·s/rad (—–, black), c = 0.3473 N·m·s/rad (—–, red), c = 1.0 N·m·s/rad (—–,green), and c = 2.0 N·m·s/rad (—–, blue); (e) comparison of time-averaged powers from the interval t = 10 s to t = 20 s foreach value of c; (f) yaw torques Mφ and Mz1 produced in the inertial reference frame and gyroscope reference frame (inset),respectively.

33

0 5 10 15 20

-10

-5

0

5

10

15

14.6 14.8 15 15.2

-10

-8

-6

-4

-2

0


0 5 10 15 20

-100

-75

-50

-25

0

25

50

75

100

14.6 14.8 15 15.2

-80

-60

-40

-20

0


0 5 10 15 20

-30

-20

-10

0

10

20

30

14.4 14.6 14.8 15

-25

-20

-15

-10

-5

0

(c) Pitch torque unloaded on the hull

0 5 10 15 20

-2

-1

0

1

2

13.8 14 14.2 14.4

-2

-1.5

-1

-0.5

0

(d) Precession torque on the PTO axis

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

(e) Time-averaged powers in the system

0 0.5 1 1.5 20

10

20

30

40

(f) Relative capture width (RCW)

Figure 17: Dynamics of the 2D ISWEC model for four different values of PTO damping coefficient c, with irregular waveproperties Hs = 0.1 m and Tp = 1 s and 50 wave components with frequencies ωi in the range 3.8 rad/s to 20 rad/s. Temporalevolution of (a) hull pitch angle δ, (b) gyroscope precession angle ε, (c) pitch torqueMδ, and (d) precession torqueMε for c =0.05 N·m·s/rad (—–, black), c = 0.1724 N·m·s/rad (—–, red), c = 1.0 N·m·s/rad (—–, green), and c = 2.0 N·m·s/rad (—–,blue); (e) comparison of time-averaged powers from the interval t = 10 s to t = 20 s for each value of c; (f) relative capturewidth η for each value of c.

34


0 2000 4000 6000 80000

0.5

1

1.5

2

2.5

3

3.5

(c) Powers

Figure 18: Dynamics of the 2D ISWEC model for four different values of flywheel speed φ. The regular wave properties are H= 0.1 m and T = 1 s. Temporal evolution of (a) hull pitch angle δ, and (b) gyroscope precession angle ε for φ = 100 RPM(—–, black), φ = 1000 RPM (—–, red), φ = 4000 RPM (—–, green), and φ = 8000 RPM (—– , blue); (c) comparison oftime-averaged powers from the interval t = 10 s to t = 20 s for each value of φ.

φ = 4000 RPM, which can be physically explained as follows. As Jφ increases, the gyroscopic system is ableto generate significant precession torque which, increases the absorption capacity of the PTO unit. However,this increased angular momentum also increases the pitch torque opposing the hull, thereby limiting itspitching motion and reducing the power absorbed from the waves. These two competing factors leads to anoptimum value of φ.

Similar dynamics are obtained when the ISWEC interacts with irregular waves for varying values of φ.The results are shown in Fig. 19. The comparison of pitch angle for various φ values is shown in Fig. 19(a)and of precession angle is shown in Fig. 19(b). Eq. 23 is again satisfied as seen from the time-averagedpowers in Fig. 19(c).

7.3.3. Flywheel moment of inertia J and I

The angular momentum Jφ generated in the gyroscope can also be modified by varying the flywheel sizevia its moment of inertia components J and I. First, we consider three different values J = 0.0005 kg·m2,0.0058 kg·m2 and 0.5 kg·m2, which correspond to light, medium, and heavy weight gyroscopes, respectively.The J = 0.0058 value is obtained from theoretical estimates based on the prescribed δ0 and ε0 values. Avalue of I = 0.94× J is set for each case, and the remaining gyroscope parameters are prescribed based onTable 5.

The results for a hull interacting with regular waves are shown in Fig. 20. It is seen that the light gyroscopeproduces insignificant precession angles and torques due to the lack of angular momentum generated by theflywheel. Moreover, the heavy gyroscope produces even smallerMε torque as it slowly drifts around the PTOaxis; the proportional component of the control torque (kε) is not strong enough to return the gyroscope to

35

0 5 10 15 20

-10

-5

0

5

10

15

14.6 14.8 15 15.2

-12

-10

-8

-6

-4

-2

0


0 5 10 15 20

-75

-50

-25

0

25

50

75

14.8 15 15.2

-50

-40

-30

-20

-10

0


0 2000 4000 6000 80000

0.5

1

1.5

2

(c) Powers

Figure 19: Dynamics of the 2D ISWEC model for four different values of flywheel speed φ. The irregular wave properties areHs = 0.1 m and Tp = 1 s and 50 wave components with frequencies ωi in the range 3.8 rad/s to 20 rad/s. Temporal evolution

of (a) hull pitch angle δ, and (b) gyroscope precession angle ε for φ = 100 RPM (—–, black), φ = 1000 RPM (—–, red), φ =4000 RPM (—–, green), and φ = 8000 RPM (—– , blue); (c) comparison of time-averaged powers from the interval t = 10 sto t = 20 s for each value of φ.

36

0 5 10 15 20-10

-5

0

5

10


0 5 10 15 20

-40

-20

0

20

40


0 5 10 15 20

-20

-10

0

10

20


0 5 10 15 20-2

-1

0

1

2

(d) Precession torque on the PTO axis

Figure 20: Dynamics of 2D ISWEC model for three different values of flywheel moment of inertia J . The regular wave propertiesare H = 0.1 m and T = 1 s. Temporal evolution of (a) hull pitch angle δ, (b) gyroscope precession angle ε, (c) pitch torqueMδ, and (d) precession torqueMε for J = 0.0005 kg·m2 (—–, black), J = 0.0058 kg·m2 (—–, red), and J = 0.5 kg·m2 (—–,green). For all cases, I = 0.94× J .

its mean position of ε = 0. Additionally, the light (heavy) weight gyroscope produces small (large) pitchtorques Mδ opposing the hull, which explains the large (small) pitch amplitudes exhibited by the device.Finally, it is seen that the medium weight gyroscope, with J = 0.0058 kg·m2 calculated from the proceduredescribed in Sec. 2.3, produces the largest precession amplitudes ε and velocities ε, leading to high powerabsorption by the PTO unit.

We also study the effect of varying I while keeping J = 0.0058 kg·m2 fixed. We consider four differentvalues I = 0.5× J , I = 0.75× J , I = 0.94× J and I = 1.0× J , and the results for a device interacting withregular waves are shown in Fig. 21. It is seen that the dynamics of the hull and gyroscope and the systempowers are not significantly affected by the choice of I.

Similarly, ISWEC dynamics with irregular waves are studied for three different values of J . Results forvarying J values are compared in Fig. 22, which are qualitatively similar to the results obtained with regularwaves. The effect of varying I with respect to J is also simulated, and the results are shown in Fig. 23. It isseen that the hull pitch and the gyroscope precession angles are relatively insensitive to variations in I. It isseen that the powers are relatively constant across different I values under irregular wave conditions as well.

7.3.4. PTO stiffness coefficient k

Finally, we study the effect of varying the PTO stiffness coefficient k on the dynamics of the ISWECdevice. This term appears as a restoring torque kε in the precession angle Eq. (16) and acts to drive thegyroscope’s oscillation about its mean position ε = 0. The oscillation frequency is directly influenced by kand can be chosen to ensure a resonant condition is attained between the gyroscope and the incoming waves,thus maximizing the power absorbed by the system.

37


3 4 5 6

10-3

0

1

2

3

4

5

(c) Powers

Figure 21: Dynamics of 2D ISWEC model for four different values of I. The regular wave properties are H = 0.1 m and T = 1s. Temporal evolution of (a) hull pitch angle δ, and (b) gyroscope precession angle ε for I = 0.5× J (—–, black), I = 0.75× J(—–, red), I = 0.94× J (—–, green), and I = 1.0× J (—– , blue). (c) Comparison of time-averaged powers from the intervalt = 10 s to t = 20 s for each value of I. For all cases, J = 0.0058 kg·m2.

38

0 5 10 15 20-15

-10

-5

0

5

10

15


0 5 10 15 20

-60

-40

-20

0

20

40

60


0 5 10 15 20

-30

-20

-10

0

10

20


0 5 10 15 20-2

-1

0

1

2

(d) Precession torque produced by the gyroscope

Figure 22: Dynamics of 2D ISWEC model for three different values of flywheel moment of inertia J . The irregular waveproperties are Hs = 0.1 m and Tp = 1 s and 50 wave components with frequencies ωi in the range 3.8 rad/s to 20 rad/s.Temporal evolution of (a) hull pitch angle δ, (b) gyroscope precession angle ε, (c) pitch torque Mδ, and (d) precession torqueMε for J = 0.0005 kg·m2 (—–, black), J = 0.0058 kg·m2 (—–, red), and J = 0.5 kg·m2 (—–, green). For all cases, I = 0.94×J .

39

0 5 10 15 20

-6

-4

-2

0

2

4

6

14.8 15 15.2

-6

-4

-2

0


0 5 10 15 20

-60

-40

-20

0

20

40

60

14.8 15 15.2

-50

-40

-30

-20

-10

0


3 4 5 6

10-3

0

1

2

3

4

(c) Powers

Figure 23: Dynamics of 2D ISWEC model for four different values of I. The irregular wave properties are Hs = 0.1 m and Tp= 1 s and 50 wave components with frequencies ωi in the range 3.8 rad/s to 20 rad/s. Temporal evolution of (a) hull pitchangle δ, and (b) gyroscope precession angle ε for I = 0.5× J (—–, black), I = 0.75× J (—–, red), I = 0.94× J (—–, green),and I = 1.0× J (—– , blue). (c) Comparison of time-averaged powers from the interval t = 10 s to t = 20 s for each value ofI. For all cases, J = 0.0058 kg·m2.

40

Table 6: Calculated values of PTO and gyroscope parameters for different L/λ ratios using L = 0.7665 m, δ0 = 10, ε0 = 70,and φ = 4000 RPM. I = 0.94 × J for all cases. Regular water waves with H = 0.1 m are simulated. The rated power of thedevice PR is taken to be the available wave power Pwave for these calculations. Units: λ is in m, c is in N·m·s/rad, J and I arein kg·m2 and k is in N·m/rad.

L/λ λPTO and gyroscope parametersc J k

0.25 3.0659 1.3491 0.0225 0.37050.5 1.5456 0.3473 0.0058 0.21710.75 1.0219 0.1773 0.0029 0.1679

We consider four different values of k = 0.0 N·m/rad, 0.2171 N·m/rad, 1.0 N·m/rad, and 5.0 N·m/rad,with the remaining gyroscope parameter chosen according to Table 5. The k = 0.2171 value is obtainedfrom theoretical considerations provided in Sec. 2.3. The results for a hull interacting with regular wavesare shown in Fig. 24. As k increases, the maximum precession angle ε and velocity ε decreases leading todecreased power absorption by the device. The increased PTO stiffness value tends to keep the gyroscopeclose to its zero-mean position, which reduces the hull-gyroscope coupling. This can be observed from thelowered values ofMδ torques in Fig. 24(c). As a consequence, the hull pitching motion increases, as seen inFig. 24(a).

The k = 0 case warrants additional discussion. When the PTO stiffness is zero, the gyroscope attains alarger maximum precession amplitude and generates more power than the k > 0 cases over the time periodt = 10 s and t = 20 s. However, Fig. 25 shows the long-term dynamics for k = 0; it is seen that the gyroscopeis unable to sustain its precession oscillation as it eventually falls to one side (ε = −90) and remains there.At this configuration, the gyroscope yaw axis and the hull pitch axis are aligned, and the precession effectis lost. As these gyroscopic oscillations vanish, the torques tend towards zero, the hull exhibits unrestrainedpitch oscillation, and no power is generated.

Next, we simulate ISWEC dynamics with irregular waves using four different values of k = 0 N·m/rad,0.2138 N·m/rad, 1.0 N·m/rad, and 5.0 N·m/rad. The results are compared in Fig. 26, and are qualitativelysimilar to the those obtained with regular waves. Similar behavior of the ISWEC with k = 0 is observed— the gyroscope is unable to oscillate and falls to one side (ε = −90) and produces vanishing precessioneffects. However, with irregular waves the precession effects are lost much sooner compared to the regularwaves case.

7.4. Hull length to wavelength (L/λ) variation

In this section, we study the effect of hull length to wavelength ratio (L/λ) on ISWEC dynamics. Weselect three ratios L/λ = 0.25, 0.5 and 0.75 for this analysis. The length of the hull is kept constant at L= 0.7665 m, and the wavelength of the regular water waves is varied. The PTO and gyroscope parametersused in the three simulations are presented in Table 6. Results consist of temporal evolution of the hull pitchand gyroscope precession angles in Figs. 27(a) and 27(b), respectively. It is observed that the hull pitch ismaximum when λ/3 ≤ L ≤ λ/2, as discussed in Sec. 2.4. As a consequence, the gyroscope precesses moreand the conversion efficiency of the device increases (see Fig. 27(c)).

7.5. Device protection during inclement weather conditions

The ISWEC hull houses costly electro-mechanical components that need to be protected during harsh,stormy weather conditions. During inclement weather, the hull and gyroscope dynamics can be chaotic,which may damage the system components. To protect the housed components, the gyroscope needs to beturned off. This can be done by reducing the flywheel speed to zero using remote human-machine interfaces.The combined hull-gyroscope system then behaves like a single floating entity. In this section, we simulatethe dynamics of the ISWEC device as the flywheel speed is reduced to zero amidst steady operation. Wesimulate this scenario with regular water waves of H = 0.1 m and T = 1 s. To reduce the flywheel speedfrom 4000 RPM to 0 RPM, we use the following relation

φ(t) = 4000 · (1− f(t))/2, (83)

41


(c) Pitch torque unloaded on the hull (d) Precession torque produced by the gyroscope

0 1 2 3 4 50

0.5

1

1.5

2

2.5

(e) Powers

Figure 24: Dynamics of 2D ISWEC model for four different values of PTO stiffness k. The regular wave properties are H =0.1 m and T = 1 s. Temporal evolution of (a) hull pitch angle δ, (b) gyroscope precession angle ε, (c) pitch torque Mδ, and(d) precession torqueMε for k = 0 N·m/rad (—–, black), k = 0.2171 N·m/rad (—–, red), k = 1.0 N·m/rad (—–, green), andk = 5.0 N·m/rad (—–, blue); (e) comparison of time-averaged powers from the interval t = 10 s to t = 20 s for each value of k.

42

0 10 20 30 40 50 60-10

-5

0

5

10


0 10 20 30 40 50 60-100

-80

-60

-40

-20

0

20

40


0 10 20 30 40 50 60

-10

-5

0

5

10


0 10 20 30 40 50 60-2

-1

0

1

2


Figure 25: Long-term dynamics of the 2D ISWEC model for k = 0 PTO stiffness: (a) hull pitch angle δ, (b) gyroscope precessionangle ε, (c) pitch torque Mδ, and (d) precession torque Mε. The regular wave properties are H = 0.1 m and T = 1 s.

43

0 5 10 15 20-15

-10

-5

0

5

10

15

14.8 15 15.2

-10

-8

-6

-4

-2

0


0 5 10 15 20

-80

-60

-40

-20

0

20

40

60

80


0 5 10 15 20

-20

-10

0

10

20


0 5 10 15 20

-2

-1

0

1

2


0 1 2 3 4 50

0.5

1

1.5

2

2.5

(e) Powers

Figure 26: Dynamics of 2D ISWEC model for four different values of PTO stiffness k. The irregular wave properties are Hs

= 0.1 m and Tp = 1 s and 50 wave components with frequencies ωi in the range 3.8 rad/s to 20 rad/s. Temporal evolution of(a) hull pitch angle δ, (b) gyroscope precession angle ε, (c) pitch torqueMδ, and (d) precession torqueMε for k = 0 N·m/rad(—–, black), k = 0.2138 N·m/rad (—–, red), k = 1.0 N·m/rad (—–, green), and k = 5.0 N·m/rad (—–, blue); (e) comparisonof time-averaged powers from the interval t = 10 s to t = 20 s for each value of k.

44

0 5 10 15 20

-5

0

5

14.6 14.8 15 15.2

-6

-4

-2

0


0 5 10 15 20

-40

-20

0

20

40

14.6 14.8 15 15.2

-40

-30

-20

-10

0


0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

(c) RCW

Figure 27: Dynamics of 2D ISWEC model for three different hull length to wavelength ratios L/λ. The regular wave heightis H = 0.1 m, while its period T is calculated based on the dispersion relation given by Eq. (36) as wavelength λ is varied.Temporal evolution of (a) hull pitch angle δ and (b) gyroscope precession angle ε for L/λ = 0.25 (—–, black), L/λ = 0.5 (—–,red), and L/λ = 0.75 (—–, green); (c) RCW η computed using time-averaged powers from the interval t = 10 s to t = 20 s foreach value of L/λ.

45

in which f(t) is a function that smoothly transitions from -1 to 1 in the transition time interval ∆T . Thefunction f is given by

f = tanh

(2π(t− Thalf)

∆T

), (84)

in which Thalf = Tstart + ∆T/2. In our simulation, we set Tstart = 15 s and ∆T = 5 s. Fig. 28(e) showsthe smooth transition of the flywheel speed towards zero in 5 s. When the gyroscope is turned off, theprecession effects cease and the system attains a mean zero position, thus protecting the device. This is seenin Figs. 28(b), 28(c), and 28(d), which show that ε, Mδ, and Mε are reduced to zero, respectively. As thegyroscopic effects vanish, the hull is observed to be oscillating with greater pitch amplitude (Fig. 28(a)).

0 5 10 15 20 25 30

-10

-5

0

5

10


0 5 10 15 20 25 30

-40

-20

0

20

40


0 5 10 15 20 25 30

-15

-10

-5

0

5

10

15


0 5 10 15 20 25 30-2

-1

0

1

2

(d) Precession torque on PTO axis

0 5 10 15 20 25 30

0

1000

2000

3000

4000

(e) Flywheel speed

Figure 28: Dynamics of 2D ISWEC model as the flywheel speed φ is reduced from 4000 RPM to 0 RPM amidst steady operation.The regular wave properties are H = 0.1 m and T = 1 s. Temporal evolution of (a) hull pitch angle δ; (b) gyroscope precessionangle ε; (c) pitch torque Mδ; (d) precession torque Mε; and (e) temporal variation of flywheel speed.

46

8. Conclusions

In this study, we systematically investigated the wave-structure interaction dynamics of the inertial seawave energy converter (ISWEC) technology. Our computational model is based on the incompressible Navier-Stokes equations and employs a fictitious domain Brinkman penalization (FD/BP) approach to handle thefluid-structure coupling. The dynamics of the ISWEC hull and gyroscope system were coupled to this CFDsolver to enable fully-resolved 1-DOF simulations of the device. To emulate realistic operating conditions ofthe device, a numerical wave tank was used to generate both regular waves based on fifth-order Stokes theoryand irregular waves based on the JONSWAP spectrum. We performed Froude scaling analysis of the full-scaleISWEC model to determine the required parameters for our 1:20 scaled-down two- and three-dimensionalsimulations.

Our numerical investigation demonstrated that the 2D model was sufficient to accurately simulate thehull’s pitching motion, and to predict the power generation/absorption capability of the converter. Weshowed that setting the prescribed hull pitch angle parameter δ0 close to the maximum wave steepness willmaximize the device’s relative capture width (i.e. power generation efficiency). A comprehensive parametersweep demonstrated that the device achieves peak performance when the gyroscope specifications are chosenbased on the reactive control theory described in Sec. 2.3. It was also shown that a proportional control ofthe PTO control torque is required to generate continuous precession effects of the gyroscope, without whichthe gyroscope tends to align with the hull pitch axis. Under this scenario, the device does not generateany power. We also showed that although the yaw torque in the gyroscope reference frame is small, itis of the same order of magnitude as the pitch torque induced on the hull in an inertial reference frame.Therefore, the yaw torque on the hull should be considered in the design phase of these devices to avoidany misalignment of the converter from the main wave direction. Our simulations also verify that the hulllength to wavelength ratio should be between one-half and one-third to achieve high conversion efficiency.Throughout our parameter study, we numerically verify the theoretical power transfer pathway between thewater waves and the hull, the hull and the gyroscope, and the gyroscope to the PTO unit for both regularand irregular wave environments. Although the power transfer is derived for ISWEC devices in this work,an analogous relationship could be derived for heaving or surging point absorbers. Finally, we investigatedthe dynamics of the ISWEC system as the flywheel speed is reduced to zero to emulate device protectionduring inclement weather conditions.

By making use of high performance computing, our work demonstrates that it is feasible to use fully-resolved simulations to interrogate the device physics and dynamics of wave energy converters. They canalso be used as a design tool to explore the parametric space for further optimization of such devices.

Acknowledgements

A.P.S.B acknowledges helpful discussions with Giovanni Bracco and Giuliana Mattiazzo over the courseof this work. A.P.S.B. acknowledges support from NSF award OAC 1931368. NSF XSEDE and SDSU Fermicompute resources are particularly acknowledged.

A. Two degrees of freedom ISWEC model

We compare the hull and gyroscope dynamics obtained using two degrees of freedom (pitch and heave)and one degree of freedom (pitch only) ISWEC models. The same case from Sec. 6.1 is simulated using thetwo models on a medium grid resolution. Figs. 29(a) and 29(b) show the comparison of hull pitch angle δand gyroscope precession angle ε, respectively. As can be seen in Fig. A.29, including an additional heavedegree of freedom only marginally affects the rotational motion of the hull and gyroscope, and consequentlythe power output of the device. Fig. 29(c) shows the heave dynamics of the hull about its mean z-location.The heave amplitude is approximately one-tenth of the hull height for the prescribed wave characteristics.Although the heave motion is not negligible in this case, it nonetheless does not significantly affect therotational dynamics. Moreover, in a real device, the heave (along with the surge and sway) motion isconstrained to a certain extent by the mooring system. Finally, Fig. 29(d) shows that the power transferequation is satisfied even for the 2-DOF ISWEC model.

47

0 5 10 15 20-4

-2

0

2

4

(a) Hull pitch

0 5 10 15 20

-40

-20

0

20

40

(b) Gyroscope precession

0 5 10 15 20-0.04

-0.02

0

0.02

0.04

(c) Hull heave

0 5 10 15 20-2

0

2

4

6

8

10

(d) Powers in the system (2-DOF model)

Figure A.29: Comparison of 2-DOF (pitch and heave) and 1-DOF (pitch only) ISWEC models for (a) hull pitch angle δ,and (b) gyroscope precession angle ε. (c) Hull heave displacement, and (d) power at various levels for the 2-DOF ISWECmodel. Fifth-order regular water waves are generated with H = 0.1 m, T = 1 s and λ = 1.5456 m, satisfying the dispersionrelation given by Eq. 35. A maximum ISWEC pitch angle δ0 = 5 and a maximum gyroscope precession angle of ε0 = 70

are used. The gyroscope parameters are: damping coefficient c = 0.3473 N·m·s/rad, moment of inertia J = 0.0116 kg·m2, andPTO stiffness k = 0.4303 N·m/rad. The speed of the flywheel is φ = 4000 RPM, and I = 0.94× J = 0.0109 kg ·m2.

48

Bibliography

References

[1] K. Gunn, C. Stock-Williams, Quantifying the global wave power resource, Renewable Energy 44(2012) 296–304.

[2] U. A. Korde, J. Ringwood, Hydrodynamic control of wave energy devices, Cambridge UniversityPress, 2016.

[3] G. Bracco, E. Giorcelli, G. Mattiazzo, ISWEC: A gyroscopic mechanism for wave power exploitation,Mechanism and machine theory 46 (10) (2011) 1411–1424.

[4] A. Cagninei, M. Raffero, G. Bracco, E. Giorcelli, G. Mattiazzo, D. Poggi, Productivity analysis of thefull scale inertial sea wave energy converter prototype: A test case in Pantelleria Island, Journal ofRenewable and Sustainable Energy 7 (2015) 061703. doi:10.1063/1.4936343.

[5] G. Vissio, ISWEC toward the sea - Development, Optimization and Testing of the Device ControlArchitecture, Ph.D. thesis, Politecnico di Torino (2017).

[6] G. Bracco, E. Giorcelli, G. Mattiazzo, M. Pastorelli, J. Taylor, ISWEC: Design of a prototype modelwith gyroscope, IEEE, 2009, pp. 57–63.

[7] G. Bracco, E. Giorcelli, G. Mattiazzo, D. Poggi, J. Taylor, ISWEC: experimental tests on a small scaleprototype model, 2010.

[8] G. Bracco, E. Giorcelli, G. Mattiazzo, ISWEC: Design of a prototype model for wave tank test,American Society of Mechanical Engineers, 2010.

[9] G. Bracco, E. Giorcelli, G. Mattiazzo, M. Pastorelli, M. Raffero, Testing of a gyroscopic wave energysystem, IEEE, 2012.

[10] J. Medeiros, S. Brizzolara, Mathematical framework for hydromechanical time-domain simulation ofwave energy converters, Mathematical Problems in Engineering 2018.

[11] N. Faedo, Y. Pena-Sanchez, J. V. Ringwood, Moment-matching-based identification of wave energyconverters: the iswec device, IFAC-PapersOnLine 51 (29) (2018) 189–194.

[12] G. Bracco, E. Giorcelli, G. Giorgi, G. Mattiazzo, B. Passione, M. Raffero, G. Vissio, Performanceassessment of the full scale ISWEC system, in: 2015 IEEE International Conference on IndustrialTechnology (ICIT), IEEE, 2015, pp. 2499–2505.

[13] G. Bracco, M. Canale, V. Cerone, Energy harvesting optimization of an inertial sea wave energyconverter through model predictive control, IEEE, 2019, pp. 85–90.

[14] M. Raffero, M. Martini, B. Passione, G. Mattiazzo, E. Giorcelli, G. Bracco, Stochastic Control ofInertial Sea Wave Energy Convertor, The Scientific World Journal.

[15] G. Bracco, A. Cagninei, E. Giorcelli, G. Mattiazzo, D. Poggi, M. Raffero, Stochastic Control ofInertial Sea Wave Energy Convertor, Ocean Engineering 120 (2016) 40–51.

[16] K. Ruehl, C. Michelen, S. Kanner, M. Lawson, Y.-H. Yu, Preliminary verification and validation ofWEC-Sim, an open-source wave energy converter design tool, in: ASME 2014 33rd InternationalConference on Ocean, Offshore and Arctic Engineering, American Society of Mechanical EngineersDigital Collection.

[17] Y.-H. Yu, Y. Li, Reynolds-Averaged Navier–Stokes simulation of the heave performance of a two-bodyfloating-point absorber wave energy system, Computers & Fluids 73 (2013) 104–114.

49

http://dx.doi.org/10.1063/1.4936343

[18] M. Bergmann, G. Bracco, F. Gallizio, A. Giorcelli, E. Iollo, G. Mattiazzo, M. Ponzetta, A two-waycoupling CFD method to simulate the dynamics of a wave energy converter., in: OCEANS 2015 -Genova, Italy, IEEE, pp. 1–6.

[19] N. Nangia, N. A. Patankar, A. P. S. Bhalla, A DLM immersed boundary method based wave-structureinteraction solver for high density ratio multiphase flows, Journal of Computational Physics 398 (2019)108804.

[20] A. P. S. Bhalla, N. Nangia, P. Dafnakis, G. Bracco, G. Mattiazzo, Simulating water-entry/exitproblems using Eulerian-Lagrangian and fully-Eulerian fictitious domain methods within theopen-source IBAMR library, Applied Ocean Research 94 (2020) 101932.

[21] M. Raffero, Design of a Wave Energy Converter-A case of application: ISWEC, Ph.D. thesis,Politecnico di Torino (2014).

[22] J. M. J. Journee, W. W. Massie, Offshore Hydromechanics, Delft University of Technology, 2001.

[23] J. D. Fenton, A fifth-order stokes theory for steady waves, Journal of waterway, port, coastal, andocean engineering 111 (2) (1985) 216–234.

[24] J. D. Fenton, The numerical solution of steady water wave problems, Computers & Geosciences 14 (3)(1988) 357–368.

[25] A. M. Miquel, A. Kamath, M. Alagan Chella, R. Archetti, H. Bihs, Analysis of different methods forwave generation and absorption in a CFD-based numerical wave tank, Journal of Marine Science andEngineering 6 (2) (2018) 73.

[26] C. Windt, J. Davidson, J. V. Ringwood, High-fidelity numerical modelling of ocean wave energysystems: A review of computational fluid dynamics-based numerical wave tanks, Renewable andSustainable Energy Reviews 93 (2018) 610–630.

[27] C. Windt, J. Davidson, P. Schmitt, J. V. Ringwood, On the assessment of numerical wave makers inCFD simulations, Journal of Marine Science and Engineering 7 (2) (2019) 47.

[28] N. G. Jacobsen, D. R. Fuhrman, J. Fredsøe, A wave generation toolbox for the open-source cfd library:Openfoam R©, International Journal for Numerical Methods in Fluids 70 (9) (2012) 1073–1088.

[29] P. Higuera, J. L. Lara, I. J. Losada, Realistic wave generation and active wave absorption forNavier–Stokes models: Application to OpenFOAM R© , Coastal Engineering 71 (2013) 102–118.

[30] P. Frigaard, M. Brorsen, A time-domain method for separating incident and reflected irregular waves,Coastal Engineering 24 (3-4) (1995) 205–215.

[31] H. A. Schaffer, G. Klopman, Review of multidirectional active wave absorption methods, Journal ofwaterway, port, coastal, and ocean engineering 126 (2) (2000) 88–97.

[32] J. Choi, S. B. Yoon, Numerical simulations using momentum source wave-maker applied to RANSequation model, Coastal Engineering 56 (10) (2009) 1043–1060.

[33] T. Ha, P. Lin, Y.-S. Cho, Generation of 3D regular and irregular waves using Navier–Stokes equationsmodel with an internal wave maker, Coastal Engineering 76 (2013) 55–67.

[34] A. Ghasemi, A. Pathak, M. Raessi, Computational simulation of the interactions between moving rigidbodies and incompressible two-fluid flows, Computers & Fluids 94 (2014) 1–13.

[35] Z. Dong, J. Zhan, Comparison of existing methods for wave generating and absorbing in VOF-basednumerical tank, Journal of Hydrodynamics, Ser. A 24 (1) (2009) 15–21.

[36] N. G. Jacobsen, M. R. van Gent, G. Wolters, Numerical analysis of the interaction of irregular waveswith two dimensional permeable coastal structures, Coastal Engineering 102 (2015) 13–29.

50

[37] Z. Hu, W.-y. Tang, H.-x. Xue, X.-y. Zhang, Numerical wave tank based on a conserved wave-absorbingmethod, China Ocean Engineering 30 (1) (2016) 137–148.

[38] N. Nangia, B. E. Griffith, N. A. Patankar, A. P. S. Bhalla, A robust incompressible Navier-Stokessolver for high density ratio multiphase flows, Journal of Computational Physics 390 (2019) 548–594.

[39] P. Dafnakis, A. P. S. Bhalla, S. A. Sirigu, M. Bonfanti, G. Bracco, G. Mattiazzo, Comparison ofwave-structure interaction dynamics of a submerged cylindrical point absorber with three degrees offreedom using potential flow and CFD models, arXiv preprint arXiv:1908.04981.

[40] M. Cai, A. Nonaka, J. B. Bell, B. E. Griffith, A. Donev, Efficient variable-coefficient finite-volumestokes solvers, Communications in Computational Physics 16 (5) (2014) 1263–1297.

[41] B. E. Griffith, An accurate and efficient method for the incompressible Navier-Stokes equations usingthe projection method as a preconditioner, Journal of Computational Physics 228 (20) (2009)7565–7595.

[42] A. P. S. Bhalla, R. Bale, B. E. Griffith, N. A. Patankar, A unified mathematical framework and anadaptive numerical method for fluid–structure interaction with rigid, deforming, and elastic bodies,Journal of Computational Physics 250 (2013) 446–476.

[43] W. J. Rider, J. A. Greenough, J. R. Kamm, Accurate monotonicity-and extrema-preserving methodsthrough adaptive nonlinear hybridizations, Journal of Computational Physics 225 (2) (2007)1827–1848.

[44] M. Gazzola, O. V. Vasilyev, P. Koumoutsakos, Shape optimization for drag reduction in linked bodiesusing evolution strategies, Computers & Structures 89 (11-12) (2011) 1224–1231.

[45] N. M. Newmark, A method of computation for structural dynamics, Journal of the engineeringmechanics division 85 (3) (1959) 67–94.

[46] I. Robertson, L. Li, S. Sherwin, P. Bearman, A numerical study of rotational and transverse gallopingrectangular bodies, Journal of fluids and structures 17 (5) (2003) 681–699.

[47] J. Yang, F. Stern, A simple and efficient direct forcing immersed boundary framework forfluid–structure interactions, Journal of Computational Physics 231 (15) (2012) 5029–5061.

[48] J. Yang, F. Stern, A non-iterative direct forcing immersed boundary method for strongly-coupledfluid–solid interactions, Journal of Computational Physics 295 (2015) 779–804.

[49] E. M. Kolahdouz, A. P. Bhalla, B. A. Craven, B. E. Griffith, An Immersed Lagrangian-EulerianMethod for Fluid-Structure Interaction, arXiv preprint arXiv:2003.12046.

[50] IBAMR: An adaptive and distributed-memory parallel implementation of the immersed boundarymethod, https://github.com/IBAMR/IBAMR.

[51] R. D. Hornung, S. R. Kohn, Managing application complexity in the SAMRAI object-orientedframework, Concurrency Comput Pract Ex 14 (5) (2002) 347–368.

[52] SAMRAI: Structured Adaptive Mesh Refinement Application Infrastructure,http://www.llnl.gov/CASC/SAMRAI.

[53] S. Balay, W. D. Gropp, L. C. McInnes, B. F. Smith, Efficient management of parallelism in objectoriented numerical software libraries, in: E. Arge, A. M. Bruaset, H. P. Langtangen (Eds.), ModernSoftware Tools in Scientific Computing, Birkhauser Press, 1997, pp. 163–202.

[54] S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout,W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, S. Zampini,H. Zhang, PETSc users manual, Tech. Rep. ANL-95/11 - Revision 3.6, Argonne National Laboratory(2015).URL http://www.mcs.anl.gov/petsc

51

https://github.com/IBAMR/IBAMR

http://www.llnl.gov/CASC/SAMRAI

http://www.mcs.anl.gov/petsc


[55] S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout,W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, S. Zampini,H. Zhang, PETSc Web page, http://www.mcs.anl.gov/petsc (2015).URL http://www.mcs.anl.gov/petsc

52




Documents

arXiv:2005.06108v1 [physics.flu-dyn] 13 May 2020 · The inertial sea wave energy converter (ISWEC) technology: device-physics, multiphase modeling and simulations Kaustubh Khedkara,