# Evaluation of the Viscous Drag for a Domed Cylindrical Moored Wave Energy Converter

^{*}

## Abstract

**:**

## 1. Introduction and Background

## 2. Methodology

#### 2.1. Introduction

- –
- $\rho $ = fluid density,
- –
- U = maximum velocity,
- –
- D = relevant characteristic dimension of the rigid structure,
- –
- $\mu $ = dynamic viscosity of the fluid.

- –
- ${U}_{m}$ = amplitude of the sinusoidal velocity,
- –
- T = time period of the sinusoidal velocity.

- –
- force due to the inertia; an effect of the irrotational (potential) assumption, i.e., $\rho V{C}_{I}\ddot{X}$;
- –
- force due to the viscous drag; effect of the skin friction and flow separation, i.e., $\frac{1}{2}\rho {A}_{r}{C}_{d}\dot{X}\mid \dot{X}\mid $

$\rho $ | = fluid density, | ${A}_{r}$ | = relevant cross-sectional area, |

V | = volume of the structure, | ${C}_{d}$ | = drag coefficient, |

${C}_{I}$ | = inertia coefficient, | $\dot{X}$ | = fluid velocity, |

$\ddot{X}$ | = fluid acceleration, | U | = body velocity, |

$\dot{U}$ | = body acceleration. |

- (i).
- a submerged horizontal cylindrical part which has domed ends;
- (ii).
- two rectangular columns attached at the top surface of the domed cylinder.

#### 2.2. Frequency Domain Model And Results

$\omega $ | = angular frequency in [rad/s], | m | = mass of the structure, |

$A\left(\omega \right)$ | = frequency-dependent added mass, | $B\left(\omega \right)$ | = frequency-dependent wave damping, |

C | = linear viscous damping (if any), | ${k}_{h}$ | = hydrostatic stiffness, |

K | = additional stiffness (if any), | $Fe\left(\omega \right)$ | = frequency-dependent excitation force amplitude |

(complex variable). |

#### 2.3. Time Domain Model

- –
- ${\mu}_{\infty}$ = added mass at infinite frequency,
- –
- $\ddot{Z}\left(t\right)$ = acceleration of structure,
- –
- ${F}_{ex}\left(t\right)$ = wave excitation force,
- –
- ${F}_{d}\left(t\right)$ = non-linear viscous drag force.

- –
- ${\left({I}_{i}\right)}_{i=1,{N}_{prony}}$= complex velocity-dependent variables,
- –
- $\alpha $ = real part of the Prony coefficient,
- –
- $\beta $ = imaginary part of the Prony coefficient.

- –
- ${A}_{wave}$ = wave amplitude of ${j}^{th}$ frequency wave,
- –
- $Fe\left({\omega}_{j}\right)$ = frequency-dependent wave excitation force (complex variable),
- –
- ${\phi}_{j}$ = phase variable used to generate random wave field’
- –
- ${N}_{wave}$ = total number of wave frequencies.

#### 2.4. Evaluation of the Drag Coefficient (${C}_{d}$)

## 3. Results

#### 3.1. The Drag Coefficient (${C}_{d}$)

#### 3.2. Time Domain Model Results And Analysis

- –
- ${S}_{r}$ = Spectral density of the response,
- –
- ${S}_{w}$ = Spectral density of the wave signal.

#### 3.3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Perez, T. Ship Motion Control: Course Keeping and Roll Stabilisation Using Rudder and Fins; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Bhinder, M.A.; Babarit, A.; Gentaz, L.; Ferrant, P. Potential time domain model with viscous correction and CFD analysis of a generic surging floating wave energy converter. Int. J. Mar. Energy
**2015**, 10, 70–96. [Google Scholar] [CrossRef] - Penalba, M.; Giorgi, G.; Ringwood, J. A Review of Non-Linear Approaches for Wave Energy Converter Modelling. In Proceedings of the 11th European Wave and Tidal Energy Conference, Nantes, France, 6–11 September 2015. [Google Scholar]
- Palm, J.; Eskilsson, C.; Paredes, G.M.; Bergdahl, L. Coupled mooring analysis for floating wave energy converters using CFD: Formulation and validation. Int. J. Mar. Energy
**2016**, 16, 83–99. [Google Scholar] [CrossRef] - Thilleul, O.; Babarit, A.; Drouet, A.; Le Floch, S. Validation of CFD for the determination of damping coefficients for the use of Wave Energy Converters modelling. In Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering-OMAE, Nantes, France, 9–14 June 2013; Volume 9. [Google Scholar] [CrossRef]
- Bhinder, M.A.; Babarit, A.; Gentaz, L.; Ferrant, P. Assessment of Viscous Damping via 3D-CFD Modelling of a Floating Wave Energy Device. In Proceedings of the 9th European Wave and Tidal Energy Conference EWTEC, Southampton, UK, 5–9 September 2011. [Google Scholar]
- Bhinder, M.A.; Babarit, A.; Gentaz, L.; Ferrant, P. Effect of viscous forces on the performance of a surging wave energy converter. In Proceedings of the Twenty-Second International Offshore and Polar Engineering Conference, Rhodes, Greece, 17–22 June 2012. [Google Scholar]
- Pistidda, A.; Ottens, H.; Zoontjes, R. Using CFD to assess low frequency damping. In Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering-OMAE, Rio de Janeiro, Brazil, 1–6 July 2012; Volume 5, pp. 657–665. [Google Scholar]
- Jin, S.; Patton, R.J.; Guo, B. Viscosity effect on a point absorber wave energy converter hydrodynamics validated by simulation and experiment. Renew. Energy
**2018**, 129, 500–512. [Google Scholar] [CrossRef] - Ananthakrishnan, P. Viscosity and nonlinearity effects on the forces and waves generated by a floating twin hull under heave oscillation. Appl. Ocean Res.
**2015**, 51, 138–152. [Google Scholar] [CrossRef] - Nematbakhsh, A.; Michailides, C.; Gao, Z.; Moan, T. Comparison of Experimental Data of a Moored Multibody Wave Energy Device With a Hybrid CFD and BIEM Numerical Analysis Framework. In Proceedings of the 34th International Conference on Ocean, Offshore and Arctic Engineering, St. John’s, NF, Canada, 31 May–5 June 2015; Volume 9. [Google Scholar] [CrossRef]
- Palm, J.; Eskilsson, C.; Bergdahl, L.; Bensow, R.E. Assessment of Scale Effects, Viscous Forces and Induced Drag on a Point-Absorbing Wave Energy Converter by CFD Simulations. J. Mar. Sci. Eng.
**2018**, 6, 124. [Google Scholar] [CrossRef] - Davis, A.F.; Fabien, B.C. Systematic identification of drag coefficients for a heaving wave follower. Ocean Eng.
**2018**, 168. [Google Scholar] [CrossRef] - Zangeneh, R.; Thiagarajan, K.; Urbina, R.; Tian, Z. Viscous damping effects on heading stability of turret-moored ships. In Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering-OMAE, San Francisco, CA, USA, 7–12 June 2014; Volume 8A. [Google Scholar] [CrossRef]
- Sumer, B.; Fredsøe, J. Hydrodynamics around Cylindrical Strucures; World Scientific Pub Co Inc.: Hackensack, NJ, USA, 2006; Volume 26. [Google Scholar]
- Faltinsen, O. Sea Loads on Ships and Offshore Structures; Cambridge University Press: Cambridge, UK, 1993; Volume 1. [Google Scholar]
- Morison, J.; O’Brien, M.; Johnson, J.; Schaaf, S. The force exerted by surface waves on piles. J. Pet. Trans.
**1950**, 189, 149–157. [Google Scholar] [CrossRef] - Costello, R.; Padeletti, D.; Davidson, J.; Ringwood, J. Comparison of numerical simulations with experimental measurements for the response of a modified submerged horizontal cylinder moored in waves. In Proceedings of the ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering OMAE, San Francisco, CA, USA, 8–13 June 2014. [Google Scholar]
- Cummins, W. The Impulse Response Function and Ship Motions; Technical Report; DTIC Document; David Taylor Model Basin: Washington, DC, USA, 1962. [Google Scholar]
- Babarit, A.; Duclos, G.; Clément, A. Comparison of latching control strategies for a heaving wave energy device in random sea. Appl. Ocean Res.
**2004**, 26, 227–238. [Google Scholar] [CrossRef] - Sheng, W.; Alcorn, R.; Lewis, A. A new method for radiation forces for floating platforms in waves. Ocean Eng.
**2015**, 105, 43–53. [Google Scholar] [CrossRef] - Journee, J.; Massie, W. Offshore Hydromechanics; Delft University of Technology: Delft, The Netherlands, 2001. [Google Scholar]
- Bhinder, M.A. 3D Non-Linear Numerical Hydrodynamic Modelling of Floating Wave Energy Converters. Ph.D. Thesis, Ecole Centrale de Nantes, Nantes, France, 2013. [Google Scholar]

**Figure 1.**Geometry of the doomed cylinder body—(

**a**) schematic of the mooring setup and (

**b**) perspective outline view of the submerged surface of geometry with dimensions; here, D is the diameter, and the overall length of the cylindrical body is 0.8 m.

**Figure 3.**Evaluation of the drag coefficient ${C}_{d}$ from the curve fitting exercise. Here, Exp represents experimental results adapted from [18], with permission from ASME, 2014, and TD represents the time domain computational model with ${C}_{d}=0.5$.

**Figure 4.**Evaluation of the drag coefficient ${C}_{d}$ from the curve fitting exercise for surge mode. Here, Exp represents experiment data adapted from [18], with permission from ASME, 2014, and TD represent the time domain computational model with ${C}_{d}=2.4$.

**Figure 8.**Spectral density of heave response (root mean squared error (RMSE) of computed and experimental results = 3.9 × 10${}^{-5}$).

**Figure 9.**Spectral density of surge response (RMSE of computed and experiment results = 2.3 × 10${}^{-5}$).

**Figure 10.**Heave response amplitude operator (RAO) (from the frequency domain) alongside the normalised response from the time domain results of the panchromatic waves ($R{N}_{input\phantom{\rule{3.33333pt}{0ex}}wave-TD}$). Experimental measurements for the regular waves of amplitude 12 mm, 25 mm, and 75 mm are also shown.

**Figure 11.**Surge RAO (fromthe frequency domain) alongside the normalised response from time domain results of the panchromatic waves ($R{N}_{input\phantom{\rule{3.33333pt}{0ex}}wave-TD}$). Experimental measurements for the regular waves of amplitude 12 mm, 25 mm, and 75 mm are also shown.

**Table 1.**Maximum and minimum values of the Keulegan–Carpenter (KC) number for the heave and the surge modes corresponding to experiment results for the regular wave of amplitude 75mm—corresponding Re number is also shown.

Heave | |||||

Period | $RA{O}_{n}$ | $a=RA{O}_{n}\times {A}_{w}$ | Um | $KC=2\pi a/D$ | $Re=\rho UD/\mu $ |

2.29595 | 0.16901025 | 2.25347 | 0.462519968 | 7.01 | 4.63 × 10${}^{5}$ |

0.89581 | 0.01740397 | 0.23205 | 0.122070975 | 0.72 | 1.22 × 10${}^{5}$ |

Surge | |||||

Period | $RA{O}_{n}$ | $a=RA{O}_{n}\times {A}_{w}$ | Um | $KC=2\pi a/D$ | $Re=\rho UD/\mu $ |

3.7 | 2.84164 | 0.21312 | 0.36303 | 8.84 | 5.50 × 10${}^{4}$ |

0.90 | 0.66324 | 0.04974 | 0.34634 | 2.06 | 5.24 × 10${}^{4}$ |

Parameter | Symbol & Units | Value |
---|---|---|

Heave mode | ||

Total stiffness | ${K}_{s}$ [N/m] | 369.5 |

Estimated drag coefficient. | ${C}_{d}$ [-] | 0.5 |

Mass | m [kg] | 8.9 |

Mass of clump | ${m}_{c}$ [kg] | 19.75 |

Surge mode | ||

Total stiffness | ${K}_{s}$ [N/m] | 132 |

Estimated drag coefficient. | ${C}_{d}$ [-] | 2.4 |

Mass | m [kg] | 8.9 |

Simulation—Time Domain Model | RMSE |
---|---|

From time series | |

Surge (time 0 s–512 s) | 0.016 |

Surge (time 256 s–512 s) | 0.017 |

Heave(time 0 s–512 s) | 0.010 |

Heave (time 256 s–512 s) | 0.012 |

From spectral densities | |

Surge (for time series of 256 s–512 s) | 2.3 × 10${}^{-5}$ |

Heave (for time series of 256 s–512 s) | 3.9 × 10${}^{-5}$ |

Response | Correlation Coefficient (r) |
---|---|

Surge | 0.92 |

Heave | 0.91 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bhinder, M.A.; Murphy, J. Evaluation of the Viscous Drag for a Domed Cylindrical Moored Wave Energy Converter. *J. Mar. Sci. Eng.* **2019**, *7*, 120.
https://doi.org/10.3390/jmse7040120

**AMA Style**

Bhinder MA, Murphy J. Evaluation of the Viscous Drag for a Domed Cylindrical Moored Wave Energy Converter. *Journal of Marine Science and Engineering*. 2019; 7(4):120.
https://doi.org/10.3390/jmse7040120

**Chicago/Turabian Style**

Bhinder, Majid A, and Jimmy Murphy. 2019. "Evaluation of the Viscous Drag for a Domed Cylindrical Moored Wave Energy Converter" *Journal of Marine Science and Engineering* 7, no. 4: 120.
https://doi.org/10.3390/jmse7040120