# 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

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**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