# Numerical Study of an Oscillating-Wing Wingmill for Ocean Current Energy Harvesting: Fluid-Solid-Body Interaction with Feedback Control

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

#### The Feedback Loop

## 3. Numerical Simulations

#### 3.1. The Mesh

#### 3.2. Mesh Validation

#### 3.3. Control Scheme

#### 3.4. Limitations

## 4. Dimensionless Parameters

- The efficiency $\eta =\frac{\overline{P}/s}{1/2\rho {U}^{3}{h}_{0}}$
- The Reynolds number $Re=\frac{\rho Uc}{\mu}$
- Dimensionless heaving $\frac{{h}_{0}}{c}$
- Inertia 1 ${\Pi}_{1}=\frac{M}{\rho {c}^{3}}$
- Inertia 2 (rotational) ${\Pi}_{2}=\frac{J}{\rho {c}^{5}}$
- Dimensionless damping ratio ${\Pi}_{3}=\frac{b}{\rho {c}^{2}U}$
- Dimensionless gain ${\Pi}_{4}=\frac{K}{\rho {U}^{2}{c}^{3}}$
- Strouhal number $St=\frac{fc}{U}$
- Density ratio between the fluid and solid body $\frac{\rho}{{\rho}_{a}}$

#### Reynolds Number and Turbulence

## 5. Results

#### 5.1. Effects of the Closed-Loop Control

- A noisy high frequency oscillation can occur on the reference signal for ${\alpha}_{ref}$ shortly after the foil has changed direction. This is an effect of the control action; since the dynamic angle $\alpha $ depends on $\dot{y}$, the latter derivative amplifies small oscillations that are caused by the control torque. Even if the torque is applied at the center of mass, linear and angular motions are coupled by hydrodynamic forces, so a sudden torque may cause a small jump in $\dot{y}$. This effect is only observable when the heaving speed is close to zero (and it can be avoided while using a properly tuned PI controller; this added complexity is out of the scope of the present report).
- The control effort (energy spent in the control action) may be very important in the vicinity of $y={h}_{0}$. The net power could be substantially increased if this energy could be spared (following the sign change in the reference angle). Three possible ways of doing so would be: (a) a passive mechanism that turns the airfoil once the threshold $\pm {h}^{\prime}$ is reached, (b) instead of pitching of the whole airfoil, set an aileron near the trailing edge, so the pitching effort decreases substantially, (c) to saturate the controller output to limit the spent energy (this would have consequences on the response time), and (d) to implement a control strategy that can be proved to be optimal in the sense of net power extraction. Of course, some of these suggestions may be combined in order to increase the efficiency.
- The power curve shows a maximum before reaching the time where the new ${\alpha}_{ref}$ is assigned. This should not happen if the lift coefficient was that of a static polar for the airfoil (maximum). The power should not decay until the controller sets a new reference (${h}^{\prime}$ is reached). Even if the reference angle was chosen, such as to have maximum static lift, the actual lift is not parallel to the y axis, multiplying, by $cos(\Delta \theta )$, where $\Delta \theta =arctan(\dot{y}/U)$. This causes the lift to decrease whenever the heaving speed becomes comparable to U. When this happens, a plot of the factor $cos(\Delta \theta )$ versus power (not presented here for brevity) shows that the maximum power coincides with the cosine crest. This correlation is clear when $\dot{y}/U<1$, as $cos(\Delta \theta )\simeq 1-{(\dot{y}/U)}^{2}$ and the power that is extracted from the damping is proportional to ${\dot{y}}^{2}$. This is a very important consideration, because this is a limit to the velocity (and power) that can be extracted while using oscillating foils whose motion is constrained to the y axis.
- Dynamic stall: even though the reference angle was chosen to be below the maximum lift (5% smaller) in order to avoid stall, a more detailed inspection of the boundary layer separation and vortices detachment made clear that unsteady stall is taking place for some cases. Figure 6 shows the vorticity of the velocity field for three cases ($B=0.4$) at different times and ${h}_{0}/c$: (a) for ${h}_{0}/c=2.5$, $t=239\mathrm{s}$, the boundary layer remains attached to the foil, with the exception of a small perturbation (incipient eddy) forming near the leading edge lower surface. This eddy will eventually detach from the trailing edge. One can observe that a pair of counter-rotating vortices detached from the foil when it switched direction in the vicinity of $y=\pm {h}_{0}/c$; (b) for ${h}_{0}/c=1.25$, $t=239\mathrm{s}$ the boundary layer clearly stays attached to the foil, and the only remnant eddies are again forming a vortex pair, close to the turning point $y=\pm {h}_{0}/c$; however, in (c), there is dynamic stall (${h}_{0}/c=3.75$, $B=0.2$, $t=235$). The image shows the airfoil heaving downwards (halfway towards $-{h}_{0}$). There is an alternating sign vortex street left from the previous cycle. There is clear detachment of the boundary layer in the lower surface of the airfoil. The time t corresponds to the horizontal axis shown in Figure 5. The exact nature of the dynamic stall, in our case, is out of the scope of this work; however, the reader may refer to [47,53,54] for a comprehensive characterization of the phenomenon. Moreover, we observed that, whenever the controller overshoots (${\alpha}_{ref}$ crosses $\theta $), there is a detachment of eddies and oscillations in $\dot{y}$, so dynamic stall may, in some cases, be caused by the control action itself (if that is the case, the optimal control problem is further complicated).

#### 5.2. Effect of B on the Efficiency

## 6. Conclusions and Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Tables and Figures

Authors | Year | AR | Type | Foil | $\mathit{Re}$ | ${\mathit{\eta}}_{\mathit{max}}$ |
---|---|---|---|---|---|---|

Kinsey and Dumas | 2012 | 2D | Prescribed & tandem | NACA0015 | $5.0\times {10}^{5}$ | 0.63 |

Platzer et al. | 2010 | 2D | Fully Passive prescribed y tandem | NACA0014 | $2.0\times {10}^{4}$ | 0.54 |

Ashraf et al. | 2011 | 2D | Prescribed & tandem | NACA0014 | $2.0\times {10}^{4}$ | 0.54 |

Young et al. | 2013 | 2D | Fully passive | NACA0012 | $1.1\times {10}^{3}$–$1.1\times {10}^{6}$ | 0.41 |

Campobasso et al. | 2013 | 2D | Prescribed | NACA0015 | $1.1\times {10}^{3}$–$1.5\times {10}^{6}$ | 0.40 |

Le et al. | 2013 | 2D | Prescribed | Biomimetic | $9\times {10}^{4}$ | 0.39 |

Ashraf et al. | 2009 | 2D | Prescribed | NACA0012 | 1100 | 0.38 |

Shimizu et al. | 2008 | 2D | Semi-passive open-loop | NACA0012 | $4.62\times {10}^{5}$ | 0.35 |

This study | 2019 | Semi-passive closed-loop | NACA0015 | $2.0\times {10}^{5}$ | 0.12 |

Authors | Year | Type | Foil | $\mathit{Re}$ | ${\mathit{\eta}}_{\mathit{max}}$ |
---|---|---|---|---|---|

Kinsey et al. | 2011 | Fully passive | NACA0015 | 5 ×${10}^{5}$ | 0.4 |

Kinsey and Dumas | 2010 | Fully passive | NACA0015 | 5 ×${10}^{5}$ | 0.4 |

Simpson et al. | 2009 | Prescribed | NACA0012 | 1.38 ×${10}^{4}$ | 0.32 |

Huxham et al | 2012 | Semi-passive | NACA0015 | 4.5 ×${10}^{4}$ | 0.24 |

Lindsey, Jones et al. | 2003 | Fully passive | NACA0014 | 2.2 ×${10}^{4}$ | 0.23 |

McKinney and DeLaurier | 1981 | Fully passive | NACA0012 | 8.5 ×${10}^{4}$–1.1 ×${10}^{5}$ | 0.17 |

**Table A3.**Mesh parameters. The initial-final-refinement level indicates how refinement evolves on the different mesh parts. Larger values indicate more refinement. The first and second digits indicate the two levels of refinement closest to the foil surface. the last digit indicates how many levels of refinement exist in the refinement box (large square containing the foil in Figure 2. The test parameter was energy output averaged through a large number of cycles (more than 25).

Name | Number of Cells | Initial-Final-Refinement Level | Average Energy (J) |
---|---|---|---|

M1 | 18,040 | 3-4-1 | $5.65$ |

M2 | 47,440 | 3-4-3 | $5.649$ |

M3 | 290,326 | 4-4-1 | $5.654$ |

M4 | 291,286 | 4-4-3 | $5.654$ |

M5 | 322,606 | 4-5-3 | $5.655$ |

ID | $\mathit{\eta}\phantom{\rule{3.33333pt}{0ex}}[\%]$ | $\mathit{Re}$ | $\frac{{\mathit{h}}_{0}}{\mathit{c}}$ | ${\mathsf{\Pi}}_{1}$ | ${\mathsf{\Pi}}_{2}$ | ${\mathsf{\Pi}}_{3}$ | ${\mathsf{\Pi}}_{4}$ | $\mathit{St}$ | B |
---|---|---|---|---|---|---|---|---|---|

1 | 4.03 | $2\times {10}^{4}$ | 2.5 | 0.0937 | $5.12\times {10}^{-6}$ | 0.0437 | 0.25 | $8.32\times {10}^{-2}$ | 0.175 |

2 | 9.03 | $2\times {10}^{4}$ | 2.5 | 0.0937 | $5.12\times {10}^{-6}$ | 0.0625 | 0.25 | $7.36\times {10}^{-2}$ | 0.25 |

3 | 10.97 | $2\times {10}^{4}$ | 2.5 | 0.0937 | $5.12\times {10}^{-6}$ | 0.0937 | 0.25 | $7.04\times {10}^{-2}$ | 0.375 |

4 | 7.17 | $2\times {10}^{4}$ | 2.5 | 0.0937 | $5.12\times {10}^{-6}$ | 0.1250 | 0.25 | $4.48\times {10}^{-2}$ | 0.5 |

5 | 6.29 | $2\times {10}^{4}$ | 2.5 | 0.0937 | $5.12\times {10}^{-6}$ | 0.1562 | 0.25 | $3.84\times {10}^{-2}$ | 0.625 |

6 | 5.61 | $2\times {10}^{4}$ | 2.5 | 0.0937 | $5.12\times {10}^{-6}$ | 0.1875 | 0.25 | $3.52\times {10}^{-2}$ | 0.75 |

7 | 2.25 | $2\times {10}^{4}$ | 1.25 | 0.0937 | $5.12\times {10}^{-6}$ | 0.0437 | 0.25 | $11.84\times {10}^{-2}$ | 0.175 |

8 | 6.62 | $2\times {10}^{4}$ | 1.25 | 0.0937 | $5.12\times {10}^{-6}$ | 0.0625 | 0.25 | $9.76\times {10}^{-2}$ | 0.25 |

9 | 6.86 | $2\times {10}^{4}$ | 1.25 | 0.0937 | $5.12\times {10}^{-6}$ | 0.0937 | 0.25 | $9.28\times {10}^{-2}$ | 0.375 |

10 | 7.38 | $2\times {10}^{4}$ | 1.25 | 0.0937 | $5.12\times {10}^{-6}$ | 0.1250 | 0.25 | $8.64\times {10}^{-2}$ | 0.5 |

11 | 7.32 | $2\times {10}^{4}$ | 1.25 | 0.0937 | $5.12\times {10}^{-6}$ | 0.1562 | 0.25 | $7.92\times {10}^{-2}$ | 0.625 |

12 | 6.62 | $2\times {10}^{4}$ | 1.25 | 0.0937 | $5.12\times {10}^{-6}$ | 0.1875 | 0.25 | $7.68\times {10}^{-2}$ | 0.75 |

13 | 2.39 | $2\times {10}^{4}$ | 3.75 | 0.0937 | $5.12\times {10}^{-6}$ | 0.0250 | 0.25 | $6.97\times {10}^{-2}$ | 0.1 |

14 | 3.75 | $2\times {10}^{4}$ | 3.75 | 0.0937 | $5.12\times {10}^{-6}$ | 0.0437 | 0.25 | $6.15\times {10}^{-2}$ | 0.175 |

15 | 3.40 | $2\times {10}^{4}$ | 3.75 | 0.0937 | $5.12\times {10}^{-6}$ | 0.0625 | 0.25 | $4.64\times {10}^{-2}$ | 0.25 |

16 | 2.91 | $2\times {10}^{4}$ | 3.75 | 0.0937 | $5.12\times {10}^{-6}$ | 0.0937 | 0.25 | $3.36\times {10}^{-2}$ | 0.375 |

17 | 2.66 | $2\times {10}^{4}$ | 3.75 | 0.0937 | $5.12\times {10}^{-6}$ | 0.1250 | 0.25 | $2.88\times {10}^{-2}$ | 0.5 |

18 | 2.54 | $2\times {10}^{4}$ | 3.75 | 0.0937 | $5.12\times {10}^{-6}$ | 0.1562 | 0.25 | $2.56\times {10}^{-2}$ | 0.625 |

19 | 1.74 | $2\times {10}^{4}$ | 3.75 | 0.0937 | $5.12\times {10}^{-6}$ | 0.1875 | 0.25 | $1.92\times {10}^{-2}$ | 0.75 |

20 | 1.46 | $2\times {10}^{4}$ | 3.75 | 0.0937 | $5.12\times {10}^{-6}$ | 0.2187 | 0.25 | $1.60\times {10}^{-2}$ | 0.875 |

1 | 0.009 | $1\times {10}^{3}$ | 2.5 | 0.0937 | $9.8\times {10}^{-3}$ | 0.0437 | 0.25 | $7.84\times {10}^{-2}$ | 0.175 |

2 | 0.028 | $1\times {10}^{3}$ | 2.5 | 0.0937 | $9.8\times {10}^{-3}$ | 0.0625 | 0.25 | $8.0\times {10}^{-2}$ | 0.25 |

3 | 0.016 | $1\times {10}^{3}$ | 2.5 | 0.0937 | $9.8\times {10}^{-3}$ | 0.0937 | 0.25 | $5.2\times {10}^{-2}$ | 0.375 |

4 | 0.015 | $1\times {10}^{3}$ | 2.5 | 0.0937 | $9.8\times {10}^{-3}$ | 0.1250 | 0.25 | $3.6\times {10}^{-2}$ | 0.5 |

5 | 0.028 | $1\times {10}^{3}$ | 2.5 | 0.0937 | $9.8\times {10}^{-3}$ | 0.1562 | 0.25 | $2.56\times {10}^{-2}$ | 0.625 |

6 | 0.024 | $1\times {10}^{3}$ | 2.5 | 0.0937 | $9.8\times {10}^{-3}$ | 0.1875 | 0.25 | $2.4\times {10}^{-2}$ | 0.75 |

7 | -0.005 | $1\times {10}^{3}$ | 1.25 | 0.0937 | $9.8\times {10}^{-3}$ | 0.0437 | 0.25 | $10.66\times {10}^{-2}$ | 0.175 |

8 | 0.012 | $1\times {10}^{3}$ | 1.25 | 0.0937 | $9.8\times {10}^{-3}$ | 0.0625 | 0.25 | $9.44\times {10}^{-2}$ | 0.25 |

9 | 0.021 | $1\times {10}^{3}$ | 1.25 | 0.0937 | $9.8\times {10}^{-3}$ | 0.0937 | 0.25 | $8.96\times {10}^{-2}$ | 0.375 |

10 | 0.016 | $1\times {10}^{3}$ | 1.25 | 0.0937 | $9.8\times {10}^{-3}$ | 0.1250 | 0.25 | $11.2\times {10}^{-2}$ | 0.5 |

11 | 0.004 | $1\times {10}^{3}$ | 1.25 | 0.0937 | $9.8\times {10}^{-3}$ | 0.1562 | 0.25 | $4.8\times {10}^{-2}$ | 0.625 |

12 | 0.006 | $1\times {10}^{3}$ | 1.25 | 0.0937 | $9.8\times {10}^{-3}$ | 0.1875 | 0.25 | $4.64\times {10}^{-2}$ | 0.75 |

13 | 0.008 | $1\times {10}^{3}$ | 3.75 | 0.0937 | $9.8\times {10}^{-3}$ | 0.0250 | 0.25 | $6.88\times {10}^{-2}$ | 0.1 |

14 | 0.013 | $1\times {10}^{3}$ | 3.75 | 0.0937 | $9.8\times {10}^{-3}$ | 0.0437 | 0.25 | $5.44\times {10}^{-2}$ | 0.175 |

15 | 0.014 | $1\times {10}^{3}$ | 3.75 | 0.0937 | $9.8\times {10}^{-3}$ | 0.0625 | 0.25 | $3.52\times {10}^{-2}$ | 0.25 |

16 | 0.012 | $1\times {10}^{3}$ | 3.75 | 0.0937 | $9.8\times {10}^{-3}$ | 0.0937 | 0.25 | $3.2\times {10}^{-2}$ | 0.375 |

17 | 0.009 | $1\times {10}^{3}$ | 3.75 | 0.0937 | $9.8\times {10}^{-3}$ | 0.1250 | 0.25 | $1.76\times {10}^{-2}$ | 0.5 |

18 | 0.011 | $1\times {10}^{3}$ | 3.75 | 0.0937 | $9.8\times {10}^{-3}$ | 0.1562 | 0.25 | $1.92\times {10}^{-2}$ | 0.625 |

19 | 0.011 | $1\times {10}^{3}$ | 3.75 | 0.0937 | $9.8\times {10}^{-3}$ | 0.1875 | 0.25 | $1.6\times {10}^{-2}$ | 0.75 |

20 | 0.010 | $1\times {10}^{3}$ | 3.75 | 0.0937 | $9.8\times {10}^{-3}$ | 0.2187 | 0.25 | $1.44\times {10}^{-2}$ | 0.875 |

$\frac{{\mathit{h}}_{0}}{\mathit{c}}$ | Unused Energy | Net Harnessed Energy | Control Energy Consumption |
---|---|---|---|

2.5 | 83.8% | 10.6% | 5.7% |

1.25 | 87.3% | 6.9% | 5.8% |

3.75 | 95.9% | 3.2% | 0.9% |

**Figure A1.**System response for different ${h}_{0}/c$. (

**a**) ${h}_{0}/c=1.25$, (

**b**) ${h}_{0}/c=3.25$, as in Figure 5.

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**Figure 2.**Mesh configuration: to the center, the airfoil and the first two internal mesh regions, which are are zoomed-out in the box to the lower right. The circular mesh deformation area is also shown (deformable mesh). The domain boundaries are indicated in units of chord size. The deformable region is contained in a refinement region, which can contain different levels of refinement.

**Figure 3.**Variability of average energy production as a function of the number of cycles taken into account for averaging.

**Figure 4.**Close loop active control scheme. Dashed box groups the coupled modules that conform the equivalent to the plant of the system. $T\left(t\right)$ stands for torque.

**Figure 5.**System response behavior. ${h}_{0}/c=2.5$, $B=0.375$. From top to bottom: (

**a**) Reference tracking. (

**b**) Power output and energy consumption. (

**c**) Position of the rotation center of the foil. (

**d**) Lift Coefficient ${C}_{L}$.

**Figure 6.**Vorticity field, $B=0.4$ for different times: (

**a**) for ${h}_{0}/c=2.5$, $t=239\mathrm{s}$; (

**b**) ${h}_{0}/c=1.25$, $t=239\mathrm{s}$; (

**c**) ${h}_{0}/c=3.75$, $t=235$. Red to blue color scale represents positive to negative vorticity.

**Figure 7.**$Re=2\times {10}^{4}$. $\eta $ vs. B. Markers correspond to different values of ${h}_{0}/c$: circles: ${h}_{0}/c=1.25$; squares: ${h}_{0}/c=3.75$; asterisks: ${h}_{0}/c=2.5$. Gray markers correspond to same cases with turbulence model applied.

**Figure 9.**$\eta $ vs. B. Same as in Figure 7, if the “turn around” of the foil was made passively ($Re=2\times {10}^{4}$). The markers correspond to different values of ${h}_{0}/c$: circles: ${h}_{0}/c=1.25$; squares: ${h}_{0}/c=3.75$; and, asterisks: ${h}_{0}/c=2.5$.

**Figure 10.**$Re=1000$. $\eta $ vs. B. Markers correspond to different values of ${h}_{0}/c$: circles: ${h}_{0}/c=1.25$; squares: ${h}_{0}/c=3.75$; asterisks: ${h}_{0}/c=2.5$.

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**MDPI and ACS Style**

Balam-Tamayo, D.; Málaga, C.; Figueroa-Espinoza, B.
Numerical Study of an Oscillating-Wing Wingmill for Ocean Current Energy Harvesting: Fluid-Solid-Body Interaction with Feedback Control. *J. Mar. Sci. Eng.* **2021**, *9*, 23.
https://doi.org/10.3390/jmse9010023

**AMA Style**

Balam-Tamayo D, Málaga C, Figueroa-Espinoza B.
Numerical Study of an Oscillating-Wing Wingmill for Ocean Current Energy Harvesting: Fluid-Solid-Body Interaction with Feedback Control. *Journal of Marine Science and Engineering*. 2021; 9(1):23.
https://doi.org/10.3390/jmse9010023

**Chicago/Turabian Style**

Balam-Tamayo, David, Carlos Málaga, and Bernardo Figueroa-Espinoza.
2021. "Numerical Study of an Oscillating-Wing Wingmill for Ocean Current Energy Harvesting: Fluid-Solid-Body Interaction with Feedback Control" *Journal of Marine Science and Engineering* 9, no. 1: 23.
https://doi.org/10.3390/jmse9010023