# Numerical Analysis of Leading-Edge Vortex Effect on Tidal Current Energy Extraction Performance for Chord-Wise Deformable Oscillating Hydrofoil

^{*}

## Abstract

**:**

## 1. Introduction

_{c}= 0, 0.05c, 0.1c, 0.15c). Then, the vortex that attached to the hydrofoil’s surface is quantified, and the distribution of the pressure coefficient along the hydrofoil’s surface is analyzed in detail to better reveal the effect of the chord-wise flexure on the energy extraction performance.

## 2. Hydrofoil Motion and Deformation Equation

_{0}and θ

_{0}are the amplitude of the oscillating hydrofoil heave and pitch motion separately, f is the oscillating frequency, φ is the phase angle difference between the heave and pitch motion, and φ is set here as 90°.

_{h}(t) and the pitching angular velocity γ(t) of the oscillating hydrofoil obtained by the derivate Equations (1) and (2) are expressed as:

_{c}is the starting point of hydrofoil deformation. Since the hydrofoil starts to deform from the pitching axis, and the pitching axis is located at c/3 from the leading-edge of the hydrofoil, the final deformable equation is as follows:

## 3. Numerical Calculation Method

_{∞}is set to 1.8 m/s. Besides, user-defined functions (UDFs) are used to control the motion of the hydrofoil.

## 4. Energy Extraction Parameter

_{y}(t) and P

_{θ}(t) denote the energy extraction power from the heave motion and pitch motion respectively, while the F

_{y}(t) and M(t) represent the force component in heave direction and the moment about the pitch axis. The average power during an oscillating cycle is as follows:

_{l}(t) and C

_{m}(t) are the instantaneous lift coefficient and momentum coefficient, respectively. They are defined as:

_{1}is the overall vertical extent of the hydrofoil motion, as seen in Figure 1.

_{1}for the two-dimensional simulation of the oscillating hydrofoil.

## 5. Validation of the Numerical Results

## 6. Results and Discussion

#### 6.1. Quantification of the Attached Vortex and Its Relationship with Hydrofoil Pressure Distribution

_{L}is the lift force on the hydrofoil, $\rho $ is the fluid density, U is the fluid velocity, and $\Gamma $ is the vortex circulation.

#### 6.2. The Effect of Chord-Wise Flexure on Hydrofoil Lift

#### 6.3. The Effect of Chord-Wise Flexure on Hydrofoil Energy Extraction

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Comparison of lift, momentum and power coefficient in the same cycle at f* = 0.14 (the solid red lines show simulation results with the Spalart-Allmaras (SA) model and the solid black lines show results from Reference [8]). (

**a**) Lift coefficient. (

**b**) Momentum coefficient. (

**c**) Power coefficient.

**Figure 6.**(

**a**) Contours of vortices with streamlines and pressure of flexible hydrofoil (n = 5, δc = 0.1c, f* = 0.13) with θ = 45°. (

**b**) The curve diagram of −Γ/Uc change with pitch angle in the case n = 5, δc = 0.1c and f* = 0.13.

**Figure 7.**Contours of vortices with streamlines and pressure of flexible hydrofoil (n = 5, δ

_{c}= 0.1c, f* = 0.13) in the half-cycle. (

**a**) θ = −72°, (

**b**) θ = −65°, (

**c**) θ = −56°, (

**d**) θ = −45°, (

**e**) θ = −31°, (

**f**) θ = −16°, (

**g**) θ = 16°, (

**h**) θ = 31°, (

**i**) θ = 45°, (

**j**) θ = 56°, (

**k**) θ = 65°, (

**l**) θ = 72°.

**Figure 8.**Distribution of the pressure coefficient along the flexible hydrofoil surfaces (n = 5, δ

_{c}= 0.1c, f* = 0.13) in the half-cycle. (

**a**) θ = −72°, (

**b**) θ = −65°, (

**c**) θ = −56°, (

**d**) θ = −45°, (

**e**) θ = −31°, (

**f**) θ = −16°, (

**g**) θ = 16°, (

**h**) θ = 31°, (

**i**) θ = 45°, (

**j**) θ = 56°, (

**k**) θ = 65°, (

**l**) θ = 72°.

**Figure 9.**Contours of vortices with streamlines and pressure (Pa) of flexible hydrofoil in the case δ

_{c}= 0.05c and f* = 0.13 (the thicker the blue, the stronger the vortex). (

**a**) rigid, θ = −45°, (

**b**) rigid, θ = −16°, (

**c**) rigid, θ = 16°, (

**d**) rigid, θ = 31°, (

**e**) n = 2, θ = −45°, (

**f**) n = 2, θ = −16°, (

**g**) n = 2, θ = 16°, (

**h**) n = 2, θ = 31°, (

**i**) n = 5, θ = −45°, (

**j**) n = 5, θ = −16°, (

**k**) n = 5, θ = 16°, (

**l**) n = 5, θ = 31°, (

**m**) n = 10, θ = −45°, (

**n**) n = 10, θ = −16°, (

**o**) n = 10, θ = 16°, (

**p**) n = 10, θ = 31°.

**Figure 10.**Contours of vortices with streamlines and pressure (Pa) of flexible hydrofoil in the case n = 5 and f *= 0.13 (the thicker the blue, the stronger the vortex). (

**a**) rigid, θ = −45°, (

**b**) rigid, θ = −16°, (

**c**) rigid, θ = 16°, (

**d**) rigid, θ = 31°, (

**e**) δ

_{c}= 0.05c, θ = −45°, (

**f**) δ

_{c}= 0.05c, θ = −16°, (

**g**) δ

_{c}= 0.05c, θ = 16°, (

**h**) δ

_{c}= 0.05c, θ = 31°, (

**i**) δ

_{c}= 0.1c, θ = −45°, (

**j**) δ

_{c}= 0.1c, θ = −16°, (

**k**) δ

_{c}= 0.1c, θ = 16°, (

**l**) δ

_{c}= 0.1c, θ = 31°, (

**m**) δ

_{c}= 0.15c, θ = −45°, (

**n**) δ

_{c}= 0.15c, θ = −16°, (

**o**) δ

_{c}= 0.15c, θ = 16°, (

**p**) δ

_{c}= 0.15c, θ = 31°.

**Figure 11.**(

**a**,

**b**) Comparison of $-\Gamma /{U}_{c}$ with different pitch angles at f* = 0.13. (

**c**,

**d**) Comparison of the pressure coefficient of the hydrofoil surface with $\theta $ = 0 at f* = 0.13. (

**e**,

**f**) Comparison of lift coefficient with different chord-wise flexible hydrofoils at f* = 0.13. Here, (

**a**,

**c**,

**e**) Fixed δ

_{c}= 0.05c, (

**b**,

**d**,

**f**) Fixed n = 5.

**Figure 12.**Comparison of average power coefficient among different chord-wise flexible hydrofoil under different reduced frequencies. (

**a**) Fixed δ

_{c}= 0.05c. (

**b**) Fixed n = 5.

Hydrofoil | ${\mathit{\chi}}_{\mathit{p}}$ | $\mathit{\nu}\hspace{0.17em}{(\mathbf{m}}^{2}/\mathbf{s})$ | $\mathit{\varphi}\hspace{0.17em}(\mathbf{rad})$ | c (m) | ${\mathit{U}}_{\mathit{\infty}}\hspace{0.17em}(\mathbf{m}/\mathbf{s})$ | ${\mathit{h}}_{0}$ | Re | θ_{o} | n | δ_{c} |
---|---|---|---|---|---|---|---|---|---|---|

NACA0015 | c/3 | 10^{−6} | 0.5π | 0.22 | 1.8 | c | 500,000 | 72^{0} | 1, 2, 5, 10 | 0, 0.05c, 0.1c, 0.15c |

Hydrofoil | ${\mathit{\chi}}_{\mathit{p}}$ | $\mathit{\nu}\hspace{0.17em}{(\mathbf{m}}^{2}/\mathbf{s})$ | $\mathit{\varphi}\hspace{0.17em}(\mathbf{rad})$ | c (m) | ${\mathit{U}}_{\mathit{\infty}}\hspace{0.17em}(\mathbf{m}/\mathbf{s})$ | f* | ${\mathit{h}}_{0}$ | Re | θ_{o} |
---|---|---|---|---|---|---|---|---|---|

NACA0015 | c/3 | 10^{−6} | 0.5π | 0.25 | 2.0 | 0.14 | c | 500,000 | 75^{0} |

**Table 3.**Average power coefficient of hydrofoil under different flexibility coefficients and trailing edge offsets.

Hydrofoil | Total Average Power Coefficient | Average Power Coefficient of the Heave Motion | Average Power Coefficient of the Pitch Motion |
---|---|---|---|

Rigid hydrofoil | 0.8731 | 0.9744 | −0.1013 |

Flexibility coefficient n = 2 | 0.9743 | 1.0947 | −0.1204 |

Flexibility coefficient n = 5 | 0.9972 | 1.1304 | −0.1332 |

Flexibility coefficient n = 10 | 1.0168 | 1.1547 | −0.1379 |

Maximum offset ${\delta}_{c}$ = 0.05c | 0.9972 | 1.1304 | −0.1332 |

Maximum offset ${\delta}_{c}$ = 0.1c | 1.0808 | 1.2345 | −0.1537 |

Maximum offset ${\delta}_{c}$ = 0.15c | 1.1441 | 1.3258 | −0.1817 |

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**MDPI and ACS Style**

Xu, J.; Zhu, H.; Guan, D.; Zhan, Y.
Numerical Analysis of Leading-Edge Vortex Effect on Tidal Current Energy Extraction Performance for Chord-Wise Deformable Oscillating Hydrofoil. *J. Mar. Sci. Eng.* **2019**, *7*, 398.
https://doi.org/10.3390/jmse7110398

**AMA Style**

Xu J, Zhu H, Guan D, Zhan Y.
Numerical Analysis of Leading-Edge Vortex Effect on Tidal Current Energy Extraction Performance for Chord-Wise Deformable Oscillating Hydrofoil. *Journal of Marine Science and Engineering*. 2019; 7(11):398.
https://doi.org/10.3390/jmse7110398

**Chicago/Turabian Style**

Xu, Jianan, Haiyang Zhu, Daitao Guan, and Yong Zhan.
2019. "Numerical Analysis of Leading-Edge Vortex Effect on Tidal Current Energy Extraction Performance for Chord-Wise Deformable Oscillating Hydrofoil" *Journal of Marine Science and Engineering* 7, no. 11: 398.
https://doi.org/10.3390/jmse7110398