# Comparing Models of Lateral Station-Keeping for Pitching Hydrofoils

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Water Channel Setup

#### 2.1.1. Tethered Hydrofoil Tests

#### 2.1.2. Maneuvering Experiments

## 3. Water Channel Results

## 4. Theoretical Modelling

#### 4.1. Theodorsen Model

#### 4.2. Theodorsen with Finite Aspect Ratio Corrections

#### 4.3. Theodorsen with Vectored Garrick

#### 4.4. Semiempirical Model

#### 4.5. Testing the Theoretical Models

#### 4.5.1. High Frequency Predictions

#### 4.5.2. Low Frequency Predictions

#### 4.5.3. Summary of Model Testing

- 2D Theodorsen No Wake
- High frequency solution is from Equation (24) with $F=1$ and $G=0$, and 2D coefficients ${C}_{\mathrm{am}}$ and ${C}_{\mathrm{circ}}$ are used.

- 3D Theodorsen No Wake
- High frequency solution is the same as 2D Theodorsen No Wake, except the 3D coefficients ${C}_{\mathrm{am}}^{\prime}$ and ${C}_{\mathrm{circ}}^{\prime}$ are used.
- Low frequency solution is the same as 2D Theodorsen No Wake, except the 3D coefficients ${C}_{\mathrm{am}}^{\prime}$ and ${C}_{\mathrm{circ}}^{\prime}$ are used.

- 3D Theodorsen Wake
- High frequency solution is from Equation (24), where the 3D coefficients ${C}_{\mathrm{am}}^{\prime}$ and ${C}_{\mathrm{circ}}^{\prime}$ are used.

- Vectored Garrick
- High frequency solution is the same as 3D Theodorsen Wake.
- Low frequency solution is from Equation (21) using the same ${C}_{L,\mathrm{Theo}}$ as the 3D Theodorsen Wake model.

- Semiempirical
- High frequency solution is the same as 3D Theodorsen Wake.
- Low frequency solution from Equation (23) using the same ${C}_{L,\mathrm{Theo}}$ as the 3D Theodorsen Wake model.

## 5. Model Testing and Discussion

#### 5.1. High Frequency Model Predictions

#### 5.2. Low Frequency Model Predictions

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Coordinate system used by Sedov. The plate is moving through the water to the left at a speed u. The y direction represents the lateral direction in the water channel reference frame, and the x direction represents the streamwise direction in the water channel reference frame. A negative streamwise force ${F}_{x}$ would represent thrust. The directions ${\overrightarrow{\mathrm{e}}}_{\Vert}$ and ${\overrightarrow{\mathrm{e}}}_{\perp}$ are fixed to the plate and represent the tangential and normal directions, respectively.

## Appendix B

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**Figure 1.**A pitch-biased hydrofoil suspended from air bushings performed lateral maneuvers in a water channel. (

**A**) Channel: Rolling Hills 1520; test section: 1.52 m long, 0.38 m wide, and 0.45 m deep. Compressed air at 4500 psi. (

**B**) Carriage is free to move laterally ($\pm y$). The hydrofoil is driven with pitch oscillations ($\theta ={\theta}_{\mathrm{bias}}\left(t\right)+\alpha \phantom{\rule{0.166667em}{0ex}}sin\left(2\mathsf{\pi}ft\right)$), leading to passive surging (amplitude ${h}_{0}$). (

**C**–

**E**) The hydrofoil’s response during a 6 cm lateral maneuver was decoupled into its low ($<f$) and high ($>f$) frequency components using a moving average filter.

**Figure 2.**Lateral and streamwise forces deviate from linear theory at high Strouhal numbers and pitch bias angles. (

**A**) The time-averaged force perpendicular to the incoming flow (lift) increases linearly with pitch bias (${\theta}_{\mathrm{bias}}$) with a slope that increases with Strouhal number. Colored circles show experimental data; solid lines show linear fits (average ${R}^{2}$ = 0.998). The Vectored Garrick model (dashed lines; Section 4.3) predicts the linear trend but underpredicts the slope. At $St=0$, the experimental data, the Vectored Garrick model, and the 3D Theodorsen model overlap and so cannot be distinguished on the plot. (

**B**) The time-averaged force parallel to the incoming flow (thrust) decreases with pitch bias. Colored circles show experimental data; solid lines show quadratic fits (average ${R}^{2}$ = 0.994). The inviscid Garrick model always predicts positive thrust and a small decrease with pitch bias of only 3% over 15 degrees of pitch bias. At $St=0$, the Garrick model is equivalent to the 2D and 3D Theodorsen model.

**Figure 3.**The response in lateral position (y) depends on Strouhal number. (

**A**) The lowpass-filtered lateral position settles into a new equilibrium after around 5 s. The overshoots and settling times (see Figure 1D) are larger for lower Strouhal numbers. Inset: average, dark lines; average $\pm \sigma $ ($n=5$), shaded bands. (

**B**) The high-pass filtered lateral position oscillates at the pitching frequency. The phases and amplitudes of oscillation (see Figure 1E) decrease with higher Strouhal numbers.

**Figure 4.**Predictive power varies between models and changes with Strouhal number within each model. Positive % Error implies a metric was overpredicted by the model; negative % Error implies an underprediction. Markers are sometimes shifted left/right to render them distinguishable; all data were taken at exactly $St$ = 0, 0.1, 0.2, 0.3, or 0.4. Errors were calculated based on the high frequency content of the lateral position response (amplitude,

**A**; phase,

**B**) and the low frequency content of the lateral position response (overshoot,

**C**; settling time,

**D**). High frequency predictions for the Vectored Garrick and Semiempirical models are the same as the 3D Wake model.

**Figure 5.**Predictions of lateral position vary between models. The case shown ($St$ = 0.3) demonstrates typical variation between the models. The models which include the wake and 3D corrections (3D, Wake,

**C**; Vectored Garrick,

**D**; Semiempirical,

**E**) capture the high frequency heaving motion better than the models without both the wake and 3D corrections (2D, No Wake,

**A**; 3D, No Wake,

**B**). Only the semiempirical model accurately predicts both the low and high frequency components.

a | Tip-to-tip pitching amplitude | F | Real part of the Theodorsen function |
---|---|---|---|

$\tilde{A},\tilde{B},\tilde{C},\tilde{D}$ | Variables for the State-space Theodorsen | ${F}_{L}$ | Force perpendicular to incoming flow |

${A}_{1},{A}_{2},\dots $ | Notation for placeholders | ${F}_{T}$ | Force parallel to incoming flow |

Aspect ratio of the hydrofoil | ${F}_{y}$ | Lateral force | |

c | Chord length of the hydrofoil | G | Imaginary part of the Theodorsen function |

${C}^{*}\left(k\right)$ | Theodorsen lift deficiency function | ${h}_{0}$ | Heaving amplitude of the hydrofoil |

${C}_{\mathrm{am}}$ | Added mass coefficient | ${J}_{1},{J}_{2},{Y}_{1},{Y}_{2}$ | Bessel functions of the first and second kind |

${C}_{\mathrm{am}}^{\prime}$ | Finite added mass coefficient | k | Reduced frequency of the pitching motion |

${C}_{\mathrm{circ}}$ | Circulatory force coefficient | ${K}_{p}$ | Proportional controller gain |

${C}_{\mathrm{circ}}^{\prime}$ | Finite circulatory force coefficient | m | Mass of the airfoil and attached rig |

${C}_{L}$ | Lift coefficient | s | Span of the airfoil |

${C}_{L,\mathrm{am}}$ | Added mass component of Theodorsen lift | $St$ | Strouhal number of the pitching motion |

${C}_{L,\mathrm{circ}}$ | Circulatory component of Theodorsen lift | u | Incoming flow speed |

${C}_{L,\mathrm{exp}}$ | Empirical lift coefficient | $\overrightarrow{x}$ | State-space Theodorsen wake downwash |

${C}_{L,\mathrm{Theo}}$ | Theodorsen model lift coefficient | y | Lateral position of the airfoil |

${C}_{T}$ | Thrust coefficient | ${y}^{*}$ | Complex heaving motion for Theodorsen |

${C}_{T,\mathrm{exp}}$ | Empirical thrust coefficient | $\alpha $ | Angular amplitude of flapping motion |

${C}_{T,\mathrm{Gar}}$ | Garrick model thrust coefficient | $\theta $ | Hydrofoil angle relative to water channel |

${C}_{y}$ | Lateral force coefficient | ${\theta}^{*}$ | Complex pitching motion for Theodorsen |

${C}_{y,\mathrm{exp}}$ | Empirical lateral force coefficient | ${\theta}_{\mathrm{bias},\mathrm{eff}}$ | Pitch bias relative to incoming flow |

${C}_{y,\mathrm{Gar}}$ | Garrick model lateral force coefficient | ${\theta}_{\mathrm{bias}}$ | Hydrofoil pitch bias |

${C}_{y,\mathrm{semiemp}}$ | Semiempirical lateral force coefficient | $\rho $ | Density of water |

${C}_{y,\mathrm{Theo}}$ | Theodorsen model lateral force coefficient | $\varphi $ | Phase of the heaving relative to pitching |

f | Flapping frequency (Hz) | ${\widehat{\omega}}_{\mathrm{LE}}$ | Dimensionless leading edge vorticity |

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## Share and Cite

**MDPI and ACS Style**

Gunnarson, P.; Zhong, Q.; Quinn, D.B.
Comparing Models of Lateral Station-Keeping for Pitching Hydrofoils. *Biomimetics* **2019**, *4*, 51.
https://doi.org/10.3390/biomimetics4030051

**AMA Style**

Gunnarson P, Zhong Q, Quinn DB.
Comparing Models of Lateral Station-Keeping for Pitching Hydrofoils. *Biomimetics*. 2019; 4(3):51.
https://doi.org/10.3390/biomimetics4030051

**Chicago/Turabian Style**

Gunnarson, Peter, Qiang Zhong, and Daniel B. Quinn.
2019. "Comparing Models of Lateral Station-Keeping for Pitching Hydrofoils" *Biomimetics* 4, no. 3: 51.
https://doi.org/10.3390/biomimetics4030051