# Unsteady Lift Prediction with a Higher-Order Potential Flow Method

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Background on Unsteady Lift Prediction

## 2. Method

## 3. Results

#### 3.1. Panel and Time-Step Size Constraint

#### 3.2. Sharp-Edged Gust Verification

#### 3.3. Sinusoidal Gust Verification

#### 3.4. Finite Wing Comparison

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$A,B,C$ | Circulation coefficients |

c | Wing chord |

${c}_{l}$ | Two-dimensional lift coefficient |

${C}_{L}$ | Three-dimensional lift coefficient |

$C\left(k\right)$ | Theodorsen function |

D | Propeller diameter |

$DVE$ | Distributed vorticity element |

$HOFW$ | Higher-order fixed-wake |

k | Reduced frequency |

m | Number of lifting lines |

n | Number of spanwise panels |

q | Freestream dynamic pressure |

s | Semi-chord of wing or airfoil ($c/2$) |

$S\left(k\right)$ | Sears function |

$\hat{s}$ | Unit vector aligned with lifting line |

$SDVE$ | Surface DVE |

u,v,w | Components of the velocity in the global reference frame |

$\overline{V}$ | Velocity vector |

${\overline{V}}_{\infty}$ | Freestream velocity vector |

${w}_{0}$ | Magnitude of the vertical gust |

$\Delta {x}_{w}/\Delta {x}_{c}$ | Ratio of the distance traversed by a wing in a time-step to the length of a surface DVE |

$\gamma $ | Vorticity |

$\Gamma $ | Circulation |

$\zeta $ | Coordinate in DVE reference frame orthogonal to $\xi $ and $\eta $ |

$\eta $ | Spanwise coordinate in the DVE reference frame |

$\xi $ | Streamwise coordinate in the DVE reference frame |

$\rho $ | Air density |

$\omega $ | Frequency of oscillations in unsteady flow |

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Sample Availability: The potential flow method is available from the authors. |

**Figure 1.**A visualization of a distributed vorticity element (DVE) [1].

**Figure 2.**Higher-order fixed-wake response to a sharp-edged gust as compared with the Küssner function.

**Figure 3.**Comparison of the unsteady lift response as a function of non-dimensional time due to a sinusoidal vertical gust as predicted using the unsteady Kutta–Joukowski approach with specified terms and discretization methods.

**Figure 4.**Comparison of the unsteady lift response due to a sinusoidal vertical gust as calculated with the unsteady Kutta–Joukowski approach.

**Figure 5.**Comparison of predictions made with the current approach (abbreviated as HOFW for higher-order fixed-wake) and two other methods for the finite wing unsteady lift response to 1-cosine gust.

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## Share and Cite

**MDPI and ACS Style**

Cole, J.A.; Maughmer, M.D.; Bramesfeld, G.; Melville, M.; Kinzel, M. Unsteady Lift Prediction with a Higher-Order Potential Flow Method. *Aerospace* **2020**, *7*, 60.
https://doi.org/10.3390/aerospace7050060

**AMA Style**

Cole JA, Maughmer MD, Bramesfeld G, Melville M, Kinzel M. Unsteady Lift Prediction with a Higher-Order Potential Flow Method. *Aerospace*. 2020; 7(5):60.
https://doi.org/10.3390/aerospace7050060

**Chicago/Turabian Style**

Cole, Julia A., Mark D. Maughmer, Goetz Bramesfeld, Michael Melville, and Michael Kinzel. 2020. "Unsteady Lift Prediction with a Higher-Order Potential Flow Method" *Aerospace* 7, no. 5: 60.
https://doi.org/10.3390/aerospace7050060