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**Unsteady lifting-line theory as a singular-perturbation problem.**
*(English)*
Zbl 0589.76027

Summary: Unsteady lifting-line theory is developed for a wing of large aspect ratio oscillating at low frequency in inviscid incompressible flow. The wing is assumed to have a rigid chord but a flexible span. Use of the method of matched asymptotic expansions reduces the problem from a singular integral equation to quadrature. The pressure field and airloads, for a prescribed wing shape and motion, are obtained in closed form as expansions in inverse aspect ratio. A rigorous definition of unsteady induced downwash is also obtained. Numerical calculations are presented for an elliptic wing in pitch and heave; compared with numerical lifting-surface theory, computation time is reduced significantly. The present work also identifies and resolves errors in the unsteady lifting-line theory of E. C. James [ibid. 70, 753-771 (1975; Zbl 0363.76006)], and points out a limitation in that of T. Van Holten [e.g.: ibid. 77, 561-579 (1976; Zbl 0338.76009)].

### MSC:

76B10 | Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing |

76B25 | Solitary waves for incompressible inviscid fluids |

76M99 | Basic methods in fluid mechanics |

### Keywords:

Unsteady lifting-line theory; wing of large aspect ratio oscillating at low frequency; matched asymptotic expansions; singular integral equation; airloads; unsteady induced downwash; elliptic wing in pitch and heave
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\textit{A. R. Ahmadi} and \textit{S. E. Widnall}, J. Fluid Mech. 153, 59--81 (1985; Zbl 0589.76027)

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### Digital Library of Mathematical Functions:

§11.12 Physical Applications ‣ Applications ‣ Chapter 11 Struve and Related Functions### References:

[1] | DOI: 10.1017/S0022112071000570 · Zbl 0242.76009 |

[2] | DOI: 10.1017/S0022112071000685 |

[3] | Van Holten, Vertica 1 pp 239– (1977) |

[4] | DOI: 10.1017/S0022112076002255 · Zbl 0338.76009 |

[5] | Van Holten, Delft Univ. Tech., Dept Aerosp. Engng Rep. 8 pp 104– (1975) |

[6] | Sears, J. Aero. Sci. 8 pp 104– (1941) |

[7] | Landahl, AIAA J. 6 pp 345– (1968) |

[8] | DOI: 10.1007/BF02086522 · JFM 63.0774.04 |

[9] | Guiraud, Rech. Aerospat 1 pp 327– (1981) |

[10] | DOI: 10.1017/S0022112075002339 · Zbl 0363.76006 |

[11] | Cheng, J. Fluid Mech. 143 pp 327– (1984) |

[12] | DOI: 10.1017/S0022112080002686 · Zbl 0444.76042 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.