# Airfoils and Airfoil Systems of Finite Span

• William Frederick Durand
Chapter

## Abstract

The subject of this Chapter is the investigation of the forces experienced by an airfoil of finite span and by airfoil systems. The basic idea for the following deductions has been indicated at the end of III 32. The only point we have to add is that in the case of a single wing of ordinary type we may substitute the lift per unit span l for Λ (see III 31), and consequently take l as the principal unknown quantity. Then III (27.7) becomes:
$$w = \frac{1}{{{e^V}}}\int\limits_{ - b}^{ + b} {dn\frac{{{{dl} \mathord{\left/ {\vphantom {{dl} {dn}}} \right. \kern-\nulldelimiterspace} {dn}}}}{{4\pi \left( {y - \eta } \right)}}}$$
(1.1)
and III (31.9), neglecting the difference between φ and tan φ:
$$\varphi = \frac{w}{V} = \frac{1}{{{e^{{V^2}}}}}\int\limits_{ - b}^{ + b} {dn\frac{{{{dl} \mathord{\left/ {\vphantom {{dl} {dn}}} \right. \kern-\nulldelimiterspace} {dn}}}}{{4\pi \left( {y - n} \right)}}}$$
(1.2)
Now suppose that the geometrical angle of incidence of an airfoil section. defined with reference to the x axis, or what comes to the same, with reference to the horizontal velocity V, has the value α; then the effective angle of incidence i, measured from the downward sloping effective velocity, is given by1:i = αφ

## Chapter IV A. Single Wing

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## Chapter IV B. Multiplane Systems

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## Chapter IV C. Influence of Boundaries (Interference)

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22. 1.
The reader is referred to Th. von Kábmán, Beitrag zur Theorie des Auftriebes, Vorträge aus dem Gebiete der Aerodynamik usw. (Aachen 1929), p. 98. Also for cases treated with the aid of the theory of two-dimensional motion to T. Sasaki, On the effect of the walls of a wind tunnel upon the lift coefficient of a model, Rep. Aeron. Res. Instit. Tokyo No. 77 (Vol. VI, p. 315, 1931; Japanese with English abstract).Google Scholar
23. 2.
Cases of two-dimensional motion have been considered by various authors, who nearly always make use of the methods of conformal transformation. By way of example the following papers are mentioned: T. Sasaki, On the effect of the wall of a wind tunnel upon the lift coefficient of a model, Rep. Aeron. Res. Instit. Tokyo No. 46 (Vol. IV, p. 149, 1928; Japanese with English abstract) and No. 77 (see footnote above)Google Scholar
24. 2a.
L. Rosenhead, The lift on a flat plate between parallel walls, Proc. Roy. Soc. London A 132, p. 127, 1931
25. 2b.
S. Tomotika, On the moment of the force acting on a flat plate placed in a stream between two parallel walls, Rep. Aeron. Res. Instit. Tokyo No. 94 (Vol. VII, p. 357, 1933)Google Scholar
26. 2c.
S. Tomotika, T. Nagamiya and Y. Takenouti, The lift on a flat plate near a plane wall etc., ibidem No. 97 (Vol. VIII, p. 1, 1933; further papers are following).Google Scholar
27. 1.
In the case of an airfoil of finite span (see 32) there is also a change in the angle of incidence, causing an increase of the lift. Other effects causing an increase of lift come into play as soon as H is of the order of the chord, or smaller than the chord of the airfoil [see: Tomotika and others, Rep. Aeron. Res. Instit. Tokyo No. 97 (Vol. VIII, p. 1, 1933)Google Scholar
28. 1a.
further: G. Dätwyler, Untersuchungen über das Verhalten von Tragflügelprofilen sehr nahe am Boden, Mitt. Inst. f. Aerodynamik E. T. H. Zürich, 1934].Google Scholar