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The Lift Coefficient of a Thin Jet-Flapped Wing. II. A Solution of the Integro-Differential Equation for the Slope of the Jet

D. A. Spence
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 261, No. 1304 (Apr. 11, 1961), pp. 97-118
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/2413902
Page Count: 22
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The Lift Coefficient of a Thin Jet-Flapped Wing. II. A Solution of the Integro-Differential Equation for the Slope of the Jet
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Abstract

In an earlier paper with the same general title (Spence 1956, referred to as I), a mathematical model was developed to discuss the flow past a two-dimensional wing at incidence α in a steady incompressible stream, with a jet of momentum coefficient CJ emerging from the trailing edge at an angular deflexion τ to the chordline. In linearized approximation it was shown that the slope of the jet is given by a certain singular integro-differential equation, and numerical solutions for the equation were obtained by a pivotal points method. A coordinate transformation has now been found (Spence 1959) which makes the equation independent of the jet strength for small values of 1/4CJ = μ, say, yielding a simpler equation solved by Lighthill (1959) using Mellin transforms (and by Stewartson (1959) and the present author by other methods). In this paper the expansion of the slope function is continued in ascending powers of μ and ln μ multiplied by functions of x found by solving, by Lighthill's method, a series of closely-related inhomogeneous equations. From these, expansions of the lift derivatives with respect to α and τ are found as \begin{align*}\frac{1}{4\surd(\pi\mu)}\frac{\partial C_L}{\partial\tau} &= 1 - \frac{\mu}{2\pi}\Big(\ln\frac{\mu}{\beta}\Big) - \frac{\mu^2}{8\pi^2}\Big[\Big(\ln\frac{\mu}{\beta}\Big)^2 + 4 \ln\frac{\mu}{\beta} - 4\Big]+\ldots,\\ \frac{1}{2\pi} \frac{\partial C_L}{\partial\alpha} &= 1 - \frac{\mu}{\pi}\Big(1+\frac{\mu}{\pi}\Big)\ln\Big(\frac{\mu} {\beta}-1\Big)+\ldots,\end{align*} where β = 4eγ. To this order the expressions agree closely with the numerical results found earlier, the discrepancy at μ = 1 being less than 4%.

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