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An asymptotic theory of the jet flap in three dimensions

Published online by Cambridge University Press:  29 March 2006

Naoyuki Tokuda
Naoyuki Tokuda
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Department of Mathematics, University of Southampton
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Abstract

A uniformly valid asymptotic solution has been constructed for three-dimensional jet-flapped wings by the method of matched asymptotic expansions for high aspect ratios. The analysis assumes that the flow is inviscid and incompressible and is formulated on the thin airfoil theory in accordance with the well-established Spence (1961) theory in two dimensions.

A simple method emerges in treating the bound vortices along the jet sheet which forms behind the wing with the aid of the following physical picture. Three distinct flow regions—namely inner, outer and Trefitz—exist in the problem. Close to the wing the flow approximates to that in two dimensions. Therefore, Spence's solution in two dimensions applies. In the outer region a wing shrinks to a line of singularities from which the main disturbances of flow in this region arise. In particular, we find that the shape of the jet sheet, hence the strength of vortices, is now predetermined by the strength of the singularities there. Hence a complete flow field in the outer region can now be determined first by evaluating the flow due to various degrees of singularities along this line and then adding the effect of the jet bound vortices which is now known. Far removed from the wing, the well-known Trefftz region exists in which calculations of aerodynamic forces can be most easily done.

The result has been applied to various wing planforms such as cusped, elliptic and rectangular wings. The present result breaks down for rectangular wings. However, we can apply Stewartson's (1960) solution for lifting-line theory for semi-infinite rectangular wings, because, to this second-order approximation it is established that the jet sheet in the outer region makes no contribution to lift, with the direct contribution of the deflected jet at the exit being cancelled by the reduced circulation in the jet vortices. This result for the rectangular wing gives excellent agreement with the experiments made on a rectangular wing, while the result for elliptic wings underestimates them considerably.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 46 , Issue 4 , 27 April 1971 , pp. 705 - 726
Copyright
© 1971 Cambridge University Press

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