Induced-drag characteristics of crescent-moon-shaped wings

Abstract

Results are presented of a theoretical study conducted to analyze the potential drag reduction characteristics of wings with highly tapered aft-swept tips at subsonic speeds. These planar wings, also named crescent- moon-shaped wings or wings with curved planform, can produce induced drag less than the minimal value ob- tained with the classical unswept elliptical wing for a constant lift and wing span. The induced-drag reduction is the result of the nonplanar wing and wake shape at the angle of attack. For an example wing of aspect ratio 7, the crescent-moon shape provides a reduction in cruise induced drag of 8.0% as compared to an unswept elliptical wing. equal to but not less than the minimum value obtained for the classical straight wing with elliptical loading distribution can be achieved with planar curved planforms. In order to obtain the mimimum value of induced drag for the wing with backward sweep, it may be required to introduce a substantial amount of washout, while for the wing with for- ward sweep, it may be necessary to incorporate washin. The twist requirement has a severe disadvantage, however, because now the minimum induced drag is obtained only at the angle of attack for which the wing twist distribution is optimized. At off-design conditions, drag penalties are in- curred and the induced drag produced by the twisted swept wing will be higher than for the untwisted unswept elliptical wing for a constant lift coefficient. This paper will demonstrate that an untwisted planar crescent-moon wing can be more efficient than an unswept elliptical wing. A computa- tion method that correctly accounts for the influence of the trailing wake must be applied in order to obtain this somewhat unexpected result. Zimmer3'4 demonstrated that crescent-moon planform shapes can significantly reduce the level of induced drag pro- duced by a wing. The results are obtained with a vortex- lattice method that models the lifting surface by its mean camber line and takes into account the nonlinear effect of the trailing vortex sheet. Consequently, the influence of air- foil thickness is assumed to be negligible and the leading- edge suction force is modeled mathematically. Zimmer's work resulted in the Dornier New Technology Wing (Dornier Do-228), which represents a practical (and somewhat com- promised) application of the crescent-moon planform shape. In this study, a nonlinear surface panel method is applied to study the drag characteristics of a crescent-moon-shaped lifting surface of aspect ratio 7. Consequently, no approxima- tions are required to calculate the induced-drag force. Analysis Method