A New Theory for Wings of Small Aspect Ratio

Abstract

Experiments on wings have shown that a very different kind of flow takes place for very small aspect ratios than for large aspect ratios. The lift curve continues up to about 45° before stalling occurs. During this range it has a concave curvature upward rather than downward as the lifting line or lifting surface theories predict. No theoretical explanation of this effect has yet been given since it was generally supposed to be a stalling phenomenon and thus not adaptable to perfect fluid theories. The present paper shows that this curvature effect is due to the fact that the trailing vortices leave at an angle α to the plate. For the limiting case of a plate with finite span and infinite chord it is shown that the bound vorticity and induced downwash are constant across the span, and the trailing vortices leave the wing at the half-angle of attack, α=θ/2. These results are carried over into the assumptions for the analysis of the finite rectangular flat plate of very small aspect ratio. A surface distribution of vorticity over the plate is assumed, constant across the span, and varying according to the formula γ = γ0√t/2-x/t/2+x along the chord. Straight trailing vortices are assumed leaving the plate at an undetermined angle α. The boundary condition assumed is that the mean value of the induced velocity along the center line of the span is equal to the normal component of the free-stream velocity. This determines the constant γ0 and thus the normal force coefficient CN as a function of θ. The parameter α is still undetermined; however, its limits are given. For very small aspect ratios α=θ/2, for large aspect ratios it approaches θ. Winter’s experiments on a wing of aspect ratio κ=1/30 are checked very closely by this theory assuming α=θ/2. At larger aspect ratios up to about κ=1 the experimental curves lie between the theoretically predicted curves corresponding to α=θ/2 and α=θ, moving toward the latter limit at κ=1.