The thread: Lift is circulation — page 11
96 essays carry this thread — page 11 of 11.
Ideal flow
Thickness carries no load of its own and still moves the load forward
Thin-aerofoil theory builds a section's pressures from three pieces computed apart — its thickness, its camber, its incidence — and adds them. Every piece can be computed exactly on a Joukowski section, and so can the section they add up to. Added, the parts give a suction peak a quarter too weak and a loading that sits too far aft, and they miss nine per cent of the lift: thickness, which lifts nothing by itself, multiplies what camber and incidence make by 1 + 0.77 t/c. Carry the summed velocities through Bernoulli instead of adding the pressures, and most of the error goes; the part that stays is the cross term between thickness and lift, and it is first order in the lift.
Circulation and lift
A wing that cannot hold its suction gains little from its span
The leading-edge suction is a small force concentrated at one edge, and on a finite wing it decides what the aspect ratio is worth. With it, the drag due to lift is the induced drag, which falls as the span grows. Without it, the drag due to lift is the lift tilted back by the incidence, and that falls only as fast as the lift slope rises — which stops at the section's 2π. A slender delta that loses its suction pays exactly twice its induced drag; a wing of aspect ratio eight pays more than five times it. Without the suction no span can lift a wing's best lift-to-drag ratio past a ceiling set by the section alone.
Circulation and lift
A spar that must lift itself stops the span at a quarter of the weight
Held only to a lift and a bending moment, the least-drag wing never stops growing. An aeroplane's wing stops because its spar has weight, and the weight has to be lifted. Count it, and the best span is the one at which the spar is exactly a quarter of the lift when the wing's area is fixed, and a third when its chord is — whatever the loading. The best loading then sheds some of its tip load, but not all the way to Prandtl's bell; with the chord held and nothing else charged, it is the bell exactly, and the more of the drag the wing's own area makes, the nearer it comes back to the ellipse.
Ideal flow
A NACA section multiplies its lift by nearly one plus its thickness, and a nose correction cannot supply it
On Joukowski's exact sections, thickness — which lifts nothing alone — multiplies the lift of camber and incidence by 1 + 0.77 t/c. Solved by panels, the NACA four-digit sections that real wings use do more: their lift slope grows as about 1 + 0.84 t/c, and a cambered 2412 lifts 11.5 per cent more than thin-aerofoil theory says. The more of the thickness sits near the nose, the more it multiplies. Added as thin-aerofoil theory adds them, the parts lose a quarter of the suction peak and put the centre of pressure two hundredths of the chord too far aft. Riegels's nose factor, the classical repair, makes the nose speed finite and moves the peak, and adds no lift at all: the missing lift is not a nose effect.
Circulation and lift
A stiffness floor keeps the span and gives the tips their load back
A spar sized by strength alone thins to nothing at the tips, and charged as lift it stops the best span where it weighs a quarter of everything the wing carries, with the tips a little unloaded. A real outer wing is sized by stiffness, and stiffness sets a floor under the spar's weight that no reduction of the tip load can lower. Put that floor in and the tips get their load back — the best loading slides from a sixth of the way to the bell towards the ellipse — the quarter does not survive, rising to two fifths as the floor spreads over the outer two thirds of the span, and the span itself barely moves. A weight per metre cannot stop a span.
Circulation and lift
A strut inside a box wing is invisible to the wake, and halves the spar
Prandtl's best box wing has, everywhere inside it, air moving straight down: no sideways flow at all. A strut joining its two wings anywhere along the span therefore meets no flow it could turn, carries no load, and leaves the induced drag exactly where it was — the wake cannot see it. The spar can. A strut two fifths of the way out, as stiff as a spar and rigidly joined, halves the flange material of both spars, though it is pushed and pulled by almost nothing: the two wings lift equally and bend together, so the strut works through its joints, and a pinned one does nothing. Its only aerodynamic price is friction, and a slim strut pays it.