Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Box wing — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A closed wake, and the loading of least drag on it. The wake of a box wing in the plane that decides its drag, with the circulation of least induced drag drawn as a thickness along it. The horizontal members carry a loading close to elliptic and the vertical ones carry a share that lifts nothing — they contribute no lift, since lift is Γ dy and dy is zero on a vertical, and they change the drag by changing where the wake's vorticity is. At a gap of 20 per cent of span this system costs 67.1 per cent of what a single wing of the same span and lift would.

    A wake that closes on itself

    Prandtl's best wing system is a rectangle, not a wing. Solving for the loading on a closed wake turns up a circulation that costs nothing and does nothing — a gauge freedom in the middle of an optimisation — and a drag that keeps falling with no floor under it.

    part 1 · circulation
  2. What every wing pays to bend its root less. Least induced drag against the root bending moment of the lift, both as fractions of the elliptic monoplane of the same span and lift, for the monoplane and for box wings with gaps of a tenth, a fifth and two-fifths of the span. Each curve is a parabola with its minimum at that wing's unconstrained optimum. The box wings' minima sit to the right of the monoplane's — they load their lift further out — and their parabolas are shallower: the gap-of-a-fifth box can bend its root 21 per cent less than the elliptic monoplane and still match its drag.

    A lighter spar turns a box wing into a biplane

    Prandtl's best wing system has two-thirds of a monoplane's induced drag at a gap of a fifth of the span, and part of that saving is carried by circulation turning the corner into its fins. Ask the box to bend its root less and it pays about half what a monoplane pays — but a biplane with no fins pays nearly as little, and by the time the spar is a fifth lighter the fins carry almost nothing and the box has become the biplane it was built from.

    part 2 · circulation
  3. Joined at the tips, the spars share the moment as a couple. The share of the lift's root bending moment that the box wing's two spars carry as an axial couple — one in tension, the other in compression, the gap for a lever arm — against the fins' bending stiffness as a multiple of a spar's, for gaps of a tenth, a fifth and two-fifths of the span. With floppy fins the spars bend independently and the couple is nothing. With fins as stiff as the spars, which a fin of the wing's own section is, the couple carries 29 per cent at a gap of a fifth. However stiff the fins, it stops near a third.

    A box wing's fins earn their keep in the spar

    Constrain a box wing's root bending moment and its fins stop saving drag: it becomes a biplane. But that constraint counted the lift's moment, not the spars', and the fins join the spars into one frame. Joined at the tips, the two spars hand part of the moment to a couple across the gap — tension in one, compression in the other — which costs far less material than bending. With fins as stiff as the spars that is three-tenths of the moment; with rigid fins it stops at a third, whatever the gap, because a tip joint can only guide a tip. Counted in the spar, the box has less drag and a lighter spar than the elliptic monoplane at once.

    part 3 · circulation
  4. A strut two fifths of the way out halves the spars of a box wing; near the root or the tip it buys little. The flange volume of both spars of Prandtl's best box at a gap of a fifth of the span, with one strut on each side joined rigidly to both spars, over the volume with none, against the strut's station as a fraction of the semi-span, for struts a tenth as stiff in bending as a spar, as stiff, and ten times stiffer. For a strut as stiff as a spar the least is 0.5 at 0.4 of the semi-span; at 0.1 and 0.9 it is 0.839 and 0.929.

    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.

    part 4 · circulation

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