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

Box wing — where it appears

A closed wing system: two wings one above the other joined at their tips by vertical fins, whose wake in the far field is a closed rectangle. Prandtl showed it has the least induced drag of any system of its span and height, and its fins also tie the two spars into one frame.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

circulation · Box wing 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.

circulation · Box wing

Named alongside it

The objects these essays reach for when they reach for this one.

Bending momentInduced dragModel limitOptimisationThe Trefftz planeBiplaneElliptic loadingSparStatically indeterminate

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