Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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Wave drag — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Lift and wave drag on a flat plate at Mach 2, exactly and to first order. The lift and wave-drag coefficients of a supersonic section against incidence, computed face by face from shocks and fans, with Ackeret's linear result dashed over them. The two agree to three decimal places at small angles and part company slowly — which is what a first-order theory is supposed to do, and evidence rather than tautology, since the two routes share no algebra.

    Drag with nothing to rub

    d'Alembert's paradox says a closed body in a steady, inviscid flow has no drag, and four separate arguments establish it, none of them wrong. Above Mach one it is false — the flow is still inviscid, still steady, and the drag is real, finite and quadratic in incidence.

    part 1 · compressible
  2. Four bodies of identical length and volume, and their wave drags. Each body has the same length and the same volume; only the distribution of area along it differs. The Sears–Haack body — the spindle whose area goes as the three-halves power of x(L−x) — has the least wave drag of the four, and every other shape pays between thirty-seven per cent and a hundred and seventy-five per cent more for carrying the same volume the same distance. Nothing about the cross-sections' shape enters: only the area distribution does.

    The least drag a volume can have

    A body's supersonic wave drag depends on nothing about it except how its cross-sectional area is distributed along its length. Minimising that for a given volume gives one shape — and the answer goes as the volume squared over the fourth power of the length.

    part 2 · compressible
  3. Zero wave drag at one Mach number, inside a loop it cannot enter from below. The wave drag of the pair, per unit of one element's chord, against Mach number, beside a single diamond of the pair's combined thickness. The biplane's drag is zero at its design Mach number, 1.944, and rises either side. Shaded: below 1.6 the channel cannot pass the flow at all; between 1.6 and 2.15 it can, but only if it was started above the band. Accelerating from subsonic, the biplane stays choked until Mach 2.15, past its design point.

    Busemann's biplane has to be flown past its design point

    Two half-diamonds facing across a channel cancel each other's waves at one Mach number and carry their thickness at no wave drag. The cancellation needs the channel narrowed at its middle, and a narrowed supersonic channel will not swallow the shock that forms in it on the way up. A biplane 3 per cent thick per element, designed for Mach 1.94, stays choked until Mach 2.15, and only one that is thinner than about 2.6 per cent can start by itself at any speed.

    part 3 · compressible
  4. A flap a fifth of the chord long, turned in three and a half degrees, starts the biplane at Mach 1.72 instead of 2.15. The Mach number at which the channel between the elements swallows the normal shock at its mouth, against the leading-edge flap's deflection, for flaps of 0.1, 0.2, 0.3 and 0.4 of the chord; the deflection runs from shut to the ramp's own angle, 3.43°, where the flap lies flat. Shut, the channel starts at 2.15; flat, at 1.93 (0.1), 1.72 (0.2), 1.53 (0.3), 1.32 (0.4). The design Mach number is 1.944.

    A flap that closes the mouth starts Busemann's biplane below its design point

    Busemann's biplane cancels its own waves at one Mach number and cannot get there: its channel narrows to a throat that will not swallow the shock standing in its mouth until the aeroplane is flying faster than the design point. Hinge the front of each element's inner ramp and turn it into the channel, and the mouth closes towards the throat while the throat stays put. A flap a fifth of the chord long, turned in three and a half degrees, starts the channel at Mach 1.72 instead of 2.15; once started, the flaps can shut at any speed above 1.60 and the zero at the design point returns exactly.

    part 4 · compressible
  5. At its design Mach number the biplane's channel lifts nothing: all its lift is on the two outer faces. The lift coefficient of the pair at 2° incidence against Mach number, per unit of one element's chord, beside what the two outer faces give alone, 4α/β, and two isolated flat plates, 8α/β. At the design Mach number, 1.944, the pair's lift is 0.08378 and the outer faces' 0.08378: the channel's net lift is zero to rounding. At Mach 1.7 the channel adds 13.9 per cent, at 2.2 21.1.

    Busemann's biplane lifts only on its outer faces, and its channel turns the lift into a quarter-chord load

    Tilt Busemann's biplane to make it lift and the same cancellation that removed its wave drag removes the lift of its channel: at the design Mach number the waves the incidence sends across the gap meet and cancel just as the thickness's did, and every bit of lift is carried by the two outer faces — half what two separate plates would give. The channel does not go idle. It carries a couple, lift on the front half and download on the rear, which moves the centre of pressure from mid-chord, where every supersonic plate has it, to the quarter chord. The pair pays for its lift two and a half times as steeply as flat plates do, and still beats a diamond fourfold.

    part 5 · compressible

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