Wave drag — where it appears
Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
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.
A drag that needs a discontinuity
A closed body in a steady inviscid flow has no drag, and the transonic equation obeys that as faithfully as any other — until a shock appears inside its own answer. Then the same pressure integral that was zero to five decimal places becomes four decades larger, with no viscosity added and nothing about the body changed.
The drag that can only see one curve
A slender body's far field is a line of sources of strength A′(x), and nothing else about its shape reaches it. So its supersonic wave drag depends on the cross-sectional area distribution alone — and a fuselage cut away where the wing joins it can carry the wing for nothing.
A hydrofoil loses most lift on the way up
A wing near the ground gains lift, and a hydrofoil near the surface is often described as the same thing upside down. At low speed it is: the surface acts as a lid and the foil gains lift. At high speed the surface is a boundary that cannot hold a pressure, and the foil loses lift instead. But the lift does not pass smoothly from one to the other. Between them, where the foil's waves are a few depths long, it falls well below both — to 43 per cent of its deep-water value half a chord down — and a foil boat meets that speed in the middle of its take-off run.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitArea machArea ruleChokingd'Alembert's paradoxDimensionlessDragEntropyHysteresisInviscidLiftLinearisation