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

Axisymmetric — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 1.5U at the equator, and no drag at all. The exact ideal flow past a sphere, in the meridional plane, with speed contoured behind the streamlines. The fastest fluid is at the equator at 1.5U — a cylinder's is at 2U — and the field is fore-and-aft symmetric, so the pressure integral over the surface gives a drag of -1.2e-16 against a dynamic scale of order one. The streamline spacing here does not measure speed the way it does in a plane flow: the flux between two meridional streamlines depends on the distance from the axis as well, which is why the speed is contoured rather than left to be inferred.

    Three dimensions are kinder

    The ideal flows usually drawn first are plane, and plane flow is the harsh case. Put the third dimension back and the fastest surface speed drops from twice the free stream to one and a half times, the disturbance dies as the cube of distance instead of the square, and the body cannot carry circulation at all.

    part 1 · inviscid
  2. The same thickness, drawn as a body of revolution and as a wing section. The surface speed along three prolate spheroids of thickness ratio a half, a quarter and a tenth, and along the plane ellipses of the same ratios, in units of the stream speed. Each plane section speeds the flow over its middle by exactly its own thickness ratio. The body of revolution of the same ratio does so by much less — at a tenth, by two per cent where the ellipse manages ten.

    Thickness costs a fuselage far less than a wing

    A wing section a tenth as thick as it is long speeds the air over its middle by ten per cent. A body of revolution with the same proportions speeds it by two. The difference is not a detail of shape: a plane section pays for its thickness linearly and a body of revolution pays quadratically, and that one exponent is why a fuselage reaches the speed of sound on its surface long after its wing.

    part 2 · inviscid
  3. The fuselage's overspeed reaches along the wing. The extra axial speed the fuselage alone gives the air beside it, as a fraction of the flight speed, against distance out along the wing from the fuselage's axis in fuselage radii, level with its thickest section, at Mach 0, 0.6 and 0.8. At the fuselage's side it is 2.1 per cent at low speed and 2.5 at Mach 0.8; it falls only to two-thirds of that two radii out and a quarter at five — a long body's middle " + "spreads like a line of sources, not a point — and the section beside it is already running at twelve per cent over at low speed and twenty over at Mach 0.8.

    A fuselage lowers its wing root's critical Mach number

    A slender fuselage barely disturbs the air: two per cent of overspeed at its side, against the twelve a wing section of ordinary thickness makes. At the wing root the two are added, and two per cent on top of twelve moves the root's critical Mach number down by more than a hundredth — as much as making the whole wing thirteen per cent thicker. The fuselage's overspeed also fades far more slowly along the span than its size suggests, because the middle of a long body spreads its disturbance like a line, not a point.

    part 3 · inviscid
  4. A waist moves the fuselage's overspeed off the root chord. The fuselage's axial perturbation velocity at its own surface, as a fraction of flight speed, along its length at Mach 0.8, with the wing's root chord between the two rules: the plain fuselage, which adds 0.0251 everywhere along the chord; a narrow waist 6.2 per cent deep that cancels it at mid-chord and raises it at the chord's ends; and a wide one 31 per cent deep that leaves none of it positive along the chord and lifts it on the body ahead of and behind the wing.

    A waist moves the overspeed and cannot remove it

    A fuselage adds two per cent of overspeed at the wing root, and a waist — a local narrowing of the body where the wing joins it — is the obvious cure. Slender-body theory says exactly what a waist can do. Its sources and sinks sum to nothing, so the velocity it adds along the body integrates to zero: it cannot remove the overspeed, only move it. Moved off the whole root chord it needs a third of the fuselage's cross-section at the wing, a sixth of what the transonic area rule would take, and it lands on the fuselage just ahead of and behind the wing, where there is room for it.

    part 4 · inviscid
  5. The free waist is flat-bottomed, and follows the sweep aft. The cross-section each waist removes along the body, as a fraction of the fuselage's greatest, for the root strip unswept, at 30° and at 45°. The linear programme's waists have flat bottoms and steep sides: depth is spent only where the strip needs it. With sweep they lengthen aft rather than deepen — 24.8, 24.3 and 26.9 per cent. The best single dents, dashed, must widen and deepen instead: 39.8, 45.7 and 52.6 per cent.

    A free waist needs half the depth, and the sweep costs it nothing

    A cosine-squared dent in a fuselage cancels its overspeed at an unswept wing's root for about a third of the fuselage's cross-section, and nothing can remove the overspeed, only move it. Let a linear programme choose the waist's shape instead, and half of that third turns out to have been the dent's fault. The free waist is flat-bottomed and steep-sided, and for a swept root it simply runs aft with the chord, where a single dent has to widen and deepen.

    part 5 · inviscid

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