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

Area rule — where it appears

The rule that a body's supersonic wave drag depends only on how its cross-sectional area is distributed along its length. It allows area to be moved about freely, so a fuselage can be waisted where a wing joins it and the pair given the distribution of a good body of revolution.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

The coefficient of the equation's second derivative, along a chord. The bracket multiplying the streamwise second derivative in the transonic small-disturbance equation, along a chord at Mach 0.85. Where it is positive the equation is elliptic and the flow is subsonic; where it is negative the equation is hyperbolic and the flow is supersonic. Which it is at a given point depends on the perturbation velocity there, which is the thing being solved for. Forty-two per cent of this chord is hyperbolic, and no amount of inspecting the problem beforehand could have said so.

The equation that changes type inside its own answer

Near Mach one the coefficient of the streamwise second derivative depends on the perturbation velocity, which is what is being solved for. Two solutions of the linear equation no longer add — the leftover is three times the term the linear theory keeps — and the critical Mach number approaches one as the two-thirds power of thickness.

compressible · Transonic One signature, aged four times. The pressure signature of a slender body at four distances, computed by the exact Lax formula for the nonlinear propagation. Each point of the waveform moves forward in proportion to its own overpressure, so the compression at the front catches the undisturbed air and a shock forms there, while the expansion at the rear falls behind and forms a second one. What is left is an N-wave: two discontinuities and a straight line between them, spreading and weakening.

The signature that forgets the shape

The pressure field near a supersonic aeroplane depends on every part of it. What reaches the ground has two parameters. Two bodies whose near-field signatures differ by fifty-five per cent in peak and by their whole shape age into the same N-wave, to two and a half per cent.

compressible · Sonic boom 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.

compressible · Wave drag Three aircraft, and the only thing the drag can see. The cross-sectional area against station, for a clean body, the same body with a wing added, and the body cut away where the wing is. The wave drag depends on this curve and on nothing else about the three shapes — so the middle one pays for its lump and the third does not, although the third has the same wing.

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.

inviscid · Slender body 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.

inviscid · Axisymmetric 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.

inviscid · Axisymmetric 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.

inviscid · Axisymmetric

Named alongside it

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

Model limitSuperpositionPrandtl–Glauert correctionCritical machInterferenceOptimisationSlender bodyCompressibilityFar fieldSears haackSlender body theorySonic boom

All concepts