A waist moves the overspeed and cannot remove it

A fuselage lowers its wing root’s critical Mach number superposed a slender fuselage’s field on a wing section’s and found the junction reaching sound first. The fuselage adds only about two per cent of the flight speed to the flow at the root, but it adds it to the section’s own twelve per cent, near the threshold where the air goes sonic, and a fuselage of fineness ten costs a 12-per-cent root about a hundredth and a third of critical Mach number — as much as making the wing thirteen per cent thicker.

That essay’s last figure pointed at a cure. Moving the wing along the body bought almost nothing, because along the middle of a long body the overspeed hardly changes. What the overspeed does depend on is the body’s own cross-section near the wing, and a fuselage made narrower where the wing joins it — a waist — changes it locally. The essay asked for the smallest such waist that cancels the overspeed at the root, and whether it needs as much area as the transonic area rule would take out of the fuselage there. This essay computes both, and finds a constraint on the answer that is stronger than any number.

A fuselage is a line of sources #

The tool is slender-body theory. A long, thin body of revolution pushes the stream aside as though it were a line of sources along its axis, each of strength equal to the rate at which the body’s cross-sectional area grows: where the body swells the line emits fluid, and where it tapers the line absorbs it. The axial velocity these sources add at any point, in a subsonic stream at Mach $M$, is

$$u(x, r) = \frac{1}{4\pi}\int A'(\xi)\,\frac{x - \xi}{\left[(x-\xi)^2 + \beta^2 r^2\right]^{3/2}}\,d\xi, \qquad \beta = \sqrt{1 - M^2},$$

which is the linearised subsonic field of a line of sources, with the compressibility entering through $\beta$ as it does in the Prandtl–Glauert rule.

The fuselage is the earlier essay’s prolate spheroid of fineness ten, of half-length one; its area $A = \pi\tau^2(1 - x^2)$ gives a source strength that grows linearly from the nose’s sinks — the nose pushes fluid out, the tail draws it in — through zero at the middle. Two checks establish the tool. At the equator the integral can be done by hand, $\tau^2[\operatorname{asinh}(1/r) - 1/\sqrt{1 + r^2}]$, and the quadrature reproduces it to four parts in $10^{10}$ at three thicknesses. And a short distance off the body, where the earlier essay’s exact spheroid applies through Göthert’s rule, slender-body theory is within 3 per cent of it, the size of its own approximation for a body this thick. At the wing’s station and Mach 0.8, the fuselage adds 0.025 of the flight speed to the flow at its surface — two per cent at low speed, amplified by compressibility.

The waist is a dent in the area: a cosine-squared bump subtracted from $A$, centred on the wing, of depth $\delta$ as a fraction of the fuselage’s greatest cross-section and half-width $w$ in the fuselage’s half-lengths. The wing root is the earlier essay’s: an elliptic section, of chord 0.15 of the fuselage’s length — an airliner’s proportion — sitting on the fuselage’s surface. An elliptic section’s own overspeed is uniform along its chord in thin-aerofoil theory, so the root is exactly as good as the isolated wing when the fuselage’s addition is nowhere positive along the root chord, and worse wherever it is.

The waist moves the overspeed #

The first figure is the result in one picture. The plain fuselage adds 0.025 along the whole root chord and slowly less towards its ends. A narrow waist, of half-width equal to half the chord and only 6.2 per cent deep, drives the overspeed to zero at mid-chord. It does not clear the chord: at the chord’s leading and trailing ends the overspeed is not reduced but raised, to 0.034. A wide waist, of half-width half the fuselage and 31 per cent deep, leaves none of the overspeed positive anywhere along the chord, and the price is visible on either side of the wing, where the fuselage’s own overspeed climbs to nearly twice what it was.

In both, the overspeed has gone somewhere. That is not an accident of the shapes chosen.

Why a waist can only move it #

The second figure isolates the waist’s own contribution. Every waist adds a velocity that is negative over its middle and positive at its two shoulders, and the positive and negative parts have equal areas. Integrated along a line parallel to the axis, from far ahead to far behind, the waist’s added velocity is zero to five parts in $10^5$ of its magnitude.

The reason is short. A waist removes area and then gives it back: its source strength $-\Delta A'$ is a sink where the dent deepens and an equal source where it fills again, and its total strength is zero. The axial velocity integrated along any line parallel to the axis is the change in velocity potential between the line’s two ends, and a set of sources summing to nothing leaves the potential the same far ahead as far behind. So

$$\int_{-\infty}^{\infty} u_{\text{waist}}(x, r)\,dx = 0 \quad\text{at every radius.}$$

A waist cannot remove overspeed from the fuselage’s field; it can only move it along the body. Every design question about a waist is therefore a question about where the overspeed it displaces should go. The narrow waist put it on the root chord’s ends, which is useless. The wide one put it on the fuselage ahead of and behind the wing, which is the whole point.

The same bookkeeping holds for any local reshaping of a body that leaves its total volume’s rate of change unaltered at the ends — a fairing, a bulge, a fillet — and it is the subsonic, local cousin of the statement the area rule makes about wave drag: a body’s far field sees only the distribution of its cross-section, and rearranging that distribution moves disturbances without creating or destroying them.

Why curvature, not depth, sets the local overspeed #

The line of sources is a fiction the body’s exterior flow cannot tell from the real thing: the part of the flow inside the body continues the outside flow inwards until it meets singularities, and for a slender body they crowd onto the axis. The inside a flow does not decide is the warning that goes with it — many interiors give one exterior — and the line of sources is simply the interior slender-body theory chooses, the one a body with no lift and a moment anyway used to find Munk’s couple.

Close to a slender body the integral above separates into two parts. One is local, proportional to the second derivative of the area at the point and to the logarithm of the radius over the body’s length; the other is an average of the area’s slope over the whole body, which varies slowly. The local part is why a waist works at all, and why a narrow one works best at its centre: the curvature of a cosine-squared dent grows as the inverse square of its width, so halving the width quadruples the local effect of a given depth. It is also why the narrow waist’s shoulders are strong — the curvature there is just as large and of the opposite sign — and why the zero integral cannot be escaped by making the waist sharper. Sharpening it concentrates the moved overspeed; it does not reduce it.

The Mach number enters only through $\beta r$ inside the logarithm and through the Göthert scaling of the whole field, which amplifies the fuselage’s overspeed and the waist’s correction almost equally. That is why the depth needed hardly changes with speed — 34 per cent of the cross-section at Mach 0.6 and 31 at Mach 0.8 — while the overspeed being cancelled grows from 0.022 to 0.025.

Mid-chord is cheap, and the chord is not #

The third figure prices the two criteria. Cancelling at mid-chord alone — where the earlier essay added the fuselage’s overspeed to the section’s — is cheap and gets cheaper as the waist narrows: 6 per cent of the fuselage’s cross-section at a half-width equal to half the chord, 11 per cent at the chord’s full width, because a narrow dent is sharply curved and the overspeed a slender body makes at its surface is dominated by the local curvature of its area. But the narrow waist’s shoulders land on the chord. The shallowest waist that clears the whole chord has to be wider than the chord, so that its shoulders fall outside it, and then it needs about a third of the cross-section: 31 per cent at a half-width of half the fuselage, rising again for wider waists, whose curvature weakens.

So the question the earlier essay posed has two answers, and only one of them is useful. The smallest local reduction that cancels the overspeed at a point is a few per cent. The smallest that restores the root to the wing alone is a third of the fuselage’s area at the wing.

A sixth of the area rule #

The fourth figure makes the comparison the question asked for. At transonic speed the area rule asks that the fuselage be narrowed by the wing’s whole cross-sectional area at each station, so that the aircraft’s total area varies smoothly. For an unswept wing of 12 per cent thickness, root chord 0.15 of the fuselage’s length and span 0.9 of it, that is 1.8 times the fuselage’s own greatest cross-section — more than the fuselage has, which is why the area rule is applied to swept wings, whose area is spread along the body, and in part rather than in full.

The subsonic waist removes 17 per cent of that at Mach 0.8, 18 at Mach 0.6. About a sixth. The two operations answer different questions: the area rule smooths a total that the far field sees and wave drag charges for; the waist moves a local overspeed off one chord. The second is much cheaper because it only has to act at one place, and it has somewhere to put what it moves.

Where the overspeed goes, and why there is room for it #

The fifth figure follows the displaced overspeed. On the fuselage itself, ahead of and behind the wing, the waist’s shoulders raise the fuselage’s greatest overspeed from 0.025 to 0.048 — nearly double. That would be a poor trade at the wing root, where the section’s own twelve per cent is already near the edge. On the fuselage it is harmless: the linearised overspeed at which the fuselage’s surface reaches sound at Mach 0.8 is 0.217, and the doubled shoulders reach less than a quarter of it. Thickness costs a fuselage far less than it costs a wing, and a waist spends that slack.

That is the design logic in one line: the waist moves the fuselage’s overspeed from where it adds to the wing’s to where it adds to nothing. The shoulders must not land on anything else — a nacelle, a fairing, the tail — and the calculation puts them about 0.15 of the fuselage’s length ahead of and behind the wing’s mid-chord.

What a real junction does instead #

No airliner has a waist a third of its fuselage’s area deep, and the reason is plain in numbers. A fuselage four metres across has a cross-section of 12.3 square metres; removing 31 per cent of it at the wing means narrowing its radius from 1.98 to 1.64 metres, a 17 per cent cut in diameter over some ten metres of cabin, exactly where the wing box passes through and the cabin floor is widest. The payload cannot give that up for a hundredth of critical Mach number, and the same effect can be bought where it is cheaper. The earlier essay listed what designers do at the root: a thinner, more strongly cambered root section, a larger root sweep, a fillet that changes the cross-section locally. The last is a small, shaped change of area near the junction — a local rearrangement of the kind the zero-integral rule governs — and a root section designed for its actual local Mach number takes the fuselage’s addition into its own shape instead of removing it. The waist is the clean version of the problem; the fairing and the section are the practical one.

Where waists do appear is where the area rule is being applied at transonic speed, and there the argument above explains why they are wide and smooth. The pocket on top of the wing is where the local overspeed turns supersonic, and a waist sized to its root chord’s whole length is what moves the fuselage’s contribution out of it.

What was checked #

The ledger holds four checks: the quadrature against the equator’s closed form; slender-body theory against the exact spheroid through Göthert’s rule, within the slender approximation’s own few per cent; the zero integral of a waist’s added velocity, to $5\times10^{-5}$ of its magnitude; and a waist’s field vanishing at twenty half-lengths from the body, to $10^{-8}$.

What slender-body theory leaves out #

The wing’s own field on the body. The fuselage’s overspeed is added to the section’s; the section’s overspeed is not added to the fuselage’s surface, where the wing’s thickness and lift also act. The waist’s shoulders sit on the fuselage near the wing’s leading and trailing edges, where the section’s field is not negligible.

Lift. The wing’s circulation carries flow round the fuselage and changes the root’s local incidence; the calculation is at zero lift.

Sweep. A swept wing’s root chord runs diagonally along the body, the fuselage’s overspeed varies along it, and the wing’s own isobars straighten at the root, losing the benefit a swept wing feels exactly where the fuselage adds to the section. A swept root would need a waist sheared along the chord.

Accuracy near the surface. Slender-body theory’s field is accurate to a few per cent next to a body of fineness ten; the ratios the argument rests on — cancellation, the zero integral — are exact within the theory.

The convention: depth against the body’s area #

The fuselage’s half-length is the unit of length. The waist’s depth is the area it removes at its centre as a fraction of the fuselage’s greatest cross-section; its half-width is the half-length of the cosine-squared dent. Perturbation velocities are fractions of flight speed at the surface of the waisted body. The wing’s cross-sectional area at the root station is its thickness at mid-chord times its exposed span.

Who worked it out #

Slender-body theory for bodies of revolution is Munk’s, from the 1920s airship work, and its compressible form is Göthert’s and Ward’s from the 1940s. The area rule was found by Whitcomb at the NACA in 1952, anticipated by Otto Frenzl in Germany in the 1940s, and the first aircraft built to it, the Convair F-102A, had its fuselage waisted beside the wing in 1954. The subsonic use of local body shaping at wing junctions — fillets and root fairings designed from the local pressure field — belongs to the same decades and to every transport aircraft since.

Still open: the waist a swept root needs #

The waist here is symmetric about mid-chord because the root chord lies along the body. On a swept wing the root chord’s leading edge meets the fuselage well ahead of its trailing edge, and the root’s crest line runs across the body at the sweep angle. The next calculation gives the waist an axial shape of its own — a source distribution chosen, by least squares, to cancel the fuselage’s overspeed along a swept root chord — and asks whether its shoulders can be placed clear of both the leading edge ahead and the tailplane behind, and how deep it must be; and it compares that optimum with the fixed cosine-squared dent, to see how much of the third of the fuselage’s area is the shape’s fault and how much is the zero-integral rule’s.