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

A waist moves the overspeed and cannot remove it took a slender fuselage whose own overspeed, a few per cent of the flight speed at its sides, adds to a wing root’s and brings the root to the speed of sound first. It narrowed the fuselage at the wing with a cosine-squared dent and found two things. The shallowest dent that leaves none of the fuselage’s perturbation positive along an unswept root chord takes 31 per cent of the fuselage’s greatest cross-section. And no dent can do better than move the overspeed, because a waist removes area and gives it back, and a set of sources summing to nothing adds a velocity whose integral along any line is zero: what it takes from the root it puts on the fuselage ahead and behind.

That essay ended on two questions it could not separate. How much of the third belongs to the rule, and how much to having chosen a cosine-squared shape and only varied its width? And what does a swept wing need, whose root region runs aft as it runs outboard, so that the stretch of fuselage beside the wing is longer than the chord? Both have the same method, which is to stop choosing the shape.

A waist chosen, not drawn #

Slender-body theory makes the problem linear. The fuselage is a line of sources along its axis with strength equal to the rate at which its cross-section grows, and the axial velocity it adds at any point beside it is an integral of that strength against a kernel that depends only on distance and, through the Prandtl–Glauert factor $\beta = \sqrt{1 - M^2}$, on the Mach number. A waist is a change in the cross-section, so its velocity is the same integral of the change’s slope. Build the waist from pieces and its velocity is the sum of the pieces’ velocities.

The pieces here are twenty-four small cosine-squared dents, each a fifth of the fuselage’s length wide, spaced along the body. Their depths are the unknowns, each at least zero, since a waist only removes area. At any point near the wing the fuselage’s perturbation is the plain body’s plus the sum of the depths times each dent’s contribution, and requiring it to be zero or negative at every point of the wing’s root region is a set of linear inequalities. The quantity to make small is the waist’s peak depth — the largest fraction of the cross-section removed anywhere — and requiring every depth to stay below it is another set of linear inequalities.

Minimising one number subject to linear inequalities is a linear programme, solved here by the simplex method. One subtlety makes it iterative. The perturbation is evaluated at the fuselage’s surface, and the waist moves the surface inwards, where the fuselage’s own field is stronger; so the programme is solved, the waist’s surface computed, and the programme solved again at the new surface, until the depths settle, which takes four rounds. The programme sees the root region as a grid of points, and between them the perturbation it leaves can creep slightly above zero: on a grid four times finer it reaches at most $3\times10^{-4}$ along an unswept root and $10^{-3}$ over a swept strip, against the plain root’s 0.025.

The fuselage and flight condition are the earlier essay’s: a spheroid of fineness ten at Mach 0.8, with a root chord 0.15 of the fuselage’s length. A single piece of the basis reproduces that essay’s cosine-squared dent’s field to three parts in $10^{10}$, which is the check that the two calculations are describing the same body.

The shapes it finds #

The figure draws what the programme chooses. Its waists are flat-bottomed and steep-sided: they go down to their peak depth over the stretch of body beside the wing, stay there, and come back up sharply just outside it. The best single dents, dashed, are the opposite: a smooth dip centred on the wing, deepest in the middle, with long tapering flanks. The programme’s answer is exactly what the earlier essay’s analysis of the local term predicts. Close to a slender body the overspeed at a point is set mainly by the curvature of the area there, and a flat bottom of the right depth with its curvature concentrated at the ends puts the negative curvature where the wing is and the positive curvature — the shoulders — just clear of it. A cosine-squared dent spreads its curvature evenly and wastes depth in the middle, where a shallower flat bottom would already have been enough.

Half of the third was the shape #

The figure answers the first question directly. Cleared along the root chord alone, which was the earlier essay’s criterion, the single dent needs 31.0 per cent of the greatest cross-section — the earlier essay’s figure, recovered — and the free waist needs 16.2 per cent. Half of the third belonged to the shape. The other half belongs to the fuselage’s field and the rule together: a waist can only move overspeed, and moving 0.025 of the flight speed off a chord 0.15 of the body long, into shoulders that must fall outside it, takes a definite minimum of depth whatever the shape.

The chord at the fuselage’s side is only part of the story, because a wing’s root region extends outboard, where the fuselage’s field is weaker but not negligible: at one fuselage radius from its surface it is still two-thirds of its surface value. Cleared over a strip reaching that far out, an unswept root needs 39.8 per cent of the cross-section with a single dent and 24.8 with the free shape. The outer stations cost depth because a waist’s influence falls off faster with distance than the fuselage’s own does — a waist is a pair of opposite sources, and a pair’s field decays a power faster than a single one’s — so to reach the outboard points it has to be deeper at the body.

Sweep costs the free waist almost nothing #

A swept wing’s root region is a parallelogram rather than a rectangle: the chord at one fuselage radius outboard lies aft of the chord at the body by the radius times the tangent of the sweep. The figure draws the strip for a wing swept 30° and the two waists that clear it. The single dent has had to slide aft and widen to cover the strip’s aft corner, and it is 45.7 per cent deep. The free waist has simply lengthened aft, keeping its flat bottom, and is 24.3 per cent deep.

Across sweeps from nothing to 45°, the free waist’s depth stays between 24.3 and 26.9 per cent of the cross-section; the single dent’s rises from 39.8 to 52.6. The difference is the same geometric fact as before, applied to length rather than depth. A cosine-squared bump that must reach further has to be wider, and a wider bump is less curved, so it needs more depth to produce the same local overspeed at its centre. A flat-bottomed waist that must reach further is simply longer, and its depth is set by the strip’s width in the direction the field decays — outboard — which sweep does not change.

The earlier essay asked whether the waist for a swept root could place its shoulders clear of both the leading edge ahead and the tailplane behind. At 30° the free waist begins 0.06 of the half-length ahead of the root’s leading edge and ends 0.10 behind the strip’s aft corner, and its shoulders sit just beyond those ends; at 45° it ends 0.38 of the half-length behind the wing’s mid-chord. A tailplane on an aircraft of these proportions sits near the tail, some 0.8 of the half-length aft, so the aft shoulder lands on plain fuselage with room to spare.

The strip, cleared #

The figure shows the two waists doing their job differently. The plain fuselage puts up to 0.025 of the flight speed on the root chord and less outboard. The single dent overshoots in the middle of the strip: to bring the strip’s ends and corners to zero it has to make its centre strongly negative, which is depth spent on nothing. The free waist holds the perturbation near zero across the whole strip — that is what the programme is for — and is exactly as deep as the hardest point requires.

What the cheaper waist costs #

The zero-integral rule collects its due. Whatever a waist removes from the root it puts on the fuselage ahead and behind, and the steeper the waist’s sides, the more concentrated the shoulders. For the 30° strip the single dent raises the fuselage’s greatest overspeed from 0.025 to 0.051; the free waist raises it to 0.075, three times the plain body’s. Both are still far below the 0.217 at which the fuselage’s surface would reach sound at Mach 0.8, because, as thickness costs a fuselage far less established, a body of revolution is so much further from its own critical Mach number than a wing is.

So the trade has three terms and the programme minimised only one. The free waist takes half the depth of the single dent and puts half as much again on its shoulders, and whether that is a good trade depends on what the shoulders land on. On a bare fuselage it is. Beside a nacelle, a fairing or a canopy it might not be, and a designer who cared would add those places to the programme as more inequalities — which is the point of posing the problem as a programme rather than a family of shapes.

Two rules the waist does not break #

The free waist is still a closed set of sources, and the programme cannot change that: integrated along a line parallel to the axis, its perturbation for the 30° strip sums to four parts in $10^5$ of its absolute integral, as the rule requires. Nothing the programme chooses removes overspeed; it chooses only where the overspeed goes, and the steep-sided answer puts it closer to the wing than the gentle dent does.

And the waist is still a local statement about a subsonic field. The drag that can only see one curve is the transonic area rule, which asks the fuselage to give back the wing’s whole cross-section so that the total area varies smoothly and the wave drag is least. The earlier essay found that the subsonic waist needs about a sixth of that. The free waist needs about half as much again less. They answer different questions: the area rule smooths what the far field sees; the waist keeps one near-field overspeed off one chord. That the second is so much cheaper is because it can put what it moves somewhere harmless, which the far field’s total cannot.

Why the answer is flat-bottomed #

The flat bottom is not an accident of the basis, and the reason says something about what an optimum of this kind looks like. A linear programme’s solution sits on its constraints: at the optimum, some of the inequalities hold with equality and the rest are slack. Here the binding ones are a handful of points of the strip where the perturbation is exactly zero — the strip’s ends and outer corners, where the fuselage’s field is strongest relative to what the waist can do there — and the peak-depth constraint, which binds along the whole flat bottom. Every point of the bottom is at the peak depth because making any part of it shallower would let a binding point go positive, and making any part deeper would raise the peak. The waist is as shallow as it can be everywhere it is not forced to be deep, which is a definition of flatness.

The cosine-squared dent could not be flat, and its shape charged it twice. It wasted depth at its centre, where the strip needed least, and its gentle flanks put weak negative curvature under the strip’s ends, which needed most. The earlier essay’s observation that curvature, not depth, sets the local overspeed is exactly what the programme exploits: it buys curvature where it is needed by making the sides steep. The inside a flow does not decide is the reminder that goes with any such result — many interior source distributions give the same exterior field — and here the programme is choosing among bodies rather than interiors, so the body it returns is the one the flow outside it sees.

What was checked #

Four checks tie this calculation to the one before it and to itself. One piece of the basis, scaled to a fifth of the cross-section, reproduces the earlier essay’s cosine-squared dent’s field at three points to three parts in $10^{10}$. The single dent, searched over its centre and width, needs 31.0 per cent to clear the root chord, the earlier essay’s number. The free waist for the 30° strip is still a closed set of sources, its perturbation integrating along the body to four parts in $10^5$ of its absolute integral. And on a strip four times denser than the programme was given, its perturbation reaches at most $10^{-3}$ of the flight speed between the programme’s points, against the 0.025 it cancelled. The simplex method itself was checked on a textbook programme with a known optimum. The tests refuse a sweep of 75°, a tolerance of zero and a negative one.

What the programme cannot show #

The wing’s own field. The strip is cleared of the fuselage’s perturbation, on the earlier essay’s argument that an elliptic section’s own overspeed is uniform along its chord and the root is no worse than the isolated wing when the fuselage adds nothing. A swept wing’s root is not an isolated section: the sweep a root does not have found the isobars at a 35° wing’s root swept only fourteen degrees, the crest of the section’s own overspeed moving aft and its peak growing. A waist designed against the wing’s actual root pressures would weight the strip’s aft part more heavily.

Linear theory to Mach 0.8. Slender-body theory with the Prandtl–Glauert factor is linear. At Mach 0.8 with a 12-per-cent root section the flow is close enough to sonic that the linear superposition of fuselage and wing is where the pocket on top of the wing begins to form, and a design meant for the pocket’s own boundary needs a transonic calculation. The wind a swept wing feels is the other half of that: its normal component, which is what sets the section’s approach to sonic, is lower than the flight speed by the cosine of the sweep.

The fuselage’s interior. The line of sources on the axis is the part of the flow inside the body continued inwards; a waist a quarter of the cross-section deep is still slender enough for it, and one half deep, like the 45° single dent, is at the edge of what the theory’s local term describes accurately.

Who worked it out #

Slender-body theory is Munk’s and von Kármán’s, from the 1920s and 1930s, and its use to shape fuselages near wings is Küchemann’s, whose waisting rules for swept-wing junctions from the 1950s are the practice this calculation is a model of. Whitcomb’s area rule of 1952 is the transonic counterpart. Posing aerodynamic shaping as a linear programme — a design that satisfies inequalities at chosen points with the least of some cost — is one of the earliest forms of numerical optimisation in aerodynamics, and the simplex method is Dantzig’s, from 1947.

Still open: a waist against the root’s own pressures #

The programme cleared the fuselage’s perturbation, which is the right target only if the wing’s root behaves like an isolated section. It does not, and the next calculation joins the two: the wing’s own root overspeed from a vortex-lattice and thickness-source solution of the swept wing, the fuselage’s from slender-body theory, and a programme that asks for the least waist that keeps their sum below the isolated wing’s mid-span value everywhere on the root strip. That asks whether the waist that restores a swept root is deeper than the one that clears the fuselage alone, because the root’s own crest has moved aft to where the fuselage adds most, or shallower, because a root with less sweep is already a thinner-looking section.