The drag that can only see one curve

A body with no lift and a moment anyway solves the slender body by applying the wall condition at a place where there is no wall, and ends by naming the result that follows: if the disturbance a slender body makes depends only on its cross-sectional area distribution, then so does its wave drag, and two aircraft with the same area distribution have the same wave drag however differently they are shaped.

That is Whitcomb’s area rule. It is usually presented as a discovery, which it was, and as a surprise, which it is only if the slender-body argument has not been made first.

It is worth keeping both of those readings. As a consequence of a linear theory it is nearly immediate, and it had been available since von Kármán wrote the wave-drag integral in 1935. As a thing to do to an aeroplane it was not obvious to anybody for seventeen years, and when it was done the transonic drag rise of a fighter fell by a quarter. The gap between a theorem and its use is the whole of engineering, and this is an unusually clean example of it.

The integral, and the form that makes it obvious #

A slender body disturbs the flow like a line of sources of strength $A'(x)$, and nothing else about it reaches the far field. Von Kármán’s wave-drag formula follows:

$$D = -\frac{\rho U^2}{4\pi}\iint A''(x_1)\,A''(x_2)\,\ln|x_1 - x_2|\;\mathrm{d}x_1\,\mathrm{d}x_2.$$

In that form the sign is not obvious, the singularity on the diagonal is a nuisance, and nothing about the structure of the answer can be read off. Substituting $x = \tfrac{L}{2}(1-\cos\theta)$ and expanding the slope of the area distribution in a sine series, $A'(\theta) = \sum A_n\sin n\theta$, the double integral collapses to

$$D = \frac{\pi\rho U^2}{8}\sum_{n\ge1} n\,A_n^{2}.$$

Everything follows from that line. The drag is positive for any body whose cross-section varies at all, it is a sum of independent contributions from the harmonics of the area distribution, and each harmonic is weighted by its own index — so a wiggle at twice the frequency costs twice as much for the same amplitude.

The substitution that produced it deserves a note, because it is the same one that appears wherever a problem on a finite interval has square-root behaviour at both ends. A thin aerofoil’s camber line is expanded in the same variable, and so is a lifting line’s span loading; in every case the change of variable clusters the sample points towards the ends, where the behaviour is singular, and turns an integral equation into an orthogonal expansion.

The orthogonality is what makes the result a statement about design rather than a formula. Because the harmonics do not interfere, a change to the area distribution can be decomposed, its effect on the drag read off term by term, and the terms that cost something separated from the ones that do not. A formula that mixed the harmonics would give the same numbers and would say nothing about what to do.

One more feature is worth marking. The drag contains no Mach number. Everything about the flow’s compressibility has gone into the statement that the disturbance is a line of sources, and what is left is geometry and a dynamic pressure — which is why the rule can be stated as a fact about a shape and applied across a range of speeds.

The volume lives in exactly one harmonic #

The second consequence is the one that makes the whole thing a design rule rather than an observation.

Integrating the area distribution for the volume, in the same variable, gives

$$V = \frac{\pi L^2}{16}\,A_2.$$

The volume depends on the second harmonic and on nothing else. Every other harmonic adds drag and carries no volume whatever.

That single sentence contains the minimum-drag body as a corollary requiring no calculus of variations: the least-drag shape for a given length and volume is the one whose area distribution has a second harmonic and nothing else, which is Sears and Haack’s. The harmonic sum computed here puts 100 per cent of that body’s drag in that one harmonic, which is what it means to say the body is optimal.

Why the second and not the first is worth one sentence, because a reader may expect the lowest mode to be the cheap one. The first harmonic, $A' \propto \sin\theta$, corresponds to an area distribution that does not return to zero at the tail: it describes a body with a base rather than a closed one. Closing the body removes it, and the second harmonic is then the lowest available — so the “cheapest” mode is set by the requirement that the body be closed, not by the drag formula.

That also makes the volume relation less mysterious than it looks. The second harmonic is the only mode of a closed body’s area distribution whose integral does not cancel; every higher one adds area somewhere and removes it somewhere else in equal measure. Carrying volume and being smooth are the same requirement, and the minimum-drag body is the shape in which they coincide exactly.

Why a bump is expensive, and how expensive #

The weighting by $n$ is the quantitative heart of the rule and is worth turning into a statement about lengths rather than about harmonics.

A feature of streamwise extent $\ell$ on a body of length $L$ excites harmonics up to about $n \sim L/\ell$. Since each costs $n$ times its squared amplitude, and the amplitudes needed to build a feature of a given height fall as $1/n$, the drag of a feature of height $\delta A$ and extent $\ell$ goes as $(\delta A)^2 \times (L/\ell)$. A short bump is worse than a long one of the same height, in inverse proportion to its length.

That is the practical content, and it is what makes the rule matter for a wing rather than for, say, a canopy. A wing’s chord is a substantial fraction of an aircraft’s length, so its bump is long and its penalty is not catastrophic; a short, deep feature — an engine nacelle, a weapons bay, a canopy on a small aircraft — is far worse per unit of area added, in the way a sudden change of section is always expensive. The cure is the same in every case and it is the same cure: put a compensating dent somewhere the drag can see it, which is anywhere at all, since the drag sees only the sum.

It also says which features can be ignored. A feature contributing an area change of one per cent over a tenth of the length costs about $10^{-4}$ of the clean body’s drag, which is nothing; the same one per cent concentrated into a hundredth of the length costs ten times as much and is still nothing. The rule is about wings, and the reason is that a wing is the only thing on an aircraft whose cross-section is comparable with the fuselage’s.

And the wing, which is where the rule is #

A wing adds cross-sectional area over its own chord, so it puts a bump into $A(x)$. A bump is a lot of high harmonics, every one of them weighted by $n$, and the drag rises sharply.

The cure is to take the bump out again. Since the drag sees only $A(x)$, and since the wing’s contribution to $A(x)$ is known, cutting an equal amount out of the fuselage over the same stations restores the clean distribution — and the drag with it.

The recovery is total and not partial, which is worth stating plainly because the usual account of the area rule presents it as an improvement. Within this theory, a waisted body and a wing have exactly the wave drag the clean body had. Nothing has been traded: the wing is carried for nothing, the volume is unchanged because waisting removes from the fuselage exactly what the wing adds, and the only cost is that the fuselage is now a strange shape.

The limit is geometric rather than aerodynamic. The cut cannot be deeper than the fuselage is thick, so once the wing’s peak section exceeds the fuselage’s there is nothing left to remove, the distribution cannot be returned to the clean one, and the recovery falls away — to two thirds at one and a half times, and to a third at two and a half.

That limit is where real aircraft live, which is the honest qualification on the paragraph above. A transport’s wing carries a cross-sectional area comparable with its fuselage’s, and a fighter’s exceeds it — so the complete recovery drawn in the figure is the small-wing corner of the curve, and what is available in practice is a substantial fraction rather than all of it. Whitcomb’s measured reduction in transonic drag rise was of order a quarter to a third, which is the right order for a configuration sitting where the curve has begun to bend.

The other thing the geometric limit explains is why the waist is put where it is. If the fuselage cannot give up enough area at the wing, the distribution can still be smoothed by adding area elsewhere — bulges ahead of and behind the wing, which is what the rear-fuselage bulges on several 1950s aircraft are. Adding volume to smooth a distribution is a strange-looking thing to do and it follows directly from the harmonics: the second harmonic is free, and moving area from where it makes high harmonics to where it makes the second one is a gain.

What is actually being said about shape #

It is worth being precise about the claim, because it is stronger than it sounds and is easy to overstate.

The claim is not that shape does not matter. It is that the wave drag of a slender body at supersonic speed is a functional of one scalar function, and that two bodies agreeing in that function agree in that quantity. Everything else about the two bodies — their skin friction, their lift, their base drag, their stability, whether they can be built — differs freely.

That is the same structure as what a far field remembers: three numbers survive the journey to infinity and nothing else about a body does, so two shapes with nothing in common make the same flow a few radii away. Here one function survives, and the survivor is what the drag is computed from.

There is a caution in the comparison as well as an analogy. A far-field theorem says what survives at infinity, and a drag is computed at infinity, so the two match. A pressure distribution is not computed at infinity, and the area rule says nothing whatever about it: two aircraft with the same area distribution have the same wave drag and quite different pressures on their surfaces, different loads on their structures, and different places where their flow separates.

A rule that collapses a shape to a function has thrown away everything the function does not carry, and the discipline is in remembering what that was. Whitcomb’s rule is a statement about one number in a drag budget, and a designer who waists a fuselage has to check everything else afterwards — which is, in practice, why a waisted fuselage is a difficult structure and why the coke-bottle shapes of the 1950s were abandoned once engines were powerful enough to make the trade unnecessary.

What the picture cannot show #

The body is slender and the theory is linear. The source-line representation needs the body’s slope to be small everywhere, and a wing’s leading edge — where nothing turns a sharp corner — is not. The rule works better than that restriction suggests, which is a fact about aircraft rather than about the theory.

The area is taken on normal cuts, and supersonically it should not be. At Mach numbers above one the correct cut is oblique, at the Mach angle, and the area distribution is then a function of the Mach number and of the roll angle round the body — so there is a different area rule for every direction and the design has to compromise between them. The normal-cut rule used here is the transonic one, valid near Mach one where the Mach angle is ninety degrees.

There is no lift, so none of the drag a wing pays for it appears. A lifting body has a wave drag due to lift as well as due to volume, it is a different functional, and the area rule says nothing about it.

And nothing here is viscous or transonic. The drag computed is the inviscid wave drag of a linearised supersonic flow. What Whitcomb measured was the transonic drag rise, which a shock produces and which the linear theory does not contain at all. That the rule derived for one applies to the other is an experimental fact, and a remarkable one.

And the body is closed and pointed at both ends. The harmonic expansion assumes the area distribution starts and ends at zero with zero slope; a real aircraft has a blunt base with a jet coming out of it, and the base is treated by pretending the jet is part of the body. How good that pretence is depends on the engine setting, which makes the wave drag of a real configuration depend on its throttle.

Who found it, and when #

Von Kármán gave the wave-drag integral in 1935 and Sears and Haack the minimum-drag body at the end of the 1940s, independently. Whitcomb’s contribution, in 1952, was not the formula: it was the recognition that the formula’s consequence — that a wing and a body with a smooth combined area distribution pay no more than the body alone — was a thing that could be arranged, and the wind-tunnel programme that showed the transonic drag rise falling when it was. Hayes’ oblique-cut generalisation, of the same year, is what made it usable supersonically.

The surprising connection is with a rule about a completely different kind of drag, in the same collection. The induced drag of any system of lifting surfaces depends on one cross-section of its wake and on nothing else — Munk’s theorem, which is why a winglet works and why the vertical position of two wings does not matter. Both are far-field theorems and both have the same shape: a drag that looked like a property of a body turns out to be a property of one function measured far away from it, so anything that leaves that function alone is free and anything that changes it is paid for. The difference is which function — a span loading in a Trefftz plane, or an area distribution along a line — and the difference in which one explains why a wing may be put anywhere vertically and may not be put anywhere longitudinally.

Still open: what a lifting area rule would look like #

The rule computed here is about volume, and an aircraft’s supersonic wave drag has two parts.

Wave drag due to volume is what the area distribution governs, and it is what waisting removes. Wave drag due to lift is a separate functional — of the lift distribution rather than the area distribution, and quadratic in the lift — and the two are known not to be independent: the same oblique cuts that define the area rule define an equivalent statement for the lift, and a configuration optimised for one is not optimised for the other.

What has not been settled, and is a live question in supersonic transport design, is whether there is a combined functional with a single minimiser. The two individual minimum problems are both solved: Sears and Haack for volume, and a known loading for lift. Their sum, at a given volume and a given lift, is a quadratic functional in a pair of coupled distributions, and its minimiser is a linear algebra problem rather than a search — one that could be set up with the harmonics used here, since the volume rule is already diagonal in them. Whether the lift rule is diagonal in the same basis is the question, and if it is, the combined optimum is one matrix inversion away from a problem that has been approached by search for fifty years.