Thickness costs a fuselage far less than a wing

Three dimensions are kinder sets a sphere beside a cylinder and finds that the sphere speeds the passing flow up by half the stream speed where the cylinder manages the whole of it: 1.5 against 2. The reason given there is that fluid meeting a sphere has a second direction to go round it, so less has to squeeze past any one meridian.

That is a factor of two between two blunt bodies, and it is the smaller half of the story. Stretch both bodies out along the stream until they are slender — the cylinder into a thin elliptical wing section, the sphere into a fuselage — and the factor does not stay at two. It grows without limit as the bodies get thinner, because the two pay for their thickness at different powers.

Two exact answers #

Both bodies have exact ideal-flow solutions, and that is what makes the comparison clean: no panels, no modelling, nothing to argue about in the numbers.

The plane ellipse of semi-axes $a$ along the stream and $b$ across it, thickness ratio $\tau = b/a$, has a surface speed that peaks over its middle at

$$\frac{u_{\max}}{U} = 1 + \tau,$$

exactly. A section ten per cent thick adds ten per cent; the extra speed is linear in the thickness.

The prolate spheroid of the same semi-axes — the ellipse spun about its long axis — has an exact potential in spheroidal coordinates, built from the Legendre function $Q_1$, and on any ellipsoid the surface speed is $1 + k$ times the component of the stream along the surface, where $k$ is the body’s added-mass coefficient for motion along its axis. Lamb gave it in closed form:

$$k = \frac{\alpha_0}{2 - \alpha_0}, \qquad \alpha_0 = \frac{2(1 - e^2)}{e^3}\left(\tfrac12\ln\frac{1+e}{1-e} - e\right), \qquad e = \sqrt{1 - \tau^2}.$$

At $\tau = 1$ the spheroid is a sphere, $k = \tfrac12$, and the peak speed is the sphere’s 1.5. At $\tau = 0.1$ it is 0.0207, and the peak speed is 1.021.

The closed form was not taken on trust. The spheroidal potential was built from its definition and differenced numerically: the flow through the body’s surface is zero to $8\times10^{-8}$ at fifteen points on three bodies, Laplace’s equation holds off the body to $4\times10^{-7}$, and the speed at the equator is $1 + k$ to $9\times10^{-8}$. The sphere’s one half is recovered to $6\times10^{-8}$ as the thickness ratio goes to one.

Quadratic against linear #

For a slender body of revolution Lamb’s coefficient has the asymptote

$$k \approx \tau^2\left(\ln\frac{2}{\tau} - 1\right),$$

which the exact value approaches at the rate the figure shows: 3.8 per cent above it at a tenth, 0.06 per cent above it at a hundredth. The body of revolution pays for its thickness quadratically. Halve the thickness of a wing section and its disturbance halves; halve a fuselage’s and its disturbance falls by nearly four.

The reason is geometric and it is the sphere’s reason taken to its limit. A plane section pushes the flow aside in one direction only, so the whole of its cross-section’s worth of fluid has to go over or under it, and the extra speed is proportional to the thickness. A body of revolution pushes fluid aside in every direction round its axis. What it displaces is its cross-sectional area, which goes as the thickness squared, and that displaced flux is spread round a circumference that grows with distance from the axis. So the disturbance near the body is quadratic in the thickness, and the logarithm is the signature of a source distributed along a line in three dimensions — the same logarithm that the area rule’s drag formula carries, for the same reason.

The exponents are easy to lose in the abstract, so here are four bodies of revolution of the proportions aircraft and airships actually have. At a fineness ratio of four — an airship hull, or a stubby business-jet fuselage — the fastest fluid moves 8.2 per cent faster than the stream and the peak linearised suction is $C_p = -0.163$. At six, 4.5 per cent and $-0.090$. At eight, 2.9 per cent. At ten, 2.1 per cent and $-0.041$. A plane section of the same proportions would add 25, 17, 12.5 and 10 per cent. Going from fineness four to ten divides a fuselage’s disturbance by four; it divides a wing section’s by two and a half. The body of revolution gets more out of each increment of slenderness, and that is the quadratic law showing through.

It is also why the shapes of long bodies were settled so early. An airship designer of the 1920s choosing between fineness ratios of four and six was choosing between an overspeed of eight per cent and one of four and a half; the pressure disturbance was already small either way, and the decision was made on structure, volume and skin friction instead. The ideal-flow disturbance of a slender body of revolution stops being the argument almost as soon as the body stops being blunt.

That is the connection with the wave drag of a slender body, which depends only on how its cross-sectional area is distributed along it and on nothing else about its shape. A slender body of revolution is, to the order that matters, a line of sources whose strength is the rate of change of its area. Its disturbance is set by the area, and area is thickness squared.

The pressure, and what compressibility does to it #

A pressure coefficient is twice the excess speed, with a sign, in the linearised theory that describes slender bodies well. So a wing section a tenth thick has a peak suction of $C_p = -0.2$ and a fuselage of the same proportions $-0.041$ — five times less.

Compressibility multiplies both, and not by the same factor. In the plane, the Prandtl–Glauert rule divides the incompressible pressure by $\beta = \sqrt{1 - M^2}$. In space the corresponding statement is Göthert’s rule of 1940: the pressure on a body at Mach number $M$ is $1/\beta^2$ times the incompressible pressure on the same body with its lateral dimensions shrunk by $\beta$. For a plane section the shrinking and one power of $\beta$ cancel and Prandtl–Glauert returns. For a body of revolution the thickness enters squared, so shrinking it by $\beta$ takes out $\beta^2$ — which cancels the $1/\beta^2$ in front, and leaves $\beta$ only inside the logarithm.

The figure puts numbers to it. Between Mach 0 and 0.8, a plane section’s peak suction grows by the factor $1/\beta = 1.67$. A body of revolution of the same thickness ratio grows by 1.23. Compressibility, which dominates the design of a wing at high subsonic speed, is a secondary effect on a slender fuselage — and it is secondary precisely because the fuselage is three-dimensional, not because it is thin.

Where sound first appears #

The consequence that matters in practice is where each body first has sonic flow on its surface. The local speed reaches the local speed of sound when the pressure coefficient falls to the sonic value $C_p^{*}(M)$, which is fixed by the isentropic relations. Set the linearised peak suction of each body equal to it and solve for $M$.

At a thickness ratio of a tenth — a slim wing section, a stubby fuselage — the section reaches the sonic condition at a free-stream Mach number of 0.83 and the body of revolution at 0.96. A fuselage of fineness ten has nearly the whole transonic range to itself before its surface goes supersonic, while the wing attached to it has been dealing with shocks for thirteen hundredths of a Mach number. The wing’s critical Mach number with lift is lower still, because lift adds its own suction to the thickness’s; this comparison is at zero incidence and puts the two on the same footing.

The affine stretch in Göthert’s rule is worth one more sentence, because it says what compressibility is doing physically. At Mach 0.8 the pressure disturbance from any source reaches farther across the stream than along it — the governing equation becomes Laplace’s only after the lateral coordinates are shrunk by $\beta$ — so a body at that Mach number behaves, as far as its own pressure is concerned, like an incompressible body that is thinner by the factor $\beta = 0.6$. A plane section thinned by $\beta$ loses only a factor $\beta$ of its disturbance, and the $1/\beta^2$ in front more than restores it. A body of revolution thinned by $\beta$ loses $\beta^2$, and the $1/\beta^2$ only puts it back; what is left over is the logarithm’s small change.

That ordering is why transonic design is overwhelmingly about wings. The fuselage’s contribution to the transonic problem comes not from its own surface but from where it meets the wing, where the two bodies’ disturbances add, and that is the junction the area rule reshapes. One curve for every thickness collapses the plane section’s transonic behaviour onto a single similarity parameter built from its thickness ratio; a body of revolution has its own such parameter, with the thickness squared in it, and for the same reason.

There is a rule of thumb that air can be treated as incompressible below a Mach number of about three tenths, and where it comes from is a statement about the free stream: below that, density changes by less than about five per cent anywhere the flow is brought to rest. For a body, the question is what its own disturbance does, and a slender body of revolution raises the threshold a long way. Its pressure changes with Mach number by twenty-three per cent between 0 and 0.8, where a wing section’s changes by sixty-seven — so for the fuselage, the incompressible calculation is still within a quarter at a Mach number where the wing’s is off by two thirds. That is also why the fuselage pressure figures of the exact theory are used with so little correction at the speeds airliners cruise at, while the wing’s are not used at all without one.

The same fact in other places #

The quadratic dependence turns up wherever a slender body of revolution meets a stream.

Its added mass along its axis is small. The coefficient $k$ is the same quantity as the extra mass the fluid lends a body accelerating along its length, as a fraction of the displaced mass. A sphere borrows half its displaced mass; a fineness-ten spheroid borrows two per cent. A torpedo or an airship accelerating along its axis carries almost none of the water or air it displaces.

The transverse coefficient comes from the same closed forms, with Lamb’s second constant in place of the first, and it runs the other way: from one half for the sphere up towards one, the full displaced mass, as the body thins. A body sliding sideways through the fluid pushes fluid round its girth exactly as a cylinder does, and a cylinder’s added mass is its whole displaced mass — so a slender body of revolution, seen from the side, is a cylinder, and seen from the front it is almost nothing.

Its moment does not share the saving. Turned slightly across the stream, the same body is treated by the flow as a much blunter object: its cross-flow added mass is nearly its full displaced mass, and that is what produces the destabilising moment of a body with no lift. Slenderness buys a small disturbance along the axis and nothing across it, which is why a fuselage needs a tail.

Its far field is a line of sources. What the far field remembers about a body is its first few multipoles, and for a slender body of revolution the leading one is the line of sources whose total strength is zero and whose distribution is the rate of change of area. The disturbance at the body’s own surface and the disturbance far away are the same quantity viewed at two distances.

What the picture cannot show #

Every figure here is at zero incidence and zero lift, which is the one condition in which a fuselage and a wing section are comparable at all. A wing’s job is to carry lift, and lift brings its own suction; a fuselage at incidence carries a cross-flow the axial solution does not contain. The comparison isolates the cost of thickness alone, which is its point and its limit.

The pressure figures are linearised: the pressure coefficient is taken as twice the excess speed, which is the right approximation for slender bodies and increasingly wrong near the rounded ends of the stubby ones, where the flow stagnates and the linear pressure formula overstates the recovery. The peaks, which are what the arguments use, are over the middle where the linear formula holds.

How much the linearisation costs at the peak can be said exactly, because the peak speed is exact. The full incompressible pressure coefficient there is $1 - (1 + k)^2$: for the fineness-ten body that is $-0.0418$ against the linearised $-0.0414$, a one per cent difference, and for the ten-per-cent section it is $-0.21$ against $-0.20$, five per cent. The linearisation is itself kinder to the body of revolution, for the same reason as everything else here: its disturbance is smaller, so the square of the disturbance that the linear formula drops is smaller still.

The ends are where it is not kind. A spheroid’s nose has a radius of curvature $b^2/a$ — one hundredth of its half-length at fineness ten — and the flow comes to rest on it and accelerates to nearly the stream’s speed within a few nose radii. There the disturbance is not small, the linear pressure formula is wrong, and the figures, which are drawn across the whole length, are telling the truth about the middle and only the shape of the truth about the ends.

Where the model stops #

Ideal flow. No boundary layer, no separation. A real fuselage has a thick boundary layer on its after-body and a wake behind it, and its pressure recovery over the rear is less than ideal flow’s, which makes the real rear pressure less symmetric than the figures show.

Linearised compressibility. Göthert’s rule, like Prandtl–Glauert’s, is a small-disturbance result and overstates the growth of suction as the flow approaches sonic. The critical Mach numbers are the linearised ones; a nonlinear calculation lowers both, the section’s more than the body’s.

Ellipses and spheroids. They are chosen because they are exact. Real sections and fuselages have other thickness distributions, and a section’s peak speed depends on its nose radius as well as its thickness ratio. The exponents — one for the section, two with a logarithm for the body — do not depend on the shape; the coefficients in front of them do.

The convention the numbers depend on #

The thickness ratio $\tau$ is the ratio of the semi-axes, $b/a$: for a spheroid that is diameter over length, the reciprocal of what aircraft designers call the fineness ratio, so a fineness ratio of ten is $\tau = 0.1$. For a wing section it is the thickness-to-chord ratio. Excess speeds are in units of the free stream, pressure coefficients are linearised as $-2(u - U)/U$, and the critical Mach number is the free-stream Mach number at which the linearised peak suction reaches the isentropic sonic value.

Who found it, and when #

The spheroid’s potential and its added mass are in Lamb’s Hydrodynamics, from the 1870s onward, and the theorem that the surface speed on an ellipsoid is $1 + k$ times the stream’s tangential component follows from the ellipsoid’s special property that its interior potential is linear. The slender-body asymptote is Munk’s, from his airship work of the early 1920s. Göthert published the three-dimensional compressibility rule in 1940, and the observation that it lets a slender body of revolution keep its subsonic character much closer to sonic speed than a wing is the reason fuselages were not the transonic problem the wings were.

Still open: what the junction adds #

The two bodies here were set side by side. On an aeroplane they are joined, and at the junction the wing’s disturbance and the fuselage’s add. Near a wing root the fuselage’s overspeed adds a per cent or two to the velocity the wing already sees, which at high subsonic speed moves the wing root’s critical Mach number down, and the wing’s upwash and sidewash reach round the fuselage’s flank.

The calculation that would follow superposes the spheroid’s exact field on a wing section’s at the junction — the simplest version is the spheroid’s velocity field evaluated at the wing root’s position, added to the section’s own overspeed — and asks how far the combination moves the wing root’s critical Mach number below the isolated wing’s. That is the local, incompressible start of what the area rule does globally, and it is where the fuselage, which barely disturbs the air on its own, does most of its disturbing.