A fuselage lowers its wing root's critical Mach number

Thickness costs a fuselage far less than a wing set two exact flows side by side: a prolate spheroid, standing for a fuselage, and a plane ellipse of the same thickness ratio, standing for a wing section. A body of revolution with a thickness ratio of a tenth speeds the air at its side up by 2.1 per cent, as a sphere’s flow would lead one to expect; the section speeds it up by ten. The difference is not a detail of shape: in three dimensions the air can go round a body as well as over it, and the overspeed is quadratic in thickness where the section’s is linear. The fuselage, as a result, reaches the speed of sound at a far higher flight Mach number than any wing it carries. The essay ended by noting that on an aeroplane the two are not side by side but joined, and that at the joint the small disturbance is added to the large one.

This essay does the addition. It asks how far the fuselage’s two per cent moves the wing root’s critical Mach number — the flight speed at which the air somewhere on the root first reaches the speed of sound, and so the speed at which the pocket on top of the wing begins — how far along the span the effect reaches, and what it is worth in the terms a designer thinks in, which is thickness.

Adding two small disturbances #

In subsonic flow at small disturbances the velocity perturbations of two bodies add — superposition holds for velocities, though not for pressures — and the calculation is the sum of two that are already exact. The fuselage is a prolate spheroid of thickness ratio $\tau_f$, a tenth for a fineness ratio of ten; the essay before computed its incompressible potential exactly, in spheroidal coordinates. Its compressible perturbation at a flight Mach number $M$ follows from Göthert’s rule: the incompressible perturbation of the same body with its lateral dimensions shrunk by $\beta = \sqrt{1 - M^2}$, divided by $\beta^2$. The wing is a plane elliptic section of thickness ratio $\tau_w$, whose peak overspeed is $\tau_w$ at low speed and $\tau_w/\beta$ at Mach $M$ by the Prandtl–Glauert rule.

At the root, where the section meets the fuselage’s side, the two perturbations are added. The linearised pressure coefficient at the section’s crest is then minus twice the sum, and the critical Mach number is where it meets the sonic pressure coefficient — the correction that predicts its own failure, applied at the point where it fails first. Both pieces were checked separately: the fuselage’s surface speed at its equator reproduces Lamb’s coefficient through Göthert’s rule to two parts in $10^9$, its far field is a doublet whose strength is the body’s volume times one plus its added-mass coefficient, and with no fuselage the root’s critical Mach number is exactly the one the essay before computed for the section alone.

A long body spreads its disturbance like a line #

The first figure is the fuselage’s contribution along the span. At the fuselage’s side it is 2.1 per cent of the flight speed at low speed. A sphere’s overspeed would fall as the cube of the distance — to an eighth two radii out. The fuselage’s falls only to two-thirds of its surface value two radii out, a quarter at five, a tenth at ten.

The reason is the fuselage’s length. Seen from a point beside the middle of a body ten times longer than it is wide, the body looks less like a point than like a long line of sources and sinks, and the field of a line falls as the inverse of distance, not its cube, until the distance becomes comparable with the body’s length. Three dimensions are kinder to a body’s surface speed, because the air can go round; they are less kind to its reach, because a long body’s disturbance is spread along its length and arrives at the wing from all of it.

Compressibility makes the fuselage’s contribution grow, but slowly. By Göthert’s rule its overspeed is divided by $\beta^2$ and its thickness enters thinned by $\beta$, inside a logarithm, and the two nearly cancel: between Mach 0 and 0.8 the overspeed at its side grows by eighteen per cent, to 2.5 per cent. The section’s grows by two-thirds over the same range.

What the side-by-side comparison could not see #

The essay before compared the two bodies by their own surfaces: how fast the air runs over the fuselage, how fast over the section. By that measure the fuselage is a fifth of the section’s disturbance at low speed and less at high speed, and it was fair to call it negligible. The comparison that matters at a joint is different. The section’s crest is not on the fuselage’s surface but beside it, and what arrives there is the fuselage’s field, not its surface speed; and what the fuselage’s field is added to is not zero but the section’s own twelve per cent, already close to the edge of what the air can do before it goes sonic. A small quantity added to a large one near a threshold is not small in its consequences, and the critical Mach number is a threshold.

The arithmetic of that is worth stating once. Near the critical Mach number, each extra per cent of overspeed at the crest costs a definite amount of critical Mach number — about half a hundredth for a 12-per-cent section at Mach 0.8, where the sonic pressure coefficient and the section’s own pressure coefficient are converging at a shallow angle. Two per cent of fuselage overspeed, amplified a little by compressibility, is therefore a hundredth and a third. The fuselage’s share of the root’s disturbance is only a sixth at low speed and an eighth at Mach 0.8, but the whole of that share is spent at the threshold.

The root goes critical first #

The second figure is the answer at the root. A 12-per-cent section alone reaches sound at a flight Mach number of 0.809. Beside a fuselage of fineness ten its root reaches it at 0.796: a hundredth and a third lower. Beside a stubbier fuselage of fineness 6.7, at 0.784 — two and a half hundredths lower. The penalty is larger for thin sections, since two per cent is a larger share of a small overspeed: an 8-per-cent root beside the fineness-ten fuselage loses 0.016, a 16-per-cent one 0.012.

A hundredth of Mach number sounds small, and on the scale that matters to a transport aircraft it is not. The drag rise that follows the critical Mach number is steep, an airliner cruises within a few hundredths of it, and a hundredth at Mach 0.8 is about eight knots of cruise speed or its equivalent in thickness and weight.

A penalty measured in thickness #

The third figure converts the penalty into the currency designers trade in. For each fuselage it finds the isolated wing that would reach sound as early as the root does. A fuselage of fineness ten costs a 12-per-cent root section as much as 13 per cent more thickness — the root behaves as though it were 13.5 per cent thick. A fineness of 6.7 costs a quarter more thickness, and a stubby fuselage of fineness five more than a third. The thinner the wing, the larger the relative cost: an 8-per-cent section beside the fineness-ten fuselage behaves as though it were 9.5 per cent thick, a fifth more.

That is a large price for a body whose own disturbance the essay before found negligible, and it is the reason the wing root is the most worked-over part of a transport aircraft’s wing. The root has to carry the largest bending moment and wants to be thick for its structure; the fuselage makes it behave as though it were thicker still. Designers answer by giving the root its own section, its own twist and a fairing, and the answer is local because the problem is.

Along the span #

The fourth figure follows the penalty outwards. At the root the fineness-ten fuselage costs 0.013; two fuselage radii out it still costs 0.010; five radii out, 0.005; ten radii out, 0.002; and it takes some twenty radii to fall below a thousandth. On a transport aircraft whose fuselage is a tenth of its span, twenty fuselage radii is the whole of the wing: the inboard half of the span sees a measurable part of the root’s penalty, and the root is where it is largest, not the only place it acts.

The shape of the curve is the first figure’s slow decay, carried through the arithmetic of the critical Mach number. It is the reason a root modification — a change of section, of twist or of sweep near the fuselage — is blended over a considerable part of the span rather than applied at the joint alone.

An airliner’s root, in numbers #

A transport aircraft of the size that carries a hundred and fifty passengers has a fuselage of fineness about ten and a wing whose root section is fourteen or fifteen per cent thick, sized by the bending moment it carries. Alone, a 15-per-cent section reaches sound at Mach 0.784 in this linear estimate; at the root of a fineness-ten fuselage, at 0.771. The root behaves as though it were 16.6 per cent thick. Unswept, such a wing could not cruise at Mach 0.78 at all, and the aircraft’s sweep and supercritical sections are what let it; but the root’s penalty is carried through both of them, since the fuselage’s overspeed is there whatever the section does.

The consequence is a familiar feature of the aircraft rather than a number on a chart. The inboard wing is given a thinner-looking, more strongly cambered section than the structure alone would want, it is faired into the fuselage with a large fillet that changes the cross-section locally, and its sweep is often increased at the root to restore the isobars’ angle. Each of those is paid for somewhere — in structure, in wetted area, in trim — and each exists because the fuselage’s two per cent arrives at the one place on the wing that can least afford it.

Moving the wing along the body buys little #

The fifth figure asks whether the wing could be put somewhere the fuselage’s overspeed is smaller. Along the middle of a long body the overspeed barely changes, so moving the wing’s thickest section forward or back along the middle half of the fuselage changes the root’s critical Mach number by a few thousandths. Only near either end, where the fuselage is thinner and its surface speed lower, does the root recover — under a hundredth at four-fifths of the way to the nose or tail, where no wing can go.

What the figure points to instead is the fuselage’s own shape at the wing. The overspeed at the root is the fuselage’s response to its own cross-section near there; make the cross-section smaller where the wing is — a waist — and the overspeed there falls. That is the local, incompressible beginning of the argument the area rule makes globally at transonic speed, where the whole aircraft’s cross-sectional area has to change smoothly and the fuselage is narrowed exactly where the wing adds area.

Sweep, and why the root loses it #

A swept wing raises its critical Mach number because the wind a swept wing feels is only the component normal to its leading edge. That argument is made for an infinite wing, whose isobars are swept with it. At the root the isobars cannot follow the sweep into the fuselage; they straighten, the root section sees more of the full flight speed than the outboard sections do, and the sweep’s benefit is partly lost there. The fuselage’s overspeed computed here is added to that loss, and the two together are why a swept wing’s root is the first part of it to go supersonic unless it is designed otherwise.

The same arithmetic wherever a body meets a surface #

The root is the largest instance of something that happens at every junction on an aircraft. An engine nacelle slung under a wing is a body of revolution a fraction of a chord below the section, and its overspeed is added to the wing’s lower surface — where the wing’s own overspeed is smaller, which is one reason nacelles hang there and not above — and to the pylon that joins them, whose own thickness adds a third term. The channel between a nacelle, its pylon and the wing can reach sound before either the wing or the nacelle would alone, which is exactly the root’s problem at a smaller scale and closer quarters.

A tailplane’s root is the opposite case, and instructive for it. The tail sits beside the fuselage’s tapering tail cone, where the body’s cross-section is shrinking, and there the fuselage’s perturbation is a deceleration rather than an overspeed: the fifth figure’s curve, followed further towards the fuselage’s end, turns into a benefit. The tailplane’s root reaches sound later than the tailplane would alone, and the fuselage that penalises the wing flatters the tail.

What was checked #

The sixth figure is the ledger. The fuselage’s surface speed at its equator, from the exact potential through Göthert’s rule, matches Lamb’s added-mass coefficient of the thinned body divided by $\beta^2$ to two parts in $10^9$ in three cases. Forty half-lengths out, the perturbation times the cube of the distance matches a doublet whose strength is the body’s volume times one plus its added-mass coefficient, to two parts in a thousand. And with the fuselage removed, the root’s critical Mach number is the isolated section’s from the essay before, exactly.

What superposition leaves out #

The wing’s effect on the fuselage. The wing’s thickness displaces the air round the fuselage as the fuselage’s does round the wing, and its lift puts upwash ahead of the fuselage’s nose and downwash behind; the calculation adds the fuselage’s field to the wing’s and not the reverse.

Nonlinearity near sound. Linear theory is least trustworthy exactly where the critical Mach number lives, and the Prandtl–Glauert rule overstates the section’s pressure near Mach one; the differences between the curves are more reliable than their positions.

The junction’s own flow. Where a wing meets a fuselage the boundary layers of both meet in a corner, the stagnation region at the root’s leading edge rolls the fuselage’s boundary layer into a horseshoe vortex, and the corner flow can separate before either surface would alone. None of that is in an inviscid superposition.

A real fuselage. A spheroid has no parallel section; a transport fuselage is a long cylinder with a nose and a tail, whose overspeed in the middle is lower and at its shoulders higher than a spheroid’s.

The convention the numbers depend on #

The fuselage’s thickness ratio is its diameter over its length, and its fineness ratio the reciprocal. Distances along the span are from the fuselage’s axis, in fuselage radii; positions along the fuselage are in half-lengths from its middle. The wing section is a plane ellipse of the stated thickness ratio, its critical Mach number the flight Mach number at which the linearised pressure coefficient at its crest reaches the sonic value, with $\gamma = 1.4$.

Who found it, and when #

Göthert gave the rule for compressible flow past three-dimensional bodies in 1940. The interference between wing and fuselage at high subsonic speed was worked on through the 1940s and 50s; Küchemann’s analysis of the swept wing’s root and centre section, from the early 1950s, is the basis of how roots are designed, and Whitcomb’s area rule of 1952 turned the fuselage’s shape at the wing into the transonic designer’s main tool.

Still open: the waist that cancels it #

The fifth figure pointed at a waist, and the size of that waist is a calculation this one sets up. The fuselage’s overspeed at the root comes from its source distribution, which is its rate of change of cross-sectional area; a waist is a change in that distribution, local to the wing. The next calculation finds, with slender-body theory, the smallest local reduction of the fuselage’s cross-section that cancels its overspeed at the root, and compares the area removed with the wing’s own cross-sectional area there — asking whether the subsonic, local cancellation needs the same area the transonic area rule would remove, or much less.