Busemann's biplane has to be flown past its design point

Drag with nothing to rub traced supersonic wave drag to its source — energy carried away to infinity along the waves a body sends out — and then described the body that sends none. Adolf Busemann’s biplane of 1935 is two half-diamond sections facing each other across a channel, flat on their outer sides, so that each element’s leading-edge compression runs across the channel and arrives at the other element’s shoulder, where the surface turns away and makes an expansion of the same strength. The two cancel, nothing leaves the channel, and the pair carries its thickness at no wave drag at all. The essay named three reasons the biplane is not an aeroplane and computed none of them. This one computes two: what happens away from the design Mach number, and what happens on the way to it.

The cancellation, drawn #

In linear supersonic theory disturbances travel along Mach lines at the angle whose tangent is $1/\beta$, $\beta = \sqrt{M^2 - 1}$. The elements here are each three per cent of their chord thick, with their peak at mid-chord, and their inner faces start a gap of 0.3 chords apart. A wave leaving the upper leading edge crosses the gap in a horizontal distance $\beta G$, and it arrives exactly on the lower element’s shoulder when $\beta G = c/2$: the design Mach number is $\sqrt{1 + (c/2G)^2} = 1.944$, set by the geometry and nothing else.

At that Mach number the drawing is the whole argument: each warm line from a leading edge meets the opposite shoulder, where the blue line of the shoulder’s expansion leaves. At Mach 2.6 the Mach lines are flatter, the compressions land on the rear faces behind the shoulders, and the expansions they should have cancelled travel on unopposed. The geometry that cancels at one Mach number cannot cancel at another, because the Mach lines’ angle is the only thing that moved.

Why a level face has no drag #

The pressure on the inner face makes the cancellation quantitative. A half-diamond alone, in Ackeret’s linear theory, carries a pressure coefficient of $2\varepsilon/\beta$ on its front face, which leans into the stream, and $-2\varepsilon/\beta$ on its rear face, which leans away: +0.072 and −0.072 at the design Mach number. The pressure on the front face pushes back and the suction on the rear face pulls back, and both are drag.

In the biplane at design the compression from the other element arrives at the shoulder and raises the rear face’s pressure by twice the half-diamond’s own suction, to +0.072. The face is now at one pressure all along. A face at one pressure has no drag, because its drag is the pressure times the integral of its slope, and the slope of a face that ends at the height it began at integrates to zero — the push on the front half and the push on the rear half are equal and opposite. That is the mechanism in one line, and it is why the cancellation cannot be partial in a useful way: off design the arriving wave lands on the wrong part of the face, and some of the rear face is left in suction.

The drag on either side #

Marching the full set of reflections across the channel gives the drag at every Mach number. Between the elements the disturbance is a sum of waves running up the Mach lines and waves running down them, each element’s surface sends one family and reflects the other, and the recursion that results has a delay of $2\beta G$: the time a wave takes to cross the channel and come back. Every reflection, however many, is in it.

The drag is zero at 1.944, and rises on both sides. At Mach 1.7 it is 0.18 of the drag of a single diamond carrying the pair’s combined thickness; at 1.6, 0.25; at 2.6, 0.44; at 3, 0.70. As the Mach number grows the channel becomes long in Mach-line terms, the waves from one element land further and further aft on the other or miss it entirely, and each element tends to a half-diamond alone — which, in linear theory, is exactly half of the combined diamond. Far from its design point the biplane is no better than the section it was meant to beat.

Below the design Mach number the failure is of a different kind. The Mach lines are steeper, so a leading-edge compression crosses the channel in less than half a chord and lands on the other element’s front face, ahead of the shoulder, where it adds to a pressure that was already pushing back; and its reflection crosses again and lands on the first element before its own shoulder too. The channel then holds its own waves for several crossings, and in linear theory the drag rises almost exactly as fast below design as above it — 0.175 of the diamond’s at 1.7 and 0.176 at 2.2, a quarter of a Mach number either side — until, below 1.60, the throat ends the linear story altogether.

The throat #

The channel between the elements is a duct. Its inlet is the gap at the leading edges, 0.3 chords; its throat is the gap at the shoulders, where each element has risen three per cent into it, $0.3 - 0.06 = 0.24$ chords. The throat is 0.8 of the inlet. And a supersonic stream entering a duct that narrows is decelerated, not accelerated, with a limit to how far it can be decelerated before the throat reaches sonic speed and can pass no more.

That limit is the isentropic one: a started supersonic flow passes a throat of area $A_t$ from an inlet of area $A_i$ only if $A_t/A_i$ is larger than $A^*/A$ at the free-stream Mach number — the ratio of the sonic area to the stream area in the area–Mach relation. Below that Mach number the throat stops listening: the flow chokes, a shock forms and moves upstream out of the channel, and the biplane is flying with a detached shock standing in front of a duct full of subsonic flow, spilling the air it cannot pass round its outer surfaces.

Two limits, and the loop between them #

There is a second limit, and it is the one that matters on the way up. A channel that is not yet started has a normal shock standing at its mouth, and behind a normal shock the stream is subsonic with a lower total pressure; what a shock costs is the loss that reduces how much the throat can pass. The channel swallows the shock and starts only if its throat can pass the flow behind the shock, which needs $A_t/A_i$ larger than $A^*/A$ evaluated at the post-shock Mach number. That is Kantrowitz and Donaldson’s starting limit, worked out in 1945 for supersonic intakes, and it is always more demanding than the isentropic one.

The biplane’s throat ratio of 0.8 crosses the isentropic limit at Mach 1.60 and Kantrowitz’s limit at 2.15. Between those two the channel has two possible states and stays in the one it was in. Its design point, 1.944, is inside that band. Accelerating from subsonic speed, the biplane arrives at 1.944 choked, with a bow shock, a subsonic channel and a drag far above even the diamond’s, and it stays choked until Mach 2.15. Only then is the shock swallowed; the flow starts, and the biplane can be slowed back to its design point, where the drag is zero, and on down to 1.60 before the channel chokes again.

That is the reason the drag figure’s design point sits in a shaded band. The zero is real, and it belongs to a state the biplane can reach only by being flown past it and brought back — a supersonic intake’s starting problem, and a cone intake designer would recognise it at once.

Why the starting limit is stricter #

The gap between the two limits is a shock’s cost, and it can be read off at the design point. At Mach 1.944 the isentropic sonic area is 0.621 of the stream area: a started channel can narrow to 62 per cent of its inlet before its throat goes sonic, and the biplane’s 80 per cent is comfortably above that. Put a normal shock at the mouth instead and the flow behind it is at Mach 0.587 with a total pressure of 0.747 of the free stream’s. The throat now has to pass the same mass flow from a reservoir that has lost a quarter of its pressure, and a sonic throat’s capacity is proportional to the total pressure feeding it, so it must be larger by the factor $1/0.747$: the sonic area becomes $0.621/0.747 = 0.831$ of the inlet. The biplane’s 0.80 is just below it, and the shock stays.

That is the structure of every starting problem, and the numbers show why it is not a detail: a few per cent of throat area decide whether the channel holds a supersonic flow at no wave drag or a subsonic one behind a shock. The shock is not swallowed until the free stream is fast enough that even after a normal shock’s loss the throat can pass the flow — Mach 2.15 for this biplane, where the shock costs more but the free stream’s own sonic area has fallen further.

What the cancellation would have bought #

The reason the biplane has been taken seriously again is not drag but noise. A section that sends no waves out of its channel sends no thickness signature to the ground, and the signature that forgets the shape showed that a boom’s strength far away is set by what the near field carries out. A started biplane at its design point carries out nothing from its thickness: the waves that would have become the front and rear shocks of an N-wave are cancelled inside the channel. Lift still makes a signature — a lifting biplane is not symmetric — but the thickness half of a boom, which for a slender transport is a large share of it, is gone. That is a prize worth a starting problem, and it explains why the starting problem has been worked on rather than abandoned.

Where the gap puts the design point #

The gap is the designer’s other choice, and it moves all three numbers. A narrower gap puts the design Mach number higher, since the waves must cross a shorter distance in the same half-chord, but it also narrows the throat relative to the inlet, raising both limits. A wider gap lowers the design Mach number and widens the throat.

The two effects do not keep pace. With elements three per cent thick the design point stays inside the loop for every gap up to 0.48 chords, and above 0.48 the design Mach number falls below the unstart line itself. There the biplane is drawn to cancel its waves at a Mach number at which its channel is choked whatever its history: the zero drag is a property of a flow that cannot exist. Linear theory does not see this — it has no throat, only waves — and a design made in linear theory alone would put the zero on a speed at which the real flow is subsonic inside the channel.

How thick a biplane can be #

Turn the question round. For each design Mach number the gap is fixed by the cancellation, so the throat ratio is fixed by the thickness, and each limit becomes a largest thickness. The overshooting limit — the isentropic one, which allows a started flow to be slowed to the design point — allows up to 6.8 per cent of the chord per element at high Mach numbers. The self-starting limit is far tighter: it is never more than 2.63 per cent of the chord per element, reached near Mach 2.3, and it falls away on both sides.

So a Busemann biplane that can accelerate to its own design point from below is a pair of elements at most about 2.6 per cent thick each — a combined section about 5.3 per cent thick carrying no wave drag. That is a real achievement; the least drag a volume can have found that every closed body of finite length must pay a wave drag, and this pair pays none. But the volume it carries for free is bounded by a starting condition and not by aerodynamics, and the elements drawn here, at three per cent, are just over the line.

What was checked #

The marching scheme is checked three ways. At the design Mach number the pair’s drag is $10^{-16}$ of a single element’s drag alone: the cancellation is exact in the discrete scheme, not merely small. The upper element’s drag, computed by its own recursion with the roles of the walls exchanged rather than assumed from symmetry, equals the lower’s. And with the gap opened to fifty chords, at Mach 2, each element’s drag comes back as Ackeret’s half-diamond, $2\tan^2\varepsilon/\beta$, to two parts in $10^{13}$. The one-dimensional limits rest on two formulas checked against tabulated values: $A/A^*$ at Mach 2 is 1.6875 and the Mach number behind a normal shock at Mach 2 is 0.57735.

The convention: linear theory for the waves, one dimension for the throat #

The drag is computed in linear supersonic theory, in which the wave angles are the Mach angles and the pressure is proportional to the local flow turning; it is per unit of one element’s chord and is for the pair. The starting limits are computed for one-dimensional flow in a channel with the biplane’s inlet and throat. The two descriptions are of the same object at different levels: linear theory sees the waves and not the blockage, the channel model sees the blockage and not the waves. The design rule $\beta G = c/2$ is linear theory’s; with real oblique shocks, which are steeper than Mach lines, the design Mach number for the same geometry is higher, and the cancellation is not exact because a shock and an expansion of the same turning are not exact opposites.

What the picture cannot show #

The figures cannot show the unstarted biplane’s drag, which needs the detached shock’s shape and the spillage round the outer surfaces — a blunt-body problem. Computations and wind-tunnel tests of the biplane at its starting condition report a drag in the choked band well above the diamond’s, and a hysteresis between start and unstart close to the one-dimensional values. They cannot show lift: a symmetric pair makes none, and tilting it to make some puts unequal waves into the channel, breaking the cancellation, so a lifting biplane pays a lift-dependent wave drag the symmetric one does not. And they cannot show the boundary layers on the inner faces, which thicken at the shoulders where the cancelling shocks land and narrow the effective throat further.

Who found it, and when #

Busemann described the biplane at the Volta conference of 1935, in the same paper that introduced the swept wing. Its starting problem was understood in the language of intakes after Kantrowitz and Donaldson’s report of 1945 on the starting of supersonic diffusers; Licher in the 1950s worked out lifting variants. Interest returned in the 2000s, when Kusunose and colleagues in Japan studied the biplane as a way to lower sonic-boom strength, found the choked-flow hysteresis in computation and in the wind tunnel, and proposed flaps on the elements that open the throat during acceleration and close it at cruise. The thickness window here is the one-dimensional statement of the constraint those flaps exist to escape.

Still open: a throat that changes with speed #

The cure that has been proposed is to change the throat: hinged leading- and trailing-edge flaps that open the channel during acceleration, so the shock is swallowed early, and close it at the design Mach number. The next calculation gives the elements a leading-edge flap whose deflection widens the inlet and narrows nothing, finds the flap schedule against Mach number that keeps the channel’s throat ratio above Kantrowitz’s limit on the way up and returns it to the design geometry at the design point, and asks what drag the flapped biplane pays on the climb through the band where an unflapped one would be choked — so that the zero at the design point can be priced against the cost of getting there.