A flap that closes the mouth starts Busemann's biplane below its design point

Busemann’s biplane has to be flown past its design point found the trap in the most elegant section supersonic aerodynamics has. Two half-diamonds facing each other across a channel, each three per cent of its chord thick and 0.3 chords apart, cancel each other’s waves at Mach 1.944: every compression from one element’s leading edge lands on the other’s shoulder and is undone there, nothing leaves the channel, and the pair carries its thickness at no wave drag. But the channel narrows from a mouth of 0.30 chords to a throat of 0.24, a ratio of 0.8, and on the climb from subsonic speed a normal shock stands in its mouth. The throat cannot pass the flow behind that shock until Mach 2.15. The biplane has to be flown past its design point to start, then slowed back to it.

The cure proposed for it is a flap, and the essay that found the trap posed the calculation and stopped. This makes it, and the answer depends on getting one thing the right way round that is easy to get wrong: what a starting channel needs is not a wider mouth but a mouth closer to its throat.

What starting asks of a channel #

A supersonic channel that contracts has two lives. Started, the flow inside it is supersonic all the way, and it survives as long as the throat is wider than the free stream’s own sonic area — the area to which the stream could be slowed isentropically to Mach one. For this channel that holds down to Mach 1.60. Unstarted, a normal shock stands at the mouth, the flow behind it is subsonic, and the throat must pass it from a reservoir whose total pressure the shock has cut. A sonic throat’s capacity is proportional to the total pressure feeding it, so the throat it needs is larger than the isentropic one by the shock’s total-pressure ratio. That is Kantrowitz’s limit, and at Mach 1.944 it asks for a throat of 0.831 of the mouth where the isentropic limit asks for 0.621.

The biplane’s 0.8 lies between them at its design point. Started, it would stay started; it cannot get started. Between Mach 1.60 and 2.15 both states exist, and which one the channel is in depends on its history — the same hysteresis an intake meets when it moves compression behind its lip, and for the same reason. The intake’s cure is to open its throat on the way up. The biplane’s throat is the elements’ shoulders, the thickest point of each, and moving it means moving the structure.

Why the shock sets the bar higher #

The two limits differ by exactly what a normal shock destroys. A supersonic channel narrows and its flow slows, as a duct does above the speed of sound, until at the narrowest point it could be sonic and no narrower; the area at which that happens is the free stream’s sonic area, a fixed fraction of the stream at each Mach number. A sonic throat passes a mass flow in proportion to the total pressure upstream of it and to nothing downstream — the throat stops listening once it is sonic — so when a normal shock sits at the mouth and cuts the total pressure, the same throat passes less, and must be bigger by the inverse of the shock’s total-pressure ratio to pass the same flow. At Mach 1.944 that ratio is 0.747, the price a normal shock charges at that strength, and the throat a starting channel needs grows from 0.621 of its mouth to 0.831.

The gap widens with speed, because the shock’s loss grows faster than the stream’s sonic area shrinks, and that is why the shut biplane, with its fixed 0.8, crosses Kantrowitz’s line only at 2.15: by then the free stream’s sonic area has fallen far enough that even after the shock the throat can cope. The flap attacks the same inequality from the other side. It cannot reduce the shock’s loss at a given Mach number; it reduces how much flow the throat is asked to pass, by reducing how much the mouth catches.

A flap that moves the mouth instead #

The ratio has two ends, and either will do. Leave the shoulders where they are and move the mouth: hinge the front of each element’s inner ramp a distance $c_f$ behind the leading edge and turn its nose into the channel by an angle $\Delta$. The ramp rises at the half-angle $\varepsilon$ of the half-diamond, 3.43° here, so turning the flap by $\varepsilon$ lays it flat at the hinge’s height and brings its nose $c_f\tan\varepsilon$ into the channel. The mouth narrows by twice that; the throat does not move.

The arithmetic is short. A 0.2-chord flap laid flat moves each lip in by $0.2 \times 0.06 = 0.012$ chords, the mouth closes from 0.300 to 0.276, and the throat-to-mouth ratio rises from 0.800 to 0.870. Kantrowitz’s limit falls to 0.870 at Mach 1.72, so the channel starts there. A 0.1-chord flap reaches 0.833 and starts it at 1.93, a hundredth below the design point — enough in principle, with no margin. A 0.3-chord flap reaches 0.909 and starts at 1.53; a 0.4-chord one 0.952 and 1.32; and a flap over the whole front ramp lays it flat level with the shoulder, makes the mouth the throat and starts the channel at any supersonic speed.

What starts a channel is a mouth closer to its throat. The tempting cure — open the inlet so it captures more easily — runs the wrong way. A wider mouth for the same throat is a larger contraction, and a larger contraction needs a faster free stream before the shock’s loss is small enough for the throat. What the flap does is take contraction out of the channel at its front. The cowl caps what an intake’s ramps can buy for the same reason: every turn inside a duct’s mouth is contraction the start must pay for.

The route to the design point #

The flaps are not needed once the shock is in. A started channel survives down to the isentropic limit, so after the start the flaps can close at any Mach number above 1.60, and with them shut the geometry is Busemann’s again: the cancellation at 1.944 returns exactly, with no wave drag from the channel. The route is open the flaps, accelerate to the start, shut them, carry on to the design point. A flap of 0.2 chords does it with a start at 1.72; nothing about the design point has to be compromised.

The order of operations matters with the longer flaps. A 0.3-chord flap starts the channel at 1.53, below the isentropic limit of the shut geometry at 1.60. Shut the flaps at 1.53 and the started flow cannot survive in the narrower channel: it unstarts. So the longer flap has to stay open until the aeroplane passes 1.60 and then close, and in between the aeroplane flies with its flaps out, paying their drag. The shortest flap that starts the channel below 1.944 and above 1.60 needs no such wait.

How much flap, when #

The flap does not have to be laid flat to start the channel; it has to be turned far enough for the ratio to clear Kantrowitz’s line at the Mach number reached. A schedule that turns it just enough at each speed traces the curves above. At Mach 1.8 a 0.2-chord flap needs 2.77° and a 0.4-chord one half that. The lengths trade against each other in the obvious way: the nose’s movement is the flap’s length times the change in its slope, so a longer flap moves the mouth further for a smaller turn.

Two things limit the trade. The flap cannot turn past flat without becoming a diverging ramp that expands the flow ahead of the throat, which does not help; and the longer flap leaves a shorter fixed ramp, whose shoulder is the one the cancellation depends on, so a flap that is most of the ramp is a hinge across most of the element’s structure. A fifth of the chord is the sort of number a designer would pick: enough to start comfortably below the design point, short enough that the rear of the ramp and the shoulder are unhinged.

What the flaps cost while they are out #

The flapped channel is not cancelled. The flap’s compression starts at a smaller angle and is spread differently, so the waves no longer land on the opposite shoulder in the proportions the cancellation needs, and the flap’s outer face now meets the outer stream at an angle and carries Ackeret’s wave drag, $2\tan^2\Delta/\beta$ over its length. Left open at the design point, 0.2-chord flaps flat cost a wave drag coefficient of 3.46 thousandths, half of it on their outer faces — two-fifths of what a plain diamond of the pair’s combined thickness pays, 8.64. That is the price if the flaps cannot be shut, and it is the reason they must be: an unflappable biplane with permanently flattened noses would have thrown away most of what Busemann’s cancellation bought, though it would still beat the diamond.

While they are out on the climb the cost is a few thousandths of drag coefficient over a band of Mach number a few tenths wide — 5.75 thousandths at the start, falling as the speed rises — against a channel that, without them, would sit unstarted with a normal shock in its mouth over that whole band and on past the design point. An unstarted channel’s drag is not in linear theory, because its flow is subsonic behind a strong shock, but it is the drag of a blunt body with its spillage, which is far larger than either curve here. The flaps swap that for a modest, calculable drag and a shorter climb.

What the start protects #

The reason anybody wants the biplane started is the zero at its design point, and that zero matters for noise more than for drag. A section that cancels its thickness waves inside its channel sends nothing outward for the thickness half of a sonic boom, and the signature that forgets the shape showed that what reaches the ground is decided by what the near field carries out. An unstarted biplane carries out the opposite: a detached normal shock across its mouth and the spillage round its leading edges, the loudest near field a thin section can make. So the band between the start and the design point is not only a drag penalty on the climb; it is a band in which a quiet-boom aeroplane would be booming louder than a conventional one.

Starting earlier also shortens the overshoot that the shut biplane needs. Without flaps it must reach 2.15 and then slow to 1.944, passing through a tenth of a Mach number that its engines must reach and its structure withstand only to get started. An aeroplane that accelerates through a band of Mach number launches its boom rays in a fold that focuses them, and every tenth of acceleration avoided is a tenth less of that. The flap buys back the overshoot, and with it the margin of engine thrust and the focused boom that the overshoot would have cost.

There is a smaller, easily missed benefit in the started band below the design point. Between 1.72 and 1.944 the flapped-then-shut pair is a started Busemann biplane flying slightly below its design Mach number, with a wave drag of a fifth of a diamond’s or less — the left-hand part of the shut curve in the drag figure — rather than an unstarted channel. The design point is a zero; the band beside it is already cheap.

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

Two models are being used and they answer different questions. The start is a one-dimensional statement about mass flow: a normal shock at the mouth, the mouth’s area the flow that must pass, the throat sonic. It ignores the shape of the flow between mouth and throat and the boundary layer, which in a real channel thickens under the shock’s pressure rise and narrows the effective throat — so the real start is later than Kantrowitz’s limit says, and the margins here are best cases. The waves are linear supersonic theory marched along the channel with every reflection kept, valid for thin elements at Mach numbers not too near one; it does not know about shocks strengthening on reflection or about the expansion fan’s curvature. The flap is a hinge with no gap; a real hinged flap leaks between the channel and the outer stream, which matters to a channel that depends on its walls.

What the picture cannot show #

It cannot show the start itself. The calculation says at what Mach number the shock can be swallowed, not how it gets swallowed: in practice the shock moves into the channel as the speed rises, sits near the throat, and jumps through it, and whether that transient is smooth depends on the boundary layer and on how fast the speed changes. Nor can it show what happens if the flow is disturbed after the start — a gust of incidence that tilts the channel, a burst of turbulence that thickens the layer — and pushes the flow below the isentropic margin. The design point sits at 0.8 against an isentropic limit of 0.621, which is a large margin, but a lifting biplane is asymmetric and its two sides’ margins differ. And none of this lifts: the pair is symmetric, the calculation is for thickness alone, and a real Busemann wing has to carry its lift by making the elements unequal, which moves the cancellation and the throat together.

Who proposed the flap #

Busemann described the biplane in 1935, as a thought experiment about cancelling waves, in the same paper in which he proposed the swept wing; both were ways round the wave drag that drag with nothing to rub computes. Its starting problem was understood from the theory of supersonic diffusers that Kantrowitz and Donaldson set out in 1945, and set aside with the biplane for half a century. The biplane came back in the 2000s at Tohoku University, where Kusunose, Matsushima, Obayashi and their students worked on it as a quiet supersonic transport’s wing — a section that sends no thickness signature to the ground — and studied flaps at its leading and trailing edges as the way through the starting hysteresis, with computations of the full flow through the start that this calculation does not attempt.

How it was checked #

With the flap shut the flapped geometry is the unflapped one in every number: the same throat ratio, zero wave drag at the design point to $10^{-19}$, the same marched drag at Mach 1.7 and the same start at 2.15 — the flap model adds nothing when it does nothing. A flap laid flat over the whole front ramp brings each nose level with its shoulder, makes the mouth the throat exactly, and starts the channel at any supersonic speed, the limit the geometry promises. Laid flat, the four flaps’ starting Mach numbers fall in order of their length, the tenth-chord flap’s just under the design point. A 0.2-chord flap cannot start the channel at Mach 1.5 at any deflection, and above 2.15 none is needed. And the model refuses a flap longer than the ramp it sits on, a deflection outward or past flat, and a throat ratio of zero or above one.

Still open: a biplane that lifts #

The cancellation and the start were both computed for a symmetric pair that carries no lift. An aeroplane’s biplane must lift, and the simplest way is to fly it at an incidence, which tilts the channel: the lower element’s ramp meets the flow at a larger angle, the upper element’s at a smaller one, the waves no longer land on the opposite shoulders, and the mouth and throat are seen at an angle. The next calculation marches the same channel at a small incidence, finds the lift and the wave drag the cancellation leaves at the design Mach number, and asks how the starting limit moves when the stream enters the mouth obliquely — whether a lifting biplane starts more easily on one side than the other, and whether the flap schedule has to be different top and bottom.