Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: Lift is circulation — page 8

Page 8 of 9, continuing through the 81 essays this motif runs through.

81 essays carry this thread — page 8 of 9.

The rudder should carry an eighth of the side force. The drag of keel and rudder together, as a percentage of the keel carrying everything, against the rudder's share of the side force, for a rudder 0.809 of the keel's depth. The induced drag is least with the keel alone — a shallower rudder makes the combined loading less elliptic for any load it takes. The profile drag is least near equal lift coefficients. Their sum is least at 12.4 per cent, 0.97 per cent below the keel alone; for the yacht drawn here, a rudder 0.809 of the keel's depth, that is 12.4 per cent. Fluids at work

The rudder pays for the keel's wake

A rudder angled to carry side force feels drag out of all proportion to its share, and that is the sensation behind the rule that weather helm is slow. The pair's drag does not care who is billed. It is least with the rudder carrying an eighth of the load, at under a degree of helm, and that share does not change with how hard the boat is pressed.

 Which bends of a vortex pair grow, and how fast. The square of the growth rate of a sinusoidal bend of a trailing vortex pair, in units of one e-fold per descent time, against the bend's wavenumber times the spacing, for a core of elliptic loading's size. Where the square is positive the bend grows. The symmetric mode, in which the two vortices bend as mirror images, grows over a long-wave band and fastest at a wavelength of 8.54 spacings, by 0.827 e-folds in the time the pair takes to sink one spacing. The antisymmetric mode, bending in step, is stable across the same band. Circulation and lift

A wake ends by bending, not by fading

The two vortices behind an aeroplane do not simply weaken until they are gone. In still air each one bends in the other's strain, the bend grows by a factor of e every thirty-three seconds behind an airliner, and after a couple of minutes the pair has pinched itself into a chain of rings. The wavelength it chooses is eight and a half times the spacing, and the vortex cores hardly enter.

 A V seen from above, and what each member pays. Nine wings in a V swept 45 degrees, tips one span apart, each carrying the same lift, seen from above with the flight direction up the page. Beside each is its induced drag as a fraction of what it would pay flying alone. The leader at the point pays 0.89; every other member pays between 0.35 and 0.39. The saving has flowed backwards through the formation. Circulation and lift

The bird at the point pays for the V

A flock's saving in a V is fixed by where its members sit across the stream, and the angle of the V cannot change it by a single per cent. What the angle changes is who gets the saving. It flows backwards through the formation, so that in a V swept forty-five degrees the leader pays nine-tenths of what it would pay alone while every bird behind it pays under four-tenths.

 The V in which every bird pays the same is curved. Nine birds one span apart, flying up the page, in plan: each short line is a wing, its height its distance behind the leader in spans. The straight V at 7.5° — the fairest straight V — and the V whose members' positions are solved so that every one pays exactly the same share of the induced drag. The equal-share arms leave the leader almost abreast and bend back ever more steeply: 1.6°, 7.6°, 17°, 32° from the apex to the tips. Circulation and lift

A fair V is a curved V

In a straight V of birds somebody always pays more than somebody else: at the fairest angle the leader and the two birds at the tips pay half again what the birds between them pay. The V in which every bird pays exactly the same can be solved for, and it is not straight. Its arms leave the leader almost abreast and bend back ever more steeply, to thirty-two degrees at the tips of a flock of nine. The share every bird then pays is the flock's average, fixed by Munk's theorem before any position is chosen — so fairness costs nothing, and the only thing that can make the flock cheaper is flying closer together sideways.

 A hydrofoil loses lift at speed and gains it slowly, with a dip between. The lift of a flat hydrofoil beneath the surface over its value in deep water, against the chord Froude number U/√(gc) on a logarithmic axis, at depths of half a chord, one chord and two. Slow, the surface is a lid and the foil gains lift, as a wing over the ground does. Fast, the surface is a pressure-release boundary and the foil loses it. Between the two it does not simply pass from one to the other: the lift dips well below its fast value where the foil's waves are longest compared with its depth, and then rises through the lid value before settling back on it. What is taught wrongly

A hydrofoil loses most lift on the way up

A wing near the ground gains lift, and a hydrofoil near the surface is often described as the same thing upside down. At low speed it is: the surface acts as a lid and the foil gains lift. At high speed the surface is a boundary that cannot hold a pressure, and the foil loses lift instead. But the lift does not pass smoothly from one to the other. Between them, where the foil's waves are a few depths long, it falls well below both — to 43 per cent of its deep-water value half a chord down — and a foil boat meets that speed in the middle of its take-off run.

 The wake a flapping bird leaves is a wave in the air. Side view, the air at rest: the path the leader's wingtip traced, where its tip vortex now lies, over two wingbeats; lengths along the flight path in wake wavelengths — the distance flown in one beat — and heights in spans, for a tip swinging a fifth of a span each way. A follower three-tenths of a wavelength behind that beats three-tenths of a beat later traces the same path and flies along the leader's vortex all the way; one that beats half a beat off that traces the mirror image and meets the vortex only twice a beat. Circulation and lift

The follower beats in time with the wake, not the bird

A gliding bird can sit in its neighbour's upwash and stay there. A flapping bird's wake is a wave left in the air — the path its wingtip traced, rising and falling with every beat — and a bird behind gains only if its own wing is where that wave is when it arrives. The best timing is a rule with no aerodynamics in it: lag the bird ahead by the time the wake took to come, so that each wingtip retraces the path of the one before. Directly behind, the rule flips by half a beat, and it buys a smaller loss rather than a gain.

 Joined at the tips, the spars share the moment as a couple. The share of the lift's root bending moment that the box wing's two spars carry as an axial couple — one in tension, the other in compression, the gap for a lever arm — against the fins' bending stiffness as a multiple of a spar's, for gaps of a tenth, a fifth and two-fifths of the span. With floppy fins the spars bend independently and the couple is nothing. With fins as stiff as the spars, which a fin of the wing's own section is, the couple carries 29 per cent at a gap of a fifth. However stiff the fins, it stops near a third. Circulation and lift

A box wing's fins earn their keep in the spar

Constrain a box wing's root bending moment and its fins stop saving drag: it becomes a biplane. But that constraint counted the lift's moment, not the spars', and the fins join the spars into one frame. Joined at the tips, the two spars hand part of the moment to a couple across the gap — tension in one, compression in the other — which costs far less material than bending. With fins as stiff as the spars that is three-tenths of the moment; with rigid fins it stops at a third, whatever the gap, because a tip joint can only guide a tip. Counted in the spar, the box has less drag and a lighter spar than the elliptic monoplane at once.

 The drag is set by the split, and stability sets the split. The least induced drag of a wing and a smaller surface together, as a multiple of the elliptic wing's alone, against the share of the lift the smaller surface carries. By Munk's stagger theorem the curve is the same whichever surface is in front. Its minimum, 0.9984, is at a share of 1.9 per cent. At a static margin of a tenth of a chord, a tail trims with 6.7 per cent of the weight on it, upwards, and pays 1.009; a canard trims with 19 per cent and pays 1.128. Circulation and lift

A canard pays for its stability in induced drag

The argument for a canard is that both of its surfaces lift upwards, while a tail pushes down and makes the wing carry the difference. Munk's stagger theorem turns the question into arithmetic: two surfaces' least induced drag depends only on how the lift is split between them, not on which is in front. Static margin sets the split. At a margin of a tenth of a chord a tail carries a small upload and costs under one per cent; a canard must carry a fifth of the weight on a third of the span and costs twelve, and the more stable it is made, the more it pays.

 A phantom biplane partner doubles the induced drag. Induced drag at a given lift, as a multiple of its deep-water value, against the foil's depth in spans. The surface's image of the foil is an identical wing, equally loaded, twice the depth above it, so the real foil pays its own induced drag plus the mutual drag of a biplane with a gap of twice its depth: 1 + σ, with σ Prandtl's biplane factor. The elliptic biplane's 1 + σ and the aspect-ratio-9 foil's lattice both run to two as the depth closes. Fluids at work

A foil under the surface flies with a phantom

At foiling speed the water's surface cannot hold a pressure, and the image that condition requires is not the reversed one a wall makes. For a horizontal foil it is an identical wing, equally loaded, above the surface — a biplane partner that takes lift and gives nothing back, doubling the induced drag as the foil rises. For the board that pierces the surface it is a reversed copy, which makes the board pay two and a half times what a keel under a hull pays. It is the board, not the foil, that decides how deep a foiling boat rides.

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