Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: Lift is circulation — page 3

Page 3 of 7, continuing through the 63 essays this motif runs through.

63 essays carry this thread — page 3 of 7.

Induced drag against aspect ratio, all at C_L = 0.6. Induced drag for wings of different aspect ratio, each flown at whatever incidence makes it carry the same lift coefficient. The drag falls as one over the aspect ratio, so the longest wing here pays a fraction of what the shortest does for exactly the same load. Circulation and lift

The span is the whole story

Induced drag is the price of having ends, and the only thing that lowers it at a given lift is putting those ends further apart. Not more area, not a better section, not a cleverer planform — span, and everything else is a correction of a few per cent to it.

 The lift does not arrive when the incidence does. Bound circulation against distance travelled, in units of the settled value. The wing starts far short of its final lift and takes tens of chords to collect the rest, because every scrap of circulation it takes has to be paid for with an opposite vortex shed behind it, and that vortex's own downwash holds the wing back until it is far away. Wagner's exact 1925 answer is drawn beside the model in the colour this site keeps for a borrowed claim: the shapes agree and the model is slower, by about a fifth at its worst. Circulation and lift

The lift that arrives late

A wing set into motion does not have its lift. Every scrap of circulation it takes has to be paid for by shedding an equal and opposite vortex behind it, and until that debt is far downstream its own downwash holds the wing back — for tens of chords, not for an instant.

 1.5U at the equator, and no drag at all. The exact ideal flow past a sphere, in the meridional plane, with speed contoured behind the streamlines. The fastest fluid is at the equator at 1.5U — a cylinder's is at 2U — and the field is fore-and-aft symmetric, so the pressure integral over the surface gives a drag of -1.2e-16 against a dynamic scale of order one. The streamline spacing here does not measure speed the way it does in a plane flow: the flux between two meridional streamlines depends on the distance from the axis as well, which is why the speed is contoured rather than left to be inferred. Ideal flow

Three dimensions are kinder

Every ideal flow solved on this site so far is plane, and plane flow is the harsh case. Put the third dimension back and the fastest surface speed drops from twice the free stream to one and a half times, the disturbance dies as the cube of distance instead of the square, and the body cannot carry circulation at all.

 A lift curve that keeps climbing to 49 degrees. The lift of a slender delta of aspect ratio 1, split into the potential term that an attached flow would give and the vortex term the separation adds, with a conventional wing's curve behind them. The delta's is nonlinear from the start and reaches 1.68 at 49 degrees, where a conventional wing stalled at fifteen. That is the whole design case for the shape: not that it is efficient — it is not — but that it still has lift at incidences where an ordinary wing has none, which is what a delta-winged aircraft needs on approach and in a turn. Circulation and lift

Lift out of a failure

Separation is what ends a wing's lift curve everywhere else on this site. A slender delta with sharp leading edges separates on purpose, rolls the shed sheet into a pair of vortices above its upper surface, and takes most of its lift from the suction they induce — with a curve that climbs to forty-nine degrees.

 Wound up, and worth exactly what it started with. A material loop in a steady cellular flow — an exact solution of Euler's equations — drawn at four times. Each streamline in the cell has its own period, so the loop is stretched steadily into a spiral: by the last frame its perimeter is 5.7 times what it started as. The circulation round it is 0.903741 at the start and 0.903666 at the end. Nothing about the curve survives except the number. Ideal flow

What survives being wound up

Draw a loop of marked fluid particles and let the flow carry it. It will be stretched, folded and wound into a spiral until nothing about its shape is recognisable, and the circulation round it will not have moved at all — provided three conditions hold, each of which can be broken on purpose.

 Four camber lines, and the angle at which each stops lifting. Four mean lines on the same chord: symmetric, a circular arc, a four-digit line with its crest at forty per cent, and a reflexed line whose tail turns up. The zero-lift angle beside each is computed from that line's own slope by quadrature and is a property of the shape alone — no incidence, no speed, no thickness enters it. The symmetric line's is exactly zero, the arc's is −2m to ten decimal places, and the reflexed line's is positive: it needs to be pointed up before it stops lifting. Circulation and lift

Where lift starts

A wing at zero incidence is not a wing making no lift. The angle at which a section stops lifting is a property of its camber line and of nothing else — not of its thickness, not of its speed, not of the air — and it is an integral anybody can take.

 Every one of these is a solution, and they lift different amounts. Lift coefficient against the circulation the aerofoil was told to carry, with the Kutta condition's own answer marked. Each point is a complete solve: the sources were found for that circulation and the surface is a wall to within 1.3e-4 at every collocation point. Every member satisfies the equations of motion and the boundary condition, and the lift runs through them at exactly 2Γ/Uc — Kutta–Joukowski appearing as a property of the family rather than as a result about any member of it. Ideal flow round a closed body does not have a unique answer, and the Kutta condition is the extra sentence that picks one. Circulation and lift

Nothing but the edge

Cut an aerofoil into panels, put a singularity on each, and require the surface to be a wall. The system that comes out has one more unknown than it has equations, and the row that is missing is not a bookkeeping slip — it is the fact that ideal flow round a closed body has no unique answer at all.

 A slotted flap at 30°, in a flow with no viscosity anywhere. Streamlines through a main element and a flap, computed by a two-body panel solve. Each element carries its own circulation and its own Kutta condition, and the two interfere through their velocity fields and through nothing else — there is no boundary layer here, no wake, no mixing region and no high-energy air. The system's lift coefficient is 2.757 against 0.698 for the main element alone at the same incidence, and the main element itself is carrying 3.98 times the circulation it carries by itself. Circulation and lift

A slot is not a nozzle

The gap between a wing and its flap is supposed to blow fast air into a tired boundary layer. A solver with no boundary layer in it at all — no viscosity, no wake, no mixing — produces most of the lift increment anyway, and produces it on the element nobody moved.

 Quasi-steady stops being true a long way before one. The magnitude of Theodorsen's function, which is the factor a quasi-steady lift calculation is wrong by, and the phase the lift lags the motion. Quasi-steady means C = 1, and the amplitude is already one per cent low at k = 0.0061 and fifteen per cent low at k = 0.1 — a reduced frequency at which nobody hesitates to call a flow quasi-steady. The lag is worse: it reaches a degree at k = 0.003, and a flutter calculation is decided by phase rather than by amplitude. Regimes and numbers

Slow enough to be steady

A wing moving slowly enough is assumed to carry the lift its instantaneous angle asks for. The reduced frequency has two thresholds — one where the apparent-mass and circulatory lifts are equal, and one where the quasi-steady answer stops being right — and they are a hundred and seventy-eight apart.

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