Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: Lift is circulation — page 6

Page 6 of 7, continuing through the 63 essays this motif runs through.

63 essays carry this thread — page 6 of 7.

Five flows past one cylinder, every one of them a solution. Surface pressure round a cylinder in a stream, at five circulations. Each satisfies Laplace's equation, the tangency condition on the body and the condition at infinity, and each has a different lift. Nothing in the problem chooses between them: the domain has a hole in it, so the potential is many-valued and the circulation is a free constant. Ideal flow

The constant a hole leaves behind

In a region without holes, Laplace's equation and the boundary values have exactly one solution. Cut a hole and they have a one-parameter family. Nothing in the mathematics chooses between its members, which is why the Kutta condition has to exist and why it cannot be derived.

 The local sweep of the isobars, across the span. The sweep of the half-load line at each spanwise station, for four geometric sweeps. Over the middle of the span it is the wing's own sweep, which is the simple theory being right. At the root it collapses — by twenty-one degrees at a geometric thirty-five — and at the tip it falls again. That root region is where the shock forms first on every swept wing ever built, and it is why they have waisted fuselages. Circulation and lift

The sweep a root does not have

Simple sweep theory is one of the cleanest arguments in aerodynamics: an infinite yawed wing cannot know about the velocity along its own span, so only the normal component matters. A real wing has a root and two tips, and at the root of a thirty-five-degree wing the isobars are swept fourteen.

 Wagner's function and Küssner's, from one solver and two inputs. Lift as a fraction of its steady value, against distance travelled in semichords. The step in incidence and the sharp-edged gust are the same unsteady problem with two different right-hand sides, and Jones's exponential fits to both are drawn over the solve. From four semichords on they are nearly the same curve — which is why they get interchanged. Circulation and lift

Two answers to one question

Unsteady aerofoil theory collapses a wing's whole history onto one function of one variable, and every quasi-steady gust calculation convolves something with it. There are two such functions, not one: a wing that is pitched changes its boundary condition everywhere at once, and a wing flying into a gust has not met most of the gust yet.

 Rolling effectiveness against dynamic pressure. The rolling moment an aileron produces, as a fraction of what it would produce on a rigid wing. It falls from one, passes through zero at the reversal pressure, and goes negative: beyond that point deflecting the aileron down rolls the aeroplane the other way. There is no oscillation anywhere in this figure and no frequency — it is a static failure. Circulation and lift

The control that works backwards

Divergence is the static aeroelastic failure everybody names, and a wing with its elastic axis at its aerodynamic centre cannot diverge at any speed. It can still reverse — deflect the aileron down above a certain dynamic pressure and the aeroplane rolls the other way — because the aileron's own nose-down moment is there whatever the elastic axis is doing.

 The six bodies, drawn at the same scale. Two circles, two ellipses and two Joukowski sections, each at the incidence that gives it a circulation of exactly two. There is no family resemblance and no common parameter; what they share is one number, and the theorem needs nothing else. Circulation and lift

One formula, and it does not ask what the shape is

Kutta–Joukowski gives the lift of any two-dimensional body from one number. Six bodies with nothing else in common are put at that number here and come out with identical lifts — and with pitching moments, load distributions and suction peaks that are not even close.

 How flat the optimum is. The same curve near its maximum, with the bands within a tenth and a half of one per cent of the best shaded. Every taper ratio from 0.31 to 0.42 is within a tenth of a per cent of the optimum, and the whole band from 0.25 to 0.51 is within half a per cent. The optimum is exact, and choosing it rather than its neighbour buys nothing a wing can measure. Circulation and lift

The optimum that does not matter

Elliptic loading gives the least induced drag, exactly, and the proof is one line: the penalty is a sum of squares. Which is also why the optimum is flat enough that a quarter of the design space sits inside a tenth of a per cent of it.

 Two camber lines, one lift and one moment. A NACA 2412 mean line and the same line with a fourth harmonic added to its slope. They are eight tenths of a per cent of the chord apart, which is forty per cent of the section's own camber, and they have the same lift and the same pitching moment at every incidence — to the last bit of double precision. Circulation and lift

Three numbers out of a camber line

Thin-aerofoil theory takes a whole function and returns a lift and a moment. Only three coefficients of that function survive: two camber lines matched in the first three, and eight tenths of a per cent of chord apart, have the same lift and the same moment at every incidence and load distributions thirty-seven per cent apart.

 A keel pulls the best beat 12 degrees closer to the wind. Speed made good as a fraction of the wind against the course, in 3 m/s: for the boat with its keel solved as a wing, and for a boat whose λ is fixed at 25.69°, the value the keeled boat has on its own best course. The search puts the keeled boat's best beat at 45.83°; the fixed-angle rule puts it at 45° + λ/2 = 57.84°. Pointing higher loads the keel towards its best lift coefficient — 0.130 at 40° against 0.076 at 60° — and lowers its drag angle, so the curve peaks early and falls away faster on the far side. The fixed-angle curve promises 0.653 of the wind; the keeled boat can make 0.553, and only by sailing twelve degrees higher than the rule says. Fluids at work

A keel flies wherever the course puts it

A keel is a wing whose lift is whatever side force the rig happens to make, so its lift coefficient is chosen by the course and the wind rather than by its designer. Solved that way, the hull's drag angle stops being a property of the boat: it pulls the best beat twelve degrees closer to the wind, halves the reaching speed a fixed angle predicts, and finally charges for reefing.

 A spun cylinder carries its whole circulation at once, and hides it until the vorticity has left. The circulation round circles of radius r about a cylinder of radius a started spinning at once, as a share of the circulation of its own surface, 2πa²Ω, against r/a on a logarithmic axis, at νt/a² = 0.01, 0.1, 1, 10 and 100. At the surface it is the whole of it from the first instant, because no slip makes the fluid there turn with the cylinder. Just outside, the spin-up has laid down an equal and opposite ring of vorticity, so the circulation round a larger circle is only what has diffused past it: at two radii 0.000, 0.035, 0.611, 0.936 and 0.993 of the surface's at the same five times. The circulation a Magnus rotor needs is in the fluid the moment it spins; the far field learns of it only as fast as the counter-vorticity moves out. Flows and fields

A wall puts in exactly its own speed

A wall sliding in its own plane makes vorticity at a rate equal to its acceleration, with no viscosity in the rate. So however a wall is started, the vorticity it has put into the fluid is its speed, to the last digit; a wall that stops takes all of it back and leaves the fluid moving; and a spinning cylinder carries its whole circulation from the first instant, hidden behind an equal and opposite ring until viscosity carries the ring away.

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