Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
The collection

Every essay — page 8

Page 8 of 39, continuing through the fields in the same order.

Flows and fields Ideal flow Circulation and lift Viscosity Regimes and numbers Compressible flow Transition and turbulence Fluids at work What is taught wrongly Series Concepts Regimes Refutations Search

Circulation and lift

Where lift actually comes from. The Kutta condition, the Joukowski aerofoil, and lift derived rather than asserted.

Deviation: how far the flow leaves from the blade angle, against solidity. The angle between the outlet flow and the blade, for a row of flat plates at 30° stagger meeting a flow at 45°. An open row barely turns the flow at all and the deviation is nearly the whole of the intended turning; a tight row guides it to within a thousandth of a degree of the blade angle. Nothing about the blade changed between the two ends of this curve.

A row is not a set of aerofoils

An isolated aerofoil's incidence is measured from the free stream. A compressor blade's cannot be, because an infinite row of identical blades above and below it has a circulation that is part of its own free stream — and the velocity the theorem uses is a vector mean that exists nowhere in the machine.

8 figures The trailing-edge speed against circulation, for a sharp edge and a round one. Sampled one grid point off the trailing edge. The sharp edge is singular at every circulation but one: the speed there is about U at the Kutta value and 11.3U half a Kutta circulation away, and it grows without bound as the sample approaches the edge. The round edge has no such point. That is the whole of the Kutta condition's justification, and it needs the corner.

The condition that can be bought

Ideal flow round a closed body has one solution for every circulation, and the Kutta condition picks one. Its justification is entirely the sharp edge: every other circulation puts an infinite velocity there. Take the corner away and nothing chooses — which is not a curiosity, it is what a circulation-control aerofoil is.

8 figures How the weight is shared between the two surfaces, against the centre of gravity. The wing's load and the tail's, for an aeroplane in level flight, against the static margin. The two must sum to the weight and their moments must cancel, and those two equations decide the split. At a forward centre of gravity the tail carries down and the wing carries more than the weight; the crossing is where the tail carries nothing.

More lift than weight

An aeroplane in level flight is drawn with one arrow up and one arrow down, equal and opposite. That equation collapses the whole configuration onto one number, and it is not true of any aeroplane with a tail behind it: there are two surfaces, two equations, and the second decides the split.

8 figures Two theories, one composite, and the aspect ratio between them. The lift-curve slope against aspect ratio. Prandtl's lifting line is exact as the aspect ratio goes to infinity and Jones's slender-wing theory is exact as it goes to zero, and each is generous outside its own limit. Helmbold's formula reduces to both with no free constant, which is what a composite expansion is, and runs under them where they disagree.

Where the line stops being a line

Prandtl's lifting line replaces a wing with a single bound vortex and its trailing sheet, and the formula that comes out is the most quoted in low-speed aerodynamics. Solved numerically at aspect ratio one it returns its own closed form to sixteen decimals — and the answer is forty-one per cent too high.

8 figures 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.

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.

8 figures 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.

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.

9 figures 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.

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.

9 figures 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.

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.

8 figures 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.

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.

8 figures The damping crosses zero once, and the crossing is the boundary. The least damping ratio of the four aeroelastic modes against speed. It falls through zero at 80.843 metres a second, and at that speed the crossing root's real part is four parts in 10¹⁸ — which is what an algebraic condition on a quartic with real coefficients looks like when it is solved numerically.

The speed where the damping is exactly zero

Flutter is an algebraic condition on a quartic: one root crosses the imaginary axis, at one speed, exactly. And the quantity that locates it is so nearly flat there that the standard way of finding it from flight test overshoots by a quarter.

8 figures 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.

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.

10 figures The wake's memory, as a gain and a phase. Theodorsen's lift deficiency against reduced frequency. It is one at zero frequency — the quasi-steady limit, where the wake has had time to convect away — and falls to a half at high frequency, with a phase lag peaking near 15 degrees in between.

The lag that makes flutter possible

This site's own flutter model set the lift deficiency to one and recorded in its notes that doing so throws away the lag which stabilises the torsion mode. Putting the lag back moves the flutter speed from 80.8 metres a second to 131, and removes the need for the structural damping that was covering the artefact.

8 figures