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

Lift curve slope — where it appears

The rate at which lift coefficient rises with angle of attack, two pi per radian for a thin section in two dimensions. A finite wing's is smaller, reduced by the downwash its own wake induces, and the reduction depends on the aspect ratio.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

A Joukowski aerofoil at 6°. A cambered aerofoil in a uniform stream. The circulation is not chosen: it is whatever value makes the flow leave the sharp trailing edge smoothly, and that single condition fixes the lift.

The lift curve, and why it is a straight line

Lift against angle of attack is a straight line, it does not pass through the origin, and its slope is very close to a number that has no business being there. All three facts fall out of the theory.

circulation · Lift Pressure recovery along the upper surface at 6°. Surface speed and the local Falkner–Skan pressure-gradient parameter, plotted along the upper surface from the nose. The speed peaks near the leading edge and then falls, which is the layer climbing back up to the pressure it started at, and the parameter crosses the separation value where that climb becomes too steep.

Where the straight line stops

Ideal flow will report a lift coefficient at forty degrees of incidence without complaint. What ends the lift curve is the boundary layer refusing to follow the surface, and the estimate of when that happens joins two solvers that have nothing else in common.

viscous · Separation A wing at h/c = 0.4. A wing close to the ground, with the mirrored vortex row that makes the ground a streamline drawn faintly beneath it. The air between wing and ground is squeezed through a narrowing gap, and the wing carries more lift than it would in free air.

The wall that pushes back

A wing near the ground carries more lift than the same wing in free air, and the usual explanation — a cushion of compressed air underneath — is not what the equations say. The ground is a mirror, and what changes is not the pressure under the wing but the circulation the wing is forced to carry.

circulation · Images The flap moves the curve and leaves its slope alone. Lift coefficient against incidence for several flap deflections. The lines are parallel: deflecting the flap gives the section lift at an incidence where it had none, and does not change how much lift each further degree of incidence buys.

What a flap does, and what it does not

Lowering a flap gives a wing lift at an incidence where it had none. It does not make the wing more responsive to being pitched — the lift curve moves sideways and its slope does not change, and those are two quite different things to buy.

circulation · Lift The correction belongs to the wing, not to the air. Lift-curve slope against Mach number for three aspect ratios, each computed by solving the lifting line for the wing the transformation actually implies — one of aspect ratio βAR — and dividing by β. The dashed curve is the two-dimensional rule, which is what an aerofoil section gets. At Mach 0.7 the aspect ratio of 20 has gained 35 per cent of slope and the aspect ratio of 4 has gained 24, against the 40 per cent the section rule promises both. The β in the finite-wing term cancels the β in front of it, so the shorter the wing the less compressibility does to it.

The wing the equation is really solving

The Prandtl–Glauert rule is usually quoted as a factor on the answer. It is a change of shape — and in three dimensions the shape it changes is the aspect ratio, so a short wing is far less affected by compressibility than a long one.

regimes · Similarity Three accounts of lift, one of them tuned to be exactly right at five degrees. Thin-aerofoil theory, which is the answer; Newtonian impact theory with its constant tuned so that it passes exactly through the truth at five degrees; and the equal-transit story, which has no free constant and sits along the bottom. At the tuning point the tuned model and the truth are indistinguishable, and no measurement there separates them.

A right total from a wrong picture

Newtonian impact theory has a free constant in it. Tuned at five degrees it reproduces a NACA 2412's lift exactly, and no measurement at five degrees can tell it from the truth. What separates them is the derivative — a lift-curve slope 2.8 times too steep — which is a second constraint rather than a better one.

misconceptions · Momentum lift Four sections, and what the shape story says about each. A flat plate, a symmetric section twelve per cent thick, a cambered one, and a section with a wavy upper skin. The first two have upper and lower surfaces of exactly equal length; the last has an upper surface two and a half per cent longer than its lower one, which is twice the cambered section's excess.

The wing that is flat, and flies

Refuting the equal-transit story by computing the parcels leaves its premise standing, and the premise is the part most readers believe: that the shape is what makes the lift. It is a claim about shapes, so it is tested with shapes — a flat plate, a symmetric section, and a cambered one flown upside down.

misconceptions · Equal transit Three answers for the lift-curve slope, with three different signs. Thin-aerofoil theory has no thickness term at all and returns 2π for every section. The exact potential solution rises: 2π(1 + 0.766 t/c), measured off the Joukowski map. Real sections do the opposite, because the boundary layer thickens towards the trailing edge and decambers the section. Two of these curves are computed here; the third is what measurement says.

The half that carries nothing

Thin-aerofoil theory splits a section into a camber line that carries all the lift and a thickness distribution that carries none — at any incidence, exactly none. That is very nearly true, and what it discards decides the peak suction, the critical Mach number and where the boundary layer gives up.

circulation · Thickness 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.

circulation · Cascade 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.

circulation · Finite wing 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.

circulation · Flutter

Named alongside it

The objects these essays reach for when they reach for this one.

Thin-aerofoil theoryCamberCirculationLift coefficientModel limitPressure distributionStallAspect ratioBound vortexBoundary conditionBoundary layerConstraint

All concepts