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

Thin-aerofoil theory — where it appears

The linearised theory of a slightly cambered section at a small angle, which replaces the body by a sheet of vorticity on its own chord. It gives a lift slope of two pi per radian and the location of the aerodynamic centre, both in closed form.

Named by 14 essays across 3 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 Finding the aerodynamic centre by sweeping the chord. How fast the pitching moment changes with incidence, plotted against where along the chord the moment is taken. The curve crosses zero once, and that crossing is the aerodynamic centre — the single point about which pitching the wing does not change the moment.

Where the lift acts

Lift gets drawn as an arrow, and an arrow has to start somewhere. The pressure is spread over the whole surface, so the choice of where to put the arrow is free — except that exactly one point on the chord has the property of not changing its answer when the wing is pitched.

circulation · Moment The circle plane and the aerofoil plane. A circle with a polar net around it, and the same net after the Joukowski map. Curves that crossed at right angles still cross at right angles everywhere except at the single point where the map's derivative vanishes, and that point is the sharp trailing edge.

From a circle to a wing

The flow past a circular cylinder is known exactly and is of no interest to anybody who wants to fly. A change of variable turns that circle into a wing section — and, because the change of variable preserves angles, it carries the whole solution across with it. Nothing is solved twice.

inviscid · Ideal flow 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 section the pressure asked for. The designed section over the one it started from, both drawn to their own chords. Asking for 28 per cent more speed over the forward 62 per cent of the upper surface produces a section 15.0 per cent thick against the original's 10, with the extra thickness forward and the camber changed — none of which was asked for, and all of which is what that pressure distribution is. The pale outline is the baseline. The one thing the method cannot be told is where along the chord any of it happens: the speed is prescribed against the circle's parameter, and where a given station ends up is an output of the same solve that produces the shape.

Ask for the pressure, and see what shape that is

A designer knows what the pressure distribution has to do long before knowing what shape does it. Running the problem that way round is possible, it is exact, and it refuses more asks than it grants.

inviscid · Inverse design 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.

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.

circulation · Thin-aerofoil 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 Why a flat plate has no drag, drawn as a triangle. A flat plate at incidence with the three forces that must balance. Pressure can only act along the plate's normal, so the pressure force is the arrow perpendicular to the plate. Kutta–Joukowski says the resultant is perpendicular to the free stream. The difference between the two directions is the suction force at the leading edge, which acts forwards along the plate and is exactly L sin α. Without it the plate would have a drag of L sin α, and an inviscid fluid does not permit one.

A finite force from an infinite speed

Pressure on a flat plate can only act along the plate's normal. Kutta–Joukowski says the force is perpendicular to the free stream. Those two directions differ by the angle of attack, and the discrepancy is made up at a single point where the velocity is infinite and the area is zero.

circulation · Leading edge suction 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 The singularity a thin aerofoil drives onto its own nose. The Joukowski map has a critical point that maps to a place inside the body, a distance 4 mu²/(1 + 2 mu) from the leading edge. Against thickness that distance is a clean square: a twelve per cent section is analytic only within eight thousandths of a chord of its own nose, and the thin-aerofoil limit is the limit in which the singularity arrives on the surface.

The part of the flow inside the body

A potential flow outside a body is an analytic function, and an analytic function does not stop at the boundary it was defined on. It continues inward until it meets a singularity — and every body in this collection has at least one inside it, in a place that decides how the flow behaves outside.

inviscid · Singularities 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 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.

circulation · Thin-aerofoil

Named alongside it

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

CamberLift coefficientLift curve slopeCirculationConformal mapKutta conditionPressure distributionAerodynamic centreModel limitPitching momentPotential flowConstraint

All concepts