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

Kutta condition — where it appears

The requirement that the flow leave a sharp trailing edge smoothly, which picks one circulation out of the infinitely many that solve the inviscid problem. It is the outcome of a viscous process rather than an axiom, and it is what makes lift calculable.

Named by 19 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.

What actually holds a wing up

Not the shape, and not the story about air meeting up again behind. A wing lifts because there is circulation round it, and the sharp trailing edge is what decides how much.

circulation · Lift A Joukowski aerofoil at 8°. 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 sharp edge decides

Ideal flow round a wing admits infinitely many solutions, each with a different lift, and all of them exact. One extra requirement — that the air leaves the trailing edge instead of whipping round it — picks a single one.

circulation · Lift 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 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 There is a constriction, and it is a consequence rather than a cause. A cambered section at 5 degrees, with one streamtube traced above it and one below. The tube above narrows by 16 per cent at mid-chord and the tube below by -112 per cent, so the venturi story's premise is true: the flow over the top really is squeezed more than the flow underneath. The difficulty is that the tube's upper boundary is a streamline, not a wall. Nothing put it there but the solution of the whole flow — the same solution that already contains the lift — so the narrowing is a way of describing the answer rather than a reason for it.

Not half a venturi

The air above a wing really is squeezed into a narrower channel, it really does speed up, and the pressure really does fall. Every step of the story is true and the whole is still not an explanation, because the channel's upper wall is a streamline — and where a streamline went is part of the answer, not part of the question.

misconceptions · Venturi 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.

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.

circulation · Unsteady 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 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.

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.

circulation · Panel 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.

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.

circulation · Slot 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 Four corners, and what the flow does in each. The local flow in corners of four different interior angles, drawn from the exact local solution ψ = r^(π/α) sin(πθ/α). The exponent of the speed is π/α − 1 and depends on nothing else: at a right angle the corner is stagnant, at a flat wall nothing happens, and at any angle greater than a straight line the speed has no bound at the corner. The last panel, at 360 degrees, is the flow round the edge of a plate.

Nothing turns a sharp corner

Near a corner the flow is fixed by the angle and by nothing else — not by the size of the corner, not by the flow far away, not by the fluid. The exponent is π/α − 1, and every sharp edge in aerodynamics is the one case where it comes out at minus one half.

inviscid · Wedge flow 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.

circulation · Circulation control 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.

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.

inviscid · Multiply-connected Six flows past one cylinder, all of them legal. The tangential speed on the surface for six values of the circulation. Every one of them solves the same equation and lets nothing through the wall; the fastest point on the surface runs from twice the free stream to eight times it.

Nothing in the present picks the flow

Six flows past one cylinder satisfy the same equation and let nothing through the wall, to the last bit of double precision. Their lifts run from zero to 37.7 and their peak suctions differ by a factor of twenty-one. The equations do not choose between them, and the thing that does is the history.

inviscid · Multiply-connected A hydrofoil loses lift at speed and gains it slowly, with a dip between. The lift of a flat hydrofoil beneath the surface over its value in deep water, against the chord Froude number U/√(gc) on a logarithmic axis, at depths of half a chord, one chord and two. Slow, the surface is a lid and the foil gains lift, as a wing over the ground does. Fast, the surface is a pressure-release boundary and the foil loses it. Between the two it does not simply pass from one to the other: the lift dips well below its fast value where the foil's waves are longest compared with its depth, and then rises through the lid value before settling back on it.

A hydrofoil loses most lift on the way up

A wing near the ground gains lift, and a hydrofoil near the surface is often described as the same thing upside down. At low speed it is: the surface acts as a lid and the foil gains lift. At high speed the surface is a boundary that cannot hold a pressure, and the foil loses lift instead. But the lift does not pass smoothly from one to the other. Between them, where the foil's waves are a few depths long, it falls well below both — to 43 per cent of its deep-water value half a chord down — and a foil boat meets that speed in the middle of its take-off run.

misconceptions · Ground cushion The camber lines that carry them. The camber lines that carry the four loads at their ideal angles, heights in chords at a design lift coefficient of one. The uniform load's line is symmetric about mid-chord, 5.52% high; tapering the load over the last fifth moves the peak to 0.515 of the chord and raises it to 6.79%; tapering it over the whole chord, a = 0, puts the peak at 0.323.

A load carried to the trailing edge needs a hook

Thin-aerofoil theory runs backwards: ask for a chordwise load and it returns the camber line that carries it. Ask for the simplest load of all, the same everywhere, and the line comes back with a vertical tangent at both ends. The one at the nose is the price of a clean entry; the one at the tail is the Kutta condition being broken, and the hook it makes is what a Gurney flap is. Every NACA a-series line is a way of not paying it.

circulation · Thin-aerofoil The parcel over the top arrives first, and never waits for the other. Two parcels released together far upstream, a whisker above and below the streamline that divides at the nose of a Joukowski section 11.8% thick at 4°. Both crawl past the nose; then the upper one is carried over the top faster and reaches the far line 1.13 time units ahead — 0.279 chord-transit times — against the 1.59 that the circulation divided by the speed squared predicts. Behind the section they travel on side by side at the stream's speed and the gap between them never closes.

The upper parcel leads by the circulation

Transit time looked like a quantity that could not be measured: a parcel released near the dividing streamline crawls past the nose for as long as one likes. The difference between two parcels' transit times, one either side of that streamline, does not crawl. It converges, and on a thin section it is the circulation divided by the square of the speed — the lift, measured in seconds. The upper parcel arrives first by that much, and behind the wing the two never close the gap.

misconceptions · Equal transit Inside the layer the upper parcel's lead grows with depth. The transit lag, lower parcel minus upper, of parcels held at a quarter, a half and three-quarters of the boundary layer's thickness and just outside it, timed from either side of the attachment point to each chordwise station, on a thin cambered section at zero incidence, up to the upper layer's laminar separation at 62% chord. At mid-chord the lag outside the layer is 0.183 chord-over-speed units; at three-quarters of the thickness 0.195, at half 0.281 and at a quarter 0.695.

Inside the layer the upper parcel leads by more

Two parcels parting at a wing's attachment point arrive at different times, and outside the boundary layer the difference is the circulation over the speed squared. Inside the layer one might expect the slow, sticky fluid to wash the difference out. It does the opposite: deep in the layer the upper parcel's lead is several times the inviscid one, because the suction side's layer is fuller than the pressure side's. Only the layer's outer quarter reads the inviscid lead, and on the suction side the laminar layer gives out before the trailing edge.

misconceptions · Equal transit

Named alongside it

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

CirculationModel limitTrailing edgeConformal mapLift coefficientThin-aerofoil theoryCamberLiftPotential flowStagnation pointJoukowskiPressure distribution

All concepts