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

The thread: The exact theory is wrong

Ideal flow is closed-form, elegant and predicts that an aeroplane needs no engines. Every honest account of this subject is organised around what that failure reveals.
Γ = 2.445C_L = 1.212ideal flow with the Kutta condition applied6° incidence Circulation and 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.

 separatedrecirculation 0.56 Dviscous flow, solved on a coarse grid — the bubble is under-resolvedRe = 40 Viscosity

Everything happens in a layer you cannot see

Air has so little viscosity that ignoring it works almost everywhere. Almost everywhere leaves out a film next to the surface, perhaps a millimetre thick, and that film decides drag, stall and whether an aircraft flies at all.

 ideal flow — inviscid, irrotational, steadyno circulation Ideal flow

The theory that solves everything

Throw away viscosity and assume nothing is spinning, and fluid mechanics collapses into a linear problem with closed-form answers. The price is one term, and the term turns out to matter more than everything kept.

 frontback+1−30angle round the bodycomputed drag-1.1e-16not “small” — zeroideal flow — inviscid, irrotational, steadyany Reynolds number Ideal flow

The exact theory says nothing has any drag

Solve the flow past a body in a fluid with no viscosity and the answer is beautiful, closed-form, and predicts that a cyclist needs no legs and an airliner no engines. This is not a small error, and it is the most useful failure in the subject.

 too little — the flow whips round the edgeΓ = 0.93the Kutta value — it leaves smoothlyΓ = 2.67too much — the rear stagnation point is on topΓ = 4.81ideal flow — three admissible solutions, one physical8° incidence Circulation and 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.

 separatedrecirculation 1.16 Dviscous flow, solved on a coarse grid — the bubble is under-resolvedRe = 100 Viscosity

When the flow lets go

Every body asks the air behind it to slow down and climb back up to the pressure it started at. Sometimes the air cannot, and the moment it refuses is separation — the source of most drag, the cause of stall, and the reason a golf ball has dimples.

 lift at zero incidenceC_Langle of attack, degreesslope6.84 / radthin-aerofoil theory: 6.28the difference is thicknessideal flow with the Kutta condition, no stall modelattached flow only Circulation and 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.

 a uniform streama doublet alonestream + doublet = a circleideal flow — superposition holds because the equations are linear Ideal flow

Flows add up

The equations of ideal flow are linear, so solutions can be laid on top of one another. A uniform stream plus a doublet produces a cylinder that nobody put there, and almost every classical result is built this way.

 ideal flow — closes up, no dragreal flow at Re 100 — separatedleft: exact closed form · right: solved on a gridRe = 100 Viscosity

The two theories, side by side

The exact solution and the real flow, for the same body in the same stream. One is beautiful and predicts nothing has drag; the other is approximate and has a wake in it. Where they agree and where they part is the whole map of the subject.

 incompressiblecompressible, subsonictransonicsupersonica cyclistdensity starts to matteran airliner cruisingshock waves everywhereMach numberspeed ÷ speed of soundMachthe ratio decides the regime, not the size or the speed alone Regimes and numbers

When air stops being incompressible

Air is a gas and can obviously be squeezed, yet most of aerodynamics treats its density as fixed. The assumption holds until the flow approaches the speed at which pressure information travels — and then everything changes at once.

 lift 3.393= ρUΓ = 3.400drag 1.5e-16 — still zeroideal flow — no shape, only circulationΓ = -3.4 Circulation and lift

Lift with no wing at all

A spinning cylinder has no camber, no aerofoil section and no trailing edge, and it lifts exactly as hard as its circulation says it should. Which settles what lift is caused by.

All themes