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

Who wins, and by how much.

Two players, no dice, nothing hidden, and the player who cannot move loses. That is a narrow enough set of rules to be worth exactly — every position has a value, the value is computed rather than estimated, and positions add. These are essays about what comes out of that, one idea at a time, with the arithmetic done rather than asserted.

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.Γ = 2.445C_L = 1.212ideal flow with the Kutta condition applied6° incidence
Fig. 1 The pressure over a lifting section, solved rather than sketched. The flow runs faster over the upper surface than the lower, and the bands are the pressure that goes with it. Every field on this site is computed and then checked — mass conserved, nothing through the wall, the lift arrived at twice by unrelated routes — because a wrong flow field is beautiful, and the picture alone cannot tell anybody which kind it is.

Start anywhere

19 essays

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.

6 figures the upper surface is 1.7% longerso equal transit needs the flow over it 1.7% fasterit is actually 76.5% fasterthe premise is false and the number it predicts is wrongsurface lengths and speeds measured on the solved field6° incidence What is taught wrongly

The story about air meeting up again

The most repeated explanation of lift says that air parting at the nose must rejoin at the tail, so the longer upper path forces a higher speed. The premise is false, and the speed it predicts is wrong by a factor of twenty.

7 figures 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.

6 figures creepingattachedseparated, sheddingturbulentbacterium swimmingshedding begins, Re ≈ 47a thrown ballan airliner winga whaleReynolds numberinertia ÷ viscositylog₁₀ Rethe ratio decides the regime, not the size or the speed alone Regimes and numbers

One number decides which physics applies

A bacterium and a whale both swim, and they are not doing the same thing at different sizes. The ratio of inertia to viscosity separates them, and crossing it changes the rules rather than the magnitudes.

6 figures thin: streamlines, frozen at one instantthick: the path one particle actually takesunsteady flow — the three families differincompressible Flows and fields

Streamlines are not the paths particles take

Three different curves get drawn through a flow and they are routinely treated as one. In steady flow they coincide, which is why the confusion survives; in unsteady flow they are as different as a photograph and a long exposure.

6 figures 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.

6 figures 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.

7 figures 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.

6 figures 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.

6 figures Bernoulli holds…along one streamline, steady, inviscid, incompressiblethe theorembetween two different streamlinesneeds irrotational flow as wellthrough a fan, pump or propellerwork is being done on the fluidinside a boundary layerviscosity is the whole story thereacross a shockentropy rises; total pressure does not survivethe hypotheses, not the algebra, are what fail What is taught wrongly

Where Bernoulli's equation applies

The equation is right. Its hypotheses are strict, and almost all misuse is a correct formula carried somewhere it does not hold — across streamlines, through a fan, or into the one layer where friction is the whole story.

6 figures 0.871.330.85ideal flow — incompressible, so the tube's area sets the speed Flows and fields

Mass has nowhere to go

Squeeze a stream of fluid and it speeds up, not because anything pushes it but because the same amount has to get through a smaller gap every second. Almost every result in the subject is that observation with more machinery attached.

6 figures creepingattachedseparated, sheddingturbulentbacterium swimmingshedding begins, Re ≈ 47a thrown ballan airliner winga whaleReynolds numberinertia ÷ viscositylog₁₀ Rethe ratio decides the regime, not the size or the speed alone Regimes and numbers

The Reynolds number, and the length in it

The most useful number in fluid mechanics has an arbitrary quantity buried in it, and quoting one without saying which length was used makes it meaningless. That detail is where most misuse comes from.

6 figures 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.

6 figures 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.

6 figures 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.

7 figures 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.

6 figures length ∝ speedideal flow past a cylinderfastest 1.91U Flows and fields

What a flow is

A fluid is made of molecules and nobody models it that way. Treating it as a continuous field with a velocity at every point is an approximation, an extremely good one, and knowing why it works is knowing where it stops.

6 figures 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.

6 figures ideal flow — inviscid, irrotational, steadyno circulation Ideal flow

Fast means low pressure

The trade between speed and pressure is the most useful relation in the subject and the most misused. Where it comes from, what it costs, and why the pressure over a wing is negative almost everywhere.

6 figures

Threads running through

themes, not chapters

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.

11 essays

One number decides the regime

The same shape in the same fluid behaves completely differently at different Reynolds numbers. The shape is not the question; the ratio is.

10 essays

Lift is circulation

Not shape, not equal transit times, not air pushed down in the way usually claimed. Lift is the circulation round the body, and the Kutta condition is what fixes it.

7 essays

Everything happens in a thin layer

Viscosity is negligible almost everywhere and decisive in a film a millimetre thick, and that film decides drag, stall and the wake.

5 essays

What is conserved

Mass, momentum, circulation, vorticity. Every result worth having is a statement that one of them cannot go anywhere, and every check in this site's solver is a test that it did not.

12 essays

A smooth picture proves nothing

Streamlines are smooth whatever nonsense produced them. A flow figure is only worth as much as the conservation laws it was checked against.

7 essays

Taught wrongly, everywhere

Some of the most repeated explanations in physics education are in this subject, and they are false. Stating them fairly and then testing them is more useful than ignoring them.

5 essays