Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: 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.
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.

 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.

 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.

 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.

 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.

 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.

 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.

 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.

 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.

 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.

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