Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
How this site is made

The figure library

Every picture here is generated from code at build time. This page lists the generators, each rendered at its defaults.

No figure on this site is a drawing that was made once and saved. Each one is a function: it takes parameters and returns SVG, so the same generator produces the 27° incline and the 5° incline without either being redrawn.

That is the reason the collection can keep growing without the illustrations drifting apart. A generator is written once, checked once, and every essay that calls it inherits the same line weights, the same colour roles, and the same behaviour in dark mode. There are 16 of them so far.

aerofoil

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

bernoulli-limits

Where Bernoulli's equation appliesThe equation is correct and its hypotheses are strict. Most misuse is not a wrong formula but a right formula carried across a streamline, through a machine, or into a region where viscosity dominates.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

continuity

A streamtube narrows and the flow speeds upTwo neighbouring streamlines bound a tube that no fluid crosses. Where the tube pinches, the same mass has to pass through a smaller gap every second, so it must move faster — which is mass conservation with no equations in sight.0.871.330.85ideal flow — incompressible, so the tube's area sets the speed

dalembert

d'Alembert's paradox, measuredSurface pressure round a cylinder in ideal flow, plotted against angle. The distribution is symmetric front to back, so every push on the front is matched by an equal push on the back, and the total force along the stream is exactly zero.frontback+1−30angle round the bodycomputed drag-1.1e-16not “small” — zeroideal flow — inviscid, irrotational, steadyany Reynolds number

equal-transit

Two parcels released together do not arrive togetherThe most repeated explanation of lift says that air parting at the leading edge must meet again at the trailing edge, so the longer upper path forces a higher speed. Released into the solved field, the upper parcel arrives long before the lower one — the premise is simply false, and the real speed difference is larger than it would require.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

ideal-cylinder

Ideal flow past a cylinderA uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.ideal flow — inviscid, irrotational, steadyno circulation

ideal-vs-real

What the ideal theory predicts, and what happensThe same cylinder, the same free stream. On the left the exact inviscid solution, closing up behind the body and exerting no drag at all. On the right the real flow at the same conditions, separated, with a wake and therefore with drag.ideal flow — closes up, no dragreal flow at Re 100 — separatedleft: exact closed form · right: solved on a gridRe = 100

kutta

The Kutta condition picks the circulationIdeal flow round an aerofoil admits any circulation at all, and each gives a different lift. Only one value lets the flow leave the sharp trailing edge without turning a corner at infinite speed, and that is the one nature selects.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

lift-curve

The lift curve, computedLift coefficient against angle of attack for a cambered Joukowski section, every point solved rather than fitted. The line is straight, it does not pass through the origin, and its slope is close to but above the thin-aerofoil value.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

magnus

Lift from a spinning cylinderA circular cylinder with circulation round it. There is no aerofoil section, no camber and no sharp trailing edge, and it lifts — which rules out shape as the explanation and leaves circulation as the thing that matters.lift 3.393= ρUΓ = 3.400drag 1.5e-16 — still zeroideal flow — no shape, only circulationΓ = -3.4

regime-axis

Reynolds number: one number, four different flowsReynolds number is inertia ÷ viscosity. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.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

same-shape-different-flow

The same shape, two different worldsA sphere at Reynolds number a ten-thousandth and a sphere at a hundred thousand are not the same problem at different speeds. In one, motion stops the instant the forcing does; in the other, the object drags a wake behind it for many diameters.a bacteriumRe 10⁻⁴everything reverses if it stopsa thrown ballRe 10⁵a wake it drags behind itschematic — the contrast, not a solved fieldtwo regimes

superposition

Flows addThe equations of ideal flow are linear, so solutions can be added. A uniform stream and a doublet, laid on top of each other, produce a flow with a circular streamline — which is to say, a cylinder appears where none was put.a uniform streama doublet alonestream + doublet = a circleideal flow — superposition holds because the equations are linear

three-curves

Streamlines and pathlines are not the same curveIn an unsteady flow the line tangent to the velocity everywhere at one instant, and the track a single particle actually follows, are different curves. They coincide only when the flow is steady, which is the hypothesis most figures forget to state.thin: streamlines, frozen at one instantthick: the path one particle actually takesunsteady flow — the three families differincompressible

velocity-field

The velocity field, arrows to scaleThe same flow drawn as arrows. Scaled to the local speed the picture is honest and crowded; drawn all the same length it is legible and hides the very variation the figure is about.length ∝ speedideal flow past a cylinderfastest 1.91U

viscous-cylinder

Flow past a cylinder at Re 40A real fluid past a circular cylinder. At low Reynolds number the flow closes up behind the body much as the ideal theory says; as it rises the flow separates and a region of reversed flow appears behind, which is where drag comes from.separatedrecirculation 0.56 Dviscous flow, solved on a coarse grid — the bubble is under-resolvedRe = 40