Tilt a wing. Change the flow.: sources & model

Run a virtual wind-tunnel test. Tilt a symmetric airfoil, probe both surfaces, and build the whole lift force from pressure around the wing.

What supports the explanation?

What this model assumes

1. Original mathematical airfoil and generic viewing fixture. Shape, velocity, pressure and integrated force share one Joukowski solution. The fixture does not reproduce a named aircraft or real wind tunnel.
2. Fixed 1 m chord, 1.225 kg/m³ supplied density, −6° to +6° and 10–30 m/s, plus air off. Moving a slider selects a new steady solution rather than simulating a pitch maneuver.
3. Ideal 2D, steady, incompressible, inviscid flow. No boundary layers, separation, stall, turbulence, compressibility, induced drag, aircraft motion or starting transient.
4. Surface velocity represents the ideal flow outside any real boundary layer. Mathematical pressure drag tends to zero here; real wings have drag.
5. Computed streamlines and speed-derived tracers. Marker motion is slowed 20 times; streamlines are not literal rows of molecules or a density measurement.
6. Pressure-deviation arrows can point outward where pressure is lower than the common reference. They do not depict negative absolute pressure or a separate suction force. Partial sums are not the whole airfoil result.
7. Ladson’s measured NACA 0012 values belong to a different airfoil at Re = 6 million, M = 0.15 with 80-grit tripped transition. They are evidence about model limits, not calibration or universal stall criteria.
8. One shape, one consistent solution: The live section is an original mathematical Joukowski airfoil, not a NACA 0012 or an aircraft replica. A circle centered at −3/121 m with radius 3/11 m is mapped by z = ζ + b²/ζ, with b = 30/121 m. The result has a 1 m chord and about 11.785% thickness. The velocity uses the derivative of that same mapping.
9. Why the trailing edge matters: An ideal flow around a sharp trailing edge needs a circulation choice. The Kutta condition selects the finite-velocity trailing-edge solution. With clockwise-positive circulation in our notation, Γ = 4πUR sin α. Lift per unit span is L′ = ρUΓ. The model uses ρ = 1.225 kg/m³ as a supplied example value.
10. Normalize before comparing: Dynamic pressure is q = ½ρU². The pressure coefficient is Cₚ = (p − p∞)/q = 1 − (local speed/U)². For this fixed geometry, Cₗ = (24π/11) sin α and L′ = q c Cₗ. At 4° and 20 m/s, Cₗ is about 0.478 and L′ about 117.144 N/m. Doubling speed at the same angle multiplies force by four while leaving normalized pressure unchanged.
11. A section force is not aircraft lift: The result is force per meter of span for a two-dimensional section. The displayed extrusion helps you recognize a wing specimen, but the calculation has no finite-wing tips, induced drag or full aircraft. It does not predict whether an aircraft climbs, trims or carries a given weight.
12. Where real flow departs: This model assumes steady, incompressible, inviscid flow. Real air has viscosity and a boundary layer near the surface. Separation can change pressure and lifting behavior; an ideal smooth solution does not predict stall. The ±6° control range is a teaching boundary, not a measured stall angle. The separate Ladson data card identifies its own airfoil and test conditions.

What has been checked

Analytical reference cases, conservation or transition invariants, finite drawing commands, bounded setup parsing, discovery and route integrity are checked automatically. These checks do not establish anatomical fidelity, learner outcomes or browser/device compatibility. Independent subject review, learner trials, comprehensive accessibility review and browser video encoding checks remain pending.

Each source supports the associated claim. Sources do not certify this implementation or its visuals.

About the cover illustration

The cover is a crop of a 1947 NACA Ames wind-tunnel photograph, credited to NASA. It shows real testing context, not the shape or measured performance of the ideal Joukowski section in our experiment. The original photograph and asset provenance are available from the lesson’s measured-data card.