Bare wingtip — the full tip vortex forms right here.
Working Cₗ
0.60
Aspect ratio AR7.5
Oswald e0.75tip-efficiency penalty
Cₘᵢ induced drag0.0203Cₗ² / (π · AR · e)
Tip vortex strengthmoderateΓ ≈ lift / span
About this lab
The Wing — Lift, Vortices, and What the Brothers Found
A browser aeronautics lab. Two panels: when does a wing fly (drag the sliders, watch the lift catch the weight), and where the air spills off the tips (three wing types compared, with three tip devices on the airliner wing — bare, blended winglet, and the split-scimitar winglet Southwest flies — so you can watch the vortex climb off the wing and the Oswald-efficiency number rise).
Companion to the water lab
The horseshoe vortex on the rear-view canvas is the same flow pattern that wraps the base of a bridge pier and scours the river bed. Both labs sit in the OPA Engineering Suite next to each other on purpose — same physics, two professions that don't usually talk to each other. See The Horseshoe Vortex (Water + Piers) for the hydraulic half.
The hidden lesson
Lift doesn't come from "air hitting the wing" — that picture predicts you'd need orders of magnitude more wing area than airplanes actually have. Lift comes from circulation around the wing: the air on top moves faster than the air on the bottom, pressure drops on top, the wing gets pulled up. The proof that this is the real picture is the trailing tip vortices — they only exist because the wing is finite, and they cost you (induced drag) exactly as much as the lift you're getting (the Prandtl equation). The vortex is the cost. There's no lift without paying it.
What isn't here
Subsonic, incompressible, attached-flow. The lab won't model post-stall separation, transonic shock waves, or supersonic compressibility — flags those regimes, refuses to lie about them. The lift coefficient curve is the linear thin-airfoil approximation (Cₗ ≈ 2πα) capped at a stall plateau; real airfoils vary ±15% off this.
The wing presets (Cessna 172, F-35, 747-400) have real published wing areas, masses, and aspect ratios, but the lab is a teaching instrument, not a flight-simulator certification tool. For airworthiness work use the actual flight manual.
Suite context
Lab of the OPA Engineering Suite · companion to The Horseshoe Vortex (Water + Piers) · sibling to the Browser Physics Suite (Chladni, Acoustic Levitation, Ripple Tank, Double-Slit). Filed under College X (Engineering). Course anchor: AERO 210. Cross-listed with HYDR 250 (the water-side horseshoe vortex).
Sources & equations
The actual math under the hood
Lift — Panel 1
L = ½ · ρ · V² · S · Cₗ
The lift equation, every undergrad aero textbook (Anderson, Introduction to Flight, §5). The lab uses English units: ρ in slug/ft³, V in ft/s, S in ft², L in lbf.
Lift coefficient — thin airfoil
Cₗ ≈ 2π · α (α in radians)
plateau at Cₗₘₐ₧ ≈ 1.4 near α ≈ 15°
collapse to ~0.4 after stall
Thin-airfoil theory. Real wings vary ±15% on the slope and stall angle; this lab uses the linear approximation up to a stall plateau, then drops past it. The "STALL" label fires when α > 15°.
Stall speed
Vₛₜₐ₄₄ = √( 2W / (ρ · S · Cₗₘₐ₧) )
The slowest speed at which L = W at maximum Cₗ. Below this speed the airplane cannot generate enough lift to support its weight, regardless of angle of attack.
Induced drag — Panel 2 (Prandtl lifting-line)
Cₘᵢ = Cₗ² / (π · AR · e)
Prandtl 1918. AR = b²/S (aspect ratio = span squared over wing area). e is the Oswald efficiency factor: 1.0 is the theoretical maximum (an infinite elliptical wing); real wings sit between 0.6 and 0.95.
Oswald efficiency values used in this lab
747-400 conventional wing, no winglet: e ≈ 0.75
747-400 with Whitcomb blended winglet: e ≈ 0.88 (~5–7% net fuel savings)
With a split-scimitar winglet (blended blade + lower ventral fin): e ≈ 0.92. The real Aviation Partners Boeing 737NG retrofit Southwest flies adds roughly another ~2% drag reduction over the blended winglet — lifting per-aircraft fuel savings from ~3.5% (blended) to ~5–5.5% (split scimitar). The lower fin recovers spill the single blade leaves behind.
Swept sport-jet wing (no winglet): e ≈ 0.78
F-35 cropped delta (low-AR, supersonic-capable shape): e ≈ 0.62 (the delta is a different optimization — supersonic stability, not subsonic cruise efficiency)
The horseshoe vortex itself
Prandtl's lifting-line model represents a finite wing as a bound vortex running spanwise (along the wing) plus two trailing vortices shed at the tips that extend infinitely downstream. The three together form a horseshoe. The trailing vortices are visible in humid air (you've seen the contrail-corkscrews behind heavy aircraft on a wet morning). They carry the induced-drag energy away as kinetic energy of rotating air, eventually dissipating as heat.
Receipts
Anderson, J.D. Fundamentals of Aerodynamics, 6th ed., McGraw-Hill, 2017. Ch. 5 covers finite-wing theory; the horseshoe-vortex derivation is §5.3. The Prandtl 1918 original paper: "Tragflügeltheorie I," Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. The Whitcomb winglet patent: NASA-Langley winglet design retrospective, NASA SP-4404 Engineer in Charge.
The long history of "first"
Five criteria, not one
The "who flew first" argument keeps going because "flight" isn't one thing. It's at least five things stacked:
Powered — the airplane carries its own engine, doesn't glide off a hill.
Airborne — both wheels (or skids, or floats) off the ground.
Sustained — long enough to be more than a hop or a bounce.
Controlled — the pilot can steer it (banking, pitch, yaw) and bring it down on purpose.
Repeatable + witnessed — it can do it again, in front of people who write it down.
The contenders
Clément Ader (France, 1890): powered + airborne + briefly sustained, but not controlled, no witnesses outside the French military, and the Army declared the next test (1897) a failure. Hits 3 of 5.
Gustave Whitehead (Connecticut, 1901): claimed powered + airborne + sustained + controlled + witnessed. No surviving photo of the alleged flight, contradictory witness accounts, decades-later disputes. The claim is in state law in Connecticut and disputed everywhere else. Hits 3–5 of 5 depending on who you ask.
The Wright Brothers (Kitty Hawk, December 17, 1903): all five, same day, four flights, photograph (John T. Daniels' shutter click). Hits 5 of 5. The clean case.
Alberto Santos-Dumont (Paris, November 12, 1906): all five, fully public, the European reference flight. The Wrights had flown three years earlier but worked in secrecy to protect their patent; many Europeans only counted the first witnessed European flight. Hits 5 of 5 + the publicity.
The honest answer
The Wrights are credited because they were the first to hit all five criteria simultaneously, on the same day, in the same machine, with a photograph. That doesn't erase the others. Each one was solving the parts of the problem that the textbook now lumps under one word: "flight."
If you're teaching this, the move is to list the five criteria first, then ask the room to score the contenders. The argument collapses into definitions, and the students discover they've learned more about the engineering problem than the trivia.