The Wing

Panel 1 · side view · airplane in free stream · lift vs weight

— "when does it fly?"

Regime: ON THE GROUND L / W = 0.00

Airspeed V

0 mph

AoA α

4.0°

Wing area S

180 ft²

Mass m

2,500 lb

Air ρ

sea level · 0.00238

Class

Preset

↳ big-iron secret — press B

Cₗ lift coef. 0.00

Lift L 0 lb

Weight W 2,500 lb

L / W ratio 0.00

Vₙₜₐₜₙ @ this α — mph stall speed at this weight

Regime ON THE GROUND

Panel 2 · rear view · 3 wing types · vortex spirals at tips

— "where the air spills off"

Wing: 747-400 conventional AR = 7.5 · e = 0.75

Wing type

Tip device

Bare wingtip — the full tip vortex forms right here.

Working Cₗ

0.60

Aspect ratio AR 7.5

Oswald e 0.75 tip-efficiency penalty

Cₘᵢ induced drag 0.0203 Cₗ² / (π · AR · e)

Tip vortex strength moderate Γ ≈ lift / span

A brief history of getting off the ground

the contested "first" · honest version

1890 · France

Clément Ader · Avion III

Steam-powered, bat-winged, hopped roughly 50 m a few inches off the ground. Powered, briefly airborne, uncontrolled. Not sustained, not controllable. The French Army watched and lost interest.

1901 · Connecticut

Gustave Whitehead · No. 21

German-American immigrant; claimed a half-mile powered flight near Bridgeport. Witnesses, but no photograph survives. Contested by most aviation historians. Connecticut still lists him in state law as the first to fly.

Dec 17, 1903 · Kitty Hawk, NC

Wright Brothers · Flyer I

Four flights that day. Longest: 59 seconds, 852 feet. Powered + sustained + controlled + witnessed + repeatable in same form on same day. All five criteria together is the case for "first." Each criterion in isolation already had claimants.

Nov 12, 1906 · Paris

Alberto Santos-Dumont · 14-bis

First public, witnessed, powered, controlled flight in Europe. 220 m, 21 seconds. Brazilian, flying in France. Many Europeans counted him as the real first because the Wrights worked in secrecy from 1903–1908 to protect the patent.

1918 · Göttingen

Ludwig Prandtl · lifting-line theory

Worked out the math the Wrights had felt their way to. Cdᵢ = Cₗ² / (π·AR·e). The tip vortices are not a side effect — they are the cost of lift. Finite wings have to pay it; only an infinite wing wouldn't.

1976 · NASA Langley

Richard Whitcomb · the winglet

Bent the tip up. Disrupts the tip vortex before it can fully form, raising effective Oswald e from ~0.75 to ~0.85–0.90. Net ~5% fuel savings on a 747. Retrofitted onto the 747-400 starting 1988. Same idea as putting a wedge nose on a bridge pier.

2014 · Aviation Partners Boeing

Split Scimitar · the winglet grows a second half

Take the blended winglet and add a downward ventral fin, so one airfoil points up and one points down-and-out — the "V" at the tip of nearly every Southwest 737. The lower fin recovers the spill the single blade leaves behind, pushing per-aircraft fuel savings from ~3.5% to ~5–5.5%. Southwest's first revenue flight on them: April 9, 2014. The blade alone was Whitcomb's wedge nose; the lower fin is the second wedge nobody thought to add for thirty years.

Honest framing: The Wrights weren't the first to leave the ground. They were the first to leave it controllably and repeatably, with witnesses and a paper trail. The other claims aren't fraud — they're missing one or two of the five criteria. Kitty Hawk gets credit because that day they hit all five.

Same horseshoe vortex, different profession. The two trailing tubes of air at every wingtip are the same flow pattern that scours sediment out from under bridge piers. The water-and-piers half of the story lives at /labs/horseshoe-vortex.html — same physics, different vocabulary.

✈ "Lift is not the wing pushing air down,
it's the wing convincing the air
to push it up."
— the Prandtl trick, restated for undergrads

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 Wing — Lift, Vortices, and What the Brothers Found

Companion to the water lab

The hidden lesson

What isn't here

Suite context