Intuition The ONE core ideaA real wing has ends , and at those ends the air leaks around from the high-pressure bottom to the low-pressure top, trailing spinning tubes of air behind the wing. Those trailing spins tilt the air the wing feels slightly downward, which tilts the lift slightly backward — and that backward tilt is a brand-new kind of drag that exists even with zero friction .
This page assumes you have seen none of the symbols in the parent note. We build each one from a picture, in an order where every symbol only uses things already defined. If you can follow line 1, you can follow the whole topic.
Before any physics, we need words for the shape we are talking about.
Read the figure: the teal outline is the wing seen from above . The orange double-arrow across the bottom is the span b chord c ( y ) y y = 0 y = ± b /2 area S
Definition The three shape numbers
Span b Chord c different chord at each spanwise position , so we write c ( y ) at location y Planform area S S = b × c
Intuition Why we care about the direction along the spanEverything interesting about a finite wing happens because things change as you walk from the middle out to the tip . So we need a coordinate that runs along the span. That coordinate is y
Definition The spanwise coordinatey y zero at the middle of the wing. The left tip sits at y = − 2 b y = + 2 b y [ − 2 b , + 2 b ]
When we write "c ( y ) stand at position y
A R = S b 2 span squared, divided by planform area. It is a single number that says how "long and skinny" a wing is.
Why squared ? Look at a rectangle where S = b c A R = b c b 2 = c b , span ÷ chord — how many chord-widths fit across the span. A glider wing (very long, very thin) has a large A R A R
Common mistake "Aspect ratio is just the wing's area."Why it feels right: both describe "how big the wing is." Fix: A R same area but wildly different A R
Aspect ratio is the star of this whole topic, so meet it early. Full treatment: Aspect ratio & wing design .
Definition Freestream velocityV ∞ V ∞ ∞ measured far away , where the wing hasn't yet disturbed anything.
ρ ∞ ρ density — how many kilograms of air sit in each cubic metre. The ∞ ρ
Intuition Why these two live in every lift/drag formulaForce from a fluid always scales with how much stuff you throw (ρ how fast you throw it (V ρ ∞ V ∞
This is the most important new idea, so we go slowly.
Read the figure: the solid black blob is the wing's cross-section, cut like a loaf of bread. The teal arrows show the air moving fast over the top and slow underneath — the signature of lift. The dashed plum loop is the path you "walk" all the way around; the orange arrowheads on it show the direction you add up the air's push along the loop. That grand total is the circulation Γ
Intuition What "circulation" is trying to captureA lifting wing bends the air: faster over the top, slower underneath. If you walk a big loop around the wing's cross-section (the dashed plum loop in the figure) and add up how much the air pushes you along your path, you get a non-zero total — the air is, on the whole, circling the wing. Circulation is that total, packaged as a single number.
Γ Γ Γ
Because the wing lifts differently at different spanwise stations, the swirl also changes along the span. So we write Γ ( y ) at spanwise station y Γ ( − 2 b ) = Γ ( + 2 b ) = 0.
Mnemonic Circulation is the wing's "spin account"More swirl banked, more lift withdrawn. Everything downstream in the topic is really about the shape of Γ ( y )
Before we can talk about the vortices that trail off the wing, we need one piece of calculus notation. We introduce it here , in plain words, before ever using it.
Intuition Why we need a "rate of change" at allThe whole story of a finite wing is about change along the span : Γ how quickly Γ "if I take one tiny step d η Γ
Definition The derivatived η d Γ d η d Γ d η d Γ = tiny step along span tiny change in Γ slope of the Γ Γ Γ η same kind of ruler as y
Intuition Reading the slope's sign off the pictureOn the right half of the wing Γ d η d Γ negative there. On the left half Γ d η d Γ positive . The steepness (size) is largest near the tips — which is exactly where tip vortices are strongest.
Definition A vortex filamentA vortex filament is an idealised thin thread of concentrated spin — picture the thin core of a tornado, or the little curl at the tip of a canoe paddle. We model the wing's swirl as such threads.
Definition Bound vs trailing vortices
The bound vortex is the spin thread that sits on the wing itself (it "lifts" the wing). Its strength is exactly Γ ( y )
Trailing vortices are threads that peel off the wing and stream backwards into the wake.
Intuition Why trailing vortices must exist (Helmholtz)A spin thread is not allowed to simply end in mid-air — spin has nowhere to go. So if the bound spin Γ ( y ) difference in spin has to escape somewhere: it turns and trails off downstream. The rule "a vortex line can't end in the fluid" is one of Helmholtz vortex theorems .
Definition How much spin is shed, and why the minus signConsider a thin strip of span between η η + d η Γ ( η ) Γ ( η + d η ) Γ ( η ) − Γ ( η + d η ) = − d η d Γ d η − d η d Γ minus sign simply says that where Γ decreases outward (d Γ/ d η < 0 positive , i.e. spin is genuinely being released. It is a bookkeeping sign, not a new physical effect. On the right half of the wing Γ − d Γ/ d η > 0
Read the figure: the thick orange bar is the wing (the bound vortex Γ ( y ) Γ trailing tip vortices , with little curls showing they spin in opposite directions. Between them, the plum arrows point straight down — that is the downwash w
w w ( y ) extra downward air velocity created at the wing by all those trailing spin threads behind it. The trailing vortices, spinning, drag air downward between them — that downward speed felt right at the wing is w
Intuition Why one vortex pushes air down at a distanceA spinning thread drags nearby air around with it. Air on one side of the wing gets nudged down. The strength of that nudge falls off with distance — a thread at spanwise station η y y − η 1 η y Biot–Savart law .
Common mistake "The integrand blows up whenη = y Why it feels right: at η = y y − η 1/ ( y − η ) Fix: this is a known, tame kind of infinity. The piece just to the left of η = y + ∞ right is − ∞ Cauchy principal value — you shrink a tiny symmetric gap around η = y
Read the figure: the horizontal teal arrow is the freestream V ∞ w actually feels — the orange arrow. The small angle between the horizontal flow and this tilted flow is the induced angle α i w V ∞
Angle of attack is the control knob of a wing, so we sort out all four flavours.
Definition Geometric angle of attackα α far-off airflow V ∞
Definition Zero-lift angleα L = 0 α L = 0 above this value.
α i w local airflow downward. That tilt angle is
α i ( y ) = tan − 1 V ∞ w ( y ) ≈ V ∞ w ( y ) . tan − 1 "which angle has this slope?" — the slope being the downward speed w V ∞ w V ∞ tan − 1 ( x ) ≈ x w / V ∞
Definition Effective angle of attackα eff actually feels , after downwash has tilted the flow:
α eff = α − α i .
Intuition This ONE subtraction is the whole plotThe wing works with α eff α less lift than a 2D airfoil would, and its lift vector tilts back by α i α eff = α − α i
Definition Making forces dimensionlessTo compare wings of different sizes and speeds, we divide out the "fuel" 2 1 ρ ∞ V ∞ 2 dynamic pressure ) and a reference area. What's left is a pure number: a coefficient .
y θ automatically vanishes at the ends: sine waves. So we relabel span position with an angle θ y = − 2 b cos θ , θ = 0 θ = π θ 0 π
Definition The Jacobian — how a step inθ y y θ y = − 2 b cos θ d θ d y = 2 b sin θ ⟹ d y = 2 b sin θ d θ . 2 b sin θ Jacobian of the change of variables — it stretches or squeezes a step d θ d y θ = 0 , π θ = 2 π θ
Definition The sine series, and where2 b V ∞ sum of sine waves :
Γ ( θ ) = 2 b V ∞ ∑ n = 1 ∞ A n sin ( n θ ) . sin ( n θ ) A n how much of each block to mix in. Why the front factor 2 b V ∞ deliberate normalization chosen so the A n dimensionless numbers. Check the units: Γ b V ∞ A n 2 C L = π A R A 1 A 1 A n
A 1 A 1 A 2 , A 3 , … A 1 Oswald efficiency factor .
Aspect ratio AR = b squared over S
Freestream V-infinity and density rho
Derivative dGamma over dy
Trailing vortices from Helmholtz rule
Effective angle = alpha minus alpha-i
Thin airfoil lift slope 2 pi
Angle swap y equals minus b over 2 cos theta
Fourier sine series with A-n
Lift C-L and induced drag C-D-i
Cover the right side; try to state each before revealing.
What does the span b y = 0 Tip-to-tip distance; y = 0 ± b /2
Write aspect ratio and say what it means in words. A R = b 2 / S
What are V ∞ ρ ∞ Airspeed far ahead of the wing, and air density far away — the "fuel" of every force formula.
What does circulation Γ ( y ) The net swirl of air around the wing at station y Γ
State Kutta–Joukowski for lift per unit span. L ′ ( y ) = ρ ∞ V ∞ Γ ( y )
What does the derivative d Γ/ d η The slope of Γ
Why must trailing vortices exist, and why the minus sign in − d Γ/ d η A vortex thread can't end in the fluid (Helmholtz); the minus sign makes shed strength positive where Γ
Where does the 1/ ( 4 π ) A trailing filament is semi-infinite = half an infinite line, so half of Γ/ ( 2 π r ) Γ/ ( 4 π r )
Why doesn't the singularity at η = y It's read as a Cauchy principal value; the ± ∞
Write the whole-wing lift coefficient C L C L = L / ( 2 1 ρ ∞ V ∞ 2 S )
What is the Jacobian of y = − 2 b cos θ d y = 2 b sin θ d θ
Why the factor 2 b V ∞ Γ A normalization giving dimensionless A n C L = π A R A 1
Which coefficient sets lift, and which only add drag? A 1 A 2 , A 3 , …
Why is α i ≈ w / V ∞ α i = tan − 1 ( w / V ∞ ) tan − 1 x ≈ x
