Thin airfoil theory — lift per unit span = πρV²(α + 2β - π c): Foundations

This page assumes you have seen nothing. We build each symbol from a picture before it is ever allowed into an equation. By the end you can read the parent note Thin airfoil theory line by line.

0 · What is a "wing seen from the side"?

Before any symbol, fix the picture in your head. Slice a wing with a knife straight across, look at the cut face. You get a thin sliver of a shape — thick-ish near the front, tapering to a sharp point at the back. That 2-D cross-section is the whole game. Everything on the parent page is about this flat-ish sliver sitting in a wind.

Figure s01 — a side-on wing slice. The lavender outline is the physical skin; the dashed slate line is the straight chord from leading edge (LE, front) to trailing edge (TE, sharp back). This picture defines what "2-D airfoil", LE, TE and the chord length $c$ mean before any algebra.

1 · The air: density $ρ$ and speed $V$

Why does the parent page always write $V^{2}$ (V squared), never plain $V$ ? Because lift comes from pressure, and the pressure a moving fluid can exert grows with the square of speed (double the speed, four times the punch). We meet this square again in §7 as the $\frac{1}{2} ρ V^{2}$ inside the lift coefficient.

2 · Angle of attack $α$ — the tilt knob

Figure s02 — the mint arrows are the freestream wind $V$ ; the lavender bar is the chord. The coral arc is the angle of attack $α$ , swept from the wind up to the chord. The curved coral rotation arrow marks the positive (nose-up about the LE) sense. The visual point: $α$ is purely the tilt of the wing relative to the wind, and positive means nose-up.

3 · Camber $z (x)$ , its slope $d z / d x$ , and $β$

A flat card can lift when tilted. But real wings are gently curved even when not tilted — like a shallow smile. That curve is camber, and it lets the wing scoop air down for free.

Figure s03 — the lavender arch is the camber line $z (x)$ ; the dashed line is the chord ( $z = 0$ ). The two coral tangent segments show the local slope $d z / d x$ : positive (uphill) near the front, negative (downhill) near the back. This is the picture behind the derivative $d z / d x$ .

Why this exact tool (a derivative)? The boundary condition on the parent page says air must flow along the surface. To know the direction the surface points at each spot, you need its local slope — and "slope of a curve at a point" is precisely what a derivative $d z / d x$ means. No other tool answers "which way is the skin facing right here?"

The angle variable $θ$ — and why it replaces $x$

Before we can write $β$ , we must meet a new bookkeeping variable $θ$ that the theory uses to walk along the chord.

Definition

$θ$ — the chord-angle variable, and the change of variable == $θ$ == ("theta") is a new coordinate that re-labels the chord: instead of the straight ruler position $x$ , we set $x = \frac{c}{2} (1 - cos θ), θ : 0 \to π .$ At $θ = 0$ : $cos 0 = 1$ , so $x = 0$ — the leading edge. At $θ = π$ : $cos π = - 1$ , so $x = c$ — the trailing edge. As $θ$ sweeps $0 \to π$ , the point $x$ walks smoothly LE→TE. Why bother? This maps the chord onto a half-circle, which is exactly the shape that makes the vortex-sheet integrals solvable (see Glauert's integral and Fourier coefficients). It bunches sample points near the sharp edges where the flow varies fastest. The Jacobian (how $d x$ becomes $d θ$ ). Differentiate the substitution: $\frac{d x}{d θ} = \frac{c}{2} sin θ ⟹ d x = \frac{c}{2} sin θ d θ .$ So any integral "along the chord" $\int_{0}^{c} (\dots) d x$ becomes $\int_{0}^{π} (\dots) \frac{c}{2} sin θ d θ$ . That is the rule that gives every " $d θ$ " integral below its meaning: they are ordinary chord integrals, re-dressed in the $θ$ costume with the factor $\frac{c}{2} sin θ$ already absorbed by the theory's algebra.

Figure s05 — the change of variable $x = \frac{c}{2} (1 - cos θ)$ . A point moving at constant speed in $θ$ around the lavender half-circle (top) projects (coral dashed drop-lines) onto the chord (bottom). Notice how equal steps in $θ$ crowd near the LE and TE — that is the "bunching near the edges" the theory wants.

Definition

$β$ — the camber parameter, defined properly With $θ$ now in hand, the Fourier coefficients of the theory are the weighted averages of the camber slope $d z / d x$ (each $d z / d x$ evaluated at the $x$ corresponding to that $θ$ ): $A_{0} = α - \frac{1}{π} \int_{0}^{π} \frac{d z}{d x} d θ, A_{1} = \frac{2}{π} \int_{0}^{π} \frac{d z}{d x} cos θ d θ .$ The only combination that enters lift is $A_{0} + \frac{A _{1}}{2}$ . Collecting its camber part defines == $β$ ==: $\frac{2 β}{π} \equiv - \frac{1}{π} \int_{0}^{π} \frac{d z}{d x} d θ + \frac{1}{2} \cdot \frac{2}{π} \int_{0}^{π} \frac{d z}{d x} cos θ d θ$ So $β$ is a specific weighted average of the camber-line slope $d z / d x$ around the chord — one number squeezed out of the whole shape $z (x)$ . A flat plate has $d z / d x = 0$ everywhere ⇒ $β = 0$ . A downward-scooping arc gives $β > 0$ . Picture: $β$ is the "smile strength" of the arch — but now you know exactly how to compute it from $z (x)$ .

Intuition WHY exactly

$A_{0} + A_{1} /2$ , and why those weightings? You are right to demand a reason rather than accept the formulas. Here is the honest sketch (full machinery lives in Glauert's integral and Fourier coefficients):

The theory writes the unknown sheet strength $γ (θ)$ as a Fourier series — a sum of simple wiggles $A_{0} (term) + A_{1} sin θ + A_{2} sin 2 θ + \dots$ . The numbers $A_{0}, A_{1}, A_{2}, \dots$ are just "how much of each wiggle" is present.
To find each $A_{n}$ you multiply the boundary condition by the matching wiggle and integrate over $θ$ . That "multiply by $cos n θ$ and integrate" is exactly why $A_{0}$ carries $\int \frac{d z}{d x} d θ$ (weight $1$ ) and $A_{1}$ carries $\int \frac{d z}{d x} cos θ d θ$ (weight $cos θ$ ): each coefficient is the overlap of the camber slope with that particular wiggle.
Total circulation $Γ = \int_{0}^{c} γ d x$ integrates the whole series. Sine wiggles $sin n θ$ for $n \geq 2$ average to zero over a full chord (they wiggle up as much as down) — this is orthogonality. Only the $A_{0}$ term and the $A_{1}$ term leave a non-zero total, and the arithmetic of those two surviving integrals produces precisely the combination $A_{0} + \frac{A _{1}}{2}$ . In one line: higher wiggles cancel themselves out when you add up the whole wing, so lift only "hears" the first two Fourier voices, $A_{0}$ and $A_{1}$ .

Recall Why does camber act like extra angle of attack?

Because a curved-down-at-the-back line already points the trailing flow downward even when the chord is level — geometrically the same as tilting a flat plate. So camber and tilt add: $α + \frac{2 β}{π}$ . ::: They add because both do the same job — deflect air down — so their effects sum inside one bracket.

4 · Circulation $Γ$ , and the Kutta condition that pins it down

Here is the heart. Imagine walking a big loop all the way around the sliver and, as you walk, keeping a running tally: is the air along my path helping me forward or fighting me? Add up (air-speed-along-my-step) × (length of step) all the way round. That total is the circulation.

Figure s04 — the dashed slate loop is the path you walk. Coral arrows (top) show fast air, mint arrows (bottom) show slow air; walking the loop clockwise, the fast top run dominates, giving a positive tally $Γ > 0$ . This picture fixes both the meaning and the sign of $Γ$ .

Definition The Kutta condition — what selects the

right amount of swirl Mathematically many different amounts of swirl $Γ$ could satisfy "air flows along the surface." Nature picks exactly one: the Kutta condition says the air must leave the sharp trailing edge smoothly, without whipping around the corner at infinite speed. In vortex-sheet language this forces the sheet strength to be finite (in fact zero) at the trailing edge, $γ (TE) = 0$ . Why we need it: without this rule the maths is under-determined — it cannot say how strong $Γ$ is. The Kutta condition is the physical extra fact (a real sharp edge cannot support infinitely fast turn-around flow) that locks in a unique $Γ$ , hence a unique lift. It is a cornerstone of the whole theory. See Kutta condition. Picture: at the sharp back point, the top stream and bottom stream meet and peel off cleanly together — not one of them curling violently around the tip.

Figure s06 — two candidate flows at the trailing edge. Left (coral, ✗): flow whips around the sharp corner at huge speed — physically impossible. Right (mint, ✓): the Kutta condition, flow leaves the TE smoothly and parallel. Only the right one occurs, and demanding it fixes the circulation $Γ$ .

5 · The prime $L^{'}$ — lift per unit span (and what "span" means)

Why work per-slice? Because our whole picture is a 2-D cross-section. Everything we compute is for the sliver, so all answers naturally come out "per metre of span."

6 · The vortex sheet $γ (x)$ — the model trick

Real wings are curved solid skin. That is hard. The clever move: replace the wing with a line of infinitely many tiny spinning points laid along the chord, each contributing a sliver of swirl.

Add up all those tiny spins and you recover the whole circulation: $Γ = \int_{0}^{c} γ (x) d x$

The parent note's Steps 3–5 are all machinery to find this $γ (x)$ — the Kutta condition fixes its value at the trailing edge, Biot–Savart law gives the downwash it induces, and Glauert's integral and Fourier coefficients solves the whole sheet — then integrate it into $Γ$ , then feed $Γ$ into $L^{'} = ρ V Γ$ . That is the entire pipeline.

7 · The lift coefficient $c_{ℓ}$ — a units-free score

The parent page also reports lift as a dimensionless number. Meet it now so it is not a stranger.

Prerequisite map

Reading the map in words (for anyone who can't render it): Start at the two air properties on the left — air density $ρ$ and freestream speed $V$ — both feed directly into Kutta–Joukowski ( $L^{'} = ρ V Γ$ ). The third input to Kutta–Joukowski is the circulation $Γ$ , and Kutta–Joukowski outputs the lift per unit span $L^{'}$ . Where does $Γ$ come from? Several streams flow into it: the angle of attack $α$ , the camber parameter $β$ , the vortex sheet $γ (x)$ , and — crucially — the Kutta condition that fixes its size. The camber parameter $β$ is itself built from the camber line $z (x)$ and its slope and the chord-angle variable $θ$ , while the vortex sheet is laid along the chord $c$ . Finally $L^{'}$ feeds both the lift coefficient $c_{ℓ}$ and the whole thin-airfoil-theory topic. So the dependency chain is: geometry knobs ( $α$ , $β$ from $z (x)$ and $θ$ , $c$ ) + the Kutta condition → circulation $Γ$ → (with $ρ$ , $V$ ) Kutta–Joukowski → lift $L^{'}$ → $c_{ℓ}$ and the topic.

Equipment checklist

Self-test: cover the right side and answer before revealing.

What does the prime in $L^{'}$ mean? :::

What does the symbol

c

measure, and in what units?

The chord length (LE to TE), in metres.

What is

ρ

and its sea-level value?

Air density, about

1.225 kg/m^{3}

Why does lift use

V^{2}

and not

V

Fluid pressure/dynamic push grows with the square of speed.

What is

α

and its sign rule?

Angle of attack;

α > 0

when the chord is nose-up about the leading edge.

Why must

α

be in radians?

The theory used

sin α \approx α

, valid only in radians.

For what range of

α

is thin-airfoil theory trustworthy?

Roughly

∣ α ∣ ≲ 0.17

rad (

\sim 10°

); beyond that the linearisation fails and the flow stalls.

Convert

6°

to radians.

6 \times π /180 = 0.1047

rad.

What does

z (x)

describe?

The height of the mean camber line above the chord at position

x

What does

d z / d x

mean physically?

The local slope (facing direction) of the camber line.

What is the variable

θ

and how does

d x

relate to

d θ

The chord-angle from

x = \frac{c}{2} (1 - cos θ)

;

d x = \frac{c}{2} sin θ d θ

How is

β

defined from the shape?

As the weighted slope average

\frac{2 β}{π} = - \frac{1}{π} \int_{0}^{π} \frac{d z}{d x} d θ + \frac{1}{π} \int_{0}^{π} \frac{d z}{d x} cos θ d θ

Why do only

A_{0}

and

A_{1}

enter lift?

Higher sine wiggles

sin n θ

(

n \geq 2

) average to zero over the chord (orthogonality).

In one sentence, what is circulation

Γ

The net swirl of air around the wing, i.e. speed-along-a-loop summed around it.

What is the sign convention for positive

Γ

Clockwise loop (wind left-to-right, lift up); positive

Γ

= faster air on top = upward lift.

What does the Kutta condition require, and why?

Smooth flow off the sharp trailing edge (

γ (TE) = 0

); it uniquely fixes

Γ

State Kutta–Joukowski.

L^{'} = ρ V Γ

What does "span" mean?

The wingtip-to-wingtip distance, perpendicular to the chord slice.