Angle of attack, lift coefficient, drag coefficient: Foundations

Before you can read the parent note, you need a vocabulary. This page introduces every symbol and idea it leans on, from the ground up — each one gets its plain meaning, its picture, and the reason the topic can't live without it. Read top to bottom; each block uses only what came before it.

1. The airfoil, its chord line $c$ , and its span $b$

The picture: figure s01 shows one airfoil slice with its chord line marked; alongside it, the whole wing seen from above, with the span $b$ running left tip to right tip and the chord $c$ running front to back.

Why the topic needs $c$ : the chord line is our "reference ruler" glued to the wing — every angle we measure is measured from this line. It is also the length that sets the wing area: for a straight wing, area $S = c \times b$ . And in the thin-airfoil derivation of the parent note, circulation is written $Γ = π c V_{\infty} α$ , so $c$ literally sets how much lift a section makes — a longer chord grips more air.

Why the topic needs $b$ : the span tells you how long-and-slender the wing is. We package it into the aspect ratio in Section 9, which controls induced drag.

Figure — Angle of attack, lift coefficient, drag coefficient

2. The relative wind $V_{\infty}$ (a velocity, drawn as an arrow)

The picture: a straight blue arrow coming in from the left, undisturbed and horizontal.

Why "relative"? Whether the plane flies through still air or the air blows past a parked plane, only the relative motion matters — the wing can't tell the difference. That's why we measure everything against $V_{\infty}$ .

3. The angle of attack $α$ — and why we use an angle at all

The picture: in figure s02 the wedge opening up between the incoming blue wind arrow and the wing's chord line is $α$ . The green wing (leading edge up) shows positive $α$ ; the faint red wing (leading edge down) shows negative $α$ .

Degrees vs radians — the two ways to name an angle

Why the topic uses both: the clean theory result " $2 π$ per radian" only looks clean in radians. Engineers quote " $0.11$ per degree" because they read wind-tunnel plots in degrees. You must be fluent switching: $radians = degrees \times \frac{π}{180} .$

4. Force $F$ , and splitting it into lift $L$ and drag $D$

The picture: in figure s03 one white total-force arrow $F$ splits into a green "up" component $L$ and a red "back" component $D$ forming a right angle — like the two sides of an L-shaped corner.

5. Air density $ρ$

The picture: imagine a $1 m \times 1 m \times 1 m$ box of air; $ρ$ is the reading on a scale weighing everything inside it.

Why the topic needs it: the force comes from deflecting mass. Thicker (denser) air has more mass to shove around, so it pushes harder. High-altitude air is thin (small $ρ$ ) → less lift at the same speed — which is exactly why airliners must fly fast up high.

6. Area $S$

The picture: the shadow the wing casts on the ground at high noon.

Why the topic needs it: a bigger wing intercepts more air, so it makes proportionally more force. $S$ is the "how much wing" factor that lets one coefficient describe wings of every size.

7. Dynamic pressure $q$ — and why the $\frac{1}{2} ρ V_{\infty}^{2}$ shape

Now we combine density and free-stream speed into one bundle the topic uses constantly.

Why this exact combination? A parcel of air of mass $m$ moving at speed $V_{\infty}$ carries kinetic energy $\frac{1}{2} m V_{\infty}^{2}$ . Divide by its volume to get energy per cubic metre: mass-per-volume is $ρ$ , so energy-per-volume is $\frac{1}{2} ρ V_{\infty}^{2}$ . That's $q$ .

The parent note also derives the same $V_{\infty}^{2}$ from momentum: air of density $ρ$ hitting area $S$ at speed $V_{\infty}$ delivers momentum at a rate $ρS V_{\infty}^{2}$ per second — again the squared speed. Both roads (energy and momentum) land on $V_{\infty}^{2}$ , which is our reassurance the shape is right.

8. Dimensionless coefficients $C_{L}$ , $C_{D}$

Why the topic needs them: they let you test a $1 m$ model and predict a $60 m$ jet. $q$ and $S$ carry the speed and size; the coefficient carries the rest, and it depends on $α$ (and, secondarily, on Reynolds Number and Mach Number) — never directly on $V_{\infty}$ .

9. Extra symbols the parent leans on (quick anchors)

Definition

$Γ$ , $α_{L = 0}$ , $a_{0}$ , AR, $e$ , $L / D$

$Γ$ (Gamma), circulation — a number measuring the net "swirl" of air looping around the wing. Picture a spinning tendency in the flow; bigger swirl → more lift (this is the heart of the Kutta-Joukowski Theorem).
$α_{L = 0}$ — the angle of attack where lift is exactly zero (see Section 3). Cambered (curved) wings make lift even at $α = 0$ , so their zero-lift angle is slightly negative.
$a_{0}$ — the lift-curve slope: how many units of $C_{L}$ you gain per unit of $α$ . On a plot of $C_{L}$ vs $α$ , it's the steepness of the straight part.
AR (aspect ratio) — how long-and-slender the wing is. Using the span $b$ (Section 1) and area $S$ , $AR = b^{2} / S$ ; for a rectangular wing this is just $b / c$ . Glider = high AR; fighter = low AR.
$e$ (Oswald efficiency) — a fudge factor (near 1) for how ideally the lift spreads across the span.
$L / D$ — lift-to-drag ratio, the glide efficiency: how far you coast per unit of height lost.

These are formally built in later deep dives; here you only need to recognise them.

10. How the foundations feed the topic

Read it top-down: the chord and span build the area and set the tilt; the wind and density define $q$ ; the total air force $F$ splits into $L$ and $D$ ; dividing by $q S$ distils the coefficients; and those, together with $α$ , are the whole topic.

Equipment checklist

Cover the right side and answer aloud; reveal to check.

What is the chord line, and what is

c

The straight line from leading edge to trailing edge;

c

is its length (metres) — the front-to-back size, and one factor of the area

S = c \times b

What is the span

b

The tip-to-tip width of the whole wing (metres); the other factor of

S

and the basis of aspect ratio.

What does

V_{\infty}

mean and why the

\infty

The free-stream (far-upstream) relative wind speed, measured where the wing hasn't disturbed the air yet — not a local speed.

Define the angle of attack

α

and its sign.

The angle between chord line and relative wind; positive when the leading edge is raised (nose-up) into the wind, negative when nose-down, zero when parallel.

Why can

α_{L = 0}

be negative?

A cambered wing lifts even at

α = 0

, so you must tilt nose-down (negative

α

) to reach zero lift.

Convert

6^{\circ}

to radians.

6 \times π /180 \approx 0.1047

rad.

Which force is perpendicular to the wind, which is parallel?

Lift

L

is perpendicular (across the flow); drag

D

is parallel (along the flow); both are pieces of the total force

F

What is

ρ

and its sea-level value?

Air density, mass per cubic metre;

\approx 1.2 kg/m^{3}

What is

S

and how does

c

enter it?

The wing plan (shadow) area in

m^{2}

; for a rectangular wing

S = c \times b

Write

q

and say which speed it uses.

q = \frac{1}{2} ρ V_{\infty}^{2}

, the kinetic energy per cubic metre of the free-stream air.

Why does

V_{\infty}

appear squared in

q

Faster flow hits harder AND more of it arrives per second — the two effects multiply.

Write

C_{L}

in terms of

L

q

S

C_{L} = L / (q S)

, dimensionless.

Does

C_{L}

depend directly on speed

V_{\infty}

No — speed lives in

q

;

C_{L}

depends on

α

(and Mach, Reynolds).

Connections

Parent: Angle of Attack, Lift & Drag Coefficients — the topic these foundations unlock.
Bernoulli's Equation — where dynamic pressure $q = \frac{1}{2} ρ V_{\infty}^{2}$ comes from.
Kutta-Joukowski Theorem — turns circulation $Γ$ into lift.
Reynolds Number & Mach Number — secondary quantities $C_{L}, C_{D}$ quietly depend on.
Induced Drag & Wingtip Vortices — where the aspect-ratio $AR$ and $e$ terms matter.