Foundations — Aerodynamic coefficients — CN, CA, CL, CD, Cm as functions of angle of attack, Mach

This page builds every symbol the parent note leans on, starting from a picture of air hitting a wing and ending with a checklist so you know you're ready.

1. The air, the body, and the wind

Before any letters, picture the scene: a wing sitting still while air streams past it from left to right. Everything downstream is named after this streaming air, the freestream.

Why the topic needs these: the force air makes depends on how fast it hits ($V_{\infty }$) and how much stuff is in it (${\rho }_{\infty }$). Any honest force formula must contain both.

2. Angle of attack $\alpha$ — the tilt

The wing has a natural straight line running nose-to-tail through it: the chord line. The air arrives along $V_{\infty }$. These two directions are usually not the same — the wing is tilted.

Why a Greek letter? By tradition angles get Greek names; $\alpha$ is just a label for "how much the wing is tilted into the wind." We measure it in degrees for intuition (like $8^{\circ }$) but in radians for formulas (why radians matters is in §7).

Why the topic needs it: tilt the wing more and it grabs more air downward, making more lift — until it tilts too far and the flow breaks away (stall). So every coefficient is really a function of $\alpha$. See Thin Airfoil Theory for where the tilt-to-lift link comes from.

3. Chord $c$ and reference area $S$ — the size

To compare a model wing with a real one we need agreed measures of "how big."

Why the topic needs them: force grows with wing size. Dividing by $S$ removes that. And a moment (a twisting effort) grows with an extra length, which is exactly what $c$ supplies — that is why only the moment coefficient carries $c$ (§6).

4. Dynamic pressure $q_{\infty }$ — the "push per area" of moving air

Here a new quantity enters, and we must justify why this exact combination and not some other.

Why not just $\rho V$ or $V^2$ alone? Because only $\frac{1}{2}\rho V^2$ has the right units and falls straight out of energy per volume. See Dynamic Pressure and Non-dimensionalization for the full derivation.

Why the topic needs it: dividing a force by $q_{\infty }S$ (pressure × area = force) cancels both the speed/density dependence and the size dependence in one shot. What remains is dimensionless.

5. The resultant force $R$, and two ways to split it

All the tiny pressure and rubbing forces over the wing add up to one arrow: the resultant $R$. One arrow, but we can name its pieces in two different frames.

6. Pitching moment $M$ — the twist

Force isn't the whole story: the air also tries to rotate the wing nose-up or nose-down.

Why the topic needs it: whether an aircraft naturally returns to level flight after a bump depends on the sign of how $M$ changes with $\alpha$ — the heart of Static Longitudinal Stability and tied to the Aerodynamic Center vs Center of Pressure.

7. Turning forces into coefficients — the punchline

Now every ingredient is on the table. We divide to erase size and speed:

Why radians for $\alpha$?

Formulas like the thin-airfoil slope $d{C}_L/d\alpha =2\pi$ per radian assume $\alpha$ is in radians. A radian is the "natural" angle unit where arc-length equals angle × radius, so calculus of trig functions comes out clean (no stray $\pi /180$). Convert with

${\alpha }_{\text{rad}}={\alpha }_{\text{deg}}\times \frac{\pi }{180}.$

8. Mach number $M_{\infty }$ — the compressibility flag

One last symbol the parent leans on heavily.

Prerequisite map

Equipment checklist