Aerodynamic coefficients — CN, CA, CL, CD, Cm as functions of angle of attack, Mach: Question bank

Symbol glossary — build every letter before you use it

Every trap below uses a small alphabet. Define it once, here, and never guess again.

Before you start, hold three anchors:

A coefficient is a force (or moment) divided by $q_{\infty} S$ — a pure number describing shape and attitude, not size or speed.
Body frame $(C_{N}, C_{A})$ and wind frame $(C_{L}, C_{D})$ describe the same resultant force, rotated by $α$ — see figure s01.
Compressibility rescales everything through a square root of $∣1 - M_{\infty}^{2} ∣$ .

Where $C_{L} = C_{N} cos α - C_{A} sin α$ comes from (the picture)

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

The parabola of the drag polar and the two compressibility corrections are worth seeing once:

Spot the error

Find the error: "Since

sin α \approx α

is small, drag is just

C_{A}

and lift is just

C_{N}

The lift approximation is fine, but drag is

C_{D} = C_{N} sin α + C_{A} cos α \approx C_{N} α + C_{A}

— the

C_{N} α

term is not negligible because

C_{N}

is large, so it dominates the induced-like part of drag.

Find the error: "

C_{L} = 4 α / 1 - M_{\infty}^{2}

for a supersonic flat plate."

Sign under the root is wrong. Supersonic uses

M_{\infty}^{2} - 1

; the subsonic form

1 - M_{\infty}^{2}

becomes imaginary above

M_{\infty} = 1

. See Supersonic Linearized (Ackeret) Theory and Prandtl-Glauert Compressibility Correction.

Find the error: "The lift-curve slope is

2 π

per degree."

It is

2 π

per radian. Per degree it is

2 π /180 \approx 0.11

; mixing the unit inflates the answer by a factor of 57.

Find the error: "A stable aircraft needs

d C_{m} / d α > 0

so the moment grows with angle."

Backwards. Static Longitudinal Stability requires

d C_{m} / d α < 0

: a nose-up disturbance must generate a restoring nose-down moment, which needs a negative slope.

Find the error: "Induced drag

C_{L}^{2} / (π e A R)

shrinks as I increase aspect ratio, so it's free lift."

The trend is right but the conclusion is wrong — higher

A R

reduces induced drag but adds structural weight and bending loads, and parasite drag

C_{D 0}

still remains; nothing is free.

Find the error: "

C_{A}

always adds to drag, so more axial force means more drag."

Not always — in

C_{D} = C_{N} sin α + C_{A} cos α

the

C_{A}

term is scaled by

cos α

and, more importantly, in

C_{L} = C_{N} cos α - C_{A} sin α

the axial force subtracts from lift; its effect depends on the frame and the sign of

α

Why questions

Why do we non-dimensionalize forces into coefficients at all?

So a wind-tunnel model and a full-scale aircraft, at different sizes, speeds, and air densities, can be compared by the same pure number that isolates shape and attitude. See Dynamic Pressure and Non-dimensionalization.

Why is the drag polar parabolic in

C_{L}

rather than linear?

Because induced drag scales as

C_{L}^{2}

(from the squared downwash energy), and since

C_{L} \propto α

, drag grows like

α^{2}

— a parabola

C_{D} = C_{D 0} + C_{L}^{2} / (π e A R)

Why does the same square root (

∣1 - M_{\infty}^{2} ∣

) appear in both subsonic and supersonic corrections?

Both come from linearizing the same compressible potential-flow equation; the character of that equation flips (elliptic → hyperbolic) at

M_{\infty} = 1

, flipping the sign under the root but keeping the structure.

Why does compressibility increase lift per degree in subsonic flow?

M_{\infty}

rises, pressure disturbances pile up ahead of the surface and density changes amplify the pressure differences, so the same shape produces a larger

Δ p

— captured by the

1/ 1 - M_{\infty}^{2}

factor.

Why must

C_{m}

be measured about a stated reference point?

A moment depends on the moment arm, so its value changes with the point you take it about; only by fixing the reference (leading edge, quarter chord, or aerodynamic center) is

C_{m}

well-defined.

Why does making lift automatically create drag on a finite wing?

Lift requires a pressure difference between upper and lower surfaces, which drives flow around the tips into trailing vortices; the energy in that swirl appears as induced drag.

Edge cases

Edge case: at

α = 0

with a symmetric airfoil, what are

C_{L}

C_{N}

, and induced drag?

All lift-related terms vanish:

C_{L} = 0

C_{N} = 0

, and induced drag

\propto C_{L}^{2} = 0

. Only the parasite drag

C_{D 0}

(and axial

C_{A}

) remains.

Edge case: as

M_{\infty} \to 1^{-}

in Prandtl–Glauert, what does the formula predict and is it physical?

It predicts

C_{L} \to \infty

because

1 - M_{\infty}^{2} \to 0

. This is unphysical — the linear theory has broken down; transonic shocks and nonlinear effects dominate well before

M_{\infty} = 1

Edge case: at

α = 9 0^{\circ}

(a flat plate face-on), what happens to the resolution relations?

With

cos 9 0^{\circ} = 0, sin 9 0^{\circ} = 1

C_{L} = - C_{A}

and

C_{D} = C_{N}

. The "normal" force now points straight into the wind and becomes pure drag — but note thin-airfoil formulas are long invalid here (deep stall).

Edge case: at exactly

M_{\infty} = 1

, which of the two compressibility formulas applies?

Neither. Subsonic

1/ 1 - M_{\infty}^{2}

diverges and supersonic

1/ M_{\infty}^{2} - 1

also diverges; the sonic point is a singularity for linearized theory and needs transonic (nonlinear/experimental) methods.

Edge case: a wing with infinite aspect ratio (

A R \to \infty

) — what happens to induced drag?

The induced-drag term

C_{L}^{2} / (π e A R) \to 0

, recovering the 2D (thin-airfoil) result where the only drag is parasite/wave drag — no tip vortices, no downwash.

Edge case: negative angle of attack on a symmetric airfoil — is drag negative?

No.

C_{L}

becomes negative (downforce) but drag stays positive:

C_{D} = C_{D 0} + C_{L}^{2} / (π e A R)

depends on

C_{L}^{2}

, so the sign of lift does not make drag negative.

Recall One-line summary of every trap

Coefficients strip out size/speed; body and wind frames differ by a rotation of $α$ ; drag is parabolic and never negative; stability needs $d C_{m} / d α < 0$ ; and both Mach corrections share $∣1 - M_{\infty}^{2} ∣$ but flip sign at $M_{\infty} = 1$ , where linear theory dies.

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

Symbol glossary — build every letter before you use it

Where $C_{L} = C_{N} cos α - C_{A} sin α$ comes from (the picture)

True or false — justify

Spot the error

Why questions

Edge cases

Symbol glossary — build every letter before you use it

Where CL​=CN​cosα−CA​sinα comes from (the picture)

True or false — justify

Spot the error

Why questions

Edge cases

Where $C_{L} = C_{N} cos α - C_{A} sin α$ comes from (the picture)