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

3.1.29 · Physics › Compressible Flow & Aerodynamics

1. DO coordinate frames KYA hain?

Resultant aerodynamic force $R$ ko resolve karne ke liye do natural directions hain:

Body frame: body axis (chord) ke saath aur uske perpendicular.
- $N$ = normal force (chord ke perpendicular)
- $A$ = axial force (chord ke saath, aft ki taraf point karta hua)
Wind frame: freestream velocity $V_{\infty}$ ke relative.
- $L$ = lift ( $V_{\infty}$ ke perpendicular)
- $D$ = drag ( $V_{\infty}$ ke parallel)

Dono frames angle of attack $α$ se rotated hain (chord aur $V_{\infty}$ ke beech ka angle).

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

2. Definitions (non-dimensional numbers)

$C_{m}$ mein extra $c$ kyun? Moment, force×length hota hai, isliye ise dimensionless banane ke liye ek extra length chahiye. Isliye sirf moment coefficient chord $c$ carry karta hai.

3. DERIVATION: body frame ↔ wind frame ko relate karna

Resultant force $R$ rakhein. Body frame mein iske components $(A, N)$ hain (chord ke saath, chord ke normal). Wind axes, body axes ko $α$ se rotate karne par milte hain. Standard 2D rotation deta hai:

$L = N cos α - A sin α$ $D = N sin α + A cos α$

Yeh step kyun? $L$ , $R$ ka component hai jo $V_{\infty}$ ke perpendicular hai. Normal force $N$ us perpendicular par $N cos α$ project karta hai; axial force $A$ (chord ke saath aft ki taraf pointing) lift direction mein backward-tilted component $- A sin α$ rakhta hai. Drag orthogonal combination hai.

Har term ko $q_{\infty} S$ se divide karne par (same denominator!) coefficient form milta hai:

4. Coefficients $α$ par KAISE depend karte hain

5. Mach number kaise enter karta hai (compressibility)

6. Worked examples

7. Common mistakes (Steel-man + fix)

8. Active recall

Force ko

q_{\infty} S

se kyun divide karte hain?

Non-dimensionalize karne ke liye taaki coefficient shape/attitude par depend kare, size, speed, ya density par nahi.

q_{\infty}

define karo.

Dynamic pressure

q_{\infty} = \frac{1}{2} ρ_{\infty} V_{\infty}^{2}

C_{L}

ka relation

C_{N}, C_{A}, α

mein.

C_{L} = C_{N} cos α - C_{A} sin α

C_{D}

ka relation

C_{N}, C_{A}, α

mein.

C_{D} = C_{N} sin α + C_{A} cos α

C_{m}

mein extra length

c

kyun hoti hai?

Moment ki units force×length hoti hain, isliye ise dimensionless banane ke liye ek extra reference length chahiye.

Thin-airfoil lift-curve slope per radian.

2 π

(toh

C_{L} = 2 π (α - α_{L = 0})

Prandtl–Glauert subsonic correction.

C_{L} = C_{L, incomp} / 1 - M_{\infty}^{2}

Supersonic flat-plate

C_{L}

C_{L} = 4 α / M_{\infty}^{2} - 1

Supersonic flat-plate wave drag.

C_{D} = 4 α^{2} / M_{\infty}^{2} - 1

Drag polar equation.

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

Static pitch stability ki condition.

d C_{m} / d α < 0

Aerodynamic center par

C_{m}

ke baare mein kya special hai?

Yeh angle of attack se independent hota hai.

Supersonically kaunsa naya drag appear karta hai jo subsonic inviscid flow mein absent hai?

Wave drag,

\propto α^{2}

, shock losses se.

Recall Feynman: 12-saal ke bacche ko samjhao

Wing par hawa push karti hai toh ek bada sa dhakka lagta hai. Hum us dhakke ko do taraon se split karte hain: wing ki flat surface ke relative upar-neeche ( $N$ aur $A$ ), ya wing jis taraf fly kar rahi hai us relative upar-peeche ( $L$ = lift jo tumhe upar rakhti hai, $D$ = drag jo tumhe slow karta hai). Yeh dono ek hi dhakka hain, bas ek chhote se angle $α$ se ghuma ke measure kiye hue — jaise apni height seedha khade ho ke measure karna vs thoda jhuk ke. Hum numbers ko "coefficients" mein turn karte hain taaki ek tiny model plane aur ek giant jet same number share kar sakein. Zyada fast jaana (high Mach) hawa ko springy bana deta hai: yeh tilt per zyada lift deti hai jab tak yeh bahut fast nahi ho jaata aur shock-wave drag nahi bana deta.

Connections

Thin Airfoil Theory — $d C_{L} / d α = 2 π$ aur $α_{L = 0}$ ka source.
Prandtl-Glauert Compressibility Correction — subsonic Mach scaling.
Supersonic Linearized (Ackeret) Theory — $1/ M^{2} - 1$ aur wave drag.
Induced Drag and Wingtip Vortices — drag polar mein $C_{L}^{2}$ term.
Static Longitudinal Stability — $d C_{m} / d α < 0$ kyun.
Aerodynamic Center vs Center of Pressure — jahan $C_{m}$ constant hota hai.
Dynamic Pressure and Non-dimensionalization — sabhi coefficients ki foundation.