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

3.1.29 · D5 · Physics › Compressible Flow & Aerodynamics › Aerodynamic coefficients — CN, CA, CL, CD, Cm as functions o

Symbol glossary — har letter pehle define karo, phir use karo

Neeche ke saare traps ek chhoti si alphabet use karte hain. Ise ek baar yahan define karo, aur phir kabhi guess mat karo.

Definition Force coefficients (body frame)

Wing ko aadha kaatke socho — ek side view. Chord line wo seedhi line hai jo nose se tail tak jaati hai. Hawa us section pe ek resultant force se dhakka deti hai. Body frame mein hum us dhakke ko do directions mein todte hain jo wing se hi tied hain:

$C_{N}$ = normal-force coefficient. $N$ wo force hai jo chord ke perpendicular hai (top surface se "upar" ki taraf point karti hai). $C_{N} = N / (q_{\infty} S)$ . Sign: positive jab ye upper surface ki taraf dhakka deti hai.
$C_{A}$ = axial-force coefficient. $A$ wo force hai jo chord ke along hai, aft ki taraf point karti hai (tail ki taraf). $C_{A} = A / (q_{\infty} S)$ . Sign: positive jab ye chord ke along peeche ki taraf point kare.

Yahan $q_{\infty} = \frac{1}{2} ρ_{\infty} V_{\infty}^{2}$ dynamic pressure hai aur $S$ reference (wing) area hai.

Shuru karne se pehle, teen anchors pakad lo:

Ek coefficient ek force (ya moment) hai jo $q_{\infty} S$ se divide hota hai — ek pure number jo shape aur attitude describe karta hai, size ya speed nahi.
Body frame $(C_{N}, C_{A})$ aur wind frame $(C_{L}, C_{D})$ usi resultant force ko describe karte hain, $α$ se rotated — figure s01 dekho.
Compressibility har cheez ko $∣1 - M_{\infty}^{2} ∣$ ke square root ke through rescale karta hai.

$C_{L} = C_{N} cos α - C_{A} sin α$ kahan se aata hai (picture)

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

Drag polar ka parabola aur dono compressibility corrections ek baar dekhne layak hain:

True ya false — justify karo

True ya false:

C_{L}

aur

C_{N}

har angle of attack par same cheez hain.

False. Ye sirf limit

α \to 0

mein coincide karte hain; generally

C_{L} = C_{N} cos α - C_{A} sin α

, toh ye

α

ke rotation se alag hote hain aur high

α

par clearly diverge karte hain.

True ya false: flight speed double karne se lift coefficient double ho jaata hai.

False. Lift force badhti hai (

V^{2}

ke through

q_{\infty}

se), lekin

C_{L}

ko us force ko

q_{\infty} S

se divide karke define kiya jaata hai, toh speed cancel ho jaata hai —

C_{L}

fixed

α

aur Mach par (almost) speed-independent hota hai.

True ya false: ek symmetric airfoil ka

α = 0

par

C_{L} = 0

hota hai.

True. Symmetry ka matlab hai

α_{L = 0} = 0

, toh

C_{L} = 2 π (α - 0) = 0

wahan. Ek cambered airfoil

α = 0

par lift karta hai kyunki

α_{L = 0} < 0

(dekho Thin Airfoil Theory).

True ya false: induced drag perfectly inviscid, frictionless flow mein bhi exist karta hai.

True. Induced drag trailing wingtip vortices mein shed hone wali energy se aata hai, friction se nahi — ek finite lifting wing ise zero viscosity mein bhi pay karta hai.

True ya false: wave drag (

C_{D, wave} = 4 α^{2} / M_{\infty}^{2} - 1

) har Mach number par exist karta hai.

False. Wave drag shock losses se paida hota hai aur sirf supersonic (aur transonic) flow mein appear karta hai; subsonic inviscid flow mein ye exactly zero hai — yahi d'Alembert's paradox hai.

True ya false: aerodynamic center ke baare mein,

C_{m}

tab nahi badlta jab

α

badlta hai.

True — yahi aerodynamic center ki definition hai: wo point jahan

d C_{m} / d α = 0

, toh

C_{m} = C_{m 0}

angle ke saath constant rehta hai.

True ya false: moment coefficient ka denominator

C_{L}

ke denominator se same hota hai.

False. Moment force×length hai, toh

C_{m} = M / (q_{\infty} S c)

mein dimensionless rehne ke liye ek extra reference length

c

hota hai;

C_{L}

sirf

q_{\infty} S

se divide karta hai.

True ya false: Prandtl–Glauert kehta hai ki lift smoothly

M_{\infty} = 1

tak badhti rehti hai.

False. Formula

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

predict karta hai ki

M_{\infty} = 1

par lift infinite ho jaayegi, lekin ye physics nahi balki ek breakdown hai — shocks

M_{\infty} \approx 0.7

–

0.8

ke paas form ho jaate hain aur linear theory wahan invalid hai.

Error dhundho

Error dhundho: "Kyunki

sin α \approx α

chhota hai, drag bas

C_{A}

hai aur lift bas

C_{N}

hai."

Lift approximation theek hai, lekin drag hai

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

—

C_{N} α

term negligible nahi hai kyunki

C_{N}

bada hota hai, toh ye drag ke induced-like part ko dominate karta hai.

Error dhundho: "

C_{L} = 4 α / 1 - M_{\infty}^{2}

ek supersonic flat plate ke liye."

Root ke neeche sign galat hai. Supersonic

M_{\infty}^{2} - 1

use karta hai; subsonic form

1 - M_{\infty}^{2}

M_{\infty} = 1

ke upar imaginary ho jaata hai. Dekho Supersonic Linearized (Ackeret) Theory aur Prandtl-Glauert Compressibility Correction.

Error dhundho: "Lift-curve slope

2 π

per degree hai."

2 π

per radian hai. Per degree ye

2 π /180 \approx 0.11

hai; unit mix karne se answer 57 ke factor se inflate ho jaata hai.

Error dhundho: "Ek stable aircraft ko

d C_{m} / d α > 0

chahiye taaki moment angle ke saath badhta rahe."

Ulta hai. Static Longitudinal Stability ke liye

d C_{m} / d α < 0

chahiye: ek nose-up disturbance ko ek restoring nose-down moment generate karna chahiye, jiske liye negative slope chahiye.

Error dhundho: "Induced drag

C_{L}^{2} / (π e A R)

aspect ratio badhane se shrink hota hai, toh ye free lift hai."

Trend sahi hai lekin conclusion galat hai — zyada

A R

induced drag reduce karta hai lekin structural weight aur bending loads badhata hai, aur parasite drag

C_{D 0}

phir bhi rehta hai; kuch bhi free nahi hai.

Error dhundho: "

C_{A}

hamesha drag mein add karta hai, toh zyada axial force matlab zyada drag."

Hamesha nahi —

C_{D} = C_{N} sin α + C_{A} cos α

mein

C_{A}

term

cos α

se scale hota hai aur, zyada important baat,

C_{L} = C_{N} cos α - C_{A} sin α

mein axial force lift se subtract karta hai; uska effect frame aur

α

ke sign par depend karta hai.

Why questions

Hum forces ko coefficients mein non-dimensionalize kyun karte hain?

Taaki ek wind-tunnel model aur ek full-scale aircraft, alag sizes, speeds, aur air densities par, ek hi pure number se compare kiye ja sakein jo sirf shape aur attitude isolate kare. Dekho Dynamic Pressure and Non-dimensionalization.

Drag polar

C_{L}

mein linear ki jagah parabolic kyun hota hai?

Kyunki induced drag

C_{L}^{2}

ke saath scale karta hai (squared downwash energy se), aur kyunki

C_{L} \propto α

, drag

α^{2}

ki tarah badhta hai — ek parabola

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

Same square root (

∣1 - M_{\infty}^{2} ∣

) dono subsonic aur supersonic corrections mein kyun appear karta hai?

Dono usi compressible potential-flow equation ko linearize karne se aate hain; us equation ka character

M_{\infty} = 1

par flip ho jaata hai (elliptic → hyperbolic), root ke neeche sign flip karta hai lekin structure same rehta hai.

Compressibility subsonic flow mein lift per degree kyun badhata hai?

Jaise

M_{\infty}

badhta hai, pressure disturbances surface ke aage pile up hote hain aur density changes pressure differences ko amplify karte hain, toh same shape ek bada

Δ p

produce karta hai —

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

factor se capture hota hai.

C_{m}

ko ek stated reference point ke baare mein kyun measure karna chahiye?

Ek moment moment arm par depend karta hai, toh uski value us point ke saath badlti hai jis ke baare mein tum lete ho; sirf reference fix karne se (leading edge, quarter chord, ya aerodynamic center)

C_{m}

well-defined hota hai.

Lift banana automatically ek finite wing par drag kyun create karta hai?

Lift ke liye upper aur lower surfaces ke beech pressure difference chahiye, jo tip ke around flow drive karta hai trailing vortices mein; us swirl ki energy induced drag ke roop mein appear hoti hai.

Edge cases

Edge case:

α = 0

par ek symmetric airfoil ke saath,

C_{L}

C_{N}

, aur induced drag kya hain?

Saare lift-related terms vanish ho jaate hain:

C_{L} = 0

C_{N} = 0

, aur induced drag

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

. Sirf parasite drag

C_{D 0}

(aur axial

C_{A}

) bachta hai.

Edge case: Prandtl–Glauert mein

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

jaane par, formula kya predict karta hai aur kya ye physical hai?

Ye predict karta hai

C_{L} \to \infty

kyunki

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

. Ye unphysical hai — linear theory break down kar chuki hai; transonic shocks aur nonlinear effects

M_{\infty} = 1

se kaafi pehle dominate karte hain.

Edge case:

α = 9 0^{\circ}

par (ek flat plate face-on), resolution relations ka kya hota hai?

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

ke saath:

C_{L} = - C_{A}

aur

C_{D} = C_{N}

. "Normal" force ab seedha wind mein point karta hai aur pure drag ban jaata hai — lekin note karo thin-airfoil formulas yahan kaafi pehle invalid ho chuke hain (deep stall).

Edge case: exactly

M_{\infty} = 1

par, dono compressibility formulas mein se kaun sa apply hota hai?

Dono nahi. Subsonic

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

diverge karta hai aur supersonic

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

bhi diverge karta hai; sonic point linearized theory ke liye ek singularity hai aur transonic (nonlinear/experimental) methods ki zaroorat hai.

Edge case: infinite aspect ratio (

A R \to \infty

) wala wing — induced drag ka kya hota hai?

Induced-drag term

C_{L}^{2} / (π e A R) \to 0

, 2D (thin-airfoil) result recover karta hai jahan sirf drag parasite/wave drag hai — koi tip vortices nahi, koi downwash nahi.

Edge case: ek symmetric airfoil par negative angle of attack — kya drag negative hota hai?

Nahi.

C_{L}

negative ho jaata hai (downforce) lekin drag positive rehta hai:

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

C_{L}^{2}

par depend karta hai, toh lift ka sign drag ko negative nahi banata.

Recall Har trap ka one-line summary

Coefficients size/speed strip out karte hain; body aur wind frames $α$ ke rotation se differ karte hain; drag parabolic hai aur kabhi negative nahi; stability ko $d C_{m} / d α < 0$ chahiye; aur dono Mach corrections $∣1 - M_{\infty}^{2} ∣$ share karte hain lekin $M_{\infty} = 1$ par sign flip karte hain, jahan linear theory khatam hoti hai.