Aspect ratio — effect on induced drag — Hinglish notes

3.1.23 · Physics › Compressible Flow & Aerodynamics

KYA pucha ja raha hai

KYUN matter karta hai: Cruise mein, induced drag aksar total drag ka 30–40% hota hai. Ise geometry se cut karna directly fuel burn cut karta hai — isliye sailplanes aur high-altitude drones ke enormous wingspans hote hain.

KYUN karta hai finite wing drag? (first principles)

Figure — Aspect ratio — effect on induced drag

KAISE derive karte hain induced-drag formula

Formula Step-by-step derivation

Step 1 — Jhukao ki geometry. Local lift $L$ induced angle $α_{i}$ se rotate hoti hai. Uska peeche (drag) component: $D_{i} = L sin α_{i} \approx L α_{i} (α_{i} small)$ Yeh step kyun? Drag hamesha freestream ke saath hota hai; lift vector ka us direction mein projection induced drag hai.

Step 2 — Coefficients mein express karo. $q S$ se divide karo jahan $q = \frac{1}{2} ρ V^{2}$ : $C_{D, i} = C_{L} α_{i}$ Kyun? Coefficients $ρ, V, S$ ko hata dete hain taaki geometry akele khadi rahe.

Step 3 — $α_{i}$ lifting-line theory se nikalo. Elliptical lift distribution ke liye downwash span mein uniform hoti hai, aur Prandtl ki lifting-line result deti hai $α_{i} = \frac{C _{L}}{π A R}$ Elliptical kyun? Yahi distribution hai jo, given lift ke liye, downwash constant banati hai — aur constant downwash minimum-drag case hai (calculus of variations se proved). Yeh natural benchmark hai.

Step 4 — Steps 2 aur 3 combine karo. $C_{D, i} = \frac{C _{L}^{2}}{π A R}$ Yeh headline kyun hai: induced drag $\propto C_{L}^{2}$ (toh yeh low speed / high lift par explode karta hai) aur $\propto 1/ A R$ (toh lamba patla wing ise khatam karta hai).

Step 5 — Real (non-elliptical) wings. Span efficiency $e \leq 1$ introduce karo: $C_{D, i} = \frac{C _{L}^{2}}{π e A R}$ Elliptical wing: $e = 1$ . Rectangular: $e \approx 0.7$ – $0.9$ .

Worked examples

Common mistakes

Flashcards

Span aur area ke terms mein aspect ratio define karo

A R = b^{2} / S

(aur

= b / c

constant chord ke liye)

Induced-drag coefficient (elliptical wing)

C_{D, i} = C_{L}^{2} / (π A R)

Efficiency ke saath full induced-drag coefficient

C_{D, i} = C_{L}^{2} / (π e A R)

Infinite wing ke liye induced drag zero kyun hota hai?

Tips nahi ⇒ trailing vortices nahi ⇒ downwash nahi ⇒ lift vertical rehti hai

Downwash kya hai aur yeh kya karta hai?

Tip vortices se downward velocity jo local flow ko

α_{i}

se neeche jhuka deti hai, lift ko peeche lean karti hai

Elliptical loading ke liye induced angle

α_{i} = C_{L} / (π A R)

Agar fixed

C_{L}, e

par AR double karo, induced drag kaise change hoga?

Aadha ho jaata hai (∝ 1/AR)

Agar

C_{L}

triple ho, induced drag kaise change hoga?

×9 (∝

C_{L}^{2}

)

Elliptical wing ki span efficiency

e

e = 1

(optimum)

High AR hamesha kyun nahi choose karte?

Extra wetted/parasite drag, weight, flutter — optimum induced vs parasite drag ko balance karta hai

Kya induced drag viscous hai ya inviscid?

Inviscid — energy trailing vortices mein stored hoti hai

Recall Feynman: 12-saal ke bachche ko samjhao

Socho tum ek chaudi chamach bathtub ke paani mein flat push kar rahe ho. Chamach ke edges ke paas, paani ghoom jaata hai aur chhote whirlpools banata hai jo peeche trail karte hain. Woh whirlpools banane mein energy lagti hai, aur woh energy ek tarah ka "drag" hai jo chamach ko peeche khichta hai. Agar tum chaudi chamach ki jagah lamba patla ruler use karo, toh sirf do chhote ends whirlpools banate hain — ruler ki bahut kam cheez energy waste karti hai. Isliye airplanes jo door tak glide karna chahte hain (gliders) ke bahut lambe, patle wings hote hain: lamba-aur-patla = kam waste hue whirlpools = kam drag.

Connections

Lifting-line theory (Prandtl) — $α_{i} = C_{L} / (π A R)$ ka origin
Trailing vortices & downwash — physical cause
Drag polar — $C_{D} = C_{D, 0} + C_{L}^{2} / (π e A R)$
Parasite drag — competing term jo useful $A R$ ko cap karta hai
Wingtip devices (winglets) — effective $A R$ / $e$ badhate hain
Elliptical lift distribution — minimum-drag benchmark
Glide ratio & L/D max — induced/parasite balance se set hota hai