Aspect ratio — effect on induced drag: Question bank

3.1.23 · D5 · Physics › Compressible Flow & Aerodynamics › Aspect ratio — effect on induced drag

Do headline formulas ka reminder jinpe tum rely karoge:

Sahi ya galat — justify karo

Infinite-span (2-D) wing induced drag produce karta hai

Galat. Jab tips hi nahi hain toh koi trailing vortices nahi, koi downwash nahi, isliye lift bilkul vertical rehti hai — induced drag exactly zero hota hai.

Induced drag ek perfectly frictionless (inviscid) fluid mein bhi exist karta hai

Sahi. Yeh woh kinetic energy hai jo trailing vortices mein dump hoti hai, viscous shear nahi — isliye ise inviscid drag kaha jaata hai.

Zero lift par ek wing (

C_{L} = 0

) par phir bhi kuch induced drag hota hai

Galat. Kyunki

C_{D, i} \propto C_{L}^{2}

hai, zero lift ka matlab zero induced drag hai — koi lift nahi, koi tip pressure difference nahi, koi vortices nahi.

Wingspan ko double karte hue area

S

fixed rakhne se induced drag half ho jaata hai

Sahi.

A R = b^{2} / S

hai, toh fixed

S

par

b

double karne se

A R

chaar guna ho jaata hai;

C_{L}, e

fixed rakhne par

C_{D, i} \propto 1/ A R

ek-chauthai ho jaata hai — "half" se bhi zyada.

Ek given lift coefficient ke liye, elliptical lift distribution se sabse kam possible induced drag milta hai

Sahi. Span ke across uniform downwash minimum-drag case hai, aur sirf elliptical loading hi uniform downwash produce karta hai — isliye

e = 1

Span efficiency

e

ek cleverly shaped planar wing ke liye 1 se zyada ho sakta hai

Galat.

e = 1

ek planar wing ke liye theoretical optimum hai (elliptical loading); koi bhi aur shape non-uniform downwash mein energy waste karta hai, toh

e < 1

hoga.

Ek fighter jet ko jitna possible ho utna high aspect ratio use karna chahiye

Galat. High

A R

wetted area (parasite drag), weight aur flutter risk badhata hai; fighters low

C_{L}

par fast cruise karte hain jahan induced drag already tiny hoti hai, toh low

A R

better hai.

Slow flight mein total drag ki fraction ke roop mein induced drag fast cruise se zyada hoti hai

Sahi. Slow flight ko high

C_{L}

chahiye, aur

C_{D, i} \propto C_{L}^{2}

balloon ho jaata hai, jabki parasite drag gir jaata hai — isliye takeoff aur landing ke paas induced drag dominate karta hai.

Winglets thrust add karke kaam karte hain

Galat. Winglets effective aspect ratio / span efficiency badhate hain, tip vortices ko weak karte hain — yeh drag reduce karte hain, thrust produce nahi karte.

Error dhundho

C_{D, i} = C_{L}^{2} / (π e A R)

, toh higher

C_{L}

ka matlab hamesha proportionally zyada drag hai"

Error "proportionally" mein hai. Drag

C_{L}

ke square ke saath badhta hai, linearly nahi —

C_{L}

triple karo aur induced drag nine guna ho jaata hai.

"Aspect ratio sirf span divided by area hai,

A R = b / S

Galat — yeh

b^{2} / S

hai (span squared over area). Yeh sirf

b / c

(span over chord) mein reduce hota hai jab chord constant ho.

"Induced drag air aur wingtips ke beech friction hai"

Yeh bilkul bhi friction nahi hai. Yeh inviscid hai — wasted energy wing ke peeche chhode gaye trailing vortex system mein hoti hai, skin roughness se independent.

"Kyunki induced drag ek geometry effect hai, yeh aap kitna lift banate hain uspe depend nahi karta"

Geometry (

A R, e

) sirf coefficient set karta hai; actual magnitude

C_{L}^{2}

ke saath scale hota hai, toh lift absolutely matter karti hai — koi lift nahi, koi induced drag nahi.

"Finite wing par lift bilkul upar point karti hai, same as 2-D airfoil"

Finite wing par downwash local flow ko

α_{i}

se neeche tilt karta hai, toh lift (local flow ke perpendicular) peeche jhuk jaati hai — woh backward lean hi induced drag hai.

"Ek rectangular wing ka

e = 1

hota hai ellipse ki tarah kyunki dono simple shapes hain"

Ek rectangular wing ka

e \approx 0.7

–

0.9

hota hai; iski lift tips ki taraf bahut zyada pile up hoti hai, jo non-uniform downwash aur elliptical

e = 1

ke comparison mein extra induced drag deta hai.

"Downwash woh air hai jo propeller plane ke peeche neeche blow karta hai"

Nahi — yahan downwash woh downward velocity hai jo wing ke apne trailing vortices se induce hoti hai, ek unpowered glider par bhi present hoti hai jisme koi propeller nahi hota.

Why questions

Ek finite wing apne tips ke around air leak kyun karta hai?

Lift ka matlab neeche high pressure aur upar low pressure hai; open tip par high-pressure air upar aur around curl karta hai low-pressure top ki taraf, tip vortex banata hai.

Elliptical lift distribution ko benchmark ke roop mein kyun use kiya jaata hai na ki rectangular ko?

Yeh woh unique loading hai jo span ke across downwash uniform banata hai, aur uniform downwash provably minimum-induced-drag case hai — natural best-possible reference (

e = 1

Induced drag

C_{L} sin α_{i}

ke roop mein kyun appear karta hai (lift ka backward tilt) na ki ek alag force ke roop mein?

Kyunki "drag" ka matlab freestream direction ke along force hai; jab lift

α_{i}

se peeche tilt hoti hai, toh freestream direction par uska projection exactly woh induced-drag component hai.

Aspect ratio (

b^{2} / S

) governing parameter kyun hai aur akela span kyun nahi?

Lifting-line integrals

α_{i} = C_{L} / (π A R)

par collapse hote hain;

b^{2} / S

combination exactly wahi hai jo emerge hota hai jab tum vortex strength ko lift se relate karte ho, toh

A R

— raw span nahi — downwash control karta hai.

Gliders aur high-altitude drones ke paas itne enormous, thin wings kyun hote hain?

Woh high

C_{L}

par slowly aur efficiently fly karte hain jahan induced drag dominate karta hai; huge

A R

us term ko slash karta hai, glide ratio maximize karta hai chahe extra parasite drag ki cost par.

Optimum aspect ratio simply "jitna bada ho sake" kyun nahi hai?

Ek longer thinner wing wetted area (parasite drag), weight aur structural flutter badhata hai; best

A R

drag polar par girte induced drag aur badhte parasite drag ko balance karta hai.

Edge cases

Exactly zero angle of attack par glide karne wale wing ka induced drag kya hoga lekin phir bhi lifting?

Agar woh abhi bhi lift bana raha hai (

C_{L} \neq = 0

) toh abhi bhi induced drag hai; trigger lift hai, angle of attack nahi — ek cambered wing zero geometric angle par bhi lift karta hai.

Jab aspect ratio

A R \to \infty

tab

C_{D, i}

ka kya hota hai?

Yeh zero ki taraf tend karta hai — infinitely long wing 2-D airfoil ki tarah behave karta hai bina tips ke, toh trailing vortices aur downwash vanish ho jaate hain.

Jab span efficiency

e \to 0

tab

C_{D, i}

ka kya hota hai?

Yeh infinity tak blow up ho jaata hai —

e \to 0

ka matlab hugely concentrated tip vortices ke saath infinitely wasteful lift shape hai; physically real wings kabhi iske paas nahi pahunchte,

e \approx 0.7

–

1

ke paas rehte hain.

Same speed aur lift par, kaun zyada induced drag suffer karta hai: ek stubby low-

A R

wing ya ek long high-

A R

wing?

Stubby low-

A R

wing — smaller

A R

ka matlab bada

C_{D, i} = C_{L}^{2} / (π e A R)

hai, kyunki iski zyada fraction leaky tips ke paas hoti hai.

Do identical wings same

C_{L}

par fly kar rahe hain; ek twice as fast fly karta hai. Kya unka induced-drag coefficient same hai?

Haan —

C_{D, i}

sirf

C_{L}, e, A R

par depend karta hai, sab unchanged hain. (Induced-drag force differ karta hai kyunki yeh

ρ V^{2}

ke saath scale karta hai, lekin coefficient identical hai.)

Kya small-angle approximation

sin α_{i} \approx α_{i}

kabhi formula ko break karti hai?

Normal flight ke liye

α_{i}

kuch degrees hota hai, toh yeh excellent hai; sirf extreme high-lift, low-

A R

cases mein tilt itna badh jaata hai ki linear approximation drag ko under-predict karne lagti hai.

Recall Page close karne se pehle ek-line self-test

Answers cover karo aur re-derive karo: Long-and-thin, short-and-stubby ko kyun beat karta hai? ::: Long-and-thin = high $A R$ ; sirf tiny tips leak karte hain, toh downwash aur lift ka backward tilt shrink ho jaata hai — $C_{D, i} = C_{L}^{2} / (π e A R)$ $A R$ badhne ke saath girta hai.

Connections

Aspect ratio — effect on induced drag — woh parent topic jinhe yeh traps drill karte hain
Lifting-line theory (Prandtl) — jahan $α_{i} = C_{L} / (π A R)$ prove hota hai
Trailing vortices & downwash — har "why" ke peeche physical cause
Elliptical lift distribution — $e = 1$ benchmark
Parasite drag — woh competitor jo useful $A R$ ko cap karta hai
Drag polar — jahan induced aur parasite drag add up hote hain
Wingtip devices (winglets) — "raise effective $A R$ " trap
Glide ratio & L/D max — kyun gliders $A R$ par extreme jaate hain