3.1.23 · D1Aspect ratio — effect on induced drag
Ek real wing ke ends hote hain, aur un ends par air high-pressure bottom se low-pressure top ki taraf spill karti hai, whirlpools mein curl ho jaati hai jo peeche trail karti hain aur air ko neeche push karti hain. Yeh downward push lift ko peeche ki taraf tilt karke drag mein badal deta hai — aur wing jitna lamba aur patla hoga, uska utna hi chhota fraction is leakage se suffer karega, isliye drag kam ho jaati hai.
Yeh page har letter, ratio, aur picture build karta hai jis par parent note Aspect ratio — effect on induced drag rely karta hai — ek aise smart 12-saal ke bachche ke liye shuru karta hai jisne inhe kabhi nahi dekha. Upar se neeche padho; har block sirf oopar define kiye gaye symbols use karta hai.
Kisi bhi formula se pehle humein teen simple lengths aur ek area chahiye. Wing ko ek flat board ki tarah imagine karo jo directly upar se dekhi gayi ho (bird's-eye view). Is flat outline ko planform kehte hain.
b Span b Picture: figure mein poore wing ke across stretch karta horizontal red arrow.
Kyun chahiye: leakage sirf do tips par hoti hai. Wing ka "mere paas unit wing per kitna tip hai" wala sawaal jaanne se shuru hota hai ki yeh kitni wide hai.
c Chord c Picture: figure mein nose-edge se tail-edge ki taraf point karta chhota black arrow.
Kyun chahiye: chord batata hai ki wing kitni "deep" (front-to-back) hai. Ek mota stubby wing ka bada chord hota hai; ek patli wing ka chhota chord hota hai.
S Planform area S Picture: outline ke andar shaded region.
Kyun chahiye: lift is poore area par spread hoti hai, isliye baad mein jo bhi force compute karein woh S
Ab hum topic ka star build kar sakte hain. Hum ek single number chahte hain jo kahe "kya yeh wing long-and-thin hai ya short-and-fat?" — is baat se independent ki wing overall kitni badi hai.
Intuition Sirf span ki jagahratio kyun
Ek bada airliner aur ek tiny model glider dono "long and thin" ho sakte hain. Jo matter karta hai woh raw span nahi balki span chord se compare kiya hua hai. Ek ratio overall size cancel kar deta hai aur sirf shape bacha leta hai.
A R A R = S b 2 S = b c A R = b c b 2 = c b A R kitne chord-lengths span ke along fit hote hain .
Picture: figure mein, patla red wing apne span ke across kaafi chords fit karta hai (high A R A R
Recall b / c b 2 / S kisi bhi planform shape ke liye kaam karta hai, jabki b / c c b c A R
Worked example Real numbers padhna
Ek sailplane: A R ≈ 30
Ek fighter jet: A R ≈ 3
Poora topic ek force jo tilt ho jaati hai par depend karta hai. Isliye pehle humein pin down karna hoga ki lift kya hai aur "ek arrow tilt karna" ka kya matlab hai.
Definition Freestream velocityV V Picture: left se wing ki taraf approach karte lambe horizontal black arrows.
Kyun chahiye: drag hamesha is direction ke along measure hoti hai — yeh woh "backwards" hai jisko plane fight karta hai.
L Lift woh force hai jo wing ko upar push karti hai, oncoming air ke perpendicular. Yeh wing ke upar kam air pressure aur neeche zyada pressure se aati hai.
Picture: vertical arrow. Figure mein, black lift arrow seedha upar point karta hai kyunki flow horizontal hai.
Kyun chahiye: lift hi ek wajah hai ki induced drag exist karti hai — no lift, no leak, no drag.
D D i Drag koi bhi force hai jo motion ka oppose karti ho — V Induced drag D i specific drag hai jo sirf isliye appear hoti hai kyunki ek finite wing lift banati hai .
Picture: chhota red horizontal arrow peeche ki taraf point karta hua — yeh ek tilted lift arrow ka flight direction par shadow (projection) hai.
Kis cheez ke perpendicular, exactly?
Yahan topic ka subtle core hai. Lift hamesha local airflow ke perpendicular hoti hai jo wing actually feel karti hai. Agar tips local air ko neeche tilt karne par force karein, toh "local flow ke perpendicular" seedha upar nahi rehta — lift arrow peeche ki taraf tip ho jaata hai, aur woh backward lean hi induced drag hai. Section 5 mein yeh precisely clear hoga.
Ab woh physical machinery jo tilting karti hai.
Definition Trailing vortexEk trailing vortex swirling air ki ek tube hai jo har wingtip se peel off hoti hai. Yeh isliye banta hai kyunki wing ke neeche high-pressure air tip ke around low-pressure top ki taraf sneak karti hai, aur woh sideways-then-upward escape flow ko ek spiral mein roll kar deti hai.
Picture: figure mein tips ke peeche stream karte do red spirals.
Kyun chahiye: yeh spirals induced drag ka engine hain — dekho Trailing vortices & downwash .
w Downwash w downward air velocity hai jo un do spinning vortices ke wajah se wing ke peeche create hoti hai.
Picture: vortices ke beech ke middle region mein chhote downward arrows.
Kyun chahiye: w
Common mistake "Vortex bas harmless swirling hai."Kyun sahi lagta hai: swirls wing ke peeche decoration jaisi dikhti hain.
Fix: us air ko spin karna har second energy cost karta hai. Woh continuous energy cost, wing ke dwara feel ki gayi, hi induced drag hai. Yeh ek perfectly frictionless fluid mein bhi exist karta hai — yeh inviscid hai.
Tilt ek angle hai, isliye humein ek trigonometric tool chahiye. Hum exactly woh choose karte hain jo hamare sawaal ka jawab deta hai — kuch zyada nahi.
tangent kyun aata hai
Hum do velocities jaante hain: forward speed V w right triangle banate hain. Tilt ka angle true forward direction aur thodi-si drooping local flow ke beech baitha hai. Jo ratio in do known sides ko us angle se link karta hai woh exactly tangent hai — "opposite over adjacent" — isliye tan
α i α i alpha , subscript i neeche tilt karta hai.
tan α i = V w ⟹ α i = tan − 1 ( V w ) tan − 1 "is tangent wala angle kaun sa hai?" Yeh tangent ko undo karke humein angle deta hai.
Picture: figure mein black horizontal V
Recall Kaun se cases small-angle trick ko break kar sakte hain?Bahut slow flight (landing, high lift) w α i w = 0 α i = 0
Parent achanak L D i C L C D , i
Definition Dynamic pressureq q = 2 1 ρ V 2 ρ rho ) air density hai (air ka mass per cubic metre). q Kyun chahiye: ek force ko q S
Definition Lift aur induced-drag coefficientsC L = q S L , C D , i = q S D i C L C D , i Kyun chahiye: yeh humein ek paper glider aur ek jumbo jet ko same footing par compare karne dete hain — sirf geometry (A R
Mnemonic Coefficient = force ÷ (push × area)C = q S force q S shape ki opinion hai.
Definition Span efficiencye e 0 1 ideal (elliptical) se kitna close hai. Ideal ⇒ e = 1 e ≈ 0.7 0.9 Kyun chahiye: clean formula C D , i = C L 2 / ( π A R ) best possible case hai. Real wings thoda worse karti hain, aur e Elliptical lift distribution .
Aspect ratio AR = b squared over S
CDi = CL squared over pi e AR
Har arrow yahan ek symbol hai jo tum ab padh sakte ho. Milke yeh parent ka headline result C D , i = C L 2 / ( π e A R )
Khud test karo — sirf tab reveal karo jab tumhara jawab taiyar ho.
Span b Wing ki tip-to-tip width
Planform area S Wing ki overhead (bird's-eye) outline ka area
Kisi bhi wing ke liye aspect ratio likho, phir rectangular wing ke liye A R = b 2 / S A R = b / c
Seedhe shabdon mein, A R Span ke along kitne chord-lengths fit hote hain (long-thin vs short-fat)
Lift ki direction kya hai, aur kis cheez ke relative? Local airflow ke perpendicular jo wing actually feel karti hai
Trailing vortices kya create karta hai? Neeche high-pressure air tips ke around low-pressure top ki taraf leaking
Downwash w Tip vortices ke wajah se wing ke peeche downward air velocity
Induced angle ke liye tan V w = w / V = tan α i
Small-angle shortcut batao Tiny α i α i ≈ tan α i ≈ sin α i
Forces ko q S Air density, speed aur size remove karne ke liye, ek pure shape number bachta hai
q Dynamic pressure q = 2 1 ρ V 2
Ideal (elliptical) wing ke liye span efficiency e e = 1
