3.1.22
Intuition Badi picture (WHY yeh exist karta hai)Ek infinite (2D) wing ko inviscid flow mein koi drag nahi hota (d'Alembert's paradox), aur uska fixed lift slope 2 π real wing finite hoti hai : tips par, neeche ki high-pressure air upar ki low-pressure side par leak karti hai. Isse ek trailing vortex sheet banti hai jo air ko wing ke peeche neeche ki taraf dhakelta hai.
Yeh downwash local "felt" velocity ko tilt karta hai, jo lift vector ko bhi peeche ki taraf tilt karta hai → ab lift ka ek component peeche ki taraf point karta hai = induced drag (drag-due-to-lift). Prandtl ki lifting-line theory sabse simple model hai jo yeh sab kuch ek single line of vortices se capture karti hai.
Bound vortex Γ ( y ) 1/4 L ′ ( y ) = ρ ∞ V ∞ Γ ( y ) Trailing vortices : kyunki Γ − d y d Γ Downwash w ( y ) wing par induce ki gayi vertical velocity.Aspect ratio A R = S b 2 b S
Geometry kaise baat karta hai: strength − d Γ/ d y
Intuition Downwash sab kuch kyun tilt karta haiWing V ∞ V ∞ plus w induced angle se neeche tilt ho jata hai:
α i ( y ) = tan − 1 V ∞ w ≈ V ∞ w ( y ) . local flow ke perpendicular hoti hai → vertical se α i
HOW hum ise solve karte hain: y = − 2 b cos θ Γ ( θ ) = 2 b V ∞ ∑ n = 1 ∞ A n sin ( n θ ) . Γ = 0 θ = 0 , π A n
Intuition Elliptic lift best kyun haiKyunki δ ≥ 0 = 0 A 2 = A 3 = ⋯ = 0 sirf A 1 = 0 ⇒ Γ ( θ ) = Γ 0 sin θ elliptical circulation distribution. Tab:
C D , i = π A R C L 2 constant downwash hota hai. Zyada A R
Worked example (A) Elliptic wing ka induced dragA R = 7 C L = 0.6 C D , i = π A R C L 2 = π ⋅ 7 0.36 = 0.0164. Yeh step kyun? Elliptic ⇒ δ = 0 minimum possible induced drag hai.
Worked example (B) Aspect ratio double karne ka effectSame C L A R = 7 → 14 C D , i → π ⋅ 14 0.36 = 0.0082 Aadha ho gaya.
Kyun? C D , i ∝ 1/ A R
Worked example (C) Non-elliptic penaltyEk wing mein A 2 / A 1 = 0.1 A 3 / A 1 = 0.05 δ = 2 ( 0.1 ) 2 + 3 ( 0.05 ) 2 = 0.02 + 0.0075 = 0.0275. e = 1/1.0275 = 0.973 2.75% Yeh step kyun? δ
Worked example (D) 3D lift slopeElliptic wing, A R = 8 a = 1 + 2 π / ( 8 π ) 2 π = 1 + 0.25 2 π = 1.25 2 π = 5.03 /rad Yeh step kyun? 2D a 0 = 2 π = 6.28
Common mistake "Induced drag friction/viscous drag hai."Kyun sahi lagta hai: saara drag "rub" jaisa lagta hai. Fix: induced drag bilkul inviscid flow mein bhi exist karta hai — yeh trailing vortices mein le jaai jaane wali energy hai (swirling wake ki kinetic energy). Yeh ek pressure-related, lift-coupled drag hai, skin friction aur profile drag se alag.
Common mistake "Zyada wing area hamesha induced drag reduce karti hai."Kyun sahi lagta hai: zyada area = zyada lift, efficient lagta hai. Fix: jo cheez matter karti hai woh span hai, area nahi: C D , i ∝ C L 2 / ( π A R ) = C L 2 S / ( π b 2 ) b C L badhti hai. Lamba patla > chhota mota.
Common mistake "Saare harmonics lift mein contribute karte hain."Kyun sahi lagta hai: series mein kaafi A n Fix: orthogonality ki wajah se sirf A 1 C L n ≥ 2 drag add karte hain par lift nahi — pure penalty hai.
Common mistake Trailing filaments ke liye2 π r Γ Kyun sahi lagta hai: yeh familiar vortex formula hai. Fix: trailing filaments semi-infinite hain (wing se shuru hokar ∞ 4 π r Γ
Recall Quick self-test (answers chhupao)
Ek 2D infinite wing ko inviscid flow mein kaunsa drag hota hai? → Koi nahi (d'Alembert).
Physically induced drag kya create karta hai? → Trailing tip vortices / downwash lift ko peeche tilt karta hai.
Kaunsi Γ Elliptic , C D , i = C L 2 / ( π A R )
Lift mein sirf A 1 [ 0 , π ] sin n θ
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho ek wide tray ko paani mein flat push kar rahe ho. Kinaron ke paas, paani neeche ki high-pressure se upar ki low-pressure side par round aata hai, chhote spinning tornadoes banaata hai jo peeche trail karte hain. Woh tornadoes banane mein energy lagti hai, aur woh energy "drag" wing ko thoda peeche khichne ke roop mein feel hoti hai. Ek lamba, patla tray edge-tornadoes ko beech mein ki saari acchi lifting ke comparison mein kamzor banata hai — isliye lambi, patli wings (jaise glider ki) kam energy waste karti hain. Lift ki perfect sharing, tips ke paas chhoti aur beech mein sabse badi ek oval shape mein, sabse chhote possible tornadoes banati hai.
Mnemonic Result yaad rakho"LIFT loves A-one; DRAG pays for the rest."
C L = π A R A 1 A 1 C D , i = π A R ∑ n A n 2 "Long & lean = less induced" (C D , i ∝ 1/ A R
Kutta–Joukowski theorem — L ′ = ρ V ∞ Γ Thin airfoil theory — local lift slope 2 π α L = 0 Biot–Savart law — downwash integral ka basis hai.Helmholtz vortex theorems — isliye trailing vorticity shed karni padti hai.d'Alembert's paradox — kyun 2D wing mein zero drag hota hai.Aspect ratio & wing design — gliders/airliners ke liye practical consequence.Oswald efficiency factor — drag polar C D = C D , 0 + C L 2 / ( π e A R )
Induced drag define karo. Drag-due-to-lift trailing vortices se: downwash local lift vector ko peeche tilt karta hai; inviscid flow mein bhi exist karta hai.
Prandtl ka downwash integral state karo. w ( y ) = 4 π 1 ∫ − b /2 b /2 y − η d Γ/ d η d η
Trailing filament downwash mein 4 π 2 π Trailing filaments semi-infinite hain, isliye Γ/ ( 4 π r )
Fourier coefficients ke terms mein C L C L = π A R A 1
Fourier coefficients ke terms mein C D , i C D , i = π A R ∑ n = 1 ∞ n A n 2 = π A R C L 2 ( 1 + δ )
Kaunsi lift distribution induced drag minimize karti hai aur kyun? Elliptic (Γ = Γ 0 sin θ A 1 = 0 δ = 0
Minimum induced drag formula? C D , i = C L 2 / ( π A R )
Span efficiency e e = 1/ ( 1 + δ ) ≤ 1 C D , i = C L 2 / ( π e A R ) e = 1
Finite-wing lift slope (elliptic)? a = a 0 / ( 1 + a 0 / ( π A R )) a 0 = 2 π
C D , i C D , i = C L 2 S / ( π b 2 )
Lift mein sirf A 1 Orthogonality: ∫ 0 π sin n θ sin θ d θ = 0 n = 1
Aspect ratio ki definition? A R = b 2 / S
Effective angle of attack
Aspect ratio b squared over S
