3.1.22 · D1Finite wing theory — induced drag, Prandtl's lifting line
Ek real wing ke ends hote hain, aur un ends par air high-pressure wali neeche se low-pressure wali upar ki taraf leak karti hai, wing ke peeche spinning tubes of air trail karti hai. Yeh trailing spins us hawa ko thoda neeche tilt kar deti hain jo wing feel karta hai, jisse lift thodi peeche tilt ho jaati hai — aur yeh peeche wali tilt ek bilkul naye tarah ka drag hai jo zero friction par bhi exist karta hai .
Yeh page assume karta hai ki aapne parent note ka koi bhi symbol pehle nahi dekha. Hum har ek symbol ek picture se banate hain, ek aisi order mein jahan har symbol sirf pehle se defined cheezein use karta hai. Agar aap line 1 follow kar sakte ho, toh poora topic follow kar sakte ho.
Koi bhi physics se pehle, humein un shapes ke liye words chahiye jinke baare mein hum baat kar rahe hain.
Figure padho: teal outline wing hai jo upar se dekhi gayi hai. Neeche wala orange double-arrow span b chord c ( y ) y y = 0 y = ± b /2 area S
Definition Teen shape numbers
Span b Chord c har spanwise position par alag chord ho sakta hai, isliye hum c ( y ) y par chord."Planform area S S = b × c
Intuition Kyun hum span ke saath-saath direction ki parwah karte hainEk finite wing ke baare mein sab kuch interesting isliye hota hai kyunki beech se tip tak jaate time cheezein change hoti hain . Isliye humein ek coordinate chahiye jo span ke saath chalti ho. Woh coordinate y
Definition Spanwise coordinatey y wing ke beech mein zero pe rakh di gayi hai. Left tip y = − 2 b y = + 2 b y [ − 2 b , + 2 b ]
Jab hum "c ( y ) span ke saath position y
A R = S b 2 span ka square, planform area se divide kiya. Yeh ek akela number hai jo batata hai ki wing kitna "lamba aur patla" hai.
Squared kyun? Ek rectangle dekho jahan S = b c A R = b c b 2 = c b , span ÷ chord hai — span mein kitne chord-widths fit hote hain. Ek glider wing (bahut lamba, bahut patla) ka bada A R A R
Common mistake "Aspect ratio bas wing ka area hai."Kyun sahi lagta hai: dono "wing kitna bada hai" yeh describe karte hain. Fix: A R same area ho sakta hai lekin wildly alag A R
Aspect ratio is poore topic ka star hai, isliye ise jaldi meet karo. Full treatment: Aspect ratio & wing design .
Definition Freestream velocityV ∞ V ∞ ∞ door measure kiya gaya , jahan wing ne abhi kuch disturb nahi kiya.
ρ ∞ ρ density hai — har cubic metre mein kitne kilograms air hai. ∞ ρ
Intuition Kyun yeh do har lift/drag formula mein rehte hainFluid se force hamesha scale karti hai kitna stuff throw kar rahe ho (ρ kitni fast throw kar rahe ho (V ρ ∞ V ∞
Yeh sabse important naya idea hai, isliye hum slowly jaate hain.
Figure padho: solid black blob wing ka cross-section hai, bread ki tarah kata hua. Teal arrows air ko upar fast aur neeche slow move karte dikhate hain — lift ki pehchaan. Dashed plum loop woh path hai jo tum "walk" karte ho poori taraf; us par orange arrowheads direction dikhate hain jisme tum loop ke saath air ka push add karte ho. Woh grand total circulation Γ
Intuition "Circulation" kya capture karne ki koshish kar raha haiEk lifting wing hawa ko moda deta hai: upar fast, neeche slow. Agar tum wing ke cross-section ke around ek bada loop walk karo (figure mein dashed plum loop) aur add karo ki hawa tumhe tumhare path ke saath kitna push karti hai, toh tum ek non-zero total paoge — hawa, overall, wing ke around chakkar laga rahi hai. Circulation woh total hai, ek akele number mein pack ki gayi.
Γ Γ Γ
Kyunki wing alag-alag spanwise stations par alag lift karta hai, swirl bhi span ke saath change hoti hai. Isliye hum Γ ( y ) y par circulation." Bilkul tips par wing koi load nahi uthata, isliye
Γ ( − 2 b ) = Γ ( + 2 b ) = 0.
Mnemonic Circulation wing ka "spin account" haiJitni zyada swirl bank mein, utni zyada lift withdraw. Topic mein baad ka sab kuch actually Γ ( y )
Trailing vortices ki baat karne se pehle humein ek piece of calculus notation chahiye. Hum ise yahan introduce karte hain, plain words mein, use karne se pehle.
Intuition "Rate of change" ki zaroorat kyun haiFinite wing ki poori kahani span ke saath change ke baare mein hai: Γ itni quickly Γ "agar main span ke saath ek tiny step d η Γ
d η d Γ d η d Γ d η d Γ = tiny step along span tiny change in Γ Γ slope hai — literally "rise over run." Yeh bada (size mein) hota hai jahan Γ Γ η y tarah ka ruler hai, bas alag naam diya hai kyunki thodi der mein hume ek saath do chahiye honge.
Intuition Picture se slope ka sign padhnaWing ke right half mein Γ d η d Γ negative hai. Left half mein Γ d η d Γ positive hai. Steepness (size) tips ke paas sabse zyada hai — exactly wahan tip vortices sabse strong hote hain.
Definition Ek vortex filamentVortex filament concentrated spin ka ek idealized thin thread hai — ek tornado ke thin core ki picture banao, ya canoe paddle ki tip par wali chhoti curl ki. Hum wing ki swirl ko aisi threads ke roop mein model karte hain.
Definition Bound vs trailing vortices
Bound vortex woh spin thread hai jo wing par khud baitta hai (yeh wing ko "lift" karta hai). Iska strength exactly Γ ( y ) Trailing vortices woh threads hain jo wing se peel off hote hain aur wake mein peeche stream karte hain.
Intuition Trailing vortices kyun exist karne chahiye (Helmholtz)Ek spin thread ko simply mid-air mein khatam hone ki permission nahi hai — spin ke paas jaane ki jagah nahi hai. Toh agar beech mein bound spin Γ ( y ) farq kahin escape karna hoga: woh turn karke downstream trail karta hai. Yeh rule "vortex line fluid mein end nahi ho sakti" Helmholtz vortex theorems mein se ek hai.
Definition Kitna spin shed hota hai, aur minus sign kyunη η + d η Γ ( η ) Γ ( η + d η ) Γ ( η ) − Γ ( η + d η ) = − d η d Γ d η − d η d Γ minus sign simply yeh kehta hai ki jahan Γ decrease karta hai (d Γ/ d η < 0 positive hai, yani spin genuinely release ho raha hai. Yeh ek bookkeeping sign hai, koi naya physical effect nahi. Wing ke right half mein Γ − d Γ/ d η > 0
Figure padho: mota orange bar wing hai (bound vortex Γ ( y ) Γ trailing tip vortices hain, chhoti curls dikhati hain ki woh opposite directions mein spin karti hain. Unke beech, plum arrows seedhe neeche point karte hain — woh downwash w
w w ( y ) extra downward air velocity hai jo wing par un saari trailing spin threads se create hoti hai. Trailing vortices, spin karte hue, unke beech hawa ko neeche drag karte hain — wing par feel ki gayi woh downward speed w
Intuition Kyun ek vortex door se hawa ko neeche push karta haiEk spinning thread nearby air ko apne saath drag karta hai. Wing ke ek taraf wali hawa neeche nudge hoti hai. Us nudge ki strength distance ke saath fall hoti hai — station η y y − η 1 η y Biot–Savart law hai.
Common mistake "Integrandη = y Kyun sahi lagta hai: η = y y − η 1/ ( y − η ) Fix: yeh ek jaani-pehchani, tame tarah ki infinity hai. η = y left wala piece + ∞ right wala − ∞ Cauchy principal value ke roop mein padha jaata hai — tum η = y
Figure padho: horizontal teal arrow freestream V ∞ w actually feel karta hai — orange arrow. Horizontal flow aur is tilted flow ke beech ka chhota angle induced angle α i w V ∞
Angle of attack wing ka the control knob hai, isliye hum charon flavours sort out karte hain.
Definition Geometric angle of attackα α door ki airflow V ∞
Definition Zero-lift angleα L = 0 α L = 0 upar kitna ho iske par depend karta hai.
α i w local airflow ko neeche tilt karta hai. Woh tilt angle hai
α i ( y ) = tan − 1 V ∞ w ( y ) ≈ V ∞ w ( y ) . tan − 1 "is slope wala angle kaun sa hai?" — slope downward speed w V ∞ w V ∞ tan − 1 ( x ) ≈ x w / V ∞
Definition Effective angle of attackα eff actually feel karta hai, downwash ke flow tilt karne ke baad:
α eff = α − α i .
Intuition Yeh EK subtraction hi poora plot haiWing α eff α kam lift banata hai, aur iska lift vector α i α eff = α − α i
Definition Forces ko dimensionless bananaAlag sizes aur speeds ke wings compare karne ke liye, hum "fuel" 2 1 ρ ∞ V ∞ 2 dynamic pressure kehte hain) aur ek reference area se divide kar dete hain. Jo bachta hai woh ek pure number hai: ek coefficient .
y θ automatically ends par vanish hoti hai: sine waves. Isliye hum span position ko ek angle θ y = − 2 b cos θ , θ 0 π θ = 0 θ = π
θ y y θ y = − 2 b cos θ d θ d y = 2 b sin θ ⟹ d y = 2 b sin θ d θ . 2 b sin θ Jacobian hai — yeh step d θ d y θ = 0 , π θ = 2 π θ
Definition Sine series, aur2 b V ∞ sine waves ka sum likhte hain:
Γ ( θ ) = 2 b V ∞ ∑ n = 1 ∞ A n sin ( n θ ) . sin ( n θ ) A n har block kitna mix karna hai . Front factor 2 b V ∞ deliberate normalization hai jo is liye choose kiya gaya taaki A n dimensionless numbers ban sakein. Units check karo: Γ b V ∞ A n 2 C L = π A R A 1 A 1 A n
A 1 A 1 A 2 , A 3 , … A 1 Oswald efficiency factor se tie karta hai.
Aspect ratio AR = b squared over S
Freestream V-infinity and density rho
Derivative dGamma over dy
Trailing vortices from Helmholtz rule
Effective angle = alpha minus alpha-i
Thin airfoil lift slope 2 pi
Angle swap y equals minus b over 2 cos theta
Fourier sine series with A-n
Lift C-L and induced drag C-D-i
Right side cover karo; reveal karne se pehle har ek bata ne ki koshish karo.
b y = 0 Tip-to-tip distance; y = 0 ± b /2
Aspect ratio likho aur batao words mein kya matlab hai. A R = b 2 / S
V ∞ ρ ∞ Wing ke kaafi aage airspeed, aur door ki air density — har force formula ka "fuel."
Physically Γ ( y ) Station y Γ
Lift per unit span ke liye Kutta–Joukowski state karo. L ′ ( y ) = ρ ∞ V ∞ Γ ( y )
Derivative d Γ/ d η Γ
Trailing vortices kyun exist karne chahiye, aur − d Γ/ d η Ek vortex thread fluid mein end nahi ho sakta (Helmholtz); minus sign shed strength ko positive banata hai jahan Γ
Downwash integral mein 1/ ( 4 π ) Ek trailing filament semi-infinite hai = ek infinite line ka half, isliye Γ/ ( 2 π r ) Γ/ ( 4 π r )
η = y Ise Cauchy principal value ke roop mein padha jaata hai; dono taraf se ± ∞
Whole-wing lift coefficient C L C L = L / ( 2 1 ρ ∞ V ∞ 2 S )
y = − 2 b cos θ d y = 2 b sin θ d θ
Γ 2 b V ∞ Ek normalization jo dimensionless A n C L = π A R A 1
Kaun sa coefficient lift set karta hai, aur kaun se sirf drag add karte hain? A 1 A 2 , A 3 , …
α i ≈ w / V ∞ α i = tan − 1 ( w / V ∞ ) tan − 1 x ≈ x
